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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 96, pp. 1--48.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/96\hfil A class of nonlinear differential equations]
{A class of nonlinear differential equations on the space of symmetric
matrices}

\author[V. Dragan, Gerhard Freiling, A. Hochhaus, T. Morozan\hfil EJDE-2004/96\hfilneg]
{Vasile Dragan, Gerhard Freiling, Andreas Hochhaus, Toader Morozan} % in alphabetical order


\address{Vasile Dragan \hfill\break
Institute of Mathematics of the Romanian Academy,
P.O. Box 1-764, RO-70700 Bucharest, Romania}
\email{vdragan@stoilow.imar.ro}

\address{Gerhard Freiling \hfill\break
Institut f\"ur Mathematik, Universit\"at Duisburg-Essen,
D-47048 Duisburg, Germany}
\email{freiling@math.uni-duisburg.de}

\address{Andreas Hochhaus \hfill\break
Institut f\"ur Mathematik, Universit\"at Duisburg-Essen,
D-47048 Duisburg, Germany}
\email{hochhaus@math.uni-duisburg.de}

\address{Toader Morozan \hfill\break
Institute of Mathematics of the Romanian Academy,
P.O. Box 1-764, RO-70700 Bucharest, Romania}
\email{tmorozan@stolilow.imar.ro}

\date{}
\thanks{Submitted March 18, 2003. Published August 6, 2004.}
\subjclass[2000]{34A34, 34C11, 34C25, 93E03, 93E20} 
\keywords{Rational matrix differential equations; generalized Riccati 
\hfill\break\indent 
differential equations; generalized stabilizability and detectability; comparison theorem;
\hfill\break\indent existence and convergence results}

\begin{abstract}
 In the first part of this paper we analyze the properties of 
 the evolution operators of linear differential equations generating 
 a positive evolution and provide a set of conditions which 
 characterize the exponential stability of the zero solution, which 
 extend the classical theory of Lyapunov.

 In the main part of this work we prove a monotonicity and a 
 comparison theorem for the solutions of a class of time-varying 
 rational matrix differential equations arising from stochastic 
 control and derive existence and (in the periodic case) convergence 
 results for the solutions. The results obtained are similar to those 
 known for matrix Riccati differential equations. Moreover we provide
 necessary and sufficient conditions which guarantee the existence 
 of some special solutions for the considered nonlinear differential 
 equations as: maximal solution, stabilizing solution, minimal positive 
 semi-definite solution. In particular it turns out that under the 
 assumption that the underlying system satisfies adequate generalized 
 stabilizability, detectability and definiteness conditions there
 exists a unique stabilizing solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

The extension of deterministic linear-quadratic control to the stochastic
case has been a notable and active research area and has many applications.
Let us consider a linear-quadratic stochastic optimal control problems of the
form
\begin{equation} \label{eq:6.7.1}
\begin{aligned} dx(t) & = \big[
A(t) x(t) + B(t) u(t) \big] \, dt + \big[ C(t) x(t) + D(t) u(t) \big] \,
dw(t), \\ x(t_0) & = x_0, \end{aligned}
\end{equation}
\begin{equation} \label{eq:6.7.2}
J(u) := E \Big\{ x(t_f)^T Q_f x(t_f) +
\int_{t_0}^{t_f} \begin{pmatrix} x(t) \\ u(t) \end{pmatrix}^T \begin{pmatrix}
M(t) & L(t) \\ L^T(t) & R(t) \end{pmatrix} \begin{pmatrix} x(t) \\ u(t)
\end{pmatrix} dt \Big\} \end{equation}
where the state $x(t)$ and the control $u(t)$ are stochastic processes, and
where $w(t)$ is a standard Wiener process (Brownian motion) on some
probability space $(\Omega, \mathcal{F}, P)$. Moreover, assume here that
$u(t)$ is non-anticipating with respect to $w(t)$, $x_0$ is independent of
$w(t)$ and that $A$, $B$, $C$, $D$, $M$, $L$ and $R$ are sufficiently smooth
functions defined on a right unbounded interval $\mathcal{I} \subseteq
\mathbb{R}$ with values in $\mathbb{R}^{n\times n}$, $\mathbb{R}^{n\times
m}$, $\mathbb{R}^{m \times n}$, $\mathbb{R}^{m \times m}$, $\mathbb{R}^{n
\times n}$, $\mathbb{R}^{n \times m}$ and $\mathbb{R}^{m \times m}$,
respectively. For the basic definitions from stochastic control theory see
\cite{YoZh99} and -- in the infinite dimensional case \cite{DPZa92}.

It can be shown (see \cite{YoZh99}) that the optimal control for
(\ref {eq:6.7.1}), (\ref{eq:6.7.2}) is determined by the solution
of the following terminal value problem for a generalized Riccati
matrix differential equation:

\begin{equation} \label{eq:6.7.3}
\begin{aligned} \lefteqn{\frac{d}{dt} X(t) + A^T(t)X(t) + X(t)A(t) + M(t)
+ C^T(t) X(t) C(t)} \\ & \quad {} - \big\{ X(t)B(t) + C^T(t) X(t) D(t) + L(t)
\big\} \big\{ R(t) + D^T(t) X(t) D(t) \big\}^{-1} \\ & \qquad \times \big\{
X(t)B(t) + C^T(t) X(t) D(t) + L(t) \big\}^T = 0, \quad X(t_f) = Q_f.
\end{aligned}
\end{equation}
The main goal of this paper is to show that several of the nice properties of
standard symmetric matrix Riccati differential equations (which have been
summarized in Chapter 4 of \cite{AFIJ03}) remain valid for the solutions of
rational matrix equations of the form
\begin{equation} \label{eq:6.7.4}
\begin{aligned}
\lefteqn{\frac{d}{dt} X(t) + A^T(t)X(t) + X(t)A(t) + M(t) + \Pi_1(t)[X(t)]}
\\ & \quad {} - \big\{ X(t)B(t) + \Pi_{12}(t)[X(t)] + L(t) \big\} \big\{ R(t)
+ \Pi_2(t)[X(t)] \big\}^{-1} \\ & \qquad \times \big\{ X(t)B(t) +
\Pi_{12}(t)[X(t)] + L(t) \big\}^T = 0 \end{aligned}
\end{equation}
and also of a slightly more general form. Here and below we assume that
\[
\renewcommand{\arraystretch}{1.25} \Pi(t)[X] :=
\begin{pmatrix}
\Pi_1(t)[X] & \Pi_{12}(t)[X] \\ \big( \Pi_{12}(t)[X] \big)^T & \Pi_2(t)[X]
\end{pmatrix}, \quad X \in \mathcal{S}^n,
\]
is a bounded and continuous operator valued function, which is in addition
positive, i.e. $X \geq 0$ implies $\Pi(t)[X] \geq 0$.

Note that (\ref{eq:6.7.4}) contains, as particular cases, (\ref{eq:6.7.3})
and also many other matrix Riccati differential equations arising in
connection with various types of control problems associated to linear
control systems both in deterministic and stochastic framework. In particular
in (\ref{eq:6.7.3}) $\Pi(t)$ takes the form
\[
\Pi(t)[X] = \begin{pmatrix} C(t) & D(t) \end{pmatrix}^T X
\begin{pmatrix}
C(t) & D(t) \end{pmatrix}.
\]
Equations of the form (\ref{eq:6.7.4}), the corresponding algebraic equations
(in the time-invariant case) and certain special cases have been studied
recently (see \cite{ChLZ98}, \cite{DaHi01}, \cite{FrHo6}, \cite{FrHo5},
\cite{HiPr98}, \cite {YoZh99}) and can on account of their properties be
considered as generalized Riccati-type equations.

In the case where $R $ is invertible, $\Pi_1$ is linear and
positive, $\Pi_2 \equiv 0$ and $\Pi_{12} \equiv 0$
(\ref{eq:6.7.4}) coincides with
\begin{equation} \label{eq:6.7.5}
\begin{aligned}
\lefteqn{\frac{d}{dt} X(t) + A^T(t)X(t) + X(t)A(t) + M(t) + \Pi_1(t)[X(t)]}
\\ & \quad {} - \big\{ X(t)B(t) + L(t) \big\} R^{-1}(t) \big\{ X(t)B(t) +
L(t) \big\}^T = 0.
\end{aligned}
\end{equation}
The latter class of perturbed Riccati differential equations appears among
others in control problems with stochastically jumping parameters (see \cite
{FCdS98} and \cite{AFIJ03}, Section 6.9); the corresponding algebraic
equations and inequalities play also an important role in the application of
the Lyapunov-Krasovskii method to linear time-delay systems.

First steps concerning the theory of the rational matrix
differential equation (\ref{eq:6.7.4}) have been performed by
Hinrichsen and Pritchard \cite{HiPr98}, Ait Rami et al.\
\cite{ARCMZ01,ARZh00} and Chen et al.\ \cite
{ChLZ98,ChYo01,ChZh00}, who obtained under additional assumptions
sufficient conditions for the existence of the solutions of
(\ref{eq:6.7.4}) on a given interval for certain initial values.
The algebraic equation (\ref{eq:6.7.5}) has been studied in detail
by Damm and Hinrichsen \cite{DaHi01} (see also \cite{DaHi03}).

Dragan and Morozan (see \cite{preprint00} and the papers cited therein)
considered in the case of time-varying coefficients coupled systems of
differential equations which can be transformed to the form (\ref{eq:6.7.4});
they investigate properties of stabilizing and bounded solutions of these
differential equations and provide a theorem on the existence of the maximal
solution - the case of periodic coefficients is studied as well. Moreover
\cite{preprint99} contains a rigorous analysis of linearly perturbed matrix
Riccati differential equations of the form (\ref{eq:6.7.3}). For a discussion
of the infinite-dimensional version of (1.5) see \cite{DPZa92}.

The work by Freiling and Hochhaus (see \cite{FrHo4}, \cite {FrHo6},
\cite{FrHo5} or, alternatively, Chapters 6.7 and 6.8 of \cite{AFIJ03})
contains, in particular in the time-invariant case, a unified treatment of
the class of equations (\ref{eq:6.7.4}) and of the discrete-time version of
these equations.

The main goal of the authors of this paper is to investigate several
important problems concerning the solutions of time-varying nonlinear
differential equations of the form (\ref{eq:6.7.4}) by combining the methods
developed in their above-mentioned work (see also \cite{FCdS98}, \cite
{FrJa96} and \cite{FJAK96}). Although the differential equations (\ref
{eq:6.7.4}) cannot - in contrast to standard Riccati differential equations -
be transformed by a nonlinear transformation to a linear system of
differential equations, it turns out that it is possible to show that their
solutions behave in various aspects similar to the solutions of matrix
Riccati equations.

In the first part of the paper we shall investigate the properties of the
linear evolution operators associated to a class of linear differential
equations on the space of symmetric matrices, namely, linear differential
equations generating a positive evolution. For such a class of differential
equations we provide a set of conditions which characterizes the exponential
stability of the zero solution. The stability results presented in Section 2,
which are of independent interest, can be considered as extensions of the
classical stability results of Lyapunov and their generalizations by
Schneider \cite{Schn65} and serve as a basis for the proof of the existence
results presented in the second part of this work.

Sections 3--6 contain the main results concerning differential equations of
the form (\ref{eq:6.7.4}). We provide necessary and sufficient conditions
which guarantee the existence of some special solutions for the considered
nonlinear differential equations as: maximal solution, stabilizing solution,
minimal positive semi-definite solution. In particular it turns out that
under the assumption that the system underlying (\ref{eq:6.7.4}) satisfies
adequate generalized stabilizability, detectability an definiteness
conditions there exists a unique stabilizing solution $X_s$ of (\ref
{eq:6.7.4}); moreover, in the case of periodic coefficients,
\[
\lim_{t \to - \infty} \big[ X(t) - X_s(t) \big] = 0
\]
for any solution $X$ of (\ref{eq:6.7.4}) with $X(t_f) \geq 0$.

In Section 7 we indicate that the results derived explicitly in this paper
remain valid for more general classes of nonlinear differential equations.

\section{Linear differential equations generating positive evolutions}

\subsection{Preliminaries}

Let $\mathbb{R}^{m\times n}$ be the linear space of real $m \times n $
matrices. On $\mathbb{R}^{m\times n}$ we consider the usual inner
product
\begin{equation}
\langle A, B \rangle := \mathop{\rm{Tr}} B^T A =
\mathop{\rm{trace}} B^T A \label{eqno:2.1}
\end{equation}
for all $A$ and $B\in \mathbb{R}^{m\times n}$; throughout the paper the
superscript $T$ stands for the transpose of a matrix or a vector.

The norm induced by the inner product \eqref{eqno:2.1} is
\begin{equation}
\| A \| = \langle A, A \rangle^{1/2},  \label{eqno:2.2}
\end{equation}
which is known as the Frobenius norm of a matrix. Together with
the inner product \eqref{eqno:2.1} and the norm \eqref{eqno:2.2}
on $\mathbb{R}^{m\times n}$ we consider also the Euclidian norm (or
spectral radius) of the matrix $A$, that is
\begin{equation}
\| A \|_2 = \sqrt{\lambda_{\max}(A^TA)}.  \label{eqno:2.3}
\end{equation}
Since $\mathbb{R}^{m\times n}$ is a finite dimensional linear space it
follows that the norms \eqref{eqno:2.2} and \eqref{eqno:2.3} are
equivalent.

Let $\mathcal{S}^n \subset \mathbb{R}^{n \times n}$ be the linear subspace of
symmetric $n \times n$-matrices. It is obvious that the inner product
\eqref{eqno:2.1} induces on $\mathcal{S}^n$ the structure of a Hilbert space
while the norm \eqref{eqno:2.3} induces on $\mathcal{S}^n$ the structure of a
Banach space.

Let $\mathcal{S}^n_{+} := \{ S \in \mathcal{S}^n \mid S \geq 0 \}$
be the convex cone of positive semi-definite matrices. Then
$\mathcal{S}^n_{+}$ induces an order on $\mathcal{S}^n$: $X \geq
Y$ if and only if $X-Y\in \mathcal{S}^n_{+}$.

An operator $T \colon \mathcal{S}^n \to \mathcal{S}^m$ is
a positive operator if $T(\mathcal{S}^n_{+}) \subset
\mathcal{S}^m_{+}$; in this case we shall write $T \geq 0$.

The following properties of the positive linear operators will be
used often in the paper.

\begin{lemma} \label{lm2.1}
If $T \colon \mathcal{S}^n \to \mathcal{S}^n$ is a linear
operator then the following assertions hold:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $T\geq 0$ if and only if $T^* \geq 0$, $T^*$ being the adjoint operator
of $T$ with respect to the inner product \eqref{eqno:2.1}.
\item If
\begin{equation} \| T \| := \sup \{ \| TX \|_2:  X \in \mathcal{S}^n, \|X\|_2
\leq 1 \} \label{eqno:2.4} \end{equation} then $T \geq 0$ implies
$\| T \| = \|T I_n \|_2$ where $I_n$ is the identity $n \times
n$-matrix.
\end{enumerate}
\end{lemma}

\subsection{Linear differential equations generating positive evolutions}

We note that the continuity assumptions made on the coefficients
of the differential equations appearing in this paper are not
essential -- most results remain valid for locally integrable and
bounded coefficients if we consider solutions of \eqref{eq:6.7.4}
in the sense of Caratheodory.

Let $\mathcal{L} \colon \mathcal{I} \to \mathcal{B}(\mathcal{S}^n)$
be a continuous operator valued function where $\mathcal{I} \subseteq \mathbb{R}$
is an interval and $\mathcal{B}(\mathcal{S}^n)$ denotes the space of linear
operators defined on $\mathcal{S}^n$ with values in $\mathcal{S}^n$. On
$\mathcal{S}^n$ we consider the linear differential equation
\begin{equation} \frac{d}{dt} S(t) = \mathcal{L}(t)[S(t)].  \label{eqno:2.5}
\end{equation}
For each $t_0\in \mathcal{I}$ and $H \in \mathcal{S}^n$ we denote by
$S(\cdot,t_0,H)$ the solution of \eqref{eqno:2.5} with the initial value
$S(t_0,t_0,H) = H$. Based on known uniqueness arguments we obtain that the
function $H \to S(\cdot,t_0,H)$ is linear. For each $(t,t_0)$ the
linear operator $T(t,t_0) \colon \mathcal{S}^n \to \mathcal{S}^n$,
$T(t,t_0)[H] := S(t,t_0,H)$ is well-defined.

$T(\cdot,\cdot)$ will be termed the \textbf{linear evolution operator} on
$\mathcal{S}^n$ defined by the linear differential equation \eqref{eqno:2.5}.

The next proposition summarizes well known properties of linear
evolution operators:

\begin{prop} \label{prop2.2} We have:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $T(t,s) T(s,\tau) = T(t,\tau)$ for all $t,s,\tau \in \mathcal{I}$.
\item $T(t,t) = I_{\mathcal{S}^n}$ (the identity operator on
$\mathcal{S}^n$).
\item $t \mapsto T(t,s)$ satisfies
$\frac{d}{dt} T(t,s) = \mathcal{L}(t) T(t,s)$.

\item $s \mapsto T^*(t,s)$ satisfies
$ \frac{d}{ds} T^*(t,s) + \mathcal{L}^*(s) T^*(t,s) = 0$.

\item If $\mathcal{L}(t) \equiv \mathcal{L}$ then
$$ T(t,s) = e^{\mathcal{L}(t-s)} := \sum_{k=0}^{\infty} \frac{(t-s)^k}{k!}
\mathcal{L}^k, $$ where $\mathcal{L}^k$ is the $k$-th iteration of
the operator $\mathcal{L}$.

\item If there exists $\theta > 0$ such that $\mathcal{L}(t+\theta) =
\mathcal{L}(t)$ for all $t \in \mathbb{R}$, then $T(t+\theta, s+\theta)
= T(t,s)$ for all $t, s\in \mathbb{R}$.
\end{enumerate}
\end{prop}

Let $A \colon \mathcal{I} \to \mathbb{R}^{n\times n}$ be a
continuous function. Set
\begin{equation}
\mathcal{L}_A(t)[S] := A(t) S + S A^T(t) \quad \mbox{for all}
\quad S \in \mathcal{S}^n.  \label{eqno:2.6}
\end{equation}
By direct computation we may check that the linear evolution
operator defined by the linear differential equation
\eqref{eqno:2.5} in the particular case of \eqref{eqno:2.6} is
given by
\begin{equation}
T_A(t,t_0)[H] = \Phi_A(t,t_0) H \Phi_A^T(t,t_0),  \label{eqno:2.7}
\end{equation}
where $\Phi_A(t,t_0)$ is the transition matrix of the linear
differential equation
\[
\dot x(t) = A(t)x(t).
\]
The corresponding adjoint operators $\mathcal{L}_A^*(t)$ and
$T_A^*(t,t_0)$ are given here by
\[
\mathcal{L}_A^*(t)S = A^T(t) S + S A(t)
\]
and
\begin{equation}
T_A^*(t,t_0)[H] = \Phi_A^T(t,t_0) H \Phi_A(t,t_0).
\label{eqno:2.8}
\end{equation}

\begin{remark} \label{rmk2.3} \rm
>From \eqref{eqno:2.7} and \eqref{eqno:2.8} it is seen that both $T_A(t,t_0)$
and $T_A^*(t,t_0)$ are positive operators. \end{remark}

Motivated by the above remark we define

\begin{definition} \label{def2.4} \rm
The linear differential equation \eqref{eqno:2.5} (or
equivalently, the linear operator valued function $\mathcal{L}$)
\textbf{generates a positive evolution} if
$$ T(t,t_0) \geq 0 \quad \mbox{for all} \quad t,t_0 \in \mathcal{I}, \; t
\geq t_0. $$
\end{definition}

The next result shows that the property that generates a positive
evolution is preserved under some perturbations:

\begin{prop} \label{prop2.5}
Let $\mathcal{L}(t) := \mathcal{L}_0(t) + \Pi_1(t)$ be such that
$\mathcal{L}_0(t) $ generates a positive evolution and $\Pi_1(t) \colon
\mathcal{S}^n \to \mathcal{S}^n$ is a positive linear operator. Assume
additionally that $t \mapsto \mathcal{L}_0(t)$ and $t \mapsto \Pi_1(t)$ are
continuous operator valued functions. Under these assumptions the operator
valued function $\mathcal{L}$ generates a positive evolution.
\end{prop}

\begin{proof}
Let $T_0(t,t_0)$ be the linear evolution operator defined on
$\mathcal{S}^n$ by the linear differential equation
$$ \frac{d}{dt} S(t) = \mathcal{L}_0(t)[S(t)]. $$
By assumption it follows that $T_0(t,t_0) \geq 0$ for all $t \geq
t_0$, $t,t_0 \in \mathcal{I}$. Let $S(t) = S(t,t_0,H)$ be the
solution of the initial value problem
$$ \frac{d}{dt} S(t) = \mathcal{L}(t)[S(t)], \quad S(t_0,t_0,H) = H, \quad H
\geq 0. $$ We have to show that $S(t) \geq 0$ for $t \geq t_0$, $t \in
\mathcal{I}$. The solution $S(t)$ admits the representation
$$ S(t) = T_0(t,t_0) H + \int_{t_0}^t T_0(t,s) \Pi_1(s)[S(s)] \, ds.
$$
Let $\{ S_k(t) \}_{k \geq 0}$ be the sequence of Volterra
approximations defined as
\begin{gather*}
S_0(t)  := T_0(t,t_0) H, \quad t \geq t_0, \\
S_k(t)  := T_0(t,t_0) H + \int_{t_0}^t T_0(t,s) \Pi_1(s)[S_{k-1}(s)] \, ds,
\quad t \geq t_0, \quad k \geq 1.
\end{gather*}
It is known that
\begin{equation} S(t) = \lim_{k \to \infty} S_k(t), \quad t \geq t_0.
\label{eqno:2.9} \end{equation} On the other hand we obtain
inductively that $S_k(t) \geq 0$, hence, from \eqref{eqno:2.9} we
get $S(t) \geq 0$. Since $T(t,t_0)H = S(t,t_0,H)$ it follows that
$T(t,t_0) \geq 0$ and thus the proof is complete.
\end{proof}

 From Proposition \ref{prop2.5} and Remark \ref{rmk2.3} we obtain the
 following result.

\begin{corollary} \label{cor2.6}
Let $\mathcal{L}_{A, \Pi_1}(t) \colon \mathcal{S}^n \to
\mathcal{S}^n$ be defined by
\begin{equation} \mathcal{L}_{A, \Pi_1}(t)[S] := A(t) S + S A^T(t) +
\Pi_1(t)[S], \label{eqno:2.10} \end{equation} where $A \colon
\mathcal{I} \to \mathbb{R}^{n\times n}$, $\Pi_1 \colon
\mathcal{I} \to \mathcal{B}(\mathcal{S}^n)$ are continuous
functions and $\Pi_1(t) \geq 0$ for all $t \in \mathcal{I}$. Then
$\mathcal{L}$ generates a positive evolution.
\end{corollary}

\begin{remark} \label{rmk2.7} \rm
The adjoint operator of $\mathcal{L}_{A, \Pi_1}(t)$ defined by
\eqref{eqno:2.10} is given by
\begin{equation} \mathcal{L}_{A, \Pi_1}^*(t)[S] = A^T(t) S + S A(t) +
\Pi_1^*(t)[S] \quad \mbox{for all} \quad S \in \mathcal{S}^n.
\label{eqno:2.11} \end{equation}
\end{remark}

The operators \eqref{eqno:2.10} and \eqref{eqno:2.11} contain as
particular cases the Lyapunov-type operators studied starting with
pioneering works of Wonham \cite{Wonh68,Wonh70} in connection with
the problem of exponential stability of the zero solution of a
linear It\^{o} differential equation (see also \cite[Section
4.6]{Reid72}).

For $A$ and $\Pi_1$ not depending on $t$ the operators \eqref{eqno:2.10} and
\eqref{eqno:2.11} were also considered by Damm and Hinrichsen \cite{DaHi01},
Freiling and Hochhaus \cite{FrHo5} (see also \cite{AFJ94}, \cite{DPZa92},
\cite{FCdS98}, \cite{FrJa96}, \cite{FJAK96}).

The next result can be proved following step by step the proof of
\cite[Proposition 4.4]{preprint99}.

\begin{prop} \label{prop2.8}
Let $\mathcal{L}$ be an operator valued function which generates
a positive evolution on $\mathcal{S}^n$. If $t \mapsto \|
\mathcal{L}(t) \|$ is bounded then there exist $\alpha, \beta > 0$
such that
\[ T(t,t_0) I_n \geq \beta e^{-\alpha(t-t_0)} I_n \qquad \mbox{and} \qquad
T^*(t,t_0)I_n \geq \beta e^{-\alpha(t-t_0)}I_n \] for all $t \geq
t_0$, $t,t_0 \in \mathcal{I}$. \end{prop}

\subsection{Exponential stability}

In this subsection $\mathcal{I} \subset \mathbb{R}$ is a right
unbounded interval and $\mathcal{L} \colon \mathcal{I} \to
\mathcal{B}(\mathcal{S}^n)$ is a continuous operator valued
function.

\begin{definition} \label{def2.9} \rm
We say that $\mathcal{L}$ \textbf{generates an exponentially
stable evolution} if there exist $\alpha, \beta > 0$ such that
$$ \| T(t,t_0) \| \leq \beta e^{-\alpha(t-t_0)} \quad \mbox{for} \quad t \geq
t_0, \; t, t_0 \in \mathcal{I}. $$
\end{definition}

Our goal is to provide some necessary and sufficient conditions
for exponential stability in the case when $\mathcal{L}$ generates
a positive evolution.

\begin{theorem} \label{thm2.10}
Let $\mathcal{L} \colon \mathbb{R}_+ \to
\mathcal{B}(\mathcal{S}^n)$ be a bounded and continuous operator
valued function such that $\mathcal{L}$ generates a positive
evolution. Then the following are equivalent:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $\mathcal{L}$ defines an exponentially stable evolution.
\item There exists some $\delta > 0$ such that
$$ \int_{t_0}^t \| T(t,s) \| \, ds \leq \delta \quad \mbox{for all} \quad t
\geq t_0, \; t_0 \in \mathbb{R}_+. $$
\item There exists some $\delta > 0$ being independent of $t$ and $t_0$ with
\begin{equation} \int_{t_0}^t T(t,s) I_n \, ds\leq \delta I_n \quad \mbox{for
all} \quad t \geq t_0, \; t_0 \in \mathbb{R}_+. \label{eqno:2.12}
\end{equation}
\end{enumerate}
\end{theorem}

\begin{proof} (i) $\Rightarrow$ (ii) obvious (from definition).

