\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 58, pp. 1--8.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/58\hfil Exponential stability]
{Exponential stability criteria of linear non-autonomous
systems with multiple delays}
\author[V. N. Phat,  P. T. Nam\hfil EJDE-2005/58\hfilneg]
{Vu Ngoc Phat,  Phan  T. Nam}  % in alphabetical order

\address{Institute of Mathematics \\
18 Hoang Quoc Viet Road, Hanoi, Vietnam}
\email[Vu N. Phat]{vnphat@math.ac.vn}

\date{}
\thanks{Submitted November 12, 2004. Published June 6, 2005.}
\subjclass[2000]{34K20}
\keywords{Exponential stability; time-delay; Lyapunov function; \hfill\break\indent
 Riccati equation}

\begin{abstract}
 In this paper, we study the exponential stability of linear
 non-au\-tonomous systems with multiple delays. Using
 Lyapunov-like function, we find sufficient conditions for the
 exponential stability in terms of the solution of a
 Riccati differential equation. Our results are illustrated
 with numerical examples.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

The topic of Lyapunov stability of linear systems has been an
interesting research area in the past decades. An integral part of
the stability analysis of differential equations is the existence
of inherent time delays. Time delays are frequently encountered in
many physical and chemical processes as well as in the models of
hereditary systems, Lotka-Volterra systems, control of the growth
of global economy, control of epidemics, etc. Therefore, the stability
problem of time-delay systems has been received considerable
attention from many researchers (see; e.g. \cite{j1,h1,p1,p3,s2}
and references therein). One of the extended stability properties
is the concept of the $\alpha$-stability, which relates to the
exponential stability with a convergent rate $\alpha > 0$. Namely,
a retarded system
\begin{gather*}
\dot x = f(t,x(t), x(t-h)), \quad t\geq 0, \\
x(t) = \phi (t),  \quad t \in [-h, 0],
\end{gather*}
is $\alpha$-stable, with $\alpha> 0$, if there is a function $\xi(.)$
such that for each $\phi(.)$, the solution $x(t,\phi)$ of the system
satisfies
$$
\|x(t, \phi)\|\leq \xi(\|\phi\|) e^{-\alpha t},\quad \forall
t\geq 0,
$$
where $\|\phi\| = \max \{\|\phi(t)\|: t\in [-h,0]\}$. This implies
that for $\alpha > 0$, the system can be made exponentially stable
with the convergent rate $\alpha$. It is
well known that there are many different methods to study the
stability problem of time-delay linear autonomous systems. The widely used method is
the approach of Lyapunov functions with Razumikhin techniques
 and the asymptotic stability conditions are presented in terms of the
solution of either linear matrix inequalities or Riccati equations
\cite{b1,k1,k2}.  By using both the time-domain and the frequency-domain
techniques, the paper \cite{x1} derived sufficient conditions for the
asymptotic stability of a linear autonomous  system with multiple
time delays of the form
\begin{equation} \label{e1}
\begin{gathered}
\dot x(t) = A_0x(t) + \sum_{i=1}^mA_ix(t-h_i), \quad t\geq 0, \\
x(t) = \phi (t), \quad t \in [-h, 0],
\end{gathered}
\end{equation}
where $A_i$ are given constant matrices, $h = \max\{h_i: i =1, 2,\dots, m\}$.
These conditions depend only on the eigenvalues of $A_0$ and the
norm values of $A_i$ of the system. For studying the
$\alpha$-stability problem, based on the asymptotic stability of the linear
undelayed part, i.e. $A_0$ is a Hurwitz matrix, the papers \cite{s1,s2}
 proposed sufficient conditions for the $\alpha$-stability of
system \eqref{e1} in terms of the solution of a scalar inequality
involving the eigenvalues, the matrix measures and the spectral
radius of the system matrices. It is worth noticing that although
the approach used in these papers allows us to derive the less
conservative stability conditions, but it can not be applied to
non-autonomous delay systems. The reason is that, the
assumption $A_0(t)$ to be a Hurwitz matrix for each $t\geq 0$,
i.e. $\mathop{\rm Re}\lambda(A(t)) < 0$, for each $t$, does not implies the
exponential stability of the linear non-autonomous  system $\dot x
= A_0(t)x$. It is the purpose of this paper to search sufficient
conditions for the $\alpha$-stability of non-autonomous delay
systems. Using the Lyapunov-like function method, we develop the
results obtained in \cite{c1,s2} to the non-autonomous systems
with multiple delays. Do not using any Lyapunov stability
theorem, we establish sufficient conditions for the
$\alpha$-stability of system \eqref{e2}, which are given in terms of the
solution of a  Riccati differential equation (RDE). These
conditions do not involve any stability property of the system
matrix $A_0(t)$. Although the problem of solving of RDEs
is in general still not easy, various effective approaches for
finding the solutions of RDEs  can be found in \cite{a1,g1,l1,w1}.


