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\markboth{\hfil Carleman estimates \hfil EJDE--2000/53}
{EJDE--2000/53\hfil Paolo Albano \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~53, pp.~1--13. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
  Carleman estimates for the Euler--Bernoulli plate operator 
\thanks{ {\em Mathematics Subject Classifications:} 93B07, 74F05.
\hfil\break\indent
{\em Key words:} Carleman estimates, observability, thermo--elasticity.
\hfil\break\indent
\copyright 2000 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted March 14, 2000. Published July 12, 2000.} }
\date{}
%
\author{ Paolo Albano }
\maketitle

\begin{abstract} 
 We present some a--priori estimates of Carleman type for the 
 Euler--Bernoulli plate operator.  As an application, we consider 
 a problem of boundary observability for the Euler--Bernoulli 
 plate coupled  with the heat equation.  
\end{abstract}


\newtheorem{Theorem}{Theorem}[section]
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{Remark}[Theorem]{Remark}
\newtheorem{Example}[Theorem]{Example}


\section{Introduction}
In \cite{AT} Tataru and the author proved some estimates with 
singular weights for the heat and for the wave equations. 
These estimates are a powerful tool for the study of
observability (and controllability) of pde's. 
The present paper is concerned with a problem of boundary observability for 
the following system    
\begin{equation}\label{e:state}
\begin{array}{c}
w_{tt}+ \Delta^2 w=\alpha \Delta \theta \quad \mbox{in}\quad ]0,T[\times \Omega 
\\[1mm]
\theta _t-\Delta \theta =\beta \Delta w\quad \mbox{in}\quad ]0,T[\times \Omega  
\\[1mm]
w=\partial_\nu w=\Delta w=\theta =0\quad \mbox{on}\quad [0,T]\times \partial\Omega  
\end{array}
\end{equation}
here  $\alpha ,\beta \in {\mathbb R} $,   $\Omega$ is an open domain in ${\mathbb R}^n$ 
with smooth boundary $\partial\Omega $,  $T$ is a positive number and  
by $\nu =(\nu_0 ,\dots ,\nu_n )$ we denote the unit outer 
normal vector to $[0,T]\times \partial\Omega $. 

We are concerned with the following question.  
Knowing that  $\partial_\nu \Delta w $ and $\partial_\nu \theta$ are equal to $0$ on
$[0,T]\times \partial\Omega $ can we conclude   
that $w$ and $\theta$ are  $0$ in $[0,T]\times \Omega $?  
More precisely, for any solution $(w,\theta )$ of problem
(\ref{e:state}), we want to prove an observability estimate of the
form
$$
\| (w,\theta )\| \le C \| (\partial_\nu \Delta w,\partial_\nu \theta )
\|_\partial
$$
where $\| \cdot \|$ and $\| \cdot \|_\partial$ are suitable interior and
boundary Sobolev norms. In other words, this would say that the
solution $(w,\theta)$ can be reconstructed in a stable fashion if one
observes some derivatives on the boundary. Note that the
initial data cannot be recovered stably due to the parabolic
regularizing effect. However, the final data at time $T$ can be
obtained. By duality this implies an exact null controllability result
for the adjoint problem. 

 
The affirmative answer to the previous questions  is given 
using a--priori estimates of Carleman type for the heat equation and for 
the following problem   
\begin{equation}\label{eq:plate}
\begin{array}{c}
w_{tt}+\Delta^2  w=f \quad \mbox{in}\quad ]0,T[\times \Omega  \\[1mm]
w=\partial_\nu w =\Delta w=0\quad \mbox{on}\quad [0,T]\times
\partial\Omega \, . 
\end{array}
\end{equation}
Our goal is to derive an estimate of the form 
\[
\| e^{\tau \phi }w\| \le C 
( \| \partial_\nu \Delta w\|_\partial +\| e^{\tau \phi }f\| ) 
\]
for solutions of (\ref{eq:plate}).