(ii) $\Rightarrow$ (iii) follows from $0 \leq T(t,s) I_n \leq \|
T(t,s) \| I_n$.

It remains to prove (iii) $\Rightarrow$ (i). Therefore let $H
\colon \mathbb{R}_+ \to \mathcal{S}^n$ be a bounded and
continuous function. This means that there exist $\gamma_1,
\gamma_2 \in \mathbb{R}$ such that $\gamma_1 I_n \leq H(s) \leq
\gamma_2 I_n$ for $s\geq 0$. Since $T(t,s)$ is a positive operator
we obtain
$$ \gamma_1 T(t,s) I_n \leq T(t,s) H(s) \leq \gamma_2 T(t,s) I_n, $$
hence
$$ \gamma_1 \int_0^t T(t,s) I_n \, ds \leq \int_0^t T(t,s)H(s) \, ds \leq
\gamma_2 \int_0^t T(t,s) I_n \, ds. $$
Using Proposition \ref{prop2.8} and
inequalities \eqref{eqno:2.12} we get
$$ \tilde{\gamma}_1 I_n \leq \int_0^t T(t,s)H(s) \, ds \leq \tilde{\gamma}_2
I_n \quad \mbox{for} \quad t\geq 0 $$ with some constants
$\tilde{\gamma}_1, \tilde{\gamma}_2 \in \mathbb{R}$. So we have
obtained that $t \to \int_0^t T(t,s)H(s) \, ds$ is bounded
for arbitrary bounded and continuous functions $H(s)$. Applying
Perron's theorem (see \cite{Hala66}) we conclude that there exist
$\alpha, \beta > 0$ such that
$$ \| T(t,t_0) \| \leq \beta e^{-\alpha(t-t_0)} \quad \mbox{for all} \quad t
\geq t_0 \geq 0, $$ and the proof is complete. \end{proof}

Throughout the paper a function $H \colon \mathcal{I} \to
\mathcal{S}^n$ is termed \textbf{uniformly positive} and we write $H(t) \gg
0$ if there is a constant $c > 0$ such that $H(t) \geq c I_n > 0$ for all $t
\in \mathcal{I}$.

The next theorem is a time-varying version of results that have
been summarized e.g.\ in \cite{DaHi01} and \cite{FCdS98} and
generalizes classical results from Lyapunov theory which are in
the time-invariant case essentially due to Schneider
\cite{Schn65}.

\begin{theorem} \label{thm2.11}
Let $\mathcal{L} \colon \mathcal{I} \to
\mathcal{B}(\mathcal{S}^n)$ be a continuous operator valued
function such that $\mathcal{L}$ generates a positive evolution
and $t \mapsto \| \mathcal{L}(t) \|$ is bounded. Then the
following are equivalent:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $\mathcal{L}$ defines an exponentially stable evolution.
\item There exist $\alpha, \beta_1 > 0$ such that
$$ \| T^*(t,t_0) \| \leq \beta_1 e^{-\alpha(t-t_0)} \quad \mbox{for all}
\quad t \geq t_0, \; t, t_0 \in \mathcal{I}. $$
\item There exists $\delta > 0$ being independent of $t$ with
$$ \int_t^{\infty} T^*(s,t) I_n \, ds \leq \delta I_n \quad \mbox{for all}
\quad t \in \mathcal{I}. $$
\item The differential equation
\begin{equation} \frac{d}{dt} K(t) + \mathcal{L}^*(t)[K(t)] + I_n = 0
\label{eqno:2.13} \end{equation} has a bounded and uniformly
positive solution $K \colon \mathcal{I} \to \mathcal{S}^n$.
\item For every continuous and bounded function $H \colon \mathcal{I}
\to \mathcal{S}^n$ with $H(t)\gg 0$ the differential
equation
\begin{equation} \frac{d}{dt} K(t) + \mathcal{L}^*(t)[K(t)] + H(t) = 0
\label{eqno:2.14} \end{equation} has a bounded and uniformly
positive solution.
\item There is a continuous and bounded function $H \colon \mathcal{I}
\to \mathcal{S}^n$, $H(t)\gg 0$, such that the
corresponding differential equation \eqref{eqno:2.14} has a
bounded solution $\hat K(t)\geq 0$, $t \in \mathcal{I}$.
\item There exists a bounded $C^1$-function $K \colon \mathcal{I} \to
\mathcal{S}^n$ with bounded derivative and $K(t) \gg 0$ which fulfills the
linear differential inequality
\begin{equation} \frac{d}{dt} K(t) + \mathcal{L}^*(t)[K(t)] \ll 0.
\label{eqno:2.15} \end{equation}
\end{enumerate}
\end{theorem}

\begin{proof} First we prove (i) $\Leftrightarrow$ (ii). Let
$\|| T |\|$ be the norm of the operator $T$ induced by
\eqref{eqno:2.2} and $\| T \|$ be the norm defined by
\eqref{eqno:2.4}. Since the norms defined by \eqref{eqno:2.2} and
\eqref{eqno:2.3} are equivalent it follows that there exist
positive constants $c_1, c_2$ such that $$ c_1 \| T(t,t_0) \| \leq
\|| T(t,t_0) \|| \leq c_2 \| T(t,t_0) \| $$ and
$$ c_1 \| T^*(t,t_0) \| \leq \|| T^*(t,t_0) \|| \leq c_2 \| T^*(t,t_0)\|. $$
Taking into account that $\|| T(t,t_0) \|| = \|| T^*(t,t_0) \||$
one obtains that
$$ \frac{c_1}{c_2} \, \| T(t,t_0) \| \leq \| T^*(t,t_0)\| \leq
\frac{c_2}{c_1}\, \| T(t,t_0) \| $$ which shows that (i)
$\Leftrightarrow$ (ii).

The implication (ii) $\Rightarrow$ (iii) is obvious.

We prove now (iii) $\Rightarrow$ (iv). Let $K_0(t) := \int_t^{\infty}
T^*(s,t) I_n \, ds$. From (iii) it follows that $K_0$ is well defined and
bounded. Applying Proposition \ref{prop2.8} we deduce that there exists
$\delta_1 > 0$ such that $K_0(t) \geq \delta_1 I_n$. By direct computation,
based on Proposition \ref{prop2.2}, we get that $K_0$ is differentiable and
it solves equation \eqref{eqno:2.13}.

We prove now (iv) $\Rightarrow$ (iii). Let $K \colon \mathcal{I} \to
\mathcal{S}^n$ be a bounded and uniformly positive solution of equation
\eqref{eqno:2.13}. For all $t \leq \tau$, $t,\tau \in \mathcal{I}$ we can
write the representation formula
$$ K(t) = T^*(\tau,t) K(\tau) + \int_t^{\tau} T^*(s,t)I_n \, ds. $$
Since $T^*(\tau,t)$ is a positive operator and $K$ is a positive
and bounded function, we conclude that
$$ 0 \leq \int_t^{\tau} T^*(s,t) I_n \, ds \leq K(t) \leq \delta I_n $$
for all $t \leq \tau$, $t, \tau \in \mathcal{I}$, where $\delta >
0$ does not depend on $t$ and $\tau$. Taking the limit for $\tau
\to \infty$ we obtain that (iii) holds.

We prove now (iii) $\Rightarrow$ (v). Let $H \colon \mathcal{I}
\to \mathcal{S}^n$ be a continuous function with the
property $0 < \nu_1 I_n \leq H(s) \leq \nu_2 I_n$ for all $s \in
\mathcal{I}$ with $\nu_1, \nu_2$ being positive constants. Since
$T^*(s,t)$ is a positive operator we obtain
$$ \nu_1 T^*(s,t) I_n \leq T^*(s,t) H(s)\leq \nu_2 T^*(s,t) I_n, $$
which leads to
$$ \nu_1 \int_t^{\tau} T^*(s,t) I_n \, ds \leq \int_t^{\tau} T^*(s,t) H(s) \,
ds \leq \nu_2 \int_t^{\tau} T^*(s,t) I_n \, ds\,.
$$
Taking the limit for $\tau \to \infty$ and invoking (iii), we deduce
that for $t$ in $\mathcal{I}$,
$$ \nu_1 \int_t^{\infty} T^*(s,t) I_n \, ds  \leq
\int_t^{\infty} T^*(s,t) H(s) \, ds
\leq \nu_2 \int_t^{\infty} T^*(s,t) I_n \, ds \leq \nu_2 \delta I_n\,.
$$
We define $\tilde K(t) := \int_t^{\infty} T^*(s,t) H(s) \, ds$.
Applying Proposition \ref{prop2.8} we get
$$ \nu_1 \frac{\beta}{\alpha} I_n \leq \tilde{K}(s) \leq \nu_2 \delta I_n
\quad \mbox{for all} \quad t \in \mathcal{I}. $$ Using Proposition
\ref{prop2.2} we obtain
$$ \tilde{K}(t) = T^*(\tau,t) \tilde{K}(\tau) + \int_t^{\tau} T^*(s,t) H(s)
\, ds, $$ which shows that $\tilde{K}$ is a solution of
\eqref{eqno:2.14} and thus (v) is fulfilled.

(v) $\Rightarrow$ (vi) is obvious.

We prove now (vi) $\Rightarrow$ (ii). Let $\hat{K} \colon \mathcal{I} \to
\mathcal{S}^n$ be a bounded solution of equation \eqref{eqno:2.14} such that
$\hat{K}(t) \geq 0$, $t \in \mathcal{I}$. For each $t < \tau$, $t, \tau \in
\mathcal{I}$, we write
$$ \hat K(t) = T^*(\tau,t) \hat{K}(\tau) + \int_t^{\tau} T^*(s,t) H(s) \, ds.
$$
Since $T^*(\tau, t)\geq 0$ and $0 \leq \hat K(t)\leq \rho I_n$ for
$\rho > 0$ not depending on $t$, we deduce that
$$ 0 \leq \int_t^{\tau} T^*(s,t) H(s) \, ds \leq \hat{K}(t) \leq \rho I_n, $$
which leads to $0 \leq \int_t^{\infty} T^*(s,t) H(s) \, ds \leq \rho I_n$ for
$t \in \mathcal{I}$. Based on Proposition \ref{prop2.8} we deduce that there
exists $\rho_0 > 0$ such that
\begin{equation} \rho_0 I_n \leq \int_t^{\infty} T^*(s,t) H(s) \, ds =:
\tilde{K}(t) \leq \rho I_n. \label{eqno:2.16} \end{equation} Let
$t_0 \in \mathcal{I}$ be fixed and let us define $G(t) :=
T^*(t,t_0) \tilde{K}(t)$ for all $t \geq t_0$. It follows that
$G(t) = \int_t^{\infty} T^*(s,t_0) H(s) \, ds$. We have directly
\begin{equation} \frac{d}{dt} G(t) = - T^*(t,t_0) H(t) \quad \mbox{for all}
\quad t \geq t_0. \label{eqno:2.17} \end{equation} On the other hand there
exist positive constants $\nu_i$, $i = 1, 2$, such that
$$ \nu_1 I_n \leq H(t)\leq \nu_2 I_n, $$
which leads to
\begin{equation} \nu_1 T^*(t,t_0) I_n \leq T^*(t,t_0) H(t) \leq \nu_2
T^*(t,t_0) I_n. \label{eqno:2.18} \end{equation} From
\eqref{eqno:2.17} and \eqref{eqno:2.18} we infer
$$ \frac{d}{dt} G(t) \leq - \nu_1 T^*(t,t_0) I_n \quad \mbox{for all} \quad
t \geq t_0. $$ On the other hand from \eqref{eqno:2.16} we get
\begin{equation} \rho_0 T^*(t,t_0) I_n \leq G(t) \leq \rho T^*(t,t_0) I_n,
\label{eqno:2.19} \end{equation} which leads to $\frac{d}{dt} G(t) \leq -
\alpha G(t)$ where $\alpha = \frac{\nu_1}{\rho}$. By a standard argument we
get $G(t) \leq e^{-\alpha(t-t_0)} G(t_0)$ for all $t \geq t_0$. Using again
\eqref{eqno:2.19} we deduce that
$$ T^*(t,t_0) I_n \leq \frac{\rho}{\rho_0} e^{-\alpha(t-t_0)} I_n. $$
Hence $\|T^*(t,t_0) I_n\|_2 \leq \frac{\rho}{\rho_0}
e^{-\alpha(t-t_0)}$, therefore (ii) follows from Lemma \ref{lm2.1}, (ii).

(iv) $\Rightarrow$ (vii) is obvious since any bounded and positive
solution of \eqref{eqno:2.13} is also a solution of
\eqref{eqno:2.15}.

(vii) $\Rightarrow$ (vi). If $K \colon \mathcal{I} \to
\mathcal{S}^n$, $K(t) \gg 0$, is a bounded $C^1$-function with
bounded derivative which solves \eqref{eqno:2.15}, we define $H(t)
:= - \frac{d}{dt} K(t) - \mathcal{L}^*(t)[K(t)]$. Therefore $K$ is
a bounded and positive semi-definite solution of \eqref{eqno:2.14}
corresponding to this particular choice of $H$ and the proof ends.
\end{proof}

\begin{corollary} \label{coro2.12}
Let $\mathcal{L} := \mathcal{L}_{A, \Pi_1}$ be defined as in
\eqref{eqno:2.10} with $\Pi_1(t) \geq 0$. If $\mathcal{L}$ defines
an exponentially stable evolution then the zero state equilibrium
of the linear differential equation
$$ \dot x(t) = A(t)x(t) $$
is exponentially stable. \end{corollary}

\begin{proof} Using the equivalence (i) $\Leftrightarrow$ (vii) of
Theorem \ref{thm2.11} we deduce that there exists a bounded $C^1$-function $X
\colon \mathcal{I} \to \mathcal{S}^n$ with bounded derivative and $ X(t)\gg
0$ which satisfies \eqref{eqno:2.15}. Then it follows that
$$ \frac{d}{dt} X(t) + A^T(t) X(t) +X(t) A(t) \ll 0, $$
which guarantees the exponential stability of the zero solution of the linear
differential  equation defined by $A(t)$ and the proof ends. \end{proof}

The next result extends to the case of operators generating
positive evolutions, a well known result concerning the uniqueness
of bounded solutions of Lyapunov equations.

\begin{theorem} \label{thm2.13}
Let $\mathcal{L} \colon \mathcal{I} \to
\mathcal{B}(\mathcal{S}^n)$ be a continuous and bounded operator valued
function which generates a positive and exponentially stable evolution. Then:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item For any bounded and continuous function $H \colon \mathcal{I}
\to \mathcal{S}^n$ the differential equation
\eqref{eqno:2.14} has a unique solution bounded on $\mathcal{I}$.
This solution has the representation
\begin{equation} \tilde{K}(t) = \int_t^{\infty} T^*(s,t) H(s) \, ds \quad
\mbox{for} \quad  t \in \mathcal{I}, \label{eqno:2.20}
\end{equation} $T(s,t)$ being the linear evolution operator
defined by the linear differential equation \eqref{eqno:2.5}.
\item If $H(s) \geq 0$ then $\tilde{K}(t) \geq 0$, and if $H(s) \gg 0$
then $\tilde{K}(t) \gg 0$.
\item If $\mathcal{L}$, $H$ are periodic functions with period $\theta$, then
$\tilde{K}$ is a $\theta$-periodic function, and if $\mathcal{L}(t) \equiv
\mathcal{L}$ and $H(t) \equiv H$, then $\tilde{K}(t) \equiv \tilde{K}$ is
constant with
$$ \tilde K = - (\mathcal{L}^*)^{-1}[H] = \int_0^{\infty} e^{\mathcal{L}^*s}
H \, ds\,. $$
\end{enumerate}
\end{theorem}

\begin{proof} (a) Since $\mathcal{L}(t)$ defines a stable evolution
$\int_t^{\infty} T^*(s,t) H(s) \, ds $ is convergent for each $t
\in \mathcal{I}$. Hence $\tilde{K}(t) = \int_t^{\infty}
T^*(s,t)H(s) \, ds$ is well defined. For each $\tau \geq t$,
$\tau, t\in \mathcal{I}$ we write
$$ \tilde{K}(t) = T^*(\tau, t) \tilde{K}(\tau) + \int_t^{\tau} T^*(s,t) H(s)
\, ds. $$ Based on Proposition \ref{prop2.2} we conclude that $\tilde{K}$ is
differentiable and it solves \eqref{eqno:2.14}.

Let now $\hat{K}$ be another bounded solution of \eqref{eqno:2.14}. We may
write
\begin{equation} \hat{K}(t) = T^*(\tau, t) \hat{K}(\tau) + \int_t^{\tau}
T^*(s,t) H(s) \, ds. \label{eqno:2.21} \end{equation} Based on exponential
stability and boundedness of $\hat{K}$ we deduce
$$ \lim_{\tau \to \infty} T^*(\tau, t) \hat{K}(\tau) = 0. $$
Taking the limit for $\tau \to \infty$ in
\eqref{eqno:2.21} we get
$$ \hat{K}(t) = \int_t^{\infty} T^*(s,t) H(s) \, ds = \tilde{K}(t), $$
and thus the uniqueness of the bounded solution is proved.

(b) If $H(s) \geq 0$, then $T^*(s,t) H(s) \geq 0$, which implies
$\tilde{K}(t) \geq 0$. If $H(s) \geq \mu I_n$, then, applying
Proposition \ref{prop2.8}, we obtain that $\tilde{K}(t) \geq \tilde \mu I_n$
for all $t \in \mathcal{I}$.

(c) Let us suppose that there exists $\theta > 0$ such that
$\mathcal{L}(t+\theta) = \mathcal{L}(t)$ and $H(t+\theta) = H(t)$
for all $t \in \mathcal{I}$. We have
\begin{align*} \tilde{K}(t+\theta) & = \int_{t+\theta}^{\infty}
T^*(s,t+\theta) H(s) \, ds = \int_t^{\infty} T^*(\sigma+\theta,
t+\theta) H(\sigma+\theta) \, d\sigma \\ & = \int_t^{\infty}
T^*(\sigma, t) H(\sigma) \, d\sigma = \tilde{K}(t), \end{align*}
hence $\tilde{K}$ is a $\theta$-periodic function. Let now
$\mathcal{L}(t) \equiv \mathcal{L}$, $H(t) \equiv H$, $t \in
\mathcal{I}$. In this case the representation formula becomes
$$ \tilde{K}(t) = \int_t^{\infty} e^{\mathcal{L}^*(s-t)} H \, ds =
\int_0^{\infty} e^{\mathcal{L}^*s} H \, ds = \tilde{K}(0) \quad
\mbox{for all} \quad t \in \mathcal{I}, $$ which shows that
$\tilde{K}$ is constant.

On the other hand the fact that $\mathcal{L}$ is a linear operator
defined on a finite dimensional Hilbert space together with
exponential stability implies that the eigenvalues of the
operators $\mathcal{L}$ and $\mathcal{L}^*$ are located in the
left half-plane. Hence $\mathcal{L}^*$ is invertible. This shows
that $\check{K} := -(\mathcal{L}^*)^{-1}[H]$ is a constant
solution of \eqref{eqno:2.14} but from the uniqueness of the
bounded solution of \eqref{eqno:2.14} it follows that $\check{K} =
\tilde{K}$, thus the proof is complete.
\end{proof}

Let $H \colon \mathcal{I} \to \mathcal{S}^n$ be a fixed continuous function.
If $\mathcal{L}(t) \colon \mathcal{S}^n \to \mathcal{S}^n$ is a linear
operator and $t \mapsto \mathcal{L}(t)$ is a continuous function then we
associate with $\mathcal{L}$ two linear differential equations on
$\mathcal{S}^n$:
\begin{equation}
\frac{d}{dt} X(t) = \mathcal{L}(t)[X(t)] + H(t)  \label{eqno:2.22}
\end{equation}
and
\begin{equation}
\frac{d}{dt} Y(t) + \mathcal{L}^*(t)[Y(t)] + H(t) = 0.
\label{eqno:2.23}
\end{equation}
Any solution of equation \eqref{eqno:2.22} has the representation
\begin{equation}
X(t) = T(t,t_0) X(t_0) + \int_{t_0}^{t} T(t,s) H(s) \, ds \quad
\mbox{for} \quad t \geq t_0, \; t, t_0 \in \mathcal{I}
\label{eqno:2.24}
\end{equation}
while any solution of equation \eqref{eqno:2.23} admits the
representation
\begin{equation}
Y(t) = T^*(t_1,t) Y(t_1) + \int_t^{t_1} T^*(s,t) H(s) \, ds \quad
\mbox{for} \quad t \leq t_1, \; t \in \mathcal{I}.
\label{eqno:2.25}
\end{equation}

\begin{remark} \label{rmk2.14} \rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item The representation formulae \eqref{eqno:2.24} and \eqref{eqno:2.25}
suggest to term equation \eqref{eqno:2.22} as the \textbf{forward
equation} defined by $\mathcal{L}$ and equation \eqref{eqno:2.23}
as the \textbf{backward equation} defined by $\mathcal{L}$.
\item From \eqref{eqno:2.24} it is seen that if $H$ is bounded and
$\mathcal{L}$ generates an exponentially stable evolution then all
solutions of the forward equation \eqref{eqno:2.22} are bounded on
any subinterval $[t_0, \infty) \subset \mathcal{I}$. Under the
same assumptions the backward equation \eqref{eqno:2.23} has a
unique solution which is bounded on all subintervals $[t_0,
\infty) \subset \mathcal{I}$, namely the solution given by
\eqref{eqno:2.20}.
\item From \eqref{eqno:2.24}, \eqref{eqno:2.25} it follows that if
$\mathcal{L}$ generates a positive evolution and $H(s) \geq 0$ for
$s \in \mathcal{I}$ and if there exists $\tau \in \mathcal{I}$
such that $X(\tau) \geq 0$, $Y(\tau) \geq 0$ respectively, then
$X(t) \geq 0$ for all $t \geq \tau$ and $Y(t) \geq 0$ for all $t
\in (-\infty, \tau] \cap \mathcal{I}$, respectively.
\end{enumerate}
\end{remark}

The next result shows that, in the case when $\mathcal{I} = \mathbb{R}$, a version
of Theorem \ref{thm2.13} for the forward differential equation \eqref{eqno:2.22}
holds.

\begin{prop} \label{prop2.15}
Let $\mathcal{L} \colon \mathbb{R} \to
\mathcal{B}(\mathcal{S}^n)$ be a continuous and bounded function.
Assume that $\mathcal{L}$ generates an exponentially stable
evolution. Then:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item For each bounded and continuous function $H \colon \mathbb{R} \to
\mathcal{S}^n$ the forward differential equation \eqref{eqno:2.22}
has a unique bounded solution on $\mathbb{R}$,
$$ \tilde X(t) = \int_{-\infty}^t T(t,s) H(s) \, ds. $$
\item If $\mathcal{L}$ generates a positive evolution and $H(t) \geq
0$, then $\tilde{X}(t) \geq 0$ for all $t\in \mathbb{R}$, and if $H(t)
\gg 0$, then $\tilde{X}(t) \gg 0$.
\item If $\mathcal{L}$ and $H$ are $\theta$-periodic functions,
then $\tilde{X}$ is $\theta$-periodic function too. If $\mathcal{L}(t) \equiv
\mathcal{L}$ and $H(t) \equiv H$ for all $t \in \mathbb{R}$, then $\tilde{X}$
is constant, and it is given by
$$ \tilde{X} = -\mathcal{L}^{-1}[H]. $$
\end{enumerate}
\end{prop}

\begin{proof} The proof is similar to the one of Theorem \ref{thm2.13} and it is based on
the representation formula \eqref{eqno:2.24}. \end{proof}

Associated with $\mathcal{L} \colon \mathbb{R} \to
\mathcal{B}(\mathcal{S}^n)$ we define $\mathcal{L}^{\sharp} \colon \mathbb{R}
\to \mathcal{B}(\mathcal{S}^n)$ by $\mathcal{L}^{\sharp}(t) :=
\mathcal{L}^*(-t)$. If $t \mapsto \mathcal{L}(t)$ is continuous then $t
\mapsto \mathcal{L}^{\sharp}(t)$ is also continuous.

Let $T^{\sharp}(t,t_0)$ be the linear evolution operator on
$\mathcal{S}^n$ defined by the differential equation
\[
\frac{d}{dt} S(t) = \mathcal{L}^{\sharp}(t)[S(t)].
\]

\begin{prop} \label{prop2.16}
The following assertions hold:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $T^{\sharp}(t,t_0) = T^*(-t_0,-t)$ for all $t, t_0 \in \mathbb{R}$,
$T(\cdot, \cdot)$ being the linear evolution operator defined by
\eqref{eqno:2.5}.
\item $\mathcal{L}$ generates a positive evolution if and only if
$\mathcal{L}^{\sharp}$ generates a positive evolution.
\item $\mathcal{L}$ generates an exponentially stable evolution if and only
if $\mathcal{L}^{\sharp}$ generates an exponentially stable
evolution.
\item $X \colon \mathbb{R} \to \mathcal{S}^n$ is a solution of the forward
equation \eqref{eqno:2.22} if and only if $Y(t) = X(-t)$ defines a solution
of the backward equation
$$ \frac{d}{dt} Y(t) + [\mathcal{L}^{\sharp}(t)]^*(Y(t)) + H(-t) = 0. $$
\end{enumerate}
\end{prop}

\begin{proof} (i) follows directly from the uniqueness of the solution of a
linear initial value problem.