The paper is organized as follows. Section 2 presents notations,
mathematical definitions and an auxiliary lemma used in the next
section. The sufficient conditions for the $\alpha$-stability are
presented in Section 3. Numerical examples illustrated the
obtained result  are also given in Section 3. The paper ends with
 cited references.

\section{Preliminaries}

The following notations will be used for the remaining this paper.

\noindent $\mathbb{R}^+$ denotes the set of all real non-negative numbers;
 $\mathbb{R}^n$ denotes the $n$-dimensional space with the
scalar product $\langle.,.\rangle$ and the vector norm $\|.\|$;

\noindent $\mathbb{R}^{n\times r}$ denotes the space of all matrices
of dimension $(n\times r)$.
  $A^T$ denotes the transpose of the vector/matrix $A$;  a matrix $A$
is symmetric if $A = A^T$;
$I$ denotes the identity matrix;

\noindent $\lambda(A)$ denotes the set of all eigenvalues of $A$;
$\lambda_{\rm max}(A) = \max\{\mathop{\rm Re}\lambda: \lambda \in \lambda(A)\}$;

\noindent $\|A\|$ denotes
 the spectral norm of the matrix defined by
$$\|A\| = \sqrt{\lambda_{\max}(A^TA)};$$
$\eta(A)$ denotes the matrix measure of the
matrix $A$ given by
$$
\eta(A) = \frac{1}{2}\lambda_{\rm max}(A + A^T).
$$
$C([a, b], \mathbb{R}^n)$ denotes the set of all
$\mathbb{R}^n$-valued continuous functions on $[a, b]$;

\noindent Matrix $A$ is called semi-positive definite ($A \geq 0$) if
$\langle Ax,x\rangle \geq 0$, for all $x \in \mathbb{R}^n; A$ is positive
definite ($ A >0$) if $\langle Ax,x\rangle > 0$ for all $x\neq 0$;

In the sequel, sometimes for the sake of brevity, we will omit the
arguments of matrix-valued  functions, if it does not cause any
confusion.

Let us consider the following linear non-autonomous system with
multiple  delays
\begin{equation} \label{e2}
\begin{gathered}
\dot x(t) = A_0(t)x(t) + \sum_{i=1}^mA_i(t)x(t-h_i), \quad t\geq 0, \\
x(t) = \phi (t), \quad  t \in [-h, 0],
\end{gathered}
\end{equation}
where $h = \max\{h_i: i =1, 2,\dots, m\}, A_i(t), i=0, 1,\dots, m$, are
given matrix functions and $\phi(t)\in C([-h,0],\mathbb{R}^n)$.