 
The Carleman estimates have been introduced in \cite{C}, and were intensively
studied in \cite{H} (for hyperbolic and elliptic operators) and in
\cite{I} (in the case of anisotropic operators).  A general approach
to these estimates for boundary value problems was developed in
\cite{T1,T3}.  Carleman estimates for the initial boundary value
problem for the heat equations have been independently obtained in
\cite{EF,E} and \cite{T4} while the analogous results in the case of
hyperbolic equations were proved in \cite{T2}.  We remark that other
approaches to the observability problem for hyperbolic equations have
been developed in \cite{L} (using multipliers method) and in
\cite{BLR} (using microlocal analysis).  For what concerns
observability estimates for the heat equations, an observability
result was proved in \cite{LR} using microlocal analysis and Carleman
estimates for elliptic equations. 
A similar result was proved in \cite{AT} in the case of the wave operator 
instead of the plate operator. 
The additional difficulty here is due to the higher order of the operators 
we deal with. 
We also remark that since both the heat and the plate operators are 
anisotropic (the weights of the time and space derivatives are $2$ 
and $1$ respectively), no lower bound on the observation time, $T$,  is 
required. 
Finally, for simplicity we will limit our computations 
to the constant coefficients case, but similar results 
hold true also in the case 
of operators of the same type with $C^1$ principal parts 
and $L^\infty $ lower order terms.   


We start recalling a Carleman estimate with singular weight for the
heat equation.  Next, we deduce a similar estimate for the  plate 
operator (decoupling such an operator as the product of  two Schr\"odinger
ones).  Finally, putting together the previous estimates we get an 
observability estimate for the coupled system, which, in particular,
yields an affirmative answer to our question.  We point out that the
last step will require a precise control on the constant that appear
on the Carleman estimates.   


\section{Notation and Preliminaries} 


Denote by $(t,x)$ (or $(x_0,x_1,\dots ,x_n)$)  the coordinates in $[0,T]
\times \Omega .$ We call $t$ the ``time'' variable, while
the other $n$ coordinates are called the ``space'' variables.  
By $\langle \cdot ,\cdot \rangle $ and  $\Re \langle \cdot ,\cdot \rangle $ 
we denote the $L^2$--scalar product and its real part respectively.  
We will use the following shorter notation 
\[
 D_j=\frac 1i \frac{\partial}{\partial x^j}, \quad  u_{i_1\dots i_k}=\partial_{i_1}\dots
\partial_{i_k}u .
\]
The symbol  $u_t$ indicates the derivative of $u$ with
respect to $t$, $\nabla$ represents the gradient with respect
to the space variables while $D=i^{-1}\nabla $ is its selfadjoint version 
and $\nabla^2 u$ is the Hessian matrix of $u$ (w.r.t. the space
variables). Given two operators, $A$ and $B$,  as usual we define their
commutator as 
\[
[A,B]:=AB-BA .
\]
By $H^s$ we denote the classical Sobolev spaces, with norm $\| \cdot
\|_s$ while $\| \cdot \|$ stands for the $L^2$ norm. In the Carleman
estimates we use weighted Sobolev norms. 
Set 
$$
\alpha =(\alpha_0,\alpha_1,\dots ,\alpha_n )=(\alpha_0, \alpha ')\, .
$$
Given a nonnegative function $\eta$ define the following anisotropic norm 
$$
\| u\|_{_{k,\eta }}^2=\sum_{|\alpha|_a \leq k} \int \eta^{2(k-|\alpha|_a)}
|\partial^{\alpha} u|^2 \,dt \,dx
$$
where $\partial^\alpha =\partial^{\alpha_0}_0\dots \partial^{\alpha_n}_n$ and 
$|\alpha |_a=2\alpha_0+\alpha_1+\dots +\alpha_n .$  
 