(ii) and (iii) follow from (i), while (iv) is obtained by direct
calculation.
\end{proof}

\begin{remark} \label{rmk2.17} \rm
If $\mathcal{L}(t) \equiv \mathcal{L}$ for all $t \in \mathbb{R}$,
then $\mathcal{L}^{\sharp} = \mathcal{L}^*$. In this case
Proposition \ref{prop2.16} recovers the well-known fact from stationary
framework that the operator $\mathcal{L}$ generates an
exponentially stable evolution if and only if its adjoint operator
$\mathcal{L}^*$ generates an exponentially stable evolution. In
the time-varying case the fact that the operator $\mathcal{L}$
generates an exponentially stable evolution does not guarantee
that $\mathcal{L}^*$ generates an exponentially stable evolution.
\end{remark}

Combining the results from Theorem \ref{thm2.11} and Proposition \ref{prop2.16}
 one obtains necessary and sufficient conditions for exponential stability expressed in
terms of the forward differential equation \eqref{eqno:2.22} in the case
$\mathcal{I} = \mathbb{R}$.

\begin{theorem} \label{thm2.18}
Let $\mathcal{L} \colon \mathbb{R} \to
\mathcal{B}(\mathcal{S}^n)$ be a continuous and bounded operator
valued function which generates a positive evolution. Then the
following are equivalent:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $\mathcal{L}$ defines an exponentially stable evolution.
\item There exists some $\delta > 0$ such that
$$ \int_{-\infty}^t T(t,s) I_n \, ds \leq \delta I_n \quad \mbox{for all}
\quad t \in \mathbb{R}. $$
\item The forward differential equation
$$ \frac{d}{dt} K(t) = \mathcal{L}(t)[K(t)] + I_n $$
has a bounded and uniformly positive solution.
\item For any bounded and continuous function $H \colon \mathbb{R} \to
\mathcal{S}^n$ with $H(t) \gg 0$ the differential equation
\eqref{eqno:2.22} has a bounded and uniformly positive solution on
$\mathbb{R}$.
\item There is a bounded and continuous function $H \colon \mathbb{R} \to
\mathcal{S}^n$ with $H(t) \gg 0$ such that the corresponding forward
differential equation \eqref{eqno:2.22} has a bounded solution $\tilde{X}
\colon \mathbb{R} \to \mathcal{S}^n$ with $\tilde{X}(t) \geq 0$.
\item There exists a bounded $C^1$-function $K \colon \mathbb{R} \to
\mathcal{S}^n$ with bounded derivative and $K(t) \gg 0$ with satisfies
$$ \frac{d}{dt} K(t) - \mathcal{L}(t) K(t) \gg 0 \quad \mbox{for all} \quad t
\in \mathbb{R}. $$
\end{enumerate}
\end{theorem}

\begin{remark} \label{rmk2.19} \rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item When $\mathcal{L}$ is a periodic function with period $\theta$, then
the function $K$ in (vi) may be chosen as a $\theta$-periodic one.
\item If $\mathcal{L}(t) \equiv \mathcal{L}$ then the results stated in
Theorems \ref{thm2.11} and \ref{thm2.18} reduce to the known results from the
time-invariant case \cite{DaHi01,FrHo5}.
\end{enumerate}
\end{remark}

At the end of this section we shall prove a time-varying version
of the result in \cite[Lemma 3.7]{FrHo5}.

First we introduce the following concept of detectability which extends the
classical definition of the detectability of a linear system to this general
framework.

\begin{definition} \label{def2.20} \rm
If $\mathcal{L}_{A, \Pi_1}$ is defined as in \eqref{eqno:2.10}
and if $C \colon \mathcal{I} \to \mathbb{R}^{p \times n}$ is a
bounded and continuous function then the pair $(C, \mathcal{L}_{A,
\Pi_1})$ -- or equivalently the triple $(C, A, \Pi_1)$ -- is
called \textbf{detectable} if there exists a bounded and
continuous function $K \colon \mathcal{I} \to
\mathbb{R}^{n\times p}$ such that the operator $\mathcal{L}_{A+KC,
\Pi_1}$ generates an exponentially stable evolution. A function
$K$ having the above-mentioned properties is called a
\textbf{stabilizing injection}.
\end{definition}

Applying the implication (vii) $\Rightarrow$ (i) in Theorem \ref{thm2.11}
and taking into account the form of the adjoint operator of
$\mathcal{L}_{A+KC, \Pi_1}(t)$ we obtain immediately:

\begin{corollary} \label{coro2.21}
Let $A \colon \mathcal{I} \to \mathbb{R}^{n \times n}$, $C
\colon \mathcal{I} \to \mathbb{R}^{p \times n}$ and $\Pi_1
\colon \mathcal{I} \to \mathcal{B}(\mathcal{S}^n)$ be
continuous functions with $\Pi_1(t) \geq 0$ for all $t \in
\mathcal{I}$. Then the following are equivalent:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $(C, A, \Pi_1)$ is detectable.
\item There exist a bounded $C^1$-function $X \colon \mathcal{I} \to
\mathcal{S}^n$ with bounded derivative and $X(t) \gg 0$ and a
bounded and continuous function $D \colon \mathcal{I} \to
\mathbb{R}^{n\times p}$ which solve the linear differential inequality
\begin{align} \frac{d}{dt} X(t) + \mathcal{L}_{A, \Pi_1}^*(t)[X(t)] + D(t)
C(t) + C^T(t) D^T(t) \ll 0. \label{eqno:2.26} \end{align}
\end{enumerate}
Moreover if $(X, D)$ is a solution of \eqref{eqno:2.26} with $X(t)
\gg 0$ then $K(t) := X^{-1}(t) D(t)$ is a stabilizing injection.
\end{corollary}

Definition \ref{def2.20} of the detectability can be extended in a natural
way to the case of operators which generate a positive evolution:

\begin{definition} \label{def2.22} \rm
If $\mathcal{L} \colon \mathcal{I} \to
\mathcal{B}(\mathcal{S}^n)$ is a bounded and continuous operator
valued function such that $\mathcal{L}$ generates a positive
evolution and if $C \colon \mathcal{I} \to \mathbb{R}^{p\times
n}$ is a bounded and continuous function then the pair
$(C,\mathcal{L})$ is called \textbf{detectable} if there exists a
bounded and continuous function $K \colon \mathcal{I} \to
\mathbb{R}^{n\times p}$ such that the operator $\mathcal{L}^K(t) \colon
\mathcal{S}^n \to \mathcal{S}^n$ with
\begin{equation} \mathcal{L}^K(t)[S] := \mathcal{L}(t)[S] + K(t) C(t) S + S
C^T(t) K^T(t) \label{eqno:2.27} \end{equation} generates a
positive and exponentially stable evolution. \end{definition}

Now we prove the following result.

\begin{theorem} \label{thm2.23}
Let $\mathcal{L} \colon \mathcal{I} \to \mathcal{B}(\mathcal{S}^n)$ be a
bounded and continuous operator valued function, and let $C \colon
\mathcal{I} \to \mathbb{R}^{p \times n}$ be a bounded and continuous
function. Assume:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $\mathcal{L}$ generates a positive evolution;
\item $(C,\mathcal{L})$ is detectable;
\item The backward differential equation
\begin{equation} \frac{d}{dt} P(t) + \mathcal{L}^*(t)[P(t)] + C^T(t) C(t) = 0
\label{eqno:2.28} \end{equation}
 has a bounded and positive
semi-definite solution $P \colon \mathcal{I} \to
\mathcal{S}^n$.
\end{enumerate}
Under these assumptions the operator $\mathcal{L}$ generates an
exponentially stable evolution. \end{theorem}

\begin{proof} Our proof extends the ideas of \cite[Lemma 3.2]{FCdS98} to the
time-varying case. Let $(t_0, H) \in \mathcal{I} \times
\mathcal{S}^n_+$ be fixed and let $X(t) = X(t,t_0, H)$ be the
solution of the linear differential equation
\begin{equation} \frac{d}{dt} X(t) = \mathcal{L}(t)[X(t)], \quad X(t_0) = H.
\label{eqno:2.29} \end{equation}
Since $\mathcal{L}$ generates a
positive evolution we have $X(t) \geq 0$ for all $t \geq t_0$. If
$K$ is a stabilizing injection then equation \eqref{eqno:2.29} may
be rewritten as
\begin{equation} \frac{d}{dt} X(t) = \mathcal{L}^K(t)[X(t)] - K(t)C(t)X(t) -
X(t)(K(t)C(t))^T, \label{eqno:2.30}
\end{equation}
where $\mathcal{L}^K$ is defined in \eqref{eqno:2.27}. Moreover if
$T^K(t,s)$ is the linear evolution operator defined by
$\mathcal{L}^K$ on $\mathcal{S}^n$, then there exist $\alpha,
\beta > 0$ such that
$$ \| T^K(t,s) \| \leq \beta e^{-2\alpha(t-s)} \quad \mbox{for} \quad t \geq
s, \; t, s \in \mathcal{I}.
$$
Consider the perturbed operator $\mathcal{L}_{\varepsilon}$ defined by
$\mathcal{L}_{\varepsilon}(t)[X] := \mathcal{L}^K(t)[X] + \varepsilon^2 X$,
$t \in \mathcal{I}$, $X \in \mathcal{S}^n$, and let $T_{\varepsilon}(t,s)$ be
the linear evolution operator defined by $\mathcal{L}_{\varepsilon}$ on
$\mathcal{S}^n$. By a standard argument, based on the Gronwall-Bellman Lemma,
one obtains that there exists $\varepsilon_0>0$ such that for arbitrary
$\varepsilon \in [0, \varepsilon_0 ]$
\begin{equation} \| T_{\varepsilon}(t,s) \| \leq \beta e^{-\alpha(t-s)} \quad
\mbox{for} \quad t \geq s, \; t, s \in \mathcal{I}.
\label{eqno:2.31}
\end{equation} Let $\varepsilon > 0$ be fixed such that \eqref{eqno:2.31} is
fulfilled, and let $Y(t) = Y(t,t_0, H)$ be the solution of the initial value
problem
\begin{equation} \frac{d}{dt} Y(t) = \mathcal{L}_{\varepsilon}(t)[Y(t)] +
\frac{1}{\varepsilon^2} K(t) C(t)X(t) C^T(t) K^T(t), \quad Y(t_0)
= H. \label{eqno:2.32}
\end{equation}
Since $X \mapsto \varepsilon^2 X$ is a positive linear operator
and $\mathcal{L}^K$
generates a positive evolution then by Proposition \ref{prop2.5} one obtains
that $\mathcal{L}_{\varepsilon}$ generates a positive evolution.
Thus we may conclude that the solution of the forward differential
equation \eqref{eqno:2.32} satisfies $Y(t) \geq 0$ for all $t \geq
t_0$. Set $Z(t) := Y(t) - X(t)$. Subtracting \eqref{eqno:2.30}
from \eqref{eqno:2.32} we obtain
$$ \frac{d}{dt} Z(t) = \mathcal{L}_{\varepsilon}[Z(t)] + G(t), \quad Z(t_0) =
0, $$ where $G(t) := (\varepsilon I_n + \frac{1}{\varepsilon} K(t)
C(t)) X(t)(\varepsilon I_n + \frac{1}{\varepsilon} K(t)C(t))^T
\geq 0$ for $t \geq t_0$.

Invoking again the fact that $\mathcal{L}_{\varepsilon}$ generates a positive
evolution, one obtains that $Z(t)\geq 0$, i.e.\ $0 \leq X(t) \leq Y(t)$, $t
\geq t_0$ which leads to
\begin{equation} \|X(t)\|_2 \leq \|Y(t)\|_2 \quad \mbox{for} \quad t \geq
t_0. \label{eqno:2.33}
\end{equation}
Further we write
$$ Y(t) = T_{\varepsilon}(t,t_0)H + \frac{1}{\varepsilon^2} \int_{t_0}^t
T_{\varepsilon}(t,s) K(s) C(s) X(s) C^T(s) K^T(s) \, ds. $$ With
\eqref{eqno:2.31} we get
\begin{equation} \|Y(t)\|_2 \leq \beta e^{-\alpha(t-t_0)} \|H\|_2 +
\frac{\beta}{\varepsilon^2} \rho \int_{t_0}^t e^{-\alpha(t-s)}
\|C(s)X(s)C^T(s)\|_2 \, ds, \; t \geq t_0, \label{eqno:2.34}
\end{equation}
where $\rho := \sup_{s \in \mathcal{I}}
\|K(s)\|_2^2$. On the other hand
\begin{align*}
\|C(s)X(s)C^T(s)\|_2 &= \lambda_{\max} [C(s)X(s)C^T(s)] \\
& \leq \mathop{\rm Tr} [C(s)X(s)C^T(s)]
= \langle C^T(s)C(s), X(s)\rangle.
\end{align*}
Using \eqref{eqno:2.28} we infer
\begin{align*}
\langle C^T(s)C(s), X(s) \rangle
&= - \langle \frac{d}{ds} P(s), X(s) \rangle - \langle \mathcal{L}^*(s)[P(s)],
X(s) \rangle \\
& = - \langle \frac{d}{ds}P(s), X(s) \rangle -
\langle P(s), \frac{d}{ds} X(s) \rangle = -\frac{d}{ds} \langle
P(s), X(s) \rangle.
\end{align*}
So we have
$$ \int_{t_0}^t \|C(s)X(s)C^T(s)\|_2 \, ds \leq \langle P(t_0), X(t_0)
\rangle - \langle P(t), X(t) \rangle \quad \mbox{for} \quad t \geq
t_0.
$$
Since $\langle P(t), X(t) \rangle \geq 0$ and $P$ is bounded, we
deduce that there exists $\tilde{\rho} > 0$ such that
$$ \int_{t_0}^t \|C(s)X(s)C^T(s)\|_2 \, ds \leq \tilde{\rho} \|H\|_2 \quad
\mbox{for} \quad t \geq t_0. $$
Hence, we get for $\tau > t_0$
\begin{equation} \label{eqno:2.35}
\begin{aligned}
\lefteqn{\int_{t_0}^{\tau} \int_{t_0}^t e^{-\alpha(t-s)} \|C(s)X(s)C^T(s)\|_2
\, ds \, dt} \\ & \quad = \int_{t_0}^{\tau} \int_s^{\tau} e^{-\alpha(t-s)} \,
dt \|C(s)X(s)C^T(s)\|_2 \, ds \\ & \quad \leq \frac{1}{\alpha}
\int_{t_0}^{\tau} \|C(s)X(s)C^T(s)\|_2 \, ds \leq
\frac{\tilde{\rho}}{\alpha}\|H\|_2.
\end{aligned}
\end{equation}
Combining \eqref{eqno:2.34} with \eqref{eqno:2.35} we obtain
\begin{equation} \int_{t_0}^{\tau} \|Y(t)\|_2 \, dt \leq \delta \|H\|_2 \quad
\mbox{for} \quad \tau \geq t_0, \label{eqno:2.36}
\end{equation}
where $\delta := \frac{\beta}{\alpha}(1 + \frac{\rho
\tilde{\rho}}{\varepsilon^2})$ is independent of $(t_0, H)$.
Taking the limit for $\tau \to \infty$ in
\eqref{eqno:2.36} and using \eqref{eqno:2.33} we obtain that
\begin{equation} \int_{t_0}^{\infty} \|X(t)\|_2 \, dt \leq \delta \|H\|_2
\quad \mbox{for} \quad t_0 \in \mathcal{I}, \; H \in
\mathcal{S}^n_+. \label{eqno:2.37}
\end{equation}
For every $H \in \mathcal{S}^n$ there exist $H_i \in \mathcal{S}^n_+$, $i =
1, 2$, such that $H = H_1-H_2$ and $\|H\|_2 = \max \{ \|H_1\|_2, \|H_2\|_2
\}$. Since $X(t,t_0,H) = X(t,t_0,H_1) - X(t,t_0,H_2)$ we may infer that
\eqref{eqno:2.37} holds for arbitrary $t_0 \in \mathcal{I}$ and $H \in
\mathcal{S}^n$. The conclusion in the statement follows now from Datko type
criteria for exponential stability and the proof is complete.
\end{proof}

Finally we remark that the result proved in Theorem \ref{thm2.23} is an alternative
for implication (vi) $\Rightarrow$ (i) in Theorem \ref{thm2.11} for the case when
$H(t)$ is not uniformly positive but only positive semi-definite. Here the
loss of the uniform positivity of the forcing term of equation
\eqref{eqno:2.14} is compensated by detectability.

\section{A general class of Riccati-type differential equations}

\subsection{Notation and preliminary remarks}

In this section we deal with nonlinear differential equations of the form
\begin{equation} \label{eqno:3.1}
\begin{aligned}
\lefteqn{\frac{d}{dt} X(t) + A^T(t)X(t) + X(t)A(t) + M(t) + \Pi_1(t)[X(t)]}
\\ & \quad {} - \big\{ X(t)B(t) +\Pi_{12}(t)[X(t)] + L(t) \big\} \big\{ R(t)
+ \Pi_2(t)[X(t)]\big\}^{-1} \\ & \qquad \times \big\{ X(t)B(t) +
\Pi_{12}(t)[X(t)] + L(t) \big\}^T = 0,
\end{aligned}
\end{equation}
where $A \colon \mathcal{I} \to \mathbb{R}^{n\times n}$, $B \colon
\mathcal{I} \to \mathbb{R}^{n \times m}$, $L \colon \mathcal{I} \to
\mathbb{R}^{n \times m}$, $M \colon \mathcal{I} \to \mathcal{S}^n$ and $R
\colon \mathcal{I} \to \mathcal{S}^m$ are bounded and continuous functions on
some right unbounded interval $\mathcal{I} \subset \mathbb{R}$, and $\Pi
\colon \mathcal{I} \to C(\mathcal{S}^n, \mathcal{S}^{n+m})$,
$C(\mathcal{S}^n, \mathcal{S}^{n+m})$ being the space of continuous operators
defined on $\mathcal{S}^n$ with values in $\mathcal{S}^{n+m}$, with
\begin{equation}
\Pi(t)[X] = \begin{pmatrix} \Pi_1(t)[X] & \Pi_{12}(t)[X] \\
\big( \Pi_{12}(t)[X] \big)^T & \Pi_2(t)[X] \end{pmatrix}, \quad X \in
\mathcal{S}^n, \label{eqno:3.2} \end{equation}
is a bounded and continuous
operator valued function.

Throughout this section we assume that $X \to \Pi(t)[X]$ satisfies
the assumptions:
\begin{itemize}
\item[($\Pi_1$)]  $X \to \Pi(t)[X]$ is uniformly globally Lipschitz

\item[($\Pi_2$)] $\Pi(t)[X_1] \leq \Pi(t)[X_2]$ for $t \in \mathcal{I}$ and
$X_1, X_2 \in \mathcal{S}^n$ with $X_1 < X_2$

\item[($\Pi_3$)] $\Pi(t)[0] = 0$.
\end{itemize}

\begin{remark} \label{rmk3.1} \rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item Condition $\mathbf{(\Pi_3)}$ is satisfied without loss of generality
since if $\Pi(t)[0] \neq 0$ we can modify the values of $M(t)$, $L(t)$,
$R(t)$ to achieve  $\Pi(t)[0] = 0$.
\item If assumptions $\mathbf{(\Pi_2)}$ and $\mathbf{(\Pi_3)}$ are fulfilled
then for each $t$, $\Pi(t)$ is a positive operator.
\end{enumerate}
\end{remark}

In some particular cases of the operator $\Pi$ equation \eqref{eqno:3.1} was
investigated in connection with certain control problems for linear control
systems. Thus, if $\Pi \equiv 0$, \eqref{eqno:3.1} reduces to the well-known
Riccati differential equation intensively studied in particular since the
pioneering work of Kalman \cite{Kalm60}; we shall show that the solutions of
\eqref{eqno:3.1} have many of the nice properties of the solutions of
symmetric matrix Riccati differential equations.

In the special case
\begin{equation}
\Pi(t)[X] = \begin{pmatrix} C(t) & D(t) \end{pmatrix}^T X
\begin{pmatrix} C(t) & D(t) \end{pmatrix} \label{eqno:3.3}
\end{equation}
the corresponding equation \eqref{eqno:3.1} was considered in several papers,
in connection with the linear-quadratic optimization problem with indefinite
sign for linear stochastic system with multiplicative white noise (see
\cite{ARCMZ01,ARZh00,ChLZ98}). Systems of coupled rational differential
equations of type \eqref{eqno:3.1} with $\Pi(t)$ as in \eqref{eqno:3.3} were
studied in \cite{preprint00}.

Equation \eqref{eqno:3.1} was investigated in detail in \cite{FrHo6} and
\cite{FrHo5}, where it partially was assumed that $A$, $B$, $L$, $M$, $R$ and
$\Pi(X)$ are constant. In this paper we present various results which extend
\cite{preprint00,FrHo5} to the general case of equations of type
\eqref{eqno:3.1}. We mention that it is also possible to derive discrete-time
versions of these results obtained in this section, in the time-invariant
case they can be found in \cite{FrHo4} and \cite[Chapter 6.8]{AFIJ03}.

\begin{definition} \label{def3.2} \rm
A $C^1$-function $X \colon \mathcal{I}_1 \subset \mathcal{I}
\to \mathcal{S}^n$ is a solution of equation \eqref{eqno:3.1} if
$$ \det \big\{ R(t) + \Pi_2(t)[X(t)] \big\} \neq 0 \quad \mbox{for all} \quad
t \in \mathcal{I}_1 $$ and if $X$ verifies the relation \eqref{eqno:3.1} on
$\mathcal{I}_1$. \end{definition}

We introduce the notation
$$ \mathcal{D}(\mathcal{R}) := \left\{ (t,X) \in \mathcal{I} \times
\mathcal{S}^n \colon \det \big\{ R(t) + \Pi_2(t)[X] \big\} \neq 0\right\}$$
and $\mathcal{R} \colon \mathcal{D}(\mathcal{R}) \to \mathcal{S}^n$ by
\begin{equation} \label{eqno:3.4}
\begin{aligned}
\mathcal{R}(t,X) & = A^T(t)X + XA(t) + \Pi_1(t)[X] + M(t) \\ & \qquad {} -
\big\{ X B(t) + \Pi_{12}(t)[X] + L(t) \big\} \big\{R(t)+ \Pi_2(t)[X]
\big\}^{-1} \\ & \qquad {} \times \big\{ XB(t) + \Pi_{12}(t)[X] + L(t)
\big\}^T.
\end{aligned}
\end{equation}
So equation \eqref{eqno:3.1} can be written in a compact form as
\begin{equation} \frac{d}{dt} X(t) + \mathcal{R}(t,X(t)) = 0.
\label{eqno:3.5} \end{equation}
As we can see the operator $\mathcal{R}$ and
consequently equation \eqref{eqno:3.5} are associated to the quadruple
$\Sigma = (A, B, \Pi, \mathcal{Q})$ where $(A, B, \Pi)$ are as before and
$\mathcal{Q}$ is defined by
$$
\mathcal{Q}(t) := \begin{pmatrix} M(t) & L(t) \\ L(t)^T & R(t)
\end{pmatrix}.
 $$
We introduce the so called generalized dissipation matrix $\lambda^{\Sigma}
\colon C^1(\mathcal{I}, \mathcal{S}^n) \to \mathcal{S}^{n+m}$
associated to the quadruple $\Sigma$ by
$$ \lambda^{\Sigma}(t,X) = \left( \begin{array}{cc} \lambda_1(t) & X B(t) +
\Pi_{12}(t)[X] + L(t) \\ \big\{ X B(t) + \Pi_{12}(t)[X] + L(t)
\big\}^T & R(t) + \Pi_2(t)[X] \end{array} \right), $$ where
$$ \lambda_1(t) = \frac{d}{dt} X(t) + A^T(t)X(t) + X(t)A(t) + \Pi_1(t)[X(t)]
+ M(t) $$ and $C^1(\mathcal{I}, \mathcal{S}^n)$ is the space of
$C^1$-functions defined on the interval $\mathcal{I}$ taking values in
$\mathcal{S}^n$. Notice that
\[ \frac{d}{dt} X(t) + \mathcal{R}(t,X(t)) \]
is the Schur complement of $R(t) + \Pi_2(t)[X(t)]$ in
$\lambda^\Sigma(t,X(t))$.

The following two subsets of
$$
C^1_b(\mathcal{I}, \mathcal{S}^n) = \big\{ X \in C^1(\mathcal{I},
\mathcal{S}^n) \colon X, \frac{d}{dt} X \mbox{ are bounded} \big\}$$ will
play an important role in the next developments:
\begin{gather*}
\Gamma^{\Sigma}  = \left\{ X \in C^1_b(\mathcal{I},\mathcal{S}^n) \colon
\lambda^{\Sigma}[X(t)] \geq 0, \; R(t) +\Pi_2(t)[X(t)] \gg 0, \; t \in
\mathcal{I} \right\}, \\
\tilde \Gamma^{\Sigma}  = \left\{ X \in C^1_b(\mathcal{I},\mathcal{S}^n)
\colon \lambda^{\Sigma}[X(t)] \gg 0, \; t \in \mathcal{I} \right\}.
\end{gather*}

\begin{remark} \label{rmk3.3} \rm
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item In the case when $\Pi_2(t)$ is the zero operator then in the definition
of the set $\Gamma^{\Sigma}$ we ask $R(t) \gg 0$ which is the usual condition
used in the case of Riccati differential equations of deterministic and
stochastic control. If $\Pi_2(t)$ is not the zero operator it is not
necessary to make any assumptions concerning the sign of $R(t)$.
\item We shall see later that if $A$, $B$, $\Pi$, $\mathcal{Q}$ are
$\theta$-periodic functions and if $\Gamma^{\Sigma}$ is not empty
($\tilde{\Gamma}^{\Sigma}$ is not empty, respectively) then $\Gamma^{\Sigma}$
contains also a $\theta$-periodic function ($\tilde{\Gamma}^{\Sigma}$
contains also a $\theta$-periodic function). Moreover, we shall show that if
$A(t) = A$, $B(t) = B$, $\Pi(t) = \Pi$, $\mathcal{Q}(t) = \mathcal{Q}$ for
all $t \in \mathbb{R}$ and if $\Gamma^{\Sigma}$ is not empty,
($\tilde{\Gamma}^{\Sigma}$ is not empty, respectively) then there exists a
constant symmetric matrix $X \in \Gamma^{\Sigma}$ ($X \in
\tilde{\Gamma}^{\Sigma}$, respectively).
\item Based on the Schur complement one obtains that $\Gamma^{\Sigma}$
contains in particular all  bounded solutions of equation \eqref{eqno:3.1}
verifying the additional condition $R(t) + \Pi_2(t)[X(t)] \gg 0$.
\end{enumerate}
\end{remark}

The following result will be used frequently in the proofs of the remainder
of the paper, it follows easily by direct calculation (see \cite[Lemma 4.2]{FrHo5}).

\begin{lemma} \label{lm3.4}
If $W \colon \mathcal{I} \to \mathbb{R}^{m \times n}$ is a continuous
function, then for all $(t,X) \in\mathcal{D}(\mathcal{R})$
\begin{align*}
\mathcal{R}(t,X) & = [A(t)+B(t)W(t)]^T X + X[A(t)+B(t)W(t)] \\ & \qquad {} -
\big[ W(t)-F^X(t) \big]^T \big\{ R(t) + \Pi_2(t)[X] \big\} \big[ W(t)-F^X(t)
\big] \\ & \qquad {} + \begin{pmatrix} I_n \\ W(t)\end{pmatrix}^T \big[
\mathcal{Q}(t) + \Pi(t)[X] \big] \begin{pmatrix} I_n \\ W(t) \end{pmatrix}
\end{align*}
where here and below
$$ F^X(t) := - \big\{ R(t) + \Pi_2(t)[X] \big\}^{-1} \big\{ X(t)B(t) +
\Pi_{12}(t)[X] + L(t) \big\}^T $$ is the \textbf{feedback matrix}
defined by $X$. \end{lemma}

In the following we will use operators $\mathcal{L}_{A+BW}(t) \colon
\mathcal{S}^n \to \mathcal{S}^n$, where
$$\mathcal{L}_{A+BW}(t)[X]=[A(t)+B(t)W(t)]X+X[A(t)+B(t)W(t)]^T$$
and $\Pi_W(t) \colon \mathcal{S}^n \to \mathcal{S}^n$
defined by
\begin{equation}
\Pi_W(t)[X] = \begin{pmatrix} I_n \\ W(t) \end{pmatrix}^T \Pi(t)[X]
\begin{pmatrix} I_n \\ W(t) \end{pmatrix}. \label{eqno:3.6}
\end{equation}
It is clear that $\Pi_W(t)$ is a monotone increasing operator for
all $t$.