\subsection*{Definition} % 2.1.}
The system \eqref{e2} is said to be $\alpha$-stable, if there is a function
 $\xi(.):\mathbb{R}^+\rightarrow \mathbb{R}^+$ such that for each
 $\phi(t)\in C([-h,0],\mathbb{R}^n)$,
the solution $x(t,\phi)$ of the system satisfies
$$
\|x(t,\phi)\| \leq \xi(\|\phi\|)e^{-\alpha t},\quad \forall t\in
\mathbb{R}^+.
$$

The following well-known lemma, which is derived from completing the
square, will be used in the proof of our main result.

\begin{lemma} \label{lem2.1}
Assume that $S\in \mathbb{R}^{n\times n}$ is a symmetric positive definite
matrix. Then for every $P, Q\in \mathbb{R}^{n\times n}$,
$$
\langle Px,x\rangle+ 2\langle Qy,x\rangle - \langle Sy,y\rangle
\leq \langle (P+QS^{-1}Q^T)x,x\rangle, \quad \forall  x, y\in \mathbb{R}^n.
$$
\end{lemma}

\section{Main results}

Consider the linear non-autonomous  delay system \eqref{e2}, where the
matrix functions $A_i(t)$, $i=0, 1, \dots , m$, are continuous on $\mathbb{R}^+$.
Let us set
$$
A_{0,\alpha}(t) = A_0(t) +\alpha I, \quad A_{i,\alpha}(t)
 = e^{\alpha h_i}A_i(t), i=1, 2, \dots , m.
$$

\begin{theorem} \label{thm3.1}
The linear non-autonomous  system \eqref{e2} is $\alpha$-stable if there is
  a symmetric semi-positive definite matrix $P(t)$, $t\in \mathbb{R}^+$
 such that
 \begin{equation} \label{e3}
\begin{aligned}
&\dot P(t) + A_{0,\alpha}^T(t)[P(t)+I] + [P(t)+I]A_{0,\alpha}(t) \\
&+ \sum_{i=1}^m[P(t)+I]A_{i,\alpha}(t)A^T_{i,\alpha}(t)[P(t)+I] + mI = 0.
\end{aligned}
\end{equation}
\end{theorem}