With these notations $\mathcal{R}(t,X)$ can be rewritten as
\begin{equation} \label{eqno:3.7}
\begin{aligned}
\mathcal{R}(t,X) & = \mathcal{L}_{A+BW}^*(t)[X] + \Pi_W(t)[X] + M_W(t) \\ &
\qquad {} - \big[ W(t)-F^X(t) \big]^T \big\{R(t) + \Pi_2(t)[X] \big\} \big[
W(t) - F^X(t) \big]
\end{aligned}
\end{equation}
where
$$ M_W(t) = \begin{pmatrix} I_n \\ W(t) \end{pmatrix}^T \mathcal{Q}(t)
\begin{pmatrix} I_n \\ W(t) \end{pmatrix}. $$

\subsection{A comparison theorem and its consequences}

First we present a result which extends the conclusions of Remark
\ref{rmk2.14}, (c), to a nonlinear framework. The proof of this result is
based on the techniques of Volterra approximations (as in Proposition
\ref{prop2.5}) and it is omitted for shortness.

\begin{prop} \label{prop3.5}
Let $\mathcal{L} \colon \mathcal{I} \to \mathcal{B}(\mathcal{S}^n)$,
$\hat{\Pi} \colon \mathcal{I} \to C(\mathcal{S}^n, \mathcal{S}^n)$ be
bounded and continuous operator valued functions. Assume that $\mathcal{L}$
generates a positive evolution and $\hat{\Pi}$ satisfies the assumptions
$(\mathbf{\Pi_1})$ -- $(\mathbf{\Pi_3})$.

Let $H \colon \mathcal{I} \to \mathcal{S}^n_+$ be a continuous and
bounded function. Under these conditions the following hold:
\begin{enumerate}
\item[(i)] If $X \colon \mathcal{I} \to \mathcal{S}^n$ is a global
solution of the nonlinear differential equation
$$\frac{d}{dt} X(t) = \mathcal{L}(t)[X(t)] + \hat{\Pi}(t)[X(t)] + H(t)$$
such that $X(\tau) \geq 0$ for some $\tau \in \mathcal{I}$, then $X(t) \geq
0$ for all $t \geq \tau$.
\item[(ii)] If $Y \colon \mathcal{I} \to \mathcal{S}^n$ is a global
solution of the nonlinear differential equation
$$\frac{d}{dt} Y(t) + \mathcal{L}^*(t)[Y(t)] + \hat{\Pi}(t)[Y(t)] + H(t) =
0$$ such that $Y(\tau) \geq 0$ for some $\tau \in \mathcal{I}$, then $Y(t)
\geq 0$ for all $t \in (-\infty, \tau] \cap \mathcal{I}$.
\end{enumerate}
\end{prop}

As a consequence of the above Proposition one obtains the following important
result concerning the monotonic dependence of the solutions of equation
\eqref{eqno:3.1} with respect to the data (which was already proved in
\cite{FrHo5} under the additional assumption that $X \mapsto \Pi(t)[X]$ is
linear; for an elementary special case see \cite[Corollary 3.4]{ARCMZ01}).

\begin{theorem}[Comparison Theorem] \label{thm3.6}
Let $\hat{\mathcal{R}}$ be the operator
\eqref{eqno:3.4} associated to the quadruple $\hat \Sigma = (A, B, \Pi,
\hat{\mathcal{Q}})$ and $\tilde{\mathcal{R}}$ be the operator of type
\eqref{eqno:3.4} associated to the quadruple $\tilde{\Sigma} = (A, B, \Pi,
\tilde{\mathcal{Q}})$ where $A$, $B$, $\Pi$ are as before and
$\hat{\mathcal{Q}}(t) = \left( \begin{smallmatrix} \hat{M}(t) & \hat{L}(t) \\
\hat{L}(t)^T & \hat{R}(t) \end{smallmatrix} \right)$, $\tilde{\mathcal{Q}}(t)
= \left( \begin{smallmatrix} \tilde{M}(t) & \tilde{L}(t) \\ \tilde{L}(t)^T &
\tilde{R}(t) \end{smallmatrix} \right)$ with $\hat{L}(t), \tilde{L}(t) \in
\mathbb{R}^{n \times m}$, $\hat{M}(t), \tilde{M}(t) \in \mathcal{S}^n$ and
$\hat{R}(t), \tilde{R}(t) \in \mathcal{S}^m$. Let $X_i \colon \mathcal{I}_1
\subset \mathcal{I} \to\mathcal{S}^n$, $i = 1,2$, be solutions of
$$ \frac{d}{dt} X_1(t) + \hat{\mathcal{R}}(t, X_1(t)) = 0, \qquad
\frac{d}{dt} X_2(t) + \tilde{\mathcal{R}}(t, X_2(t)) = 0. $$ Assume that
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $\hat{\mathcal{Q}}(t) \geq \tilde{\mathcal{Q}}(t)$ for all $t \in
\mathcal{I}$;
\item $\tilde{R}(t) + \Pi_2(t)[X_2(t)] > 0$ for $t \in \mathcal{I}_1$;
\item there exists $\tau \in \mathcal{I}_1$ such that  $X_1(\tau) \geq
X_2(\tau)$.
\end{enumerate}
Under these conditions we have $X_1(t)\geq X_2(t)$ for all $t
\in(-\infty, \tau] \cap \mathcal{I}_1$. \end{theorem}

\begin{proof} Let
$$F_1(t) := - \big\{ \hat{R}(t) + \Pi_2(t)[X_1(t)] \big\}^{-1} \big\{
X_1(t)B(t) + \Pi_{12}(t)[X_1(t)] + \hat{L}(t) \big\}^T $$ and
$$ F_2(t) := - \big\{ \tilde{R}(t) + \Pi_2(t)[X_2(t)] \big\}^{-1} \big\{
X_2(t)B(t) + \Pi_{12}(t)[X_2(t)] + \tilde{L}(t) \big\}^T. $$
Applying Lemma \ref{lm3.4} for $W(t)=F_1(t)$ one obtains
$$ \frac{d}{dt} X_1(t) + \mathcal{L}_{A+BF_1}^*(t)[X_1(t)] +
\Pi_{F_1}(t)[X_1(t)] + \hat{M}_{F_1}(t) = 0 $$ and
\begin{align*}
\lefteqn{\frac{d}{dt} X_2(t) + \mathcal{L}_{A+BF_1}^*(t)[X_2(t)] +
\Pi_{F_1}(t)[X_2(t)] + \tilde{M}_{F_1}(t)} \\
& \quad {} - \big[ F_1(t) - F_2(t) \big]^T \big\{ \tilde{R}(t) +
\Pi_2(t)[X_2(t)] \big\} \big[ F_1(t)-F_2(t) \big] = 0.
\end{align*}
This leads to
$$ \frac{d}{dt} \big[ X_1(t) - X_2(t) \big] + \mathcal{L}_{A+BF_1}^*(t) \big[
X_1(t) - X_2(t) \big] + \hat{\Pi}(t) \big[ X_1(t) - X_2(t) \big] + H(t) = 0,
$$ where
$$\hat{\Pi}(t)[Y] := \Pi_{F_1}(t)[Y + X_2(t)] - \Pi_{F_1}(t)[X_2(t)]$$
and
\[
H(t) := \big[ F_1(t) - F_2(t) \big]^T \big\{ \tilde{R}(t) +
\Pi_2(t)[X_2(t)] \big\} \big[ F_1(t) - F_2(t) \big]
 +\hat{M}_{F_1}(t) - \tilde{M}_{F_1}(t).
\]
Since
$$ \hat{M}_{F_1}(t) - \tilde{M}_{F_1}(t) = \begin{pmatrix} I_n \\ F_1(t)
\end{pmatrix}^T \big[ \hat{\mathcal{Q}}(t) - \tilde{\mathcal{Q}}(t)\big]
\begin{pmatrix} I_n \\ F_1(t) \end{pmatrix} \geq 0 $$
it follows that $H(t)\geq 0$ for all $t \in \mathcal{I}_1$.

On the other hand it is not difficult to see that the above operator
$\hat{\Pi}(t)$ verifies the hypotheses ($\mathbf{\Pi_1}$) --
($\mathbf{\Pi_3}$).

Applying Proposition \ref{prop3.5}, (ii), to the backward equation verified by $X_1 -
X_2$ one gets that $X_1(t) - X_2(t) \geq 0$ for all $t \in (-\infty, \tau]
\cap \mathcal{I}_1$ and thus the proof is complete. \end{proof}

Using the above theorem we prove the following result concerning the maximal
interval of definition of a solution of \eqref{eqno:3.1} with given terminal
conditions.

\begin{theorem} \label{thm3.7}
Assume that $\Sigma = (A, B, \Pi, \mathcal{Q})$ satisfies $\Gamma^{\Sigma}
\neq \emptyset$. Let
$$ \tilde{\mathcal{D}}(\mathcal{R}) := \left\{ (\tau, X) \in
\mathcal{D}(\mathcal{R}) \colon \exists \hat{X} \in \Gamma^{\Sigma} \mbox{
such that } X \geq \hat{X}(\tau) \right\} $$ and let $X(\cdot, \tau, X_0)$ be
the solution of equation \eqref{eqno:3.1} with $X(\tau, \tau, X_0) = X_0$.

If $(\tau, X_0)\in \tilde{\mathcal{D}}(\mathcal{R})$ then the
solution $X(\cdot, \tau, X_0)$ is well defined on $(-\infty,
\tau]\cap \mathcal{I}$.
\end{theorem}

\begin{proof} Let $\mathcal{I}_{\tau, X_0} \subset (-\infty, \tau]$ be the
maximal interval on which $X(\cdot, \tau, X_0)$ is defined and let $\hat{X}
\in \Gamma^{\Sigma}$ be such that $X_0 \geq \hat{X}(\tau)$. Obviously there
exists a bounded and continuous function $M_{\ell} \colon \mathcal{I}
\to \mathcal{S}^n$ such that $M_{\ell}(t) \leq 0$ and
$$\frac{d}{dt} \hat{X}(t) + \mathcal{R}(t, \hat{X}(t)) + M_{\ell}(t) = 0.
$$
Applying Theorem \ref{thm3.6} for the quadruples $\hat{\Sigma} = \Sigma$ and
$\tilde{\Sigma} = (A, B, \Pi, \tilde{\mathcal{Q}})$ with
$$ \tilde{\mathcal{Q}}(t) := \begin{pmatrix} M(t) + M_{\ell}(t) & L(t) \\
L^T(t) & R(t) \end{pmatrix} = \mathcal{Q}(t) + \begin{pmatrix} M_{\ell}(t) &
0 \\ 0 & 0 \end{pmatrix} $$
we conclude that
\begin{equation} X(t, \tau, X_0) \geq \hat{X}(t) \quad \mbox{for all} \quad t
\in \mathcal{I}_{\tau, X_0}. \label{eqno:3.8}
\end{equation}
 Let $Y$ be the
solution of the terminal value problem
\begin{equation} \frac{d}{dt} Y(t) + A^T(t) Y(t) + Y(t) A(t) + \Pi_1(t)[Y(t)]
+ M(t) = 0, \quad Y(\tau) = X_0. \label{eqno:3.9} \end{equation} Since $Y \to
\Pi_1(t)[Y]$ is uniformly globally Lipschitz it follows that $Y(t)$ is well
defined for all $t \in \mathcal{I}$. By direct calculation we obtain that
\begin{equation}
\frac{d}{dt} \big[ Y(t) - X(t) \big] + \mathcal{L}_A^*(t) \big[ Y(t) - X(t)
\big] + \hat{\Pi}(t) \big[ Y(t) - X(t) \big] + \hat{H}(t) = 0
\label{eqno:3.10} \end{equation}
for $t \in \mathcal{I}_{\tau, X_0}$ where
$X(t) = X(t, \tau, X_0)$. Here
$$ \hat{H}(t) := F^T(t) \big\{ R + \Pi_2(t)[X(t)] \big\} F(t) $$
with $F(t) := F^X(t)$, and the map $Y \to \hat{\Pi}(t)[Y]$ is defined
by
$$ \hat{\Pi}(t)[Y] = \Pi_1(t)[Y + X(t)] - \Pi_1(t)[X(t)]. $$
>From \eqref{eqno:3.8} and assumption ($\mathbf{\Pi_2}$) we deduce that
$$ R(t) + \Pi_2(t)[X(t)] \geq R(t) + \Pi_2(t)[\hat{X}(t)] \gg 0. $$
So $\hat{H}(t) \geq 0$ for $t \in \mathcal{I}_{\tau, X_0}$.
Proposition \ref{prop3.5}, (ii), applied in the case of
equation \eqref{eqno:3.10}, gives
\begin{equation} X(t) \leq Y(t) \label{eqno:3.11}
\end{equation}
for all $t \in \mathcal{I}_{\tau, X_0}$. From \eqref{eqno:3.8},
\eqref{eqno:3.11} and $\hat{X} \in \Gamma^\Sigma$ it follows easily that
$X(t,\tau,X_0)$ is defined for all $t \in (-\infty, \tau] \cap \mathcal{I}$
and thus the proof ends. \end{proof}

The proof of Theorem \ref{thm3.7} shows that -- as a consequence of the Comparison
Theorem -- each element $\hat{X} \in \Gamma^\Sigma$ is providing a lower
bound for the solution $X(\cdot, \tau, X_0)$ of \eqref{eqno:3.1} (see
\eqref{eqno:3.8}), whereas the solution $Y$ of \eqref{eqno:3.9} gives an
upper bound.

\begin{corollary} \label{coro3.8}
Assume that $0 \in \Gamma^{\Sigma}$. Then for all $(\tau, X_0)
\in \mathcal{I} \times \mathcal{S}^n_+$ the solution $X(\cdot, \tau, X_0)$ of
equation \eqref{eqno:3.1} is defined on the whole interval $(- \infty, \tau]
\cap \mathcal{I}$ and fulfills there the inequality
$$ 0 \leq X(t, \tau, X_0) \leq Y(t), $$
where $Y$ is the solution of \eqref{eqno:3.9}. Moreover if $X_0 > 0$ then
$X(t, \tau, X_0) > 0$ for all $t \in \mathcal{I}$ with $t \leq \tau$.
\end{corollary}

\begin{proof} Since $0 \in \Gamma^{\Sigma}$ it follows that $\mathcal{I} \times
\mathcal{S}_+^n \subset \tilde{\mathcal{D}}(\mathcal{R})$. So from the above
theorem one obtains that $X(t, \tau, X_0)$ is well defined for all $t \in
(-\infty, \tau] \cap \mathcal{I}$ for arbitrary $(\tau, X_0) \in \mathcal{I}
\times \mathcal{S}_+^n$.

The inequality $X(t, \tau, X_0) \geq 0$ is just \eqref{eqno:3.8} for
$\hat{X}(t) \equiv 0$. On account of \eqref{eqno:3.11} it remains to prove
that $X(t, \tau, X_0) > 0$ if $X_0 > 0$. To this end we set $X(t) = X(t,
\tau, X_0)$ and $F(t) = F^X(t)$ for $t \in \mathcal{I}$ with $t \leq \tau$.
Applying Lemma \ref{lm3.4} for $W(t) = F(t)$ one obtains
$$ \frac{d}{dt} X(t) + \mathcal{L}_{A+BF}^*(t)[X(t)] + \Pi_F(t)[X(t)] +
M_F(t) = 0. $$ Further we write the representation formula
$$ X(t) = \Phi_{A+BF}^T(\tau, t) X_0 \Phi_{A+BF}(\tau, t) + \int_t^\tau
\Phi_{A+BF}^T (s,t) H(s) \Phi_{A+BF}(s,t) \, ds, $$ where $\Phi_{A+BF}(s,t)$
is the fundamental matrix solution defined by
$$ \frac{d}{ds} \Phi_{A+BF}(s,t) = \big[ A(s) + B(s) F(s) \big]
\Phi_{A+BF}(s,t), \quad \Phi_{A+BF}(t,t) = I $$ and where
$$ H(s) = \Pi_F(s)[X(s)] + M_F(s) \quad \mbox{for} \quad s \in \mathcal{I}
\quad \mbox{with} \quad s \leq \tau. $$ The assumption $0 \in
\Gamma^{\Sigma}$ is equivalent to
\begin{equation} R(t) \gg 0 \quad \mbox{and} \quad \begin{pmatrix} M(t) &
L(t) \\ L^T(t) & R(t) \end{pmatrix} \geq 0. \label{eqno:3.13}
\end{equation}
 From \eqref{eqno:3.13} and the monotonicity of $\Pi(s)[\cdot]$ we
conclude that $H(s) \geq 0$ for all $s \in \mathcal{I}$ with $s
\leq \tau$. From the representation formula we obtain that
$$X(t) \geq \Phi_{A+BF}^T(\tau,t) X_0 \Phi_{A+BF}(\tau,t) > 0. $$
For the last inequality we have taken into account that $X_0 > 0$ and
$\Phi_{A+BF}(\tau, t)$ is invertible, thus the proof is complete. \end{proof}

The next result provides a set of sufficient conditions which guarantee the
existence of the minimal positive semi-definite solution $\tilde{\tilde{X}}$
of equation \eqref{eqno:3.1}.

\begin{theorem} \label{thm3.9}
Assume that:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $0 \in \Gamma^{\Sigma}$;
\item The nonlinear differential equation \eqref{eqno:3.9} has a bounded
solution $\tilde{X} \colon \mathcal{I} \to \mathcal{S}^n$ such that
$\tilde{X}(t) \geq 0$.
\end{enumerate}
Under these conditions the differential equation \eqref{eqno:3.1} has a
solution $\tilde{\tilde{X}} \colon \mathcal{I} \to \mathcal{S}^n$
with the additional property $0 \leq \tilde{\tilde{X}}(t) \leq \bar{X}(t)$
for all $t \in \mathcal{I}$, for any bounded and positive semi-definite
solution $\bar{X} \colon \mathcal{I} \to \mathcal{S}^n$ of equation
\eqref{eqno:3.1}.

Additionally if $A$, $B$, $\Pi$, $\mathcal{Q}$ are periodic functions with
period $\theta > 0$, then $\tilde{\tilde{X}}$ is a periodic function with the
same period $\theta$. Moreover if $A(t) \equiv A$, $B(t) \equiv B$, $\Pi(t)
\equiv \Pi$, $\mathcal{Q}(t) \equiv \mathcal{Q}$ for all $t \in \mathcal{I}$,
then $\tilde{\tilde{X}}$ is constant and it solves the nonlinear algebraic
equation
\begin{equation} \label{eqno:3.14}
\begin{aligned} \lefteqn{A^T X + X A + \Pi_1(X) + M - [X B + \Pi_{12}(X) +
L]} \\ & \quad \times [R + \Pi_2(X)]^{-1} [X B + \Pi_{12}(X) + L]^T = 0.
\end{aligned}
\end{equation}
\end{theorem}

\begin{proof} For each $\tau \in \mathcal{I}$ let $X_{\tau}(t) = X(t,\tau,0)$ be
the solution of (3.1) such that $X_{\tau}(\tau)=0$. From Corollary
\ref{coro3.8} we have that $X_{\tau}$ is well defined on $(-\infty, \tau]
\cap \mathcal{I}$ and $X_{\tau}(t) \geq 0$ for all $t \in \mathcal{I}$ with
$t \leq \tau$.

We show that:
\begin{enumerate}
\item[($\alpha)$] $\quad  X_{\tau_1}(t)\leq X_{\tau_2}(t)$ for all $t \in
\mathcal{I}$ with $t \leq \tau_1 < \tau_2$.
\item[($\beta)$] $\quad  X_{\tau}(t) \leq \tilde{Y}(t)$ for all $t \in
\mathcal{I}$ with $t \leq \tau$, where $\tilde{Y}$ is a bounded and positive
semi-definite solution of equation \eqref{eqno:3.9}.
\end{enumerate}
If $\tau_1 < \tau_2 \in \mathcal{I}$, then $X_{\tau_2}(\tau_1) \geq 0 =
X_{\tau_1}(\tau_1)$. Then, based on Theorem \ref{thm3.6} we obtain that $X_{\tau_2}(t)
\geq X_{\tau_1}(t)$ for $t \in (-\infty, \tau_1] \cap \mathcal{I}$ and thus
item $\alpha)$ is valid.

On the other hand if $\tilde{Y} \colon \mathcal{I} \to \mathcal{S}^n$
is a bounded and positive semi-definite solution of equation \eqref{eqno:3.9}
then for arbitrary $\tau \in \mathcal{I}$ we can write
$$\frac{d}{dt} \left[ \tilde{Y}(t) - X_{\tau}(t) \right] + \mathcal{L}_A^*(t)
\left[ \tilde{Y}(t) - X_{\tau}(t) \right] + \hat{\Pi}(t) \left[ \tilde{Y}(t)
- X_{\tau}(t) \right] + H_{\tau}(t) =0 $$ where
$$ \hat{\Pi}(t)[Z] = \Pi_1(t)[Z + X_{\tau}(t)] - \Pi_1(t)[X_{\tau}(t)] $$
and
$$ H_{\tau}(t) = F_\tau^T(t) \big\{ R(t) + \Pi_2(t)[X_{\tau}(t)] \big\}
F_\tau(t) $$ where $F_\tau(t) := F^{X_\tau}(t)$. It is easy to verify that $Z
\to \hat{\Pi}(t)[Z]$ satisfies the assumptions ($\mathbf{\Pi_1}$) --
($\mathbf{\Pi_3}$) and that $H_\tau(t) \geq 0$.

Taking into account that $\tilde{Y}(\tau) - X_{\tau}(\tau) = \tilde{Y}(\tau)
\geq 0$ we obtain via Proposition \ref{prop3.5}, (ii), that $\tilde{Y}(t) - X_{\tau}(t)
\geq 0$ for all $t \in \mathcal{I}$ with $t \leq \tau$, and thus item
$\beta)$ holds.

 From $(\alpha)$ and $(\beta)$ we deduce that $\tilde{\tilde{X}}
\colon \mathcal{I} \to \mathcal{S}^n$ is well defined by
\begin{equation}
\tilde{\tilde X}(t) := \lim_{\tau \to \infty} X_{\tau}(t) \quad \mbox{for}
\quad t \in \mathcal{I}.  \label{eqno:3.15} \end{equation} Obviously $0 \leq
\tilde{X}(t) \leq \tilde{Y}(t)$. In a standard way we can show that
$\tilde{\tilde{X}}(t)$ is a solution of equation \eqref{eqno:3.1}. If
$\bar{X} \colon \mathcal{I} \to \mathcal{S}^n$ is a bounded and positive
semi-definite solution of \eqref{eqno:3.1}, then, using Theorem \ref{thm3.6}, we
obtain that $X_{\tau}(t) \leq \bar{X}(t)$ for all $t \in \mathcal{I}$ with $t
\leq \tau$.

So, invoking \eqref{eqno:3.15} we conclude that $0 \leq \tilde{\tilde{X}}(t)
\leq \bar{X}(t)$ for all $t \in \mathcal{I}$, which shows that
$\tilde{\tilde{X}}$ is the minimal solution in the class of bounded and
positive semi-definite solutions of the differential equation
\eqref{eqno:3.1}.

Let us assume that $A$, $B$, $\Pi$, $\mathcal{Q}$ are periodic functions with
period $\theta > 0$. We have to show that the minimal solution
$\tilde{\tilde{X}}$ is also a periodic function with period $\theta$.

Let $\hat{X}_{\tau}(t) = X_{\tau + \theta}(t + \theta)$. It is easy to see
that $\hat{X}_{\tau}$ is a solution of \eqref{eqno:3.1} which satisfies
$\hat{X}_{\tau}(\tau) = 0$. From the uniqueness of the solution of this
terminal value problem it follows that $\hat{X}_{\tau}(t) = X_{\tau}(t)$ for
all $t \in (-\infty, \tau] \cap \mathcal{I}$. We have
$$ \tilde{\tilde{X}}(t) = \lim_{\tau \to \infty} X_{\tau}(t) = \lim_{\tau \to
\infty} \hat{X}_{\tau}(t) = \lim_{\tau \to \infty} X_{\tau + \theta}(t +
\theta) = \tilde{\tilde{X}}(t + \theta) $$ for all $t \in \mathcal{I}$ which
shows that $\tilde{\tilde{X}}$ is a periodic function with period $\theta$.
Finally, if $A(t) \equiv A$, $B(t) \equiv B$, $\Pi(t) \equiv \Pi$,
$\mathcal{Q}(t) \equiv \mathcal{Q}$ then $\tilde{\tilde{X}}$ is a periodic
function of arbitrary period. Therefore it is a constant function and thus
the proof ends. \end{proof}

\begin{lemma} \label{lm3.10}
Assume that $A$, $B$, $\mathcal{Q}$ and
$\Pi$ are periodic functions with period $\theta > 0$. If the
symmetric solution $X_1 = X_1(\cdot, t_0, X_0)$ of
\eqref{eqno:3.1} exists on an interval $\mathcal{ I}_1$ containing
the interval $[t_0 -2 \theta, t_0]$ such that
$R(t)+\Pi_2(t)[X_1(t)]>0$ for all $t\in \mathcal{I}_1$  and if
\[ X_1(t_0 - \theta) \; \Box \; X_1(t_0) \quad \mbox{for some} \quad \Box \in
\{ <, \leq, =, \geq, > \}, \] then $X_1(t - \theta) \; \Box
\;X_1(t)$ holds on the maximal common interval $\mathcal{
I}_1\subset \mathcal{I} \cap (- \infty, t_0]$ of existence of $X$
and $X(\cdot - \theta)$; in this case $X_1$ is called {\bf
cyclomonotonic} on $\mathcal{ I}_1$.

If in addition $\mathcal{I}_1 = (- \infty, t_0]$ and $X_1$ is
bounded on $\mathcal{ I}_1$ with $R(t) + \Pi_2(t)[X_1(t)] \gg 0$,
then
\begin{eqnarray*}  X_\infty \colon \mathbb{R}
%set
\to \mathcal{S}^n \quad \mbox{with} \quad X_\infty(t) := \lim_{k
\to \infty} X_1(t - k \theta), \quad t_0 - \theta < t \leq t_0,
\quad(3.15) \label{eqno:3.16} \end{eqnarray*} exists and defines a
$\theta$-periodic solution of \eqref{eqno:3.1}.
\end{lemma}

\begin{proof}
Let $\Box \, = \, \geq$ (the other cases are treated similarly). As in the
proof of Theorem \ref{thm3.6} it follows that $\Delta(t) := X_1(t - \theta) -
X_1(t)$ satisfies (on its maximal interval of existence $\mathcal{I}_1
\subset (- \infty, t_0]$)
\[ \frac{d}{dt} \Delta(t) + \mathcal{L}_{A+BF_1}^*(t)[\Delta(t)]+
\hat{\Pi}(t)[\Delta(t)]+ H(t) = 0, \quad \Delta(t_0) \geq 0, \] where $H(t)
\geq 0$ for $t \in \mathcal{I}_1$. Therefore Proposition \ref{prop3.5}, (ii), yields
that $\Delta(t) \geq 0$ on $\mathcal{I}_1$ -- this proves the first assertion
of the lemma.

If in addition $\mathcal{I}_1 = (- \infty, t_0]$ and $X_1$ is bounded on
$\mathcal{I}_1$ then the limits in \eqref{eqno:3.16} exist since the
sequences $\big( X_1(t - k \theta) \big)_{k \in \mathbb{N}}$ are monotonic
and bounded. Therefore the $\theta$-periodic function $X_\infty$ defined by
\eqref{eqno:3.16} is obviously a solution of \eqref{eqno:3.1} with $R(t) +
\Pi_2(t)[X_\infty(t)] \gg 0$. \end{proof}

\begin{theorem} \label{thm3.11}
The following statements are equivalent:
\begin{enumerate}
\item[(i)] Equation \eqref{eqno:3.1}
has a $\theta$-periodic solution $\hat X$ with the additional
property $R(t)+\Pi_2(t)[\hat X(t)]>0$ for all $t\in\mathbb{R}$.
\item[(ii)] There exist two $\theta$-periodic functions $X_\ell, X_u$ and
$t_0 \in \mathbb{R}$ with
\begin{enumerate}
\renewcommand{\labelenumii}{\alph{enumii})}
\renewcommand{\itemsep}{5pt}
\item[(a)] $X_\ell(t) \leq X_u(t)$ for $t \in \mathbb{R}$,
\item[(b)] $\frac{d}{dt} X_{\ell}(t) + \mathcal{R}(t, X_{\ell}(t)) \geq
0$ and $R(t)+\Pi_2(t)[X_{\ell}(t)]>0$ for all $t \leq t_0$,
\item[(c)] $\frac{d}{dt} X_u(t)+ \mathcal{R}(t, X_u(t)) \leq 0$ for
$t \leq t_0$.
\end{enumerate}
\item[(iii)] There exist symmetric matrices $X_0, X_1$ with:
\begin{enumerate}
\item[($\alpha$)] $X_0 \leq X_1$.
\item[($\beta$)] The  solution $X(\cdot, t_0,X_0)$ of \eqref{eqno:3.1} is
defined on an interval  $\mathcal{I}_1$ containing $[t_0-2\theta,
t_0]$ such that $R(t)+\Pi_2(t)[X(t,t_0,X_0)]>0$ for all $t\in
\mathcal{I}_1$ and $X_0 \leq X(t_0-\theta, t_0, X_0)$.
\item[($\gamma$)] The  solution $X(\cdot, t_0,X_1)$ of \eqref{eqno:3.1} is
defined on an interval $\mathcal{I}_1$ containing $[t_0-2\theta,
t_0]$ such that $R(t)+\Pi_2(t)[X(t,t_0,X_1)]>0 $ for all $t\in
\mathcal{ I}_1$ and $X_1 \geq X(t_0 - \theta, t_0, X_1)$.
\end{enumerate}
\end{enumerate}
\end{theorem}

\begin{proof} (i) $\Longrightarrow$ (ii) is trivial since a), b), c) are
fulfilled with $X_\ell = X_u = \hat{X}$ for any $\theta$-periodic
solution $\hat{X}$ of \eqref{eqno:3.1}.

(ii) implies that there are functions $Q_\ell, Q_u$ with
$Q_\ell(t) \leq 0 \leq Q_u(t)$ and
\begin{gather*} \frac{d}{dt} X_\ell(t) + \mathcal{R}(t, X_\ell(t)) +
Q_\ell(t) = 0, \\
\frac{d}{dt} X_u(t) + \mathcal{R}(t, X_u(t)) + Q_u(t) = 0
\end{gather*}
for $t \leq t_0$. From (ii), a) and the Comparison Theorem we infer that for
$t \leq t_0$
\[ X_\ell(t) \leq X(t, t_0, X_\ell(t_0)) \leq X (t, t_0, X_u(t_0)) \leq
X_u(t). \]
Since $X_\ell$ and $X_u$ are periodic, the last
inequalities imply in particular that
\[
X_\ell(t_0 - \theta) = X_\ell(t_0) =: X_0 \leq X(t_0 - \theta, t_0 , X_0)
\]
and
\[ X_u(t_0 - \theta) = X_u(t_0) =: X_1 \geq X(t_0 - \theta, t_0, X_1). \]
Hence (ii) implies (iii).

Assume that (iii) is valid. Then it follows from Lemma \ref{lm3.10} and
Theorem \ref{thm3.6} that $X(\cdot, t_0, X_0)$ and $X(\cdot, t_0, X_1)$ are
cyclomonotonic with
\begin{align*} X(t, t_0, X_0) & \leq X(t - \theta, t_0, X_0) \\ & \leq X(t -
\theta, t_0, X_1) \leq X(t, t_0, X_1) \quad \mbox{for} \quad t
\leq t_0.
\end{align*}
Hence the limits
\begin{align*}
X_{j \infty}(t) & = \lim_{k \to \infty} X (t - k \theta, t_0, X_j) \\
& = \lim_{k \to \infty} X(t - (k \mp 1) \theta), t_0, X_j) = X_{j
\infty}(t \pm \theta) \end{align*}
 exist for $j = 0, 1$ and $t \leq t_0$.
Consequently (i) holds, since $X_{0 \infty}$ and $X_{1 \infty}$ are obviously
both periodic solutions of \eqref{eqno:3.1} (which may coincide). \end{proof}

\begin{remark} \label{rmk3.12}\rm
Although it is in general difficult to prove that
\eqref{eqno:3.1} has a $\theta$-periodic solution we can use
criterion (iii) of Theorem \ref{thm3.11} (in connection with Lemma \ref{lm3.10} and
the Comparison Theorem) to test if such an equilibrium exists.
Notice that it follows from Corollary \ref{coro5.11} (below) and Corollary
\ref{coro3.8} that the conditions of Theorem \ref{thm3.11}, (iii), are fulfilled with
$X_0(t) \equiv 0$ and the (stabilizing) function $X_1$ constructed
in the first step of the proof of Theorem \ref{thm4.7} (below) if
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $(A,B,\Pi)$ is stabilizable;
\item $0\in \Gamma^{\Sigma}$.
\end{enumerate}
Here b) ensures the existence of $X_0$ and a) the existence of
$X_1 \geq X_0$. \end{remark}

\section{Maximal solutions}

Throughout this section we assume that $X\to \Pi(t)[X]$ is
a linear operator.

\begin{definition} \label{def4.1} \rm
A solution $\tilde{X} \colon \mathcal{I} \to
\mathcal{S}^n$ of equation \eqref{eqno:3.1} is said to be the \textbf{maximal
solution with respect to $\Gamma^{\Sigma}$} (or \textbf{maximal solution} for
shortness) if $\tilde{X}(t) \geq \hat{X}(t)$ for arbitrary $\hat{X} \in
\Gamma^{\Sigma}$. \end{definition}

In this section we prove a result concerning the existence of the maximal
solution with respect to $\Gamma^{\Sigma}$ of equation \eqref{eqno:3.1}.
First, we give a definition which will play a crucial role in the next
developments. That is the concept of stabilizability for the triple $(A, B,
\Pi)$.

\begin{definition} \label{def4.2} \rm
We say that the triple $(A, B, \Pi)$ is \textbf{stabilizable} if
there exists a bounded and continuous function $F \colon \mathcal{I}
\to \mathbb{R}^{m \times n}$ such that the operator $\mathcal{L}_{A+BF,
\Pi_F^*}$ generates an exponentially stable evolution. The function $F$ will
be termed a \textbf{stabilizing feedback gain}. \end{definition}

We shall show later (Corollary \ref{coro5.11}) that if $A$, $B$, $\Pi$ are periodic
functions with period $\theta$ and if the triple $(A, B, \Pi)$ is
stabilizable then there exists a stabilizing feedback gain which is a
periodic function with period $\theta$. Moreover if $A(t) \equiv A$, $B(t)
\equiv B$, $\Pi(t) \equiv \Pi$ for $t \in \mathbb{R}$, and if the triple $(A,B,
\Pi)$ is stabilizable, then there exists a stabilizing feedback gain which is
constant.

In the particular case when $\Pi(t)$ is of the form \eqref{eqno:3.3} then the
above definition of stabilizability (see also Section 7.3) reduces to the
standard definition of stabilizability for stochastic systems (mean-square
stabilizability -- see \cite{FCdS98}).

Applying Theorem \ref{thm2.11} we have the following result.

\begin{corollary} \label{coro4.3}
The triple $(A, B, \Pi)$ is stabilizable if and only if there
exists some $X \in C^1_b(\mathcal{I}, \mathcal{S}^n)$ with $X(t) \gg 0$ and a
bounded and continuous function $F \colon \mathcal{I} \to \mathbb{R}^{m
\times n}$ such that
$$ \frac{d}{dt} X(t) + \mathcal{L}_{A+BF, \Pi_F^*}^*(t)[X(t)] \ll 0 \quad
\mbox{for all} \quad t \in \mathcal{I}. $$
\end{corollary}

In the case $\mathcal{I} = \mathbb{R}$, using Theorem \ref{thm2.18}, we get the following
corollary.

\begin{corollary} \label{coro4.4}
For $\mathcal{I} = \mathbb{R}$ the following are equivalent:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item The triple $(A,B,\Pi)$ is stabilizable.
\item There exists a bounded $C^1$-function $X \colon \mathbb{R} \to
\mathcal{S}^n$ with bounded derivative, $X(t) \gg 0$ and a bounded and
continuous function $F \colon \mathbb{R} \to \mathbb{R}^{m\times n}$ which
satisfies
\begin{equation} \frac{d}{dt} X(t) - \mathcal{L}_{A+BF, \Pi_F^*}(t)[X(t)] \gg
0. \label{eqno:4.1} \end{equation}
\end{enumerate}
\end{corollary}

\begin{remark} \label{rmk4.5} \rm
In the particular case when the coefficients do not depend on
$t$, \eqref{eqno:4.1} can be converted in an LMI which can be solved using an
LMI solver (see \cite{ARZh00}). \end{remark}

Now we state an auxiliary result which together with Lemma \ref{lm3.4} plays a
crucial role in the proof of the main result of this section and follows
directly from \eqref{eqno:3.7}.

\begin{lemma} \label{lm4.6}
If $W \colon \mathcal{I} \to \mathbb{R}^{m \times n}$ is a
continuous function and if $X$ is a solution of the differential equation
$$ \frac{d}{dt} X(t) + \mathcal{L}_{A+BW, \Pi_W^*}^*(t)[X(t)] + M_W(t) = 0 $$
and if $\det \big\{ R(t) + \Pi_2(t)[X(t)] \big\} \neq 0$ then $X$ satisfies
also the differential equation
\begin{align*} \lefteqn{\frac{d}{dt} X(t) + \mathcal{L}_{A+BF,
\Pi_F^*}^*(t)[X(t)] + M_F(t)} \\ & \quad {} + \big[ F(t) - W(t)\big]^T
\Theta(t,X(t)) \big[ F(t) - W(t) \big] = 0 \end{align*} where $F(t) :=
F^{X}(t)$ and $\Theta(t,X(t)) := R(t)+\Pi_2(t)[X(t)]$. \end{lemma}

The main result of this section is as follows:

\begin{theorem} \label{thm4.7}
Assume that $(A, B, \Pi)$ is stabilizable. Then the following are
equivalent:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $\Gamma^{\Sigma} \neq \emptyset$.
\item Equation \eqref{eqno:3.1} has a maximal and bounded solution $\tilde{X}
\colon \mathcal{I} \to \mathcal{S}^n$ with $R(t) + \Pi_2(t)[\tilde{X}(t)] \gg
0$.
\end{enumerate}
If $A$, $B$, $\Pi$, $\mathcal{Q}$ are $\theta$-periodic functions, then
$\tilde{X}$ is also a $\theta$- periodic function.

Moreover, if $A(t) \equiv A$, $B(t) \equiv B$, $\Pi(t) \equiv \Pi$,
$\mathcal{Q}(t) \equiv \mathcal{Q}$, then the maximal solution of equation
\eqref{eqno:3.1} is constant and it solves the nonlinear algebraic equation
\eqref{eqno:3.14}.
\end{theorem}

\begin{proof} (ii) $\Rightarrow$ (i) is obvious, since $\tilde{X} \in
\Gamma^{\Sigma}$.

It remains to prove the implication (i) $\Rightarrow$ (ii). Since $(A, B,
\Pi)$ is stabilizable it follows that there exists a bounded and continuous
function $F_0 \colon \mathcal{I} \to \mathbb{R}^{m \times n}$ such that
the operator $\mathcal{L}_{A+BF_0, \Pi_{F_0}^*}$ generates an exponentially
stable evolution.

Let $\varepsilon > 0$ be fixed. Using Theorem \ref{thm2.13} one obtains that equation
\begin{equation} \frac{d}{dt} X(t) + \mathcal{L}_{A+BF_0,\Pi_{F_0}^*}^*(t)
[X(t)] + M_{F_0}(t) + \varepsilon I_n = 0 \label{eqno:4.2}
\end{equation}
has a unique bounded solution $X_1 \colon \mathcal{I} \to\mathcal{S}^n$. We
shall show that $X_1(t) \gg \hat X(t)$ for arbitrary $\hat{X} \in
\Gamma^{\Sigma}$. If $\hat{X} \in \Gamma^{\Sigma}$ then we obtain immediately
that $\hat{X}$ fulfills
$$ \frac{d}{dt} \hat{X}(t) + \mathcal{R}(t,\hat X(t)) \geq 0 \quad \mbox{for}
\quad t \in \mathcal{I}; $$ consequently $\hat{X}$ solves the equation
\begin{equation} \frac{d}{dt} \hat{X}(t) + \mathcal{R}(t,\hat{X}(t)) -
\hat{M}(t) = 0, \label{eqno:4.3}
\end{equation}
where $\hat{M}(t)=
\frac{d}{dt} \hat{X}(t) + \mathcal{R}(t,\hat{X}(t)) \geq 0$.
Applying Lemma \ref{lm3.4}, \eqref{eqno:4.3} may be written as
\begin{equation} \label{eqno:4.4}
\begin{aligned}
\lefteqn{\frac{d}{dt} \hat{X}(t) + \mathcal{L}_{A+BF_0,
\Pi_{F_0}^*}^*(t)[\hat{X}(t)] + M_{F_0}(t)} \\ & \quad {} -\big[ F_0(t) -
\hat{F}(t) \big]^T \Theta(t,\hat{X}(t)) \big[F_0(t) - \hat{F}(t) \big] -
\hat{M}(t) = 0,
\end{aligned}
\end{equation}
where $\hat{F}(t) := F^{\hat{X}}(t)$. From \eqref{eqno:4.2} and
\eqref{eqno:4.4} we deduce that $t \mapsto X_1(t) - \hat{X}(t)$ is a bounded
solution of the differential equation
$$ \frac{d}{dt} Y(t) + \mathcal{L}_{A+BF_0, \Pi_{F_0}^*}^*(t)[Y(t)] + H_1(t)
= 0 $$ with
$$ H_1(t) = \varepsilon I_n + [F_0(t) - \hat{F}(t)]^T \Theta(t,\hat{X}(t))
[F_0(t) - \hat{F}(t)] + \hat{M}(t). $$ Clearly $H_1(t) \geq
\varepsilon I_n> 0$. Hence Theorem \ref{thm2.13} implies $X_1(t) -
\hat{X}(t) \gg 0$. Therefore $R(t)+ \Pi_2(t)[X_1(t)] \geq R(t) +
\Pi_2(t)[\hat{X}(t)] \gg 0$ for $t\in \mathcal{I}$. Thus we obtain
that $F_1(t) := F^{X_1}(t)$ is well defined. We show that $F_1$ is
a stabilizing feedback gain for the triple $(A, B, \Pi)$. To this
end, based on Lemma \ref{lm4.6}, we rewrite equation \eqref{eqno:4.2} as
\begin{equation} \label{eqno:4.5}
\begin{aligned}
\lefteqn{\frac{d}{dt} X_1(t) + \mathcal{L}_{A+BF_1, \Pi_{F_1}^*}^*(t)[X_1(t)]
+ M_{F_1}(t) + \varepsilon I_n} \\ & \quad {} + \big[ F_1(t) - F_0(t) \big]^T
\Theta(t,X_1(t)) \big[ F_1(t) - F_0(t) \big] = 0.
\end{aligned}
\end{equation}
On the other hand, based on Lemma \ref{lm3.4}, equation \eqref{eqno:4.3} can be
rewritten as
\begin{align*} \lefteqn{\frac{d}{dt} \hat{X}(t) +
\mathcal{L}_{A+BF_1, \Pi_{F_1}^*}^*(t)[\hat{X}(t)] + M_{F_1}(t) - \hat{M}(t)}
\nonumber \\ & \quad {} - \big[ F_1(t) - \hat{F}(t) \big]^T
\Theta(t,\hat{X}(t)) \big[ F_1(t) - \hat{F}(t) \big] = 0.
\end{align*}
Subtracting the last equation from \eqref{eqno:4.5} we obtain
$$ \frac{d}{dt} \big[ X_1(t) - \hat{X}(t) \big] + \mathcal{L}_{A+BF_1,
\Pi_{F_1}^*}^*(t) \big[ X_1(t) - \hat{X}(t) \big] + \tilde{H}(t)=0
$$
where
\begin{align*} \tilde{H}(t) & = \varepsilon I_n + \big[ F_1(t) - F_0(t)
\big]^T \Theta(t, X_1(t)) \big[ F_1(t) - F_0(t) \big] \\ & \qquad {} + \big[
F_1(t) - \hat F(t) \big]^T \Theta(t,\hat X(t)) \big[ F_1(t) - \hat{F}(t)
\big] + \hat{M}(t) \gg 0. \end{align*} Applying the implication (vi)
$\Rightarrow$ (i) in Theorem \ref{thm2.11} we infer that
$\mathcal{L}_{A+BF_1, \Pi_{F_1}^*}$ generates an exponentially stable
evolution. This means that $F_1 = F^{X_1}$ \textit{is a stabilizing feedback
gain}; notice that, as a consequence of Theorem \ref{thm2.13}, $F_1$ is
constant (or periodic) if the coefficients of \eqref{eqno:3.1} are constant
(or periodic, respectively).

Taking $X_1$, $F_1$ as a first step we construct two sequences $\{
X_k \}_{k \geq 1} $ and $\{ F_k \}_{k \geq 1}$, where $X_k$ is the
unique bounded solution of the differential equation
\begin{equation} \frac{d}{dt} X_k(t) + \mathcal{L}_{A+BF_{k-1},
\Pi_{F_{k-1}}^*}^*(t)[X_k(t)] + M_{F_{k-1}}(t) +
\frac{\varepsilon} {k} I_n = 0 \label{eqno:4.6}
\end{equation}
 and
$F_k(t) := F^{X_k}(t)$. We show inductively that the following
items hold:
\begin{enumerate}
\item[$(a_k)$] $X_k(t) - \hat{X}(t) > \mu_k I_n$ for arbitrary
$\hat{X} \in \Gamma^{\Sigma}$, $\mu_k > 0$ independent of $\hat{X}$.
\item[$(b_k)$] $F_k$ is a stabilizing feedback gain for the triple $(A, B,
\Pi)$.
\item[$(c_k)$] $X_k(t) \geq X_{k+1}(t)$ for $t \in \mathcal{I}$.
\end{enumerate}
For $k=1$, items $(a_1)$, $(b_1)$ were proved before.

To prove $(c_1)$ we subtract \eqref{eqno:4.6}, written for $k=2$,
from \eqref{eqno:4.5} and get
$$ \frac{d}{dt} \big[ X_1(t) - X_2(t) \big] + \mathcal{L}_{A+BF_1,
\Pi_{F_1}^*}^*(t) \big[ X_1(t) - X_2(t) \big] + \Delta_1(t) = 0,$$
where
$$ \Delta_1(t) := \frac{\varepsilon}{2} I_n + \big[ F_1(t) - F_0(t) \big]^T
\Theta(t,X_1(t)) \big[ F_1(t) - F_0(t) \big] \gg 0. $$ Invoking
Theorem \ref{thm2.13}, (b), one obtains that $X_1(t) - X_2(t) \gg 0$ and
thus $(c_1)$ is fulfilled. Let us assume that $(a_i)$ , $(b_i)$,
$(c_i)$ are fulfilled for $i \leq k-1$ and let us prove them for
$i=k$.

Based on $(b_{k-1})$ and Theorem \ref{thm2.13} we deduce that equation
\eqref{eqno:4.6} has a unique solution $X_k \colon \mathcal{I}
\to \mathcal{S}^n$. Applying Lemma \ref{lm3.4} with $W(t)
:=F_{k-1}(t)$ one obtains that \eqref{eqno:4.3} may be rewritten
as
\begin{align*} \lefteqn{\frac{d}{dt} \hat{X}(t) +
\mathcal{L}_{A+BF_{k-1}, \Pi_{F_{k-1}}^*}^*(t)[\hat{X}(t)] + M_{F_{k-1}}(t)}
\\ & \quad {} - \big[ F_{k-1}(t) - \hat{F}(t) \big]^T \Theta(t,\hat{X}(t))
\big[ F_{k-1}(t) - \hat{F}(t) \big] - \hat{M}(t) = 0. \end{align*}
Subtracting this equation from \eqref{eqno:4.6} one obtains that $t \mapsto
X_k(t) - \hat X(t)$ is a bounded solution of the equation
$$ \frac{d}{dt} X(t) + \mathcal{L}_{A+BF_{k-1}, \Pi_{F_{k-1}}^*}^*(t)[X(t)] +
H_k(t) = 0,
$$
where
$$ H_k(t) = \frac{\varepsilon}{k} I_n + [F_{k-1}(t) - \hat{F}(t)]^T
\Theta(t,\hat{X}(t)) [F_{k-1}(t) - \hat{F}(t)] + \hat{M}(t) \gg
0.$$
Since $\mathcal{L}_{A+BF_{k-1}, \Pi_{F_{k-1}}^*}$ generates
an exponentially stable evolution, we obtain from Theorem \ref{thm2.13},
(b), that there exist
\begin{equation} \mu_k > 0 \quad \mbox{such that} \quad X_k(t) - \hat{X}(t)
\geq \mu_k I_n \quad \mbox{for} \quad t \in
\mathcal{I},\label{eqno:4.7}
\end{equation}
thus  $(a_k)$ is fulfilled.

Let us show that $(b_k)$ is fulfilled. Firstly from
\eqref{eqno:4.7} we have
$$ R(t) + \Pi_2(t)[X_k(t)] \gg 0, $$
therefore $F_k$ is well defined. Applying Lemma \ref{lm4.6} to equation
\eqref{eqno:4.6}, one obtains that $X_k$ solves the equation
\begin{equation} \label{eqno:4.8}
\begin{aligned}
\lefteqn{\frac{d}{dt} X_k(t) + \mathcal{L}_{A+BF_k, \Pi_{F_k}^*}^*(t)[X_k(t)]
+ M_{F_k}(t) + \frac{\varepsilon}{k} I_n} \\ & \quad {} + \big[ F_k(t) -
F_{k-1}(t) \big]^T \Theta(t,X_k(t)) \big[ F_k(t) - F_{k-1}(t) \big] = 0.
\end{aligned}
\end{equation}
On the other hand, Lemma \ref{lm3.4} applied to equation \eqref{eqno:4.3}
gives
\begin{align*}
\lefteqn{\frac{d}{dt} \hat{X}(t) + \mathcal{L}_{A+BF_k,
\Pi_{F_k}^*}^*(t)[\hat{X}(t)] + M_{F_k}(t)} \nonumber \\
& \quad {} - \big[ F_k(t) - \hat{F}(t) \big]^T \Theta(t,\hat{X}(t)) \big[
F_k(t) - \hat{F}(t) \big] - \hat{M}(t) = 0.
\end{align*}
>From the last two equations one obtains
$$ \frac{d}{dt} \big[ X_k(t) - \hat{X}(t) \big] + \mathcal{L}_{A+BF_k,
\Pi_{F_k}^*}^*(t) \big[ X_k(t) - \hat{X}(t) \big] +
\tilde{H}_k(t)= 0$$
 with
\begin{align*}
\tilde{H}_k(t) & = \frac{\varepsilon}{k} I_n + \big[ F_k(t) - F_{k-1}(t)
\big]^T \Theta(t,X_k(t)) \big[ F_k(t) - F_{k-1}(t) \big] \\ & \qquad {} +
\big[ F_k(t) - \hat F(t) \big]^T \Theta(t,\hat X(t)) \big[ F_k(t) -
\hat{F}(t) \big] + \hat{M}(t) \geq \frac{\varepsilon}{k} I_n.
\end{align*}
Implication (vi) $\Rightarrow$ (i) of Theorem \ref{thm2.11} allows us to
conclude that $\mathcal{L}_{A+BF_k, \Pi_{F_k}^*}$ generates an exponentially
stable evolution which shows that $(b_k)$ is fulfilled.