\begin{proof}
Let $P(t) \geq  0$, $t\in \mathbb{R}^+$ be a solution of the RDE \eqref{e3}. We take
the following change of the state variable
$$
y(t) = e^{\alpha t}x(t),\quad t\in \mathbb{R}^+,
$$
then the linear delay  system \eqref{e2} is
transformed to the delay system
\begin{equation} \label{e4}
\begin{gathered}
\dot y(t) = A_{0,\alpha}(t)y(t) + \sum_{i=1}^mA_{i,\alpha}(t)y(t-h_i), \\
y(t) = e^{\alpha t}\phi(t),\quad t\in [-h,0],
\end{gathered}
\end{equation}
Consider the following time-varying Lyapunov-like function
$$
V(t,y(t)) = \langle P(t)y(t),y(t)\rangle + \|y(t)\|^2 +
\sum_{i=1}^m\int_{t-h_i}^t\|y(s)\|^2ds.
$$
Taking the derivative of $V(.)$ in $t$ along the solution of
$y(t)$ of system \eqref{e4} and using the RDE \eqref{e3},  we have
\begin{equation}
\begin{aligned} \label{e5}
&\dot V(t,y(t)) \\
&= \langle \dot P(t)y(t),y(t)\rangle + 2 \langle P(t)\dot y(t), y(t)\rangle +
2\langle \dot y(t),y(t)\rangle + m\|y(t)\|^2 - \sum_{i=1}^m\|y(t-h_i)\|^2, \\
&= \langle \dot P(t)y(t),y(t)\rangle + 2\langle
  P(t)A_{0,\alpha}(t)y(t),y(t)\rangle
 +2\sum_{i=1}^m\langle P(t)A_{i,\alpha}(t)y(t-h_i),y(t)\rangle \\
&\quad + 2\langle A_{0,\alpha}(t)y(t),y(t)\rangle + 2\sum_{i=1}^m\langle
 A_{i,\alpha}(t)y(t-h_i),y(t)\rangle \\
 &\quad + m\|y(t)\|^2  -\sum_{i=1}^m\|y(t-h_i)\|^2,\\
&= \langle \dot P(t)y(t),y(t)\rangle + 2\langle
  (P(t)+I)A_{0,\alpha}(t)y(t),y(t)\rangle \\
&\quad+ 2\sum_{i=1}^m\langle (P(t)+I)A_{i,\alpha}(t)y(t-h_i),y(t)\rangle
 + m\|y(t)\|^2 - \sum_{i=1}^m\|y(t-h_i)\|^2,\\
&= - \sum_{i=1}^m\langle
[P(t)+I]A_{i,\alpha}(t)A^T_{i,\alpha}(t)[P(t)+I]y(t),y(t)\rangle\\
&\quad + 2\sum_{i=1}^m\langle [P(t)+I]A_{i,\alpha}(t)y(t-h_i),y(t)\rangle -
\sum_{i=1}^m\langle y(t-h_i), y(t-h_i)\rangle \\
&=  \sum_{i=1}^m \{- \langle
[P(t)+I]A_{i,\alpha}(t)A^T_{i,\alpha}(t)[P(t)+I]y(t),y(t)\rangle \\
&\quad + 2\langle [P(t)+I]A_{i,\alpha}(t)y(t-h_i),y(t)\rangle - \langle y(t-h_i),
y(t-h_i)\rangle\}.\end{aligned}\end{equation}
Applying Lemma \ref{lem2.1} to the  above equality, we have
\[
\dot V(t,y(t)) \leq 0, \quad \forall t\in \mathbb{R}^+.
\]
Integrating both sides of this inequality from $0$ to $t$, we find
$$
V(t,y(t)) - V(0,y(0)) \leq 0,\quad \forall t\in \mathbb{R}^+,
$$
and hence
\begin{align*}
&\langle P(t)y(t),y(t)\rangle  + \|y(t)\|^2
+ \sum_{i=1}^m\int_{t-h_i}^t\|y(s)\|^2ds\\
& \leq \langle P_0y(0),y(0)\rangle + \|y(0)\|^2 +
\sum_{i=1}^m\int_{-h_i}^0\|y(s)\|^2ds,
\end{align*}
where $P_0= P(0)\geq 0$ is any initial condition. Since
\begin{gather*}
\langle P(t)y,y\rangle \geq 0,\quad
\int_{t-h_i}^t\|y(s)\|^2ds \geq  0, \\
\int_{-h_i}^0\|y(s)\|^2ds \leq \|\phi\|\int_{-h_i}^0e^{\alpha s}ds
= \frac{1}{\alpha}(1-e^{-\alpha h_i})\|\phi\|,
\end{gather*}
it follows that
$$
\|y(t)\|^2 \leq \langle P_0y(0),y(0)\rangle + \|y(0)\|^2
+ \frac{1}{\alpha}\sum_{i=1}^m(1-e^{-\alpha h_i})\|\phi\|.
$$
Therefore, the solution $y(t,\phi)$ of the system \eqref{e4} is bounded.
Returning to the solution $x(t,\phi)$ of system \eqref{e2} and noting
that
$$
\|y(0)\| = \|x(0)\| = \phi(0) \leq \|\phi\|,
$$
we have
$\|x(t,\phi)\| \leq \xi(\|\phi\|)e^{-\alpha t}$ for all $t\in \mathbb{R}^+$,
 where
$$\xi(\|\phi\|) := \{\|P_0\|\|\phi\|^2 + \|\phi\|^2
+ \frac{1}{\alpha}\sum_{i=1}^m(1-e^{-\alpha
h_i})\|\phi\|\}^{\frac{1}{2}}.
$$
This implies system \eqref{e2} begin $\alpha$-stable and completes
the proof.
\end{proof}

\subsection*{Remark} % \label{rmk3.1} \rm
Note that the  existence of a semi-positive definite matrix
solution $P(t)$ of RDE \eqref{e3}  guarantees the boundedness of the
solution of transformed system \eqref{e4}, and hence the exponential
stability of the linear non-autonomous delay system \eqref{e2}. Also, the
stability of $A(t)$ is not assumed.