It remains to prove that $(c_k)$ holds. To this end we subtract
equation \eqref{eqno:4.6}, written for $k+1$ instead of $k$, from
equation \eqref{eqno:4.8} and get
\begin{align*}
\lefteqn{\frac{d}{dt} \big[ X_k(t) - X_{k+1}(t) \big] + \mathcal{L}_{A+BF_k,
\Pi_{F_k}^*}^*(t) \big[ X_k(t) - X_{k+1}(t) \big] +
\frac{\varepsilon}{k(k+1)} I_n} \\ & \quad {} + \big[ F_k(t) - F_{k-1}(t)
\big]^T \Theta(t,X_k(t)) \big[ F_k(t) - F_{k-1}(t) \big] = 0.
\end{align*}
Since $\mathcal{L}_{A+BF_k,\Pi_{F_k}^*}$ generates an exponentially stable
evolution one obtains, via Theorem \ref{thm2.13}, (b), that equation
\eqref{eqno:4.8} has a unique bounded solution which additionally is
uniformly positive. Therefore $X_k(t) - X_{k+1}(t) \gg 0$ and thus $(c_k)$
holds.

Now from $(a_k)$ and $(c_k)$ we deduce that the sequence $\{ X_k
\}_{k \geq 1}$ is monotonically decreasing and bounded, therefore
it is convergent.

Set $\tilde{X}(t) := \lim\limits_{k \to \infty} X_k(t)$. In a
standard way one obtains that $\tilde{X}$ is a solution of equation
\eqref{eqno:3.1}. Invoking again $(a_k)$ one obtains that $\tilde{X}(t) \geq
\hat{X}(t)$ for arbitrary $\hat{X} \in \Gamma^{\Sigma}$. Hence $\tilde{X}$ is
just the maximal solution of equation \eqref{eqno:3.1} and thus (i)
$\Rightarrow$ (ii) is proved.

To complete the proof, let us remark that if $A$, $B$, $\Pi$, $\mathcal{Q}$
are periodic functions with the same period $\theta$ then via Corollary \ref{coro5.11},
(i), it follows that there exists a stabilizing feedback gain which is a
$\theta$-periodic function. Applying Theorem \ref{thm2.13}, (c), one obtains that
$X_k$, $F_k$ are $\theta$-periodic functions for all $k$ and thus $\tilde{X}$
will be a $\theta$-periodic function. Also if $A(t) \equiv A$, $B(t) \equiv
B$, $\Pi(t) \equiv \Pi$, $\mathcal{Q}(t) \equiv \mathcal{Q}$ and $(A, B,
\Pi)$ is stabilizable, then from Corollary \ref{coro5.11}, (ii), we obtain that there
exists a stabilizing feedback gain which is constant. Applying again Theorem
\ref{thm2.13}, (c), one obtains that $X_k$ and $F_k$ are constant functions for all $k
\geq 1$ and therefore $\tilde{X}$ is constant. \end{proof}

\section{Stabilizing solutions}

In this section we deal with stabilizing solutions of equation
\eqref{eqno:3.1} in the case where $X \to \Pi(t)[X]$ is a linear
operator. We shall prove the uniqueness of a bounded and stabilizing solution
and we shall provide a necessary and sufficient condition for the existence
of a bounded and stabilizing solution of equation \eqref{eqno:3.1}.

\begin{definition}\label{def5.1} \rm
Let $X_s \colon \mathcal{I} \to \mathcal{S}^n$ be a
solution of equation \eqref{eqno:3.1} and denote by $F_s(t) :=
F^{X_s}(t)$ the corresponding feedback matrix. Then $X_s$ is
called a \textbf{stabilizing solution} if the operator
$\mathcal{L}_{A+BF_s, \Pi_{F_s}^*}$ generates an exponentially
stable evolution where $\Pi_{F_s}$ is defined as in
\eqref{eqno:3.6} for $W(t) = F_s(t)$. \end{definition}

\begin{theorem} \label{thm5.2}
Let $\Sigma = (A, B, \Pi, \mathcal{Q})$ be such that
$\Gamma^{\Sigma} \neq \emptyset$. If $X_s \colon \mathcal{I}
\to \mathcal{S}^n$ is a bounded and stabilizing solution
of equation \eqref{eqno:3.1} then $X_s$ coincides with the maximal
solution with respect to $\Gamma^{\Sigma}$ of equation
\eqref{eqno:3.1}.
\end{theorem}

\begin{proof} Applying Lemma \ref{lm3.4} we deduce that $X_s$ verifies the
equation
\begin{equation}
\frac{d}{dt} X_s(t) + \mathcal{L}_{A+BF_s, \Pi_{F_s}^*}^*(t)[X_s(t)] +
M_{F_s}(t) = 0 \label{eqno:5.1}
\end{equation}
where $F_s:= F^{X_s}$. Let $\hat{X}$ be arbitrary in $\Gamma^{\Sigma}$. As in
the proof of Theorem \ref{thm4.7} one obtains that there exists $\hat{M}(t)
\geq 0$ such that $\hat{X}$ verifies a differential equation of the form
\eqref{eqno:4.3}. Applying Lemma \ref{lm3.4} to equation \eqref{eqno:4.3} we
get
\begin{align*}
\lefteqn{\frac{d}{dt} \hat{X}(t) + \mathcal{L}_{A+BF_s,
\Pi_{F_s}^*}^*(t)[\hat{X}(t)] + M_{F_s}(t) - \hat{M}(t)} \\ & \quad {} -
\big[ F_s(t) - \hat{F}(t) \big]^T \Theta(t,\hat{X}(t)) \big[ F_s(t) -
\hat{F}(t) \big] = 0.
\end{align*}
Subtracting the last two equations we obtain that $t\mapsto X_s(t)
- \hat{X}(t)$ is a bounded solution of the backward differential
equation
$$ \frac{d}{dt} X(t) + \mathcal{L}_{A+BF_s, \Pi_{F_s}^*}^*(t)[X(t)] + H_s(t)
= 0 $$ where
$$ H_s(t) = [F_s(t) - \hat{F}(t)]^T \Theta(t,\hat{X}(t)) [F_s(t) -
\hat{F}(t)] + \hat{M}(t) \geq 0. $$ Since $\mathcal{L}_{A+BF_s,\Pi_{F_s}^*}$
generates an exponentially stable evolution one obtains, using Theorem
\ref{thm2.13},
that $X_s(t) - \hat{X}(t) \geq 0$ and thus the proof is complete. \end{proof}

\begin{remark} \label{rmk5.3} \rm
>From Theorem \ref{thm5.2} it follows that if $\Gamma^{\Sigma}$ is not empty
then a bounded and stabilizing solution of equation \eqref{eqno:3.1} (if it
exists) will satisfy the condition
$$ R(t) + \Pi_2(t)[X_s(t)] \gg 0. $$
\end{remark}

\begin{corollary} \label{coro5.4}
If $\Gamma^{\Sigma}$ is not empty then equation \eqref{eqno:3.1}
has at most one bounded and stabilizing solution. \end{corollary}

\begin{proof} Let us assume that equation \eqref{eqno:3.1} has two bounded and
stabilizing solutions $X_i$, $i = 1,2$. From the above remark we get $R(t) +
\Pi_2(t)[X_i(t)] \gg 0$. Arguing as in the proof of Theorem \ref{thm5.2}, we
obtain both $X_1(t) \geq X_2(t)$ and $X_2(t) \geq X_1(t)$ hence $X_1(t) =
X_2(t)$ and the proof ends. \end{proof}

In the particular case, where $\Pi(t)$ is of the form
\eqref{eqno:3.3}, in \cite{preprint00} the uniqueness of the
bounded and stabilizing solution of equation \eqref{eqno:3.1} was
proved without any assumption concerning $\Gamma^{\Sigma}$. In
that case $R(t) + \Pi_2(t)[X_s(t)]$ has not a definite sign.

\begin{theorem} \label{thm5.5}
Assume that $A$, $B$, $\Pi$, $\mathcal{Q}$ are periodic functions
with period $\theta$ and that $\Gamma^{\Sigma}$ is not empty. Then the
bounded and stabilizing solution of equation \eqref{eqno:3.1} (if it exists)
is $\theta$-periodic. \end{theorem}

\begin{proof} Let $X_s \colon \mathcal{I} \to \mathcal{S}^n$ be a
bounded and stabilizing solution of \eqref{eqno:3.1}. We define
$\tilde{X}(t) := X_s(t + \theta)$. By direct computation we obtain
that $\tilde{X}$ is also a solution of equation \eqref{eqno:3.1}.
We shall prove that $\tilde{X}$ is also a stabilizing solution of
equation \eqref{eqno:3.1}.

Set $\tilde{F}(t) := F^{\tilde{X}}(t)$. We show that
$\mathcal{L}_{A+B\tilde{F}, \Pi_{\tilde{F}}^*}$ generates an
exponentially stable evolution. Let $\tilde{T}(t,t_0)$ be the
linear evolution operator defined by the linear differential
equation
\begin{equation} \frac{d}{dt} S(t) = \mathcal{L}_{A+B\tilde{F},
\Pi^*_{\tilde{F}}}(t)[S(t)]. \label{eqno:5.2}
\end{equation}
Because of the periodicity we obtain that
$$ \mathcal{L}_{A+B\tilde{F}, \Pi_{\tilde{F}}^*}(t) = \mathcal{L}_{A+BF_s,
\Pi_{F_s}^*}(t+\theta) \quad \mbox{for} \quad t \in \mathcal{I}.
$$ If $\tilde{S}(t,t_0, H)$ is the solution of \eqref{eqno:5.2}
with $\tilde{S}(t_0,t_0,H) = H$, then we have
$$ \frac{d}{dt} \tilde{S}(t,t_0,H) = \mathcal{L}_{A+BF_s,
\Pi_{F_s}^*}(t+\theta) \tilde{S}(t,t_0,H). $$ From the uniqueness of the
solution of this initial value problem we infer that
$$ \tilde{S}(t,t_0,H) = S(t+\theta, t_0+\theta, H), $$
where $t \mapsto S(t,\tau, H)$ is the solution of
\begin{equation} \frac{d}{dt} S(t) = \mathcal{L}_{A+BF_s,
\Pi_{F_s}^*}(t)[S(t)], \quad S(\tau, \tau, H) = H.\label{eqno:5.3}
\end{equation}
Thus we get $\tilde{T}(t,t_0) = T_{F_s}(t+\theta, t_0+\theta)$
where $T_{F_s}(t,t_0)$ is the linear evolution operator defined by
\eqref{eqno:5.3}. The last equality leads to
$$ \|\tilde{T}(t,t_0)\| = \|T_{F_s}(t+\theta, t_0+\theta)\| \leq \beta
e^{-\alpha(t-t_0)} \quad \mbox{for} \quad t \geq t_0 $$ with
$\alpha, \beta> 0$, which shows that $\tilde{X}$ is also a bounded
and stabilizing solution of equation \eqref{eqno:3.1}.

Applying Corollary \ref{coro5.4} one obtains that $\tilde{X}(t) = X_s(t)$
for $t \in \mathcal{I}$, that means $X_s(t+\theta) = X_s(t)$ for
all $t$, which shows that $X_s$ is a $\theta$-periodic function
and thus the proof ends.
\end{proof}

\begin{corollary} \label{coro5.6}
If $\Gamma^{\Sigma} \neq \emptyset$ and $A(t) \equiv A$, $B(t)
\equiv B$, $\Pi(t) \equiv \Pi$, $\mathcal{Q}(t) \equiv
\mathcal{Q}$, $t \in \mathbb{R}$, then the stabilizing solution of
equation \eqref{eqno:3.1} (if it exists) is constant and solves
the algebraic equation \eqref{eqno:3.14}.
\end{corollary}

\begin{proof} Since the matrix coefficients of equation \eqref{eqno:3.1} are
constant functions they may be viewed as periodic functions with arbitrary
period. Applying Theorem \ref{thm5.5} it follows that the bounded and stabilizing
solution of equation \eqref{eqno:3.1} is a periodic function with arbitrary
period. Therefore it is a constant function and thus the proof ends.
\end{proof}

The following lemma which is a generalization of the invariance
under feedback transformations of standard Riccati differential
equations will be useful in the next developments. Since
\begin{align*}
\lefteqn{\big[ W(t) - F^X(t) \big]^T \big\{ R(t) + \Pi_2(t)[X(t)] \big\}} \\
& \quad = X(t)B(t) + \Pi_{12}(t)[X(t)] +W^T(t) \Pi_2(t)[X(t)] + L(t) + W^T(t)
R(t)
\end{align*}
the conclusion of this lemma follows immediately from Lemma \ref{lm3.4}.

\begin{lemma} \label{lm5.7}
Let $W \colon \mathcal{I} \to \mathbb{R}^{m \times n}$ be a
bounded and continuous function. Then $X \colon \mathcal{I}_1
\subset \mathcal{I} \to \mathcal{S}^n$ is a solution of
equation \eqref{eqno:3.1} associated to the quadruple $\Sigma =
(A, B, \Pi, \mathcal{Q})$ if and only if $X$ is a solution of the
equation of type \eqref{eqno:3.1} associated to the quadruple
$\Sigma^W = (A+BW, B, \Pi^W, \mathcal{Q}^W)$, where $\Pi^W(t)
\colon \mathcal{S}^n \to \mathcal{S}^{n+m}$ is given by
\begin{align*}
\Pi^W(t)[X]& = \begin{pmatrix} I_n & 0 \\ W(t) &
I_m \end{pmatrix}^T \begin{pmatrix} \Pi_1(t)[X] & \Pi_{12}(t)[X]
\\ \big\{ \Pi_{12}(t)[X] \big\}^T & \Pi_2(t)[X] \end{pmatrix}
\begin{pmatrix} I_n & 0\\ W(t) & I_m \end{pmatrix} \\
& =
\begin{pmatrix} \Pi_W(t)[X] &\Pi_{12}(t)[X] + W^T(t) \Pi_2(t)[X] \\ \big\{
\Pi_{12}(t)[X] +W^T(t) \Pi_2(t)[X] \big\}^T & \Pi_2(t)[X]
\end{pmatrix}
\end{align*}
and
\begin{align*}
\mathcal{Q}^W(t) & = \begin{pmatrix} I_n & 0 \\
W(t) & I_m \end{pmatrix}^T
\begin{pmatrix} M(t) & L(t) \\ L^T(t) & R(t) \end{pmatrix} \begin{pmatrix}
I_n & 0 \\ W(t) & I_m \end{pmatrix} \\
& = \begin{pmatrix} M_W(t)
& L(t) + W^T(t) R(t) \\ L^T(t) + R(t) W(t)& R(t) \end{pmatrix}.
\end{align*}
\end{lemma}

\begin{theorem} \label{thm5.8}
Under the considered assumptions the following
assertions are equivalent:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $(A,B,\Pi)$ is stabilizable and the set $\tilde{\Gamma}^{\Sigma}$ is
not empty;
\item \eqref{eqno:3.1} has a stabilizing and bounded solution $X_s \colon
\mathcal{I} \to \mathcal{S}^n$ satisfying
$$ R(t) + \Pi_2(t)[X_s(t)] \gg 0. $$
\end{enumerate}
\end{theorem}

\begin{proof} (i) $\Rightarrow$ (ii). If (i) holds then Theorem \ref{thm4.7} yields that
equation \eqref{eqno:3.1} has a bounded  maximal solution
$\tilde{X} \colon \mathcal{I} \to \mathcal{S}^n$. We show
that $\tilde{X}$ is just the stabilizing solution. If $\tilde{F}$
is the feedback matrix associated with $\tilde{X}$ then
\eqref{eqno:4.3} may be written as
\begin{align*}
\lefteqn{\frac{d}{dt} \hat{X}(t) + \mathcal{L}_{A+B\tilde{F},
\Pi_{\tilde{F}}^*}^*(t)[\hat{X}(t)] + M_{\tilde{F}}(t) - \hat{M}(t)} \\ &
\quad {} - \big[ \tilde{F}(t) - \hat{F}(t) \big]^T \Theta(t, \hat{X}(t))
\big[ \tilde{F}(t) - \hat{F}(t) \big] = 0.
\end{align*}
Since $\hat{X} \in \tilde{\Gamma}^{\Sigma}$ it is a solution of an equation
of type \eqref{eqno:4.3} with $\hat{M}(t) \gg 0$. Using again Lemma
\ref{lm3.4} one obtains that $t \mapsto \tilde{X}(t) - \hat{X}(t)$ is a
bounded and positive semi-definite solution of the backward differential
equation
$$ \frac{d}{dt} X(t) + \mathcal{L}_{A+B\tilde{F},
\Pi_{\tilde{F}}^*}^*(t)[X(t)] + H(t) = 0, $$ where
$$ H(t) := \hat{M}(t) + \big[ \tilde{F}(t) - \hat{F}(t) \big]^T \Theta(t,
\hat{X}(t)) \big[ \tilde{F}(t) - \hat{F}(t) \big]. $$
Since $\hat{M}(t) \gg 0$ it follows that $H(t) \gg 0$. Applying
implication (vi) $\Rightarrow$ (i) of Theorem \ref{thm2.11} one gets that
$\mathcal{L}_{A+B\tilde{F}, \Pi_{\tilde{F}}^*}$ generates an
exponentially stable evolution which shows that $\tilde{X}$ is a
stabilizing solution of equation \eqref{eqno:3.1}.

We prove now (ii) $\Rightarrow$ (i). If equation  \eqref{eqno:3.1}
has a bounded and stabilizing solution $X_s \colon \mathcal{I}
\to \mathcal{S}^n$ then $F_s := F^{X_s}$ is a stabilizing
feedback gain and therefore $(A, B, \Pi)$ is stabilizable.

Applying Lemma \ref{lm5.7} with $W(t) = F_s(t)$ we rewrite equation
\eqref{eqno:3.1} as
\begin{align*}
\lefteqn{\frac{d}{dt} X(t) + \mathcal{L}_{A+BF_s, \Pi_{F_s}^*}^*(t)[X(t)] +
M_{F_s}(t)} \\ & \quad {} - \mathcal{P}_{F_s}^T(t,X(t)) \Theta(t,X(t))^{-1}
\mathcal{P}_{F_s}(t,X(t)) = 0,
\end{align*}
where $X \mapsto \mathcal{P}_{F_s}(t,X) \colon \mathcal{S}^n \to
\mathbb{R}^{m \times n}$ is given by
$$
\mathcal{P}_{F_s}(t,X) = \big\{ X B(t) + \Pi_{12}(t)[X] + F_s^T(t)
\Pi_2(t)[X] + L(t) + F_s^T(t) R(t) \big\}^T $$
and
$\Theta(t,X)$ being as in Lemma \ref{lm4.6}. Let $T_{F_s}(t,t_0)$ be the linear
evolution operator defined by
$$ \frac{d}{dt} S(t) = \mathcal{L}_{A+BF_s, \Pi_{F_s}^*}(t)[S(t)].
$$
Since $F_s$ is a stabilizing feedback gain it follows that there
exist $\alpha, \beta > 0$ such that $\|T_{F_s}(t,t_0)\| \leq \beta
e^{-\alpha(t-t_0)}$.

Let $C_b(\mathcal{I}, \mathcal{S}^n)$ be the Banach space of
bounded and continuous functions $X \colon \mathcal{I} \to
\mathcal{S}^n$. Since $\Theta(t,X_s(t)) \gg 0$ for $t \in
\mathcal{I}$, it follows that there exist an open set $\mathcal{U}
\subset C_b(\mathcal{I}, \mathcal{S}^n)$ such that $X_s \in
\mathcal{U}$ and $\Theta(t,X(t)) \gg 0$ for all $X \in
\mathcal{U}$. Let $\Psi \colon \mathcal{U} \times \mathbb{R}
\to C_b$ be defined by
\begin{align*}
\Psi(X,\delta)(t) & = \int_t^{\infty} T^*_{F_s}(\sigma,t) \big[
M_{F_s}(\sigma) + \delta I_n \\ & \qquad {} - \mathcal{P}^T_{F_s}(\sigma,
X(\sigma)) \Theta^{-1}(\sigma, X(\sigma)) \mathcal{P}_{F_s}(\sigma,
X(\sigma)) \big] \, d\sigma - X(t).
\end{align*}
We apply the implicit function theorem to the equation
\begin{equation}
\Psi(X, \delta) = 0 \label{eqno:5.4}
\end{equation}
in order to obtain that there exists a function $X_{\delta} \in
\mathcal{U}$ such that
\begin{align*}
& X_{\delta}(t) \\
& = \int_t^{\infty}T^*_{F_s}(\sigma,t) \big[
M_{F_s}(\sigma) + \delta I_n  -\mathcal{P}^T_{F_s}(\sigma, X_{\delta}(\sigma))
\Theta^{-1}(\sigma, X_{\delta}(\sigma)) \mathcal{P}_{F_s}(\sigma,
X_{\delta}(\sigma) \big] \, d\sigma
\end{align*}
for $|\delta|$ small enough.

It is clear that $(X_s,0)$ is a solution of \eqref{eqno:5.4}. We
show now that
$$ d_1 \Psi(X_s(\cdot),0) \colon C_b(\mathcal{I}, \mathcal{S}^n) \to
C_b(\mathcal{I}, \mathcal{S}^n) $$ is an isomorphism, $d_1 \Psi$
being the derivative of $\Psi$ with respect to its first argument.
Since
$$ d_1 \Psi(X_s,0) Y = \lim_{\varepsilon \to 0} \frac{1}{\varepsilon}
\big[ \Psi(X_s + \varepsilon Y, 0) - \Psi(X_s, 0) \big]
$$
and $\mathcal{P}_{F_s}(\sigma, X_s(\sigma)) \equiv 0$ we obtain that $d_1
\Psi(X_s, 0) Y = -Y$ for all $Y \in C_b(\mathcal{I},\mathcal{S}^n)$.
Therefore $d_1 \Psi(X_s,0) = - I_{C_b}$, where $I_{C_b}$ is the identity
operator of $C_b(\mathcal{I}, \mathcal{S}^n)$ which is an isomorphism. Also
we see that $d_1\Psi(X,\delta)$ is continuous in $(X,\delta)=(X_s,0)$.
Applying the implicit function theorem we deduce that there exists
$\tilde{\delta} > 0$ and a smooth function $X_{\delta}(\cdot)
\colon(-\tilde{\delta}, \tilde{\delta}) \to \mathcal{U}$ which
satisfies $\Psi(X_{\delta}(\cdot), \delta) =0$  for all $\delta \in
(-\tilde{\delta}, \tilde{\delta})$. It is easy to see that if $\delta \in
(-\tilde{\delta}, 0)$ then $X_{\delta}(\cdot) \in \tilde{\Gamma}^{\Sigma}$
and the proof is complete. \end{proof}

\begin{corollary} \label{coro5.9}
Assume that $A$, $B$, $\Pi$, $\mathcal{Q}$ are periodic functions
with period $\theta > 0$. Under these conditions the following are
equivalent:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $(A, B, \Pi)$ is stabilizable and $\tilde{\Gamma}^{\Sigma}$ is not
empty.
\item Equation \eqref{eqno:3.1} has a bounded, $\theta$-periodic and
stabilizing solution $X_s \colon \mathcal{I} \to
\mathcal{S}^n$ which verifies $R(t) + \Pi_2(t)[X_s(t)] \gg 0$.
\item $(A, B, \Pi)$ is stabilizable and $\tilde{\Gamma}^{\Sigma}$ contains at
least a $\theta$-periodic function $\check{X}$.
\end{enumerate}
\end{corollary}

\begin{proof} (i) $\Leftrightarrow$ (ii) follows from Theorem \ref{thm5.8} and
Theorem \ref{thm5.5}. (iii) $\Rightarrow$ (i) is obvious.

It remains to prove (ii) $\Rightarrow$ (iii). In the proof of the implication
(ii) $\Rightarrow$ (i) in Theorem \ref{thm5.8} we have shown that there exist
$\tilde{\delta} > 0$ and a smooth function $\delta \mapsto X_{\delta}(\cdot)
\colon (-\tilde{\delta}, \tilde{\delta}) \to C_b(\mathcal{I}, \mathcal{S}^n)$
which satisfies
$$ \frac{d}{dt} X_{\delta}(t) + \mathcal{R}(t, X_{\delta}(t)) + \delta I_n =
0. $$ Let $\delta_1 \in (-\tilde{\delta}, 0)$ and set $\Sigma_1 :=
(A, B, \Pi, \mathcal{Q}_1)$ with
$$ \mathcal{Q}_1(t) := \begin{pmatrix} M(t) + \delta_1 I_n & L(t) \\ L^T(t) &
R(t) \end{pmatrix}. $$ It is easy to see that if $\delta \in
(-\tilde{\delta}, \delta_1)$ then $X_{\delta}(\cdot) \in
\tilde{\Gamma}^{\Sigma_1}$. Applying implication (i) $\Rightarrow$
(ii) of Theorem \ref{thm5.8} one obtains that the equation
$$ \frac{d}{dt} X(t) + \mathcal{R}(t,X(t)) + \delta_1 I_n = 0 $$
has a bounded and stabilizing solution $\hat{X}_{\delta_1}$.
Based on Theorem \ref{thm5.5} one obtains that $\hat{X}_{\delta_1}$ is a periodic function. The
conclusion follows since $\hat{X}_{\delta_1}(\cdot) \in
\tilde{\Gamma}^{\Sigma}$ and the proof is complete. \end{proof}

With the similar proof based on Corollary \ref{coro5.6} and Theorem \ref{thm5.8} we
obtain:

\begin{corollary} \label{coro5.10}
Assume that $A(t) \equiv A$, $B(t) \equiv B$, $\Pi(t) \equiv
\Pi$ and $\mathcal{Q}(t) \equiv \mathcal{Q}$. Then the following
are equivalent:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $(A, B, \Pi)$ is stabilizable and $\tilde{\Gamma}^{\Sigma}$ is not
empty;
\item Equation \eqref{eqno:3.1} has a bounded and stabilizing solution $X_s$
which is constant and solves the algebraic equation
\eqref{eqno:3.14}.
\item $(A, B, \Pi)$ is stabilizable and there exists at least a symmetric
matrix $\hat{X}$ such that $\hat{X} \in \tilde{\Gamma}^{\Sigma}$.
\end{enumerate}
\end{corollary}

As a simple consequence of Theorem \ref{thm5.8} we have:

\begin{corollary} \label{coro5.11}
Assume that $(A, B, \Pi)$ is stabilizable. Then:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item If $A$, $B$, $\Pi$ are periodic functions with period $\theta$ then
there exists a stabilizing feedback gain $F \colon \mathbb{R} \to \mathbb{R}^{m
\times n}$ which is a periodic function with period $\theta$.
\item If $A(t) \equiv A$, $B(t) \equiv B$, $\Pi(t) \equiv \Pi$ for all $t \in
\mathbb{R}$ then there exists a stabilizing feedback gain $F \in
\mathbb{R}^{m \times n}$.
\end{enumerate}
\end{corollary}

\begin{proof}
Consider the differential equation
\begin{equation} \label{eqno:5.5}
\begin{aligned}
\lefteqn{\frac{d}{dt} X(t) + A^T(t)X(t) + X(t)A(t) + I_n + \Pi_1(t)[X(t)]} \\
& \quad {} - \big\{ X(t)B(t) +\Pi_{12}(t)[X(t)] \big\} \big\{ I_m +
\Pi_2(t)[X(t)] \big\}^{-1} \\ & \qquad \times \big\{ X(t)B(t) +
\Pi_{12}(t)[X(t)] \big\}^T = 0,
\end{aligned}
\end{equation}
Equation \eqref{eqno:5.5} is an equation of the
type \eqref{eqno:3.1} corresponding to the quadruple $\Sigma_0 := (A, B, \Pi,
\mathcal{Q}_0)$ where $A$, $B$, $\Pi$ are as in \eqref{eqno:3.1} and
$\mathcal{Q}_0 = \left( \begin{smallmatrix} I_n & 0 \\ 0 & I_m
\end{smallmatrix} \right)$. It is seen that $\lambda^{\Sigma_0}(0) = \left(
\begin{smallmatrix} I_n & 0 \\ 0 & I_m \end{smallmatrix} \right) \gg 0$ and hence
$0 \in \tilde{\Gamma}^{\Sigma_0}$. Therefore equation \eqref{eqno:5.5} has a
bounded and stabilizing solution $X_s$ with the corresponding stabilizing
feedback gain
$$ F_s(t) = - \big\{ I_m + \Pi_2(t)[X_s(t)] \big\}^{-1} \big\{ X_s(t)B(t) +
\Pi_{12}(t)[X_s(t)] \big\}^T. $$ If the matrix coefficients of
\eqref{eqno:5.5} are periodic functions with period $\theta$ then by
Theorem \ref{thm5.5} we obtain that $F_s$ is a periodic function with the same period
$\theta$. If the matrix coefficients of the equation \eqref{eqno:5.5} are
constants then by Corollary \ref{coro5.6} one obtains that $F_s$ is constant and thus
the proof is complete. \end{proof}

The result of Corollary \ref{coro5.11}  shows that if $A$, $B$, $\Pi$ are periodic
functions then, without loss of generality, we may restrict the definition of
stabilizability working only with periodic stabilizing feedback gains. Also,
if $A$, $B$, $\Pi$ are constant functions, then, without loss of generality
the definition of stabilizability may be restricted only to the class of
stabilizing feedback gains which are constant functions.