\begin{example} \label{ex3.1} \rm
Consider the following linear non-autonomous delay system in $\mathbb{R}^2$:
$$
\dot x = A_0(t)x + A_1(t)x(t-0.5) + A_2(t)x(t-1), \quad t\in \mathbb{R}^+,
$$
with any initial function $\phi(t)\in C([-1,0],\mathbb{R}^2)$ and
\begin{gather*}
A_0(t) = \begin{pmatrix} a_0(t) & 0\\0 &-7.5\end{pmatrix},\quad
A_1(t) = \begin{pmatrix} e^{-0.5}a_1(t) &0\\0 &
e^{-0.5}\sqrt{3}\end{pmatrix},\\
 A_2(t) = \begin{pmatrix} e^{-1}a_1(t) & 0\\0 &
e^{-1}\sqrt{3}\end{pmatrix},
\end{gather*}
where
$$
a_0(t) = \frac{7e^{-9t} - 5}{2(1+e^{-9t})},\quad
a_1(t) = \frac{1}{\sqrt{2}(1+e^{-9t})}.
$$
We have $h_1 = 0.5$, $h_2 = 1$, $m=2$ and  the matrix
$A_0(t)$ is not asymptotically stable, since
$\mathop{\rm Re}\lambda(A(0)) = 0.5 > 0$.
Taking $\alpha = 1$, we have
$$
A_{0,\alpha}(t) = \begin{pmatrix} a_0(t)+1 & 0\\0&
-6.5\end{pmatrix},\quad
A_{1,\alpha}(t) = A_{2,\alpha}(t) = \begin{pmatrix} a_1(t) & 0\\0 &
\sqrt{3}\end{pmatrix}.
$$
The solution of RDE \eqref{e3} is
$$
P(t) = \begin{pmatrix} e^{-9t} & 0\\0 & 1\end{pmatrix}\geq 0,\quad
\forall t\in \mathbb{R}^+.
$$
Therefore, the system is $1$-stable.
\end{example}

For the autonomous delay systems, we have the following
$\alpha$-stability condition as a consequence.

\begin{corollary} \label{coro3.1}
The linear  delay system \eqref{e2}, where $A_i$ are constant matrices,
is $\alpha$-stable if there is  a symmetric semi-positive definite
matrix $P\in \mathbb{R}^{n\times n}$, which is a solution
of the algebraic Riccati equation
\begin{equation} \label{e8}
A_{0,\alpha}^T[P+I] + [P+I]A_{0,\alpha} +
  \sum_{i=1}^m[P+I]A_{i, \alpha}A^T_{i,\alpha}[P+I] + mI = 0.
\end{equation}
\end{corollary}

\begin{example} \label{ex3.2} \rm
  Consider the  linear autonomous delay system
$$
\dot x(t) = A_0x(t) + A_1x(t-2) + A_2x(t-4), \quad t\in \mathbb{R}^+,
$$
with any initial function $\phi(t)\in C([-4,0],\mathbb{R}^2)$
and
$$
A_0 = \begin{pmatrix}  - \frac{17}{6} & 0\\\frac{4}{3} & -3.5\cr
\end{pmatrix},\quad
A_1 = \begin{pmatrix} e^{-1} &0\\0 & e^{-1}\cr
\end{pmatrix},\quad  A_2 = \begin{pmatrix} e^{-2} & 0\\0 &
e^{-2}\end{pmatrix}.
$$
In this case, we have  $m=2$, $h_1 = 2$, $h_2 =4$. Taking $\alpha = 0.5$,
we find
$$
A_{0,\alpha}(t) =  \begin{pmatrix} -\frac{7}{3} & 0\\ \frac{4}{3} & -3\cr
\end{pmatrix},\quad
A_{1,\alpha}(t) =  A_{2,\alpha}(t) =
\begin{pmatrix} 1 & 0\\0 & 1\end{pmatrix},
$$
and  the solution of algebraic Riccati equation \eqref{e8} is
$$
P = \begin{pmatrix} 1 & -1\\-1 & 1\end{pmatrix}\geq 0.
$$
Therefore, the system is $0.5$-stable.
\end{example}

 \subsection*{Remark} %3.2
 Note that we can estimate  the value of $V(t,y)$ as follows. Since
$$
2(P+I)A_{0,\alpha} = A_0^TP +PA_0 + A_0+A_0^T + 2\alpha(P+I),
$$
from  \eqref{e5} it follows that
\begin{align*}
\dot V(t,y(t))
&= \langle [\dot P(t) + A^T_0(t)P(t) + P(t)A_0(t)
+mI]y(t),y(t)\rangle \\
&\quad + \langle [A_0(t) +
A_0^T(t)]y(t),y(t)\rangle + 2\alpha\langle(P(t)+I)y(t),y(t)\rangle \\
&\quad + \sum_{i=1}^m \Big\{2\langle [P(t)+I]A_{i,\alpha}(t)y(t-h_i),y(t)\rangle -
\|y(t-h_i)\|^2\Big\}.
\end{align*}
Using Lemma \ref{lem2.1}, we have
\begin{align*}
&\sum_{i=1}^m
\Big\{2\langle [P+I]A_{i,\alpha}y(t-h_i),y(t)\rangle - \|y(t-h_i)\|^2\Big\}\\
&\leq \sum_{i=1}^m\langle [P+I]A_{i,\alpha}
A^T_{i,\alpha}[P+I]y(t),y(t)\rangle.
\end{align*}
On the other hand, since
$$
\sum_{i=1}^m\langle [P(t)+I]A_{i,\alpha}(t)
A^T_{i,\alpha}(t)[P(t)+I]y(t),y(t)\rangle \leq
m\|P(t)+I\|^2e^{2\alpha h}\|A(t)\|^2\|y(t)\|^2,
$$
with $h = \max\{h_1, h_2, \dots , h_m\}$,
$\|A(t)\|^2 =\max\{\|A_1(t)\|^2, \|A_2(t)\|^2, \dots ,\|A_m(t)\|^2\}$,
we obtain
\begin{align*}
\dot V(t,y(t))&\leq \langle [\dot P(t) + A^T_0(t)P(t) + P(t)A_0(t) +
mI]y(t),y(t)\rangle \\
&\quad + \Big[2\eta(A_0(t)) +2\alpha\|P(t)+I\| +
m\|P(t)+I\|^2e^{2\alpha h}\|A(t)\|^2\Big]\|y(t)\|^2.
\end{align*}
Therefore,  the
$\alpha$-stability condition of Theorem \ref{thm3.1} can be
 given in terms of the solution of the following Lyapunov equation,
 which does not involve $\alpha$:
\begin{equation} \label{e9}
 \dot P(t) + A^T_0(t)P(t) + P(t)A_0(t) + mI = 0.
\end{equation}
In this case, if we assume that $P(t)$, $A_i(t)$ are bounded on
$\mathbb{R}^+$ and
\begin{equation} \label{e10}
\eta(A_0) := \sup_{t\in \mathbb{R}^+}\eta(A_0(t)) < +\infty,
\end{equation}
then the rate of convergence $\alpha > 0$ can be defined as a solution
 of the scalar inequality
\begin{equation} \label{e11}
\eta(A_0) + \alpha\|P_I\| + \frac{m}{2}e^{2\alpha h}\|P_I\|^2 \|A\|^2 \leq
0,
\end{equation}
where
$$
P_I = \sup_{t\in \mathbb{R}^+}\|P(t)+I\|,\quad
\|A\|^2 = \sup_{t\in \mathbb{R}^+}\|A(t)\|^2.
$$
Therefore, we have the following
$\alpha$-stability condition.