\section{The case $\mathbf{0 \in \Gamma^{\Sigma}}$ and $\mathbf{\Pi(t)}$ is a
linear operator} % section 6

In this section we focus our attention on those equations \eqref{eqno:3.1}
associated to the quadruple $\Sigma = (A, B, \Pi, \mathcal{Q})$ which have
the additional property $0 \in \Gamma^{\Sigma}$ and $X\to \Pi(t)[X]$
is a linear operator. This is equivalent to conditions \eqref{eqno:3.13}.


\begin{theorem} \label{thm6.1}
Assume that the quadruple $\Sigma = (A, B, \Pi, \mathcal{Q})$ satisfies the
following assumptions:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $(A, B, \Pi)$ is stabilizable;
\item $0 \in \Gamma^{\Sigma}$.
\end{enumerate}
Then equation \eqref{eqno:3.1} has two bounded solutions
$\tilde{X} \colon \mathcal{I} \to \mathcal{S}^n$,
$\tilde{\tilde{X}} \colon \mathcal{I} \to \mathcal{S}^n$
with the property $\tilde{X}(t) \geq \bar{X}(t) \geq
\tilde{\tilde{X}}(t) \geq 0$ for all $t \in \mathcal{I}$ for
arbitrary bounded and positive semi-definite solution $\bar{X}
\colon \mathcal{I} \to \mathcal{S}^n$ of equation
\eqref{eqno:3.1}.

Moreover if $A$, $B$, $\Pi$, $\mathcal{Q}$ are periodic functions with period
$\theta > 0$ then both $\tilde{X}$ and $\tilde{\tilde{X}}$ are periodic
functions with period $\theta$. If $A(t) \equiv A$, $B(t) \equiv B$, $\Pi(t)
\equiv \Pi$ and $\mathcal{Q}(t) \equiv \mathcal{Q}$ then both $\tilde{X}$ and
$\tilde{\tilde{X}}$ are constant and solve the algebraic equation
\eqref{eqno:3.14}. \end{theorem}

\begin{proof} The existence of the solution $\tilde{X}$ is guaranteed
by Theorem \ref{thm4.7}. The proof of the existence of $\tilde{\tilde X}$ is
essentially the same as the proof of the existence of the minimal solution
provided in Theorem \ref{thm3.9}. There is only one difference which consists
in proving the boundedness of the sequence $X_{\tau}, \tau \in \mathcal{I}$.
There, in Theorem \ref{thm3.9}, the boundedness of that sequence was based on
the existence of a bounded and positive semi-definite solution of equation
\eqref{eqno:3.9}. Here the boundedness of that sequence is based on the
stabilizability of the triple $(A,B,\Pi)$ and on the Comparison Theorem.

If $(A, B, \Pi)$ is stabilizable, then there exists a bounded and
continuous function $F \colon \mathcal{I} \to \mathbb{R}^{m
\times n}$ such that the corresponding operator
$\mathcal{L}_{A+BF, \Pi_F^*}$ generates an exponentially stable
evolution. Based on Theorem \ref{thm2.13} we deduce that the equation
\begin{equation}
 \frac{d}{dt} Y(t) + \mathcal{L}_{A+BF, \Pi_F^*}^*(t)[Y(t)] + M_F(t) = 0
\label{eqno:6.1}\end{equation} has a unique bounded solution
$\tilde{Y}(t) \geq 0$ on $\mathcal{I}$.

Let $X_{\tau}$ be the solution of the equation \eqref{eqno:3.1} with
$X_{\tau}(\tau)=0$. Let $F_{\tau}(t)$ be the feedback gain associated to the
solution $X_{\tau}$. Applying Lemma \ref{lm3.4}, equation \eqref{eqno:3.1}
satisfied by $X_{\tau}$ can be rewritten as:
\begin{equation}
\frac{d}{dt}X_{\tau}(t)+\mathcal{L}_{A+BF_{\tau},\Pi_{F_{\tau}}^*}(t)[X_{\tau}(t)]
+M_{F_{\tau}}(t)=0. \label{eqno:6.2}
\end{equation}
On the other hand applying Lemma \ref{lm4.6} with $W(t) = F_{\tau}$ to equation
\eqref{eqno:6.1} one obtains
\begin{equation} \label{eqno:6.3}
\begin{aligned}
\lefteqn{\frac{d}{dt} \tilde{Y}(t) + \mathcal{L}_{A+BF_\tau,
\Pi_{F_\tau}^*}^*(t)[\tilde{Y}(t)] + M_{F_{\tau}}(t)} \\ & \quad {} + \big[
F_{\tau}(t) - F(t) \big]^T \big\{ R(t) + \Pi_2(t)[\tilde{Y}(t)] \big\} \big[
F_{\tau}(t) - F(t) \big] = 0
\end{aligned}
\end{equation}
for $t \in \mathcal{I}_{\tau}$. From \eqref{eqno:6.2} and \eqref{eqno:6.3}
one obtains
\begin{align*}
\lefteqn{\frac{d}{dt} \big[ \tilde{Y}(t) - X_{\tau}(t) \big] +
\mathcal{L}_{A+BF_\tau, \Pi_{F_\tau}^*}^*(t) \big[ \tilde{Y}(t) - X_{\tau}(t)
\big]} \\ & \quad {} + \big[ F_{\tau}(t) - F(t) \big]^T \big\{ R(t) +
\Pi_2(t)[\tilde{Y}(t)] \big\} \big[ F_{\tau}(t) - F(t) \big] = 0.
\end{align*}
Since $\tilde{Y}(\tau) - X_{\tau}(\tau) = \tilde{Y}(\tau) \geq 0$, invoking
Remark \ref{rmk2.14}, (c), we conclude that
\begin{equation} \tilde{Y}(t) - X_{\tau}(t) \geq 0 \quad \mbox{for all}
\quad t \in (-\infty, \tau]\cap \mathcal{I}. \label{eqno:6.4}
\end{equation}
Inequality \eqref{eqno:6.4} together with ${\bf \alpha)}$ in the
proof of Theorem \ref{thm3.9} shows that the sequence $\{ X_{\tau}(t)
\}_{\tau \in \mathcal{I}}$ is increasing and bounded, hence it is
convergent. Define
$$ \tilde{\tilde{X}}(t) := \lim_{\tau \to \infty}
X_{\tau}(t) \quad \mbox{for} \quad t \in \mathcal{I}.
$$
In a standard way one obtains that $\tilde{\tilde{X}}$ is a solution of
\eqref{eqno:3.1}. The minimality property of $\tilde{\tilde X}$
and the periodicity follows as in the proof of Theorem \ref{thm3.9}.
\end{proof}

\begin{lemma} \label{lm6.2}
Assume that the quadruple $\Sigma = (A, B, \Pi, \mathcal{Q})$
satisfies
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $0 \in \Gamma^{\Sigma}$.
\item The triple $(C, A+BW, \Pi_W^*)$ is detectable, where $W(t) = -
R^{-1}(t) L^T(t)$ and $C$ is such that $C^T(t) C(t) = M(t) - L(t)
R^{-1}(t) L^T(t)$.
\end{enumerate}
Under these assumptions any bounded and positive semi-definite
solution of equation \eqref{eqno:3.1} is a stabilizing solution.
\end{lemma}

\begin{proof}
The proof has two stages. Firstly, the proof of the lemma is
made in the particular case $L(t) \equiv 0$. Secondly we shall
show that the general case may be reduced to the particular case
of the first step.

(i) Assume that $L(t) \equiv 0$. In this case $W(t) \equiv 0$ and $\Pi_W(t) =
\Pi_1(t)$ for $t \in \mathcal{I}$ and the assumption b) in the statement is
equivalent to the detectability of the triple $(C, A, \Pi_1^*)$ where $C$ is
such that $C^T(t) C(t) = M(t)$ for $t \in \mathcal{I}$. Let $X$ be a bounded
and positive semi-definite definite solution of equation \eqref{eqno:3.1} and
$F := F^X$. We have to show that $\mathcal{L}_{A+BF, \Pi_F^*}$ generates an
exponentially stable evolution.

Let $(t_0, H) \in \mathcal{I} \times \mathcal{S}^n_+$ be fixed and
let $S$ be the solution of the initial value problem
\begin{equation} \frac{d}{dt} S(t) = \mathcal{L}_{A+BF, \Pi_F^*}(t)[S(t)],
\quad S(t_0) = H. \label{eqno:6.5} \end{equation} We show that
$$ \int_{t_0}^{\infty} \| S(t) \|_2 \, dt \leq \delta \| H \|_2, $$ where
$\delta > 0$ is constant independent of $t_0$ and $H$.

By the detectability assumption it follows that there exists a
bounded and continuous function $K$ such that the operator
$\mathcal{L}_{A+KC, \Pi_1}$ generates an exponentially stable
evolution, where
\begin{equation}
\mathcal{L}_{A+KC, \Pi_1}(t)[X]  = \big[ A(t) + K(t)C(t)
\big] X + X \big[ A(t) + K(t)C(t) \big]^T  +\Pi_1^*(t)[X]. \label{eqno:6.6}
\end{equation}
Using \eqref{eqno:6.6}
equation \eqref{eqno:6.5} may be written as
\begin{equation} \label{eqno:6.7}
\begin{aligned}
\frac{d}{dt} S(t) & = \mathcal{L}_{A+KC, \Pi_1}(t)[S(t)] - K(t) C(t) S(t) -
S(t) C^T(t) K^T(t) \\ & \qquad + \big[\Pi_{F}^*(t)[S(t)] -
\Pi_{1}^*(t)[S(t)]\big].
\end{aligned}
\end{equation}
We introduce the following perturbed operator $X\to
\mathcal{L}_{\varepsilon}(t)[X]$ by
$$
\mathcal{L}_{\varepsilon}(t)[X] := \mathcal{L}_{A+KC, \Pi_1}(t)[X] +
\varepsilon^2 X + \varepsilon^2 \Pi_1^*(t)[X].
$$
Let $T(t,s)$ be
the evolution operator on $\mathcal{S}^n$ defined by
$$ \frac{d}{dt} S(t) = \mathcal{L}_{A+KC, \Pi_1}(t)[S(t)].
$$
Since $\mathcal{L}_{A+KC, \Pi_1}$ generates an exponentially
stable evolution, we have
$$ \| T(t,s) \| \leq \beta e^{-2\alpha(t-s)} \quad \mbox{for} \quad t \geq
s, \; t, s \in \mathcal{I}
$$
for some constants $\alpha, \beta > 0$. By a standard argument based
on Gronwall's Lemma one obtains that for $\varepsilon> 0$ small enough
\begin{equation}
\|T_{\varepsilon}(t,s)\| \leq \beta e^{-\alpha(t-s)} \quad
\mbox{for} \quad t \geq s, \; t, s \in \mathcal{I},
\label{eqno:6.8}
\end{equation}
where $T_{\varepsilon}(t,s)$ is the linear evolution operator on
$\mathcal{S}^n$ defined by the linear differential equation
$$ \frac{d}{dt} Y(t) = \mathcal{L}_{\varepsilon}(t)[Y(t)].
$$
Let $\varepsilon > 0$ be such that \eqref{eqno:6.8} is fulfilled
and let $Y(t)$ be the solution of the following forward
differential equation
\begin{equation}
\frac{d}{dt} Y(t) = \mathcal{L}_{\varepsilon}(t)[Y(t)] +
\frac{1}{\varepsilon^2} K(t) C(t) S(t) C^T(t) K^T(t)
+ \big( 1 + \frac{1}{\varepsilon^2} \big)
\hat\Pi_F^*(t)[S(t)], \label{eqno:6.9}
\end{equation}
where
$$\hat\Pi_F(t)[X]=F^T(t) \Pi_2(t)[X] F(t)$$
satisfying the initial condition $Y(t_0) = H$. Set $Z(t) := Y(t) -S(t)$. We
obtain from \eqref{eqno:6.7} and \eqref{eqno:6.9} that
$$ \frac{d}{dt} Z(t) = \mathcal{L}_{\varepsilon}(t)[Z(t)] + U(t), \quad
Z(t_0) = 0, $$
where
\[
U(t) = \Big[ \varepsilon I_n + \frac{1}{\varepsilon} K(t) C(t) \Big]
S(t) \Big[ \varepsilon I_n + \frac{1}{\varepsilon} K(t) C(t)\Big]^T  +
\Pi_{\varepsilon,F}^*(t)[S(t)]
\]
with
$$ \Pi_{\varepsilon,F}(t)[X] = \begin{pmatrix} \varepsilon I_n \\ -
\frac{1}{\varepsilon} F(t) \end{pmatrix}^T \Pi(t)[X] \begin{pmatrix}
\varepsilon I_n \\ - \frac{1}{\varepsilon} F(t) \end{pmatrix}.
$$
Taking into account that $S(t_0) = H \geq 0$ it follows that
$S(t) \geq 0$ for $t \geq t_0$ and hence $U(t) \geq 0$ for $t \geq t_0$.
On the other hand $X \mapsto \varepsilon^2 X + \varepsilon^2 \Pi_1^*(t)[X]$
is a positive linear operator.

Since $\mathcal{L}_{A+BK, \Pi_1}$ generates a positive evolution
it follows from Proposition \ref{prop2.5} that $\mathcal{L}_{\varepsilon}$
generates a positive evolution.

Based on Remark \ref{rmk2.14}, (c), we conclude that $Z(t) \geq 0$ for $t
\geq 0$, hence $0 \leq S(t) \leq Y(t)$ which leads to
\begin{equation} 0 \leq \| S(t) \|_2 \leq \| Y(t) \|_2. \label{eqno:6.10}
\end{equation}
Applying the representation formula \eqref{eqno:2.24} to equation
\eqref{eqno:6.9} we may write
\begin{align} Y(t) = T_{\varepsilon}(t,t_0) H + \int_{t_0}^t
T_{\varepsilon}(t,s) U_1(s) \, ds \quad \mbox{for} \quad t \geq
t_0, \label{eqno:6.11} \end{align}
where
\[
 U_1(s)  = \frac{1}{\varepsilon^2} K(s) C(s) S(s) C^T(s)
K^T(s)  + \big( 1 + \frac{1}{\varepsilon^2} \big)
\hat\Pi_F^*(s)[S(s)].
\]
Taking into account the
definition of the adjoint operator we obtain
$$ \hat\Pi_F^*(s)[S(s)]= \Pi_2^*(s) \big[ F(s) S(s)F^T(s) \big].
$$
This allows us to write $U_1(s)$ as
\begin{align*}
U_1(s) & = \frac{1}{\varepsilon^2} K(s) C(s) S(s) C^T(s) K^T(s) \notag \\ &
\qquad {} + \left( 1 + \frac{1}{\varepsilon^2} \right) \check{\Pi}_2(s) \big[
R^{1/2}(s) F(s) S(s) F^T(s) R^{1/2}(s) \big],
\end{align*}
where $Y \mapsto \check{\Pi}_2(s)[Y]$ is defined by
$$ \check{\Pi}_2(s)[Y] := \Pi_2^*(s) \big[ R^{-1/2}(s) Y R^{-1/2}(s) \big]. $$
Further we have
\begin{equation} \label{eqno:6.12}
\begin{aligned}
&\| U_1(s) \|_2 \leq \big( 1 + \frac{1}{\varepsilon^2}
\big) \gamma \big[  \|C(s) S(s) C^T(s)\|_2
+ \|R^{1/2}(s) F(s) S(s) F^T(s) R^{1/2}(s) \| \big],
\end{aligned}
\end{equation}
where $\gamma = \max\{\sup_{s \in\mathcal{I}} \| K(s) \|_2^2,
\sup_{s \in \mathcal{I}} \|\check{\Pi}_2(s) \| \}$. From the definition of
the norm $\| \cdot\|_2$ we deduce
\begin{align*}
\lefteqn{\| C(s) S(s) C^T(s) \|_2 + \| R^{1/2}(s) F(s) S(s) F^T(s) R^{1/2}(s)
\|_2} \\ & \quad \leq \mathop{\rm Tr}[C(s) S(s) C^T(s)] + \mathop{\rm
Tr}[R^{1/2}(s) F(s) S(s) F^T(s) R^{1/2}(s)].
\end{align*}
Using the properties of the trace together with \eqref{eqno:2.1} we have
\begin{equation} \label{eqno:6.13}
\begin{gathered}
\| C(s) S(s) C^T(s) \|_2 + \| R^{1/2}(s) F(s) S(s) F^T(s) R^{1/2}(s) \|_2 \\
\leq \langle C^T(s) C(s) + F^T(s) R(s) F(s), S(s) \rangle.
\end{gathered}
\end{equation}
Applying Lemma \ref{lm3.4}, we may write equation \eqref{eqno:3.1},
verified by the bounded and positive semi-definite solution $X$,
in the form
$$ \frac{d}{ds} X(s) + \mathcal{L}_{A+BF, \Pi_F^*}^*(s)[X(s)] + C^T(s) C(s) +
F^T(s) R(s) F(s) = 0. $$ Thus we obtain
\begin{equation} \label{eqno:6.14}
\begin{aligned}
\lefteqn{\langle C^T(s) C(s) + F^T(s) R(s) F(s), S(s) \rangle} \\ & \quad = -
\left\langle \frac{d}{ds} X(s), S(s) \right\rangle - \left\langle
\mathcal{L}_{A+BF, \Pi_F^*}^*(s)[X(s)], S(s) \right\rangle \\ & \quad = -
\left\langle \frac{d}{ds} X(s), S(s) \right\rangle - \left\langle X(s),
\mathcal{L}_{A+BF, \Pi_F^*}(s)[S(s)] \right\rangle \\ & \quad = -
\frac{d}{ds} \langle X(s), S(s) \rangle.
\end{aligned}
\end{equation}
>From \eqref{eqno:6.13}, \eqref{eqno:6.14} we get
\begin{align*}
&\int_{t_0}^t \big[ \| C(s) S(s) C^T(s) \|_2 + \|
R^{1/2}(s) F(s) S(s) F^T(s) R^{1/2}(s) \|_2 \big] ds \\
& \quad \leq \langle X(t_0), S(t_0) \rangle - \langle X(t), S(t) \rangle.
\end{align*}
Taking into account that $\langle X(t), S(t)\rangle
\geq 0$ for $t \geq t_0$ and $\| X(t) \|_2 \leq \rho$ for all $t
\in \mathcal{I}$ where $\rho > 0$ is a constant not depending on
$t$, we obtain
\begin{equation}
\int_{t_0}^t \left[ \| C(s) S(s) C^T(s) \|_2 + \| R^{1/2}(s)
F(s) S(s) F^T(s) R^{1/2}(s) \|_2 \right] ds \leq \rho \| H \|_2
\label{eqno:6.15}
\end{equation}
for $t \geq t_0$. From \eqref{eqno:6.8}, \eqref{eqno:6.11}
and \eqref{eqno:6.12} we have
\begin{align*} \| Y(t) \|_2 & \leq \beta e^{-\alpha(t-t_0)} \| H \|_2 + \beta
\gamma \left( 1 + \frac{1}{\varepsilon^2} \right) \int_{t_0}^t
e^{-\alpha(t-s)} \big[ \| C(s) S(s) C^T(s) \|_2 \\ & \qquad {} + \|
R^{1/2}(s) F(s) S(s) F^T(s) R^{1/2}(s) \|_2 \big] \, ds,
\end{align*}
which leads to
\begin{align*}
\int_{t_0}^\tau \| Y(t) \|_2 \, dt
& \leq \frac{\beta}{\alpha} \| H \|_2 + \beta \gamma
\big( 1 + \frac{1}{\varepsilon^2}\big) \int_{t_0}^{\tau}
 \int_{t_0}^t e^{-\alpha(t-s)} \big[ \|
C(s) S(s) C^T(s) \|_2 \\ & \qquad {} + \| R^{1/2}(s) F(s) S(s)
F^T(s) R^{1/2}(s) \|_2 \big] \, ds \, dt.
\end{align*}
Changing the order of integration and invoking \eqref{eqno:6.15} we obtain
$$
\int_{t_0}^\tau \| Y(t) \|_2 \, dt \leq \frac{\beta}{\alpha} \big[ 1 +
\big( 1 + \frac{1}{\varepsilon^2} \big) \gamma \rho \big] \|
H \|_2 =: \delta \| H \|_2.
$$
Taking the limit for $\tau \to \infty $ we deduce
$$ \int_{t_0}^\infty \| Y(t) \|_2 \, dt \leq \delta \| H \|_2 \quad \mbox{for
all} \quad t_0 \in \mathcal{I}, \; H \in \mathcal{S}^n_+,
$$
where
$\delta$ is independent of $t_0$ and $H$. From \eqref{eqno:6.10}
it follows now that
\begin{equation}
\int_{t_0}^\infty \| S(t) \|_2 \, dt \leq \delta \| H \|_2
\quad \mbox{for all} \quad t_0 \in \mathcal{I}, \; H \in \mathcal{S}^n_+. \label{eqno:6.16}
\end{equation}
Since for any $H
\in \mathcal{S}^n$ there are $H_i \in \mathcal{S}^n_+$, $i = 1,2$,
such that $H = H_1 - H_2$ and $\| H \|_2 = \max \{\| H_1 \|_2, \|
H_2 \|_2 \}$ one easily obtains that \eqref{eqno:6.16} holds for
arbitrary $t_0 \in \mathcal{I}$ and $H \in \mathcal{S}^n$ and thus
the proof is complete for the case $L(t) \equiv 0$.

(ii) Let us consider the general case when $L(t) \not \equiv 0$. Let $X$ be a
bounded and positive semi-definite definite solution of equation
\eqref{eqno:3.1}. Applying Lemma \ref{lm5.7} for $W(t) = -R^{-1}(t)L^T(t)$ we
obtain that $X$ is a bounded and positive semi-definite solution of the
equation
\begin{equation} \label{eqno:6.17}
\begin{aligned}
\lefteqn{\frac{d}{dt} X(t) + \mathcal{L}_{A - B R^{-1} L^T}^*(t)[X(t)] +
C^T(t) C(t) + \Pi_W(t)[X(t)]} \\ & \quad {} - \big\{ X(t)B(t) +
\Pi_{12W}(t)[X(t)] \big\} \big\{ R(t) + \Pi_2(t)[X(t)] \big\}^{-1} \\ &
\qquad \times \big\{ X(t)B(t) + \Pi_{12W}(t)[X(t)] \big\}^T = 0,
\end{aligned}
\end{equation}
where $\Pi_W(t)$ is defined as in
\eqref{eqno:3.6},
$$ \Pi_{12W}(t) := \Pi_{12}(t) - L(t) R^{-1}(t) \Pi_2(t) $$
and
$$ C^T(t) C(t) = M(t) - L(t) R^{-1}(t) L^T(t). $$
Equation \eqref{eqno:6.17} is an equation of type \eqref{eqno:3.1} with $L(t)
\equiv 0$. Applying the first part of the proof we deduce that $X$ is a
stabilizing solution of equation \eqref{eqno:6.17}. Let
$$\hat{F}(t) := - \big\{ R(t) + \Pi_2(t)[X(t)] \big\}^{-1} \big\{ X(t)B(t) +
\Pi_{12W}(t)[X(t)] \big\}^T $$ be the stabilizing feedback gain
associated to the solution $X$ regarded as a solution of
\eqref{eqno:6.17}. Then it is easy to see that $\hat{F}(t) -
R^{-1}(t) L^T(t) = F(t)$, where $F(t) := F^X(t)$ is defined as in
Lemma \ref{lm3.4}. Hence,
$$ A(t) - B(t) R^{-1}(t) L^T(t) + B(t) \hat{F}(t) = A(t) + B(t) F(t), $$
and
\begin{align*}
\lefteqn{ \begin{pmatrix} I_n \\ \hat{F}(t) \end{pmatrix}^T \begin{pmatrix}
\Pi_W(t)[X] & \Pi_{12W}(t)[X] \\ \big\{ \Pi_{12W}(t)[X] \big\}^T &
\Pi_2(t)[X] \end{pmatrix} \begin{pmatrix} I_n \\ \hat{F}(t)
\end{pmatrix}} \\ & \quad = \begin{pmatrix} I_n \\ \hat{F}(t) \end{pmatrix}^T
\begin{pmatrix} I_n & - L(t) R^{-1}(t) \\ 0 & I_m \end{pmatrix} \Pi(t)[X]
\begin{pmatrix} I_n & 0 \\ - R^{-1}(t) L^T(t) & I_m \end{pmatrix}
\begin{pmatrix} I_n \\ \hat{F}(t) \end{pmatrix} \\ & \quad = \begin{pmatrix}
I_n \\ F(t) \end{pmatrix}^T \Pi(t)[X] \begin{pmatrix} I_n \\ F(t)
\end{pmatrix} = \Pi_F(t)[X].
\end{align*}
These facts allow us to conclude that $X$ is a stabilizing solution of
equation \eqref{eqno:3.1} and the proof ends. \end{proof}

\begin{remark} \label{rmk6.3} \rm
Assume that the quadruple $\Sigma = (A, B, \Pi, \mathcal{Q})$ satisfies
$0 \in \tilde{\Gamma}^{\Sigma}$.
Then any bounded and positive semi-definite solution $X \colon \mathcal{I}
\to \mathcal{S}^n_+$ of equation \eqref{eqno:3.1} is a stabilizing solution,
and we have $X(t) \gg 0$ for $t \in \mathcal{I}$. \end{remark}

\begin{theorem} \label{thm6.4}
Assume that the quadruple $\Sigma = (A, B, \Pi, \mathcal{Q})$
satisfies the following assumptions:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $0\in \Gamma^{\Sigma}$;
\item $(A, B, \Pi)$ is stabilizable.
\item $(C, A - B R^{-1} L^T, \Pi^*_W)$ is detectable where $\Pi_W$ is defined
as in \eqref{eqno:3.6} for $W(t) = - R^{-1}(t) L^T(t)$ and $C$ is
such that
$$C^T(t) C(t) = M(t) - L(t) R^{-1}(t) L^T(t).$$
\end{enumerate}
Then \eqref{eqno:3.1} has a unique solution $X \colon \mathcal{I}
\to \mathcal{S}^n_+$ which is bounded and stabilizing.
\end{theorem}

\begin{proof} Based on Theorem \ref{thm6.1} we deduce that equation \eqref{eqno:3.1} has
both a maximal solution $\tilde{X}$ and a minimal positive
semi-definite solution $\tilde{\tilde{X}}$ such that $\tilde{X}(t)
\geq \bar{X}(t) \geq \tilde{\tilde{X}}(t) \geq 0$ for all $t \in
\mathcal{I}$, where $\bar{X}$ is an arbitrary bounded and positive
semi-definite solution of \eqref{eqno:3.1}. Applying Lemma \ref{lm6.2} it
follows that both $\tilde{X}$ and $\tilde{\tilde{X}}$ are
stabilizing solutions.

 From the uniqueness of the stabilizing and bounded solution of equation
\eqref{eqno:3.1} we conclude that $\tilde{\tilde{X}}(t) = \tilde{X}(t)$ for
all $t \in \mathcal{I}$ and thus the proof is complete. \end{proof}

\begin{remark} \label{rmk6.5}\rm
As we have seen in Theorem  \ref{thm6.1} equation \eqref{eqno:3.1} has two
remarkable solutions, namely $\tilde{X} \colon \mathcal{I} \to
\mathcal{S}^n$, which is the maximal solution, and $\tilde{\tilde{X}} \colon
\mathcal{I} \to \mathcal{S}^n$, which is the minimal solution in the
class of all bounded and positive semi-definite solutions of
\eqref{eqno:3.1}. Theorem \ref{thm6.4} shows that under the assumption of
detectability these two solutions coincide. However in the absence of
detectability these two solutions may be different. If in addition to the
assumptions of Theorem \ref{thm6.4} we assume that $0 \in \tilde{\Gamma}^\Sigma$ --
which is equivalent to $\mathcal{Q}(t) \gg 0$ -- then $\tilde{X} \gg 0$. This
results immediately from Theorem \ref{thm2.11} and formula \eqref{eqno:3.7} with $X :=
\tilde{X}$ and $W := F^{\tilde{X}}$. \end{remark}

We can see this in the following simple example. Consider equation
\eqref{eqno:3.1} with constant coefficients and $n=2$, $m=1$,
\[ A = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}, \quad B =\begin{pmatrix}
2 \\ 1 \end{pmatrix}, \quad M = \begin{pmatrix} 1 & 0 \\ 0 & 0
\end{pmatrix},
\] $L=0$, $R=1$, $\Pi_1(X) = X$, $\Pi_{12} = 0$, $\Pi_2 = 0$. One obtains
that
\[ \tilde{X} = \begin{pmatrix} 8 & -21 \\ -21 & 63 \end{pmatrix} \qquad
\mbox{and} \qquad \tilde{\tilde{X}} = \begin{pmatrix} 1 & 0 \\ 0 &
0 \end{pmatrix}. \] It can be seen that $\tilde{X}$ is just the
stabilizing solution. On the other hand if $X_{\tau}$ is the
solution of \eqref{eqno:3.1} with $X_{\tau}(\tau) = 0$ we have
\[ X_{\tau}(t) = \begin{pmatrix} x(t) & 0 \\ 0 & 0 \end{pmatrix} \quad
\mbox{with} \quad x(t) =
\frac{1-e^{-5(\tau-t)}}{1+4e^{-5(\tau-t)}}. \] Clearly
$\lim_{\tau\to \infty}X_{\tau}(t)=\tilde{\tilde{X}}$ and
therefore $\tilde{\tilde{X}}$ is the minimal positive
semi-definite solution.

\begin{theorem} \label{thm6.6}
Assume that $A$, $B$, $\mathcal{Q}$ and $\Pi$ are periodic
functions with period $\theta > 0$ and that all the assumptions of Theorem
\ref{thm6.4} are fulfilled. Then
\begin{equation} \lim_{t \to - \infty} \big( X(t, t_0, X_0) - X_s(t) \big) =
0 \quad \mbox{for all} \quad X_0 \in \mathcal{S}^n_+;
\label{eqno:6.18}
\end{equation}
here $X_s$ is the unique stabilizing $\theta$-periodic
solution of \eqref{eqno:3.1}. \end{theorem}

\begin{proof} We choose a $\theta$-periodic function $\tilde{M} \colon \mathbb{R}
\to \mathcal{S}^n$ such that $\tilde{M}(t) \geq M(t)$ and
\[ \tilde{\mathcal{Q}}(t) := \begin{pmatrix} \tilde{M}(t) & L(t) \\ L(t)^T &
R(t) \end{pmatrix} \gg 0. \] Then it follows from Corollary \ref{coro5.9}
and Remark \ref{rmk6.5} that the generalized Riccati equation
\begin{equation} \label{eqno:6.18a}
\begin{aligned}
\lefteqn{\frac{d}{dt} X(t) + A^T(t)X(t) + X(t)A(t) + \tilde{M}(t) +
\Pi_1(t)[X(t)]} \\ & \quad {} - \big\{ X(t)B(t) +\Pi_{12}(t)[X(t)] + L(t)
\big\} \big\{ R(t) + \Pi_2(t)[X(t)]\big\}^{-1} \\ & \qquad \times \big\{
X(t)B(t) +\Pi_{12}(t)[X(t)] + L(t) \big\}^T = 0
\end{aligned}
\end{equation}
has a periodic and stabilizing solution $\tilde{X}_s(t) \gg 0$. For given
$X_0 \in \mathcal{S}^n_+$ we choose $\lambda > 1$ such that $X_0 \leq \lambda
\tilde{X}_s(0)$ and consider the functions $X_\ell \equiv 0$ and $X_u =
\lambda \tilde{X}_s$. We verify that $X_\ell$ and $X_u$ satisfy the
conditions of Theorem \ref{thm3.11}, (ii). Since $0 \in \Gamma^\Sigma$ and
$X_s(t) \geq 0$ it is sufficient to show that
\begin{equation}
\frac{d}{dt} X_u(t) + \mathcal{R}(t,X_u(t)) \leq 0 \quad
\mbox{for} \quad t \leq 0. \label{eqno:6.19}
\end{equation}
 Denote by
$\mathcal{R}_\lambda(t)$ the generalized Riccati-type operator associated to
the quadruple $(A, B, \Pi, \lambda \mathcal{Q})$. Since $\mathcal{Q}(t) \leq
\lambda \mathcal{Q}(t)$ it follows from \cite[Lemma 4.4]{FrHo5} that
$\mathcal{R}(t,X_u(t)) \leq \mathcal{R}_\lambda(t,X_u(t))$. Multiplying
\eqref{eqno:6.18a} by $\lambda > 1$ we infer that
\[ \frac{d}{dt} X_u(t) + \mathcal{R}_\lambda(t, X_u(t)) = \lambda \big( M(t)
- \tilde{M}(t) \big) \leq 0. \] Therefore \eqref{eqno:6.19} is valid.

As in the proof of Theorem \ref{thm3.11} it follows that $X(\cdot, 0, 0)$ and
$X(\cdot, 0, X_u(0))$ are cyclomonotonically increasing (respectively
decreasing) as $t$ decreases, and both converge to $\theta$-periodic
solutions of \eqref{eqno:3.1}. Moreover, by Theorem \ref{thm3.6} we have
\begin{equation}
0 \leq X(t, 0, 0) \leq X(t, 0, X_0) \leq X(t, 0, X_u(0))
\quad \mbox{for} \quad t \leq 0. \label{eqno:6.20}
\end{equation}
Since under our assumptions $X_s$ is the unique positive semi-definite
$\theta$-periodic solution of \eqref{eqno:3.1}, \eqref{eqno:6.18} follows
from \eqref{eqno:6.20}. \end{proof}

Theorem \ref{thm6.6} shows that under its assumptions the dynamics defined
by \eqref{eqno:3.1} on $\mathcal{S}^n_+$ is very special and
comparable to that of symmetric matrix Riccati differential
equations -- this is the reason why we call \eqref{eqno:3.1}
(generalized) Riccati-type equation.

\begin{remark} \label{rmk6.7} \rm
The proofs of Theorems \ref{thm4.7} and \ref{thm6.1} show in connection
with Theorem \ref{thm3.6} that the maximal and the minimal positive
semi-definite solution (provided they exist) both depend
monotonically  on $\mathcal{Q}$. In the time-invariant case this
has already been mentioned in \cite{DaHi01} and \cite{FrHo5}.
\end{remark}

\section{Generalizations}

\subsection{Generalization of the linear part}

All the results for equation \eqref{eqno:3.1} obtained in Sections 3-6 are
still valid for equations of the form
\begin{align*}
\lefteqn{\frac{d}{dt} X(t) + \mathcal{L}_0^*(t)[X(t)] + M(t) +
\Pi_1(t)[X(t)]} \\ & \quad {} - \big\{ X(t)B(t) + \Pi_{12}(t)[X(t)] + L(t)
\big\} \big\{ R(t) + \Pi_2(t)[X(t)] \big\}^{-1} \\ & \qquad \times \big\{
X(t)B(t) + \Pi_{12}(t)[X(t)] + L(t) \big\}^T = 0,
\end{align*}
where $B, \Pi, \mathcal{Q}$ are as in \eqref{eqno:3.1} and $\mathcal{L}_0(t)
\colon \mathcal{S}^n \to \mathcal{S}^n$ is a linear operator with the
properties
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $\mathcal{L}_0$ generates a positive evolution and
\item the operator $X\mapsto \mathcal{L}_0(t)[X]+B(t)W(t)X+X(B(t)W(t))^T$
also generates a positive evolution for arbitrary $W \colon
\mathcal{I} \to \mathbb{R}^{m \times n}$.
\end{enumerate}
Equation \eqref{eqno:3.1} corresponds to the particular case
$\mathcal{L}_0(t)[X] = A(t)X + X A^T(t)$, and it has the above
properties.

\subsection{Generalized Riccati operators containing the Moore-Penrose
inverse}

Define $\mathcal{D}(\mathcal{R}_+)$ as the set of all $(t,X) \in \mathcal{I}
\times \mathcal{S}^n$ with $R(t) + \Pi_2(t)[X] \geq 0$ and
\[ \ker \big\{ R(t) + \Pi_2(t)[X] \big\} \subseteq \ker \big\{ L(t) + X B(t)
+ \Pi_{12}(t)[X] \big\}. \] Then we may consider
$\mathcal{R}_+\colon \mathcal{D}(\mathcal{R}_+) \to \mathcal{S}^n$
where
\begin{align*} \mathcal{R}_+(t,X) & = A^T(t)X + XA(t) + \Pi_1(t)[X] + M(t) \\
& \qquad {} - \big\{ X B(t) + \Pi_{12}(t)[X] + L(t) \big\}
\big\{R(t) \Pi_2(t)[X] \big\}^+ \\ & \qquad {} \times \big\{ XB(t)
+ \Pi_{12}(t)[X] + L(t) \big\}^T; \end{align*} here $Z^+$ denotes
the Moore-Penrose inverse of a matrix $Z$. Combining the methods
used in \cite{FrHo5} with the approach of this paper it follows
that the assertion of the Comparison Theorem \ref{thm3.6} remains
analogously valid for two rational differential equations
$$ \frac{d}{dt} X(t) + \mathcal{R}_+(t, X(t)) = 0 \quad \mbox{and} \quad
\frac{d}{dt} X(t) + \tilde{\mathcal{R}}_+(t, X(t)) = 0, $$ where
$\tilde{\mathcal{R}}_+$ is (analogously to $\mathcal{R}_+$) associated to a
quadruple $(A, B, \Pi, \tilde{\mathcal{Q}})$. In this case the assumption b)
of Theorem \ref{thm3.6} can be replaced by the weaker assumption $(t, X_2(t)) \in
\mathcal{D}(\tilde{\mathcal{R}}_+)$ for $t \in \mathcal{I}_1$. Moreover, in
generalization of Theorem \ref{thm4.7} we get

\begin{theorem} \label{thm7.1}
Assume that $(A, B, \Pi)$ is stabilizable. Then the following are
equivalent:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item The set
$$ \hat{\Gamma}^\Sigma := \{ X \in C^1_b(\mathcal{I}, \mathcal{S}^n) :
\lambda^\Sigma[X(t)] \geq 0, \; t \in \mathcal{I} \} $$ is not
empty.
\item The equation
$$ \frac{d}{dt} X(t) + \mathcal{R}_+(t, X(t)) = 0 $$
has a maximal and bounded solution $\tilde{X} \colon \mathcal{I}
\to \mathcal{S}^n$.
\end{enumerate}
\end{theorem}

\subsection{An equivalent definition of stabilizability}

Assume that the operator $X \mapsto \Pi(t)[X]$ is linear. Then as a
consequence of \cite[Lemma 5.1]{FrHo5} it follows that the Fr\'{e}chet
derivative of
$$ f(t,X) := \frac{d}{dt} X + \mathcal{R}(t,X) $$
at $X$ is given by
\[ f'_X(t,H) = \frac{d}{dt} H + \mathcal{R}'_X(t,H ) = \frac{d}{dt} H +
\mathcal{L}_{F^X}^*(t)[H]. \] Therefore $(A, B, \Pi)$ is stabilizable if and
only if there is a function $X_0 \colon \mathcal{I} \to \mathcal{S}^n$ such
that $\mathcal{R}'_X(t, X_0(t))$ (or the corresponding differential equation)
generates an asymptotic stable evolution.

It is easy to see (compare \cite{FrHo5}) that the sequence $(X_k)_{k \in
\mathbb{N}}$ of functions defined in the proof of Theorem \ref{thm3.11} are generated
by the Newton-Kantorovich procedure, applied to the operator equation $f(t,X)
= 0$. Notice that the stabilizability condition ensures the existence of a
stabilizing initial function $X_0$. The condition $\Gamma^\Sigma \neq
\emptyset$ guarantees (as a consequence of the Comparison Theorem) that the
decreasing sequence $(X_k)_{k \in \mathbb{N}}$ is bounded below and hence
convergent to $\tilde{X}$.

In the time-invariant case it has been proved recently by Damm and Hinrichsen
that the Newton-Kantorovich procedure is also the adequate tool for proving
results on the maximal solution of nonlinear operator equations of the form
$f(X) = 0$ in real Banach spaces provided $f$ is concave on some domain $D$
and certain additional conditions are fulfilled (see \cite{DaHi03} for
details).

\subsection{Sums of generalized Riccati operators}

For $1 \leq \kappa \leq k$ let quadruples $\Sigma_\kappa = (A_\kappa,
B_\kappa, \Pi_\kappa, \mathcal{Q}_\kappa)$ and associated operators
$\mathcal{R}_\kappa \colon \mathcal{D}(\mathcal{R}_k) \to \mathcal{S}^n$ of
type \eqref{eqno:3.4} be given. Define
$$ \mathcal{R}_0 \colon \mathcal{D}(\mathcal{R}_0) := \bigcap_{\kappa=1}^k
\mathcal{D}(\mathcal{R}_\kappa) \to \mathcal{S}^n, \quad (t,X) \mapsto
\sum_{\kappa=1}^k \mathcal{R}_\kappa(t,X). $$ Then (under analogous
assumptions) all results that were obtained in Sections 3 -- 6 can be derived
analogously for the differential equation
$$ \frac{d}{dt} X(t) + \mathcal{R}_0(t, X(t)) = 0. $$
For example the following Comparison Theorem can be proved analogously to
Theorem \ref{thm3.6}:

\begin{theorem} \label{thm7.2}
For $1 \leq \kappa \leq k$ let quadruples $\Sigma_\kappa =
(A_\kappa, B_\kappa, \Pi_\kappa, \mathcal{Q}_\kappa)$,
$\tilde{\Sigma}_\kappa = (A_\kappa, B_\kappa,\Pi_\kappa,
\tilde{\mathcal{Q}}_\kappa)$ and associated operators
$\mathcal{R}_\kappa \colon \mathcal{D}(\mathcal{R}_k) \to
\mathcal{S}^n$, $\tilde{\mathcal{R}}_\kappa \colon
\mathcal{D}(\tilde{\mathcal{R}}_k) \to \mathcal{S}^n$ of type
\eqref{eqno:3.4} be given. Let $X_i \colon \mathcal{I}_1 \subset
\mathcal{I} \to \mathcal{S}^n$, $i = 1,2$, be solutions of
$$ \frac{d}{dt} X_1(t) + \sum_{\kappa = 1}^k \mathcal{R}_\kappa(t, X_1(t)) =
0, \qquad \frac{d}{dt} X_2(t) + \sum_{\kappa = 1}^k
\tilde{\mathcal{R}}(t, X_2(t)) = 0. $$ Assume that
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $\mathcal{Q}_\kappa(t) \geq \tilde{\mathcal{Q}}_\kappa(t)$ for $t \in
\mathcal{I}$ and $\kappa \in \{ 1, \ldots, k \}$;
\item $\tilde{R}_\kappa(t) + \Pi_{2,\kappa}(t)[X_2(t)] \gg 0$ for $t \in \mathcal{I}_1$
and $\kappa \in \{ 1, \ldots, k \}$;
\item there exists $\tau \in \mathcal{I}_1$ such that  $X_1(\tau) \geq
X_2(\tau)$.
\end{enumerate}
Under these conditions we have $X_1(t)\geq X_2(t)$ for all $t
\in(-\infty, \tau] \cap \mathcal{I}_1$. \end{theorem}

As it was pointed out in \cite[Section 4]{FrHo5} such a Comparison
Theorem can be used in order to derive in an elegant way existence
results for nonlinear differential equations.

\begin{example} \label{ex7.3} \rm
The main results of \cite{LiZh01} on the existence of optimal
controls for certain stochastic control problems are based on the existence
of the solution of the terminal value problem
\begin{equation} \label{eqno:7.1}
\begin{gathered}
\frac{d}{dt} X(t) - A(t) X(t) - X(t) A^T(t) + B(t) R^{-1}(t) B^T(t) - X(t)
Q(t) X(t) \\ {} + C(t) X(t) \big[ S(t) X(t) + I \big]^{-1} C^T(t) = 0, \quad
X(t_f) = X_f,
\end{gathered}
\end{equation}
where the coefficients $A$, $B$, $C$ and $S$ are bounded and locally
integrable. Additionally it is assumed that $Q(t), S(t), X_f \geq 0$ and
$R(t) \gg 0$. Since $S(t)$ has a square root $D(t) := S^{1/2}(t) \geq 0$ and
\[ \big[ S(t) X(t) + I \big]^{-1} = I - D(t) \big[ I + D(t) X(t) D(t)
\big]^{-1}D(t)X(t),
\]
we can rewrite \eqref{eqno:7.1} as
\begin{equation} \label{eqno:7.2}
\begin{aligned} \frac{d}{dt} Y(t) & = A(t)Y(t) + Y(t)A^T(t) + M(t) +
\Pi_1(t)[Y(t)] \\ & \qquad {} - \big\{ Y(t)B(t) +\Pi_{12}(t)[Y(t)] + L(t)
\big\} \big\{ R(t) + \Pi_2(t)[Y(t)]\big\}^{-1} \\ & \qquad \times \big\{
Y(t)B(t) + \Pi_{12}(t)[Y(t)] + L(t) \big\}^T.
\end{aligned}
\end{equation}
where $\tilde{B}(t) := \begin{pmatrix} Q^{1/2}(t) & 0 \end{pmatrix}$,
$$ \Pi_1(t)[X(t)] := C(t) X(t) C^T(t), \quad \Pi_{12}(t)[X(t)] := \big( 0
\quad C(t) X(t) D(t) \big) $$ and
$$ \Pi_2(t)[X] := \begin{pmatrix} 0 & 0 \\ 0 & D(t) X(t) D(t) \end{pmatrix}.
$$
It is not difficult to see that
$$ \Pi(t)[X] = \begin{pmatrix} \Pi_1(t)[X] & \Pi_{12}(t)[X] \\ \big(
\Pi_{12}(t)[X] \big)^T & \Pi_2(t)[X] \end{pmatrix} = \begin{pmatrix} C(t)
\\ 0 \\ D(t) \end{pmatrix} X \begin{pmatrix} C(t) \\ 0 \\ D(t)
\end{pmatrix}^T $$
is a positive linear operator. Corollary \ref{coro3.8} shows now that the solution $X$
of \eqref{eqno:7.2} satisfies
$$ 0 \leq X(t) \leq X_u(t) \quad \mbox{for} \quad 0 \leq t \leq t_f, $$
where $X_u$ is the solution of the linear terminal value problem
\begin{align*} \lefteqn{\frac{d}{dt} X_u(t) - A(t) X_u(t) - X_u(t) A^T(t) +
B(t) R^{-1}(t) B^T(t)} \\ & \quad {} + C(t) X_u(t) C^T(t) = 0, \quad X_u(t_f)
= X_f. \end{align*} Under suitable assumptions we may derive the
corresponding results concerning the existence of the maximal solution,
stabilizing solution, minimal solution respectively of equation
\eqref{eqno:7.2} or equivalently of equation \eqref{eqno:7.1}. Notice that
Theorem \ref{thm7.2} could also be applied to equation \eqref{eqno:7.1}. \end{example}

\subsection{Forward Riccati-type differential equations}

Together with  \eqref{eqno:3.1} a qua\-druple $\Sigma = (A, B, \Pi,
\mathcal{Q})$ defines also a so-called forward Riccati-type differential
equation
\begin{equation} \label{eqno:7.3}
\begin{aligned}
\frac{d}{dt} Y(t) & = A(t)Y(t) + Y(t)A^T(t) + M(t) +
\Pi_1(t)[Y(t)]  \\
& \quad - \big\{ Y(t)B(t) +\Pi_{12}(t)[Y(t)]
+ L(t) \big\} \big\{ R(t) + \Pi_2(t)[Y(t)]\big\}^{-1}  \\
& \quad \times \big\{ Y(t)B(t) + \Pi_{12}(t)[Y(t)] + L(t) \big\}^T.
 \end{aligned}
\end{equation}
In the particular case where $\Pi(t) \equiv 0$,
$L(t) \equiv 0$ and $B^T(t)$ is the matrix coefficient of an output the
equation \eqref{eqno:7.3} appears in connection with the filtering problem
\cite{Wonh70}.

With the same proof as of Theorem \ref{thm3.6} we may obtain the following
comparison theorem for the equation \eqref{eqno:7.3}.

\begin{theorem}[Comparison Theorem] \label{thm7.4}
Let $\hat{\mathcal{R}}$ be the operator
\eqref{eqno:3.4} associated to the quadruple $\hat \Sigma = (A, B,
\Pi, \hat{\mathcal{Q}})$ and $\tilde{\mathcal{R}}$ be the operator
of type \eqref{eqno:3.4} associated to the quadruple
$\tilde{\Sigma} = (A, B, \Pi, \tilde{\mathcal{Q}})$ where $A$,
$B$, $\Pi$ are as before and
$\hat{\mathcal{Q}}(t) =\left(\begin{smallmatrix} \hat{M}(t) & \hat{L}(t) \\
\hat{L}(t)^T & \hat{R}(t) \end{smallmatrix} \right)$,
$\tilde{\mathcal{Q}}(t) =\left(\begin{smallmatrix} \tilde{M}(t) &
\tilde{L}(t)
\\ \tilde{L}(t)^T & \tilde{R}(t) \end{smallmatrix} \right)$
with $\hat{L}(t),\tilde{L}(t) \in \mathbb{R}^{n \times m}$,
$\hat{M}(t),\tilde{M}(t) \in \mathcal{S}^n$ and
$\hat{R}(t),\tilde{R}(t) \in \mathcal{S}^m$. Let $Y_i \colon
\mathcal{I}_1 \subset \mathcal{I} \to\mathcal{S}^n$, $i =
1,2$, be solutions of
$$ \frac{d}{dt} Y_1(t)=  \hat{\mathcal{R}}(t, Y_1(t)), \qquad
\frac{d}{dt} Y_2(t)= \tilde{\mathcal{R}}(t, Y_2(t)). $$ Assume
that
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $\hat{\mathcal{Q}}(t) \geq \tilde{\mathcal{Q}}(t)$ for all $t \in
\mathcal{I}$;
\item $\tilde{R}(t) + \Pi_2(t)[Y_2(t)] > 0$ for $t \in \mathcal{I}_1$;
\item there exists $\tau \in \mathcal{I}_1$ such that  $Y_1(\tau) \geq
Y_2(\tau)$.
\end{enumerate}
Under these conditions we have $Y_1(t)\geq Y_2(t)$ for all $t
\in[\tau,\infty)$. \end{theorem}

The proof of the above theorem is based on Proposition \ref{prop3.5} (i).

In order to derive results concerning the existence of the maximal
solution, stabilizing solution, minimal solution respectively for
equation \eqref{eqno:7.3} we need to assume that
$\mathcal{I}=\mathbb{R}$ and apply the results of Theorem \ref{thm2.18} and
Proposition \ref{prop2.15} instead of Theorem \ref{thm2.11} and Theorem \ref{thm2.13}. The
stabilizability concept used for the backward differential
equation \eqref{eqno:3.1} will be replaced by a dual concept of
detectability.

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