\begin{theorem} \label{thm3.2}
Assume that the matrix functions $A_i(t), i= 1, 2, \dots , m$ are bounded
on $\mathbb{R}^+$ and the conditions \eqref{e10}, \eqref{e11} hold.
The non-autonomous delay system
\eqref{e2} is $\alpha$-stable if the Lyapunov equation \eqref{e9} has a solution
$P(t)\geq 0$, which is  bounded on $\mathbb{R}^+$.
In this case, the  rate of convergence $\alpha>0$ is the solution of the
inequality \eqref{e11}.
\end{theorem}

\begin{example} \label{ex3.3} \rm
Consider the  linear non-autonomous delay system
$$
\dot x(t) = A_0(t)x(t) + A_1(t)x(t-0.5) + A_2(t)x(t-1),
\quad t\in \mathbb{R}^+,
$$
with any initial function $\phi(t)\in C([-1,0],\mathbb{R}^2)$ and
\begin{gather*}
A_0(t) = \begin{pmatrix} 0.5 -e^t & 1\\-1 &
 0.5-e^t\end{pmatrix},\quad
A_1(t)=  e^{-0.2}\sin t\begin{pmatrix}
 \frac{1}{40} &0\\0 & \frac{1}{40} \end{pmatrix},\\
A_2(t) = e^{-0.2}\cos t\begin{pmatrix} \frac{1}{40} &0\\0 &
\frac{1}{40}\end{pmatrix}.
\end{gather*}
We have $m=2$, $h_1 = 0.5$, $h_2 = 1$, $\eta(A_0) = -0.5$
and $ \|A\| = e^{-0.2}/40$.
On the other hand, the solution of Lyapunov equation \eqref{e9} is
$$
P(t) = \begin{pmatrix} e^{-t} & 0\\0 & e^{-t}\end{pmatrix},
$$
and then $\|P_I\| = 2$. The rate of convergence found from
inequality \eqref{e11} is  $\alpha = 0.2$. All conditions of Theorem \ref{thm3.2}
hold and hence the system is $0.2$-stable.
\end{example}

 For the autonomous  case,
Theorem \ref{thm3.2} gives the following $\alpha$-stability condition,
which is similar to that obtained in \cite{c1,s2}.

\begin{corollary} \label{coro3.2}
The linear delay  system \eqref{e2}, where $A_i(t)$ are constant matrices,  is
$\alpha$-stable if there is a symmetric semi-positive definite  $P$
of the algebraic Lyapunov equation
\begin{equation} \label{e12}
A^T_0P + PA_0 +  mI = 0.
\end{equation}
In this case, the convergent rate $\alpha>0$ is the solution of the
scalar inequality
\begin{equation} \label{e13}
\eta(A_0) + \alpha\|P_I\| + \frac{m}{2}\|P_I\|^2e^{2\alpha h} \|A\|^2 \leq
0,
\end{equation}
where
 $P_I = P+I,\quad \|A\|^2 = \max\{\|A_i\|^2,\; i =1, 2,\dots , m\}$.
\end{corollary}

\begin{example} \label{ex3.4} \rm
Consider the linear  autonomous delay  system
$$
\dot x(t) = A_0x(t) + A_1x(t-0.5) + A_2x(t-1),\quad t\in \mathbb{R}^+,
$$
with any initial function $\phi(t)\in C([-1,0],\mathbb{R}^2)$ and
$$
A_0 = \begin{pmatrix} -2 & 0.5\\-1 & - 4\cr \end{pmatrix},\quad
A_1 = A_2= e^{-0.4}\begin{pmatrix} 1/3  &0\\0 & 1/3 \end{pmatrix}.
$$
We have $m=2$,  $h_1 = 0.5$, $h_2 = 1$ and
$$
\eta(A_0) = -3 +0.5\sqrt{4.25},\quad \|A\|^2 = \frac{e^{-0.8}}{9}.
$$
The solution of the algebraic Lyapunov equation \eqref{e12} is
$$
P = \begin{pmatrix} 0.5 & 0\\0 & 0.25\end{pmatrix},
$$
and then $\|P+I\| = 1.5$. The rate of convergence $\alpha =0.4$ satisfies
the condition \eqref{e13}. Then, by Corollary \ref{coro3.2} the system is $0.4$-stable.
\end{example}

\subsection*{Acknowledgement} This work  was
done during the first author's visit to the Abdus Salam International
Center for Theoretical Physics (ICTP), Trieste, Italy,
under the Swedish International
Development Agency's grant (SIDA).
The authors  are also grateful to the anonymous
referee for his/her valuable remarks, which improved this paper.

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