We conclude this section recalling a Carleman estimate for the heat equation. 
Consider the parabolic initial boundary value problem
\begin{equation}\label{eq:H}
\begin{array}{c}
\theta_t (t,x)-\Delta \theta (t,x)=f(t,x) \quad\mbox{in}\quad ]0,T[\times \Omega \,,
\\[1mm]
\theta=0 \quad\mbox{on}\quad [0,T]\times \partial\Omega \,,
\\[1mm]
\theta (0,x)=\theta_0(x)\quad  \mbox{in}\quad \Omega\,,
\end{array}
\end{equation}
with $(f,\theta_0)\in L^2([0,T]\times \Omega )\times H_0^1(\Omega ).$
A Carleman estimate for the solution of the above problem looks like 
$$
 \| e^{\tau \phi }\theta \| \le \,
\| e^{\tau \phi } \partial_\nu \theta \|_{_\partial} +\| e^{\tau \phi }f\| \,,
$$
with appropriate norms and a suitable function $\phi$, uniformly
with respect to the large parameter $\tau$.  The obstruction to such
an estimate is that no Sobolev norm of the initial data can be
controlled by the right hand side.  Hence the only hope is to consider
a weight function $\phi$ which approaches $-\infty$ at time $0$.
Estimates of this type have been proved in (\cite{AT,EF,E} and
\cite{T4}).   
Thus, we introduce a   function $g$ defined as
\begin{equation}\label{g}
 g(t)=\frac 1{t(T-t)}\quad (t\in ]0,T[)\, .
\end{equation}
Notice that 
\begin{equation}\label{eq:stimag}
\Big |\Big (\frac d{dt} \Big )^k g(t)\Big | \le C_k g^k(t) \quad  \quad 
\forall t\in ]0,T[ 
\end{equation}
for a suitable positive constant $C_k.$ 
Let $\psi (x)$ be a function such that
\begin{equation}\label{A}
\nabla \psi (x)\neq  0
\quad \mbox{ for all  }x\in \Omega \, \footnote{This is a
pseudoconvexity condition with respect to the heat operator.}\,. 
\end{equation}
Define
\begin{equation}\label{d:psi}
\phi (t,x):=g(t)\big (e^{\lambda \psi (x)}- 2 e^{\lambda \Phi}), 
\quad \quad \ \  \ \Phi = \|\psi\|_{L^\infty(\Omega)} .
\end{equation}
The weight function $\phi$ thus defined approaches $-\infty$ at times 
$0,T.$ The additional parameter $\lambda$ is essential in order to obtain
the control of the constants which enables us to handle
arbitrarily large coefficients in the coupling terms. 

\begin{Theorem}\label{hpseudo} 
Let $\phi$ be given as in
  $(\ref{d:psi})$ with $\psi$ satisfying $(\ref{A})$.  Let $\theta $
  be the solution of problem $(\ref{eq:H}).$ Then there exists $\lambda_0$
so that for each $\lambda > \lambda_0$ there exists $\tau_\lambda$ so that
 for $\tau > \tau_\lambda$ the following estimate holds uniformly
in $\lambda,\tau$:
\begin{equation}\label{H}
C_1\lambda \|\tilde\tau ^{-\frac 12} e^{\tau \phi }\theta \|^2_{_{2,\tilde\tau }}
\le \|e^{\tau \phi }f\|^2+\int_{[0,T]\times \partial\Omega }
\tilde\tau e^{2\tau \phi}\partial_\nu \psi 
|\partial_\nu \theta |^2\, d\sigma \, , 
\end{equation}
here $\tilde\tau =\lambda \tau g e^{\lambda \psi }$ and $C_1$ is a suitable
positive constant. 
\end{Theorem} 

For the reader convenience we give the proof of the above
result in the appendix, for a proof in a more general setting see
\cite{AT}. 

 
\section{The Plate Operator} 

Let us consider the following problem 
\begin{equation}\label{eq:plate1}  
\begin{array}{c}
w_{tt}+\Delta^2 w=f\quad \mbox{in}\quad ]0, T[ \times \Omega
\\[1mm]
w=\partial_\nu w=\Delta w=0\quad \mbox{on}\quad [0, T]\times \partial\Omega 
\\[1mm]
(w(0,x),w_t(0,x))\in (H^4(\Omega )\cap H^2_0(\Omega ))\times H^2(\Omega )\, ,
\end{array}
\end{equation}
with $f\in L^2 ([0,T]\times \Omega ).$ 
Our goal  is to prove a Carleman estimate for a solution $w$ of the 
above problem. 
The idea of the proof is the following. 
First, we decompose the plate operator as the product of two Schr\"odinger 
ones. 
Then, we get some Carleman  estimates for such operators 
(see Theorem~\ref{l:1} and Theorem~\ref{c:2} below). 
Finally, the result follows putting together  these estimates 
(see Theorem~\ref{t:1}).  
It is clear that in order to prove a Carleman estimate for the
Schr\"odinger operator we need to assume a suitable condition on the
weight function. 
More precisely, we suppose a positive constant $\gamma$ exists so
that, for any $x\in \Omega$,  
\begin{equation}\label{A'}
\nabla^2 \psi (x) \xi \cdot \xi \geq \gamma |\xi |^2 \hspace{3 cm} \forall
\xi
\in {\mathbb R}^n\, \footnote{This is a strong pseudoconvexity condition
for the
constant coefficients Schr\"odinger equation.}\, .
\end{equation}
 
\begin{Lemma}\label{l:1}
Let $u\in C(]0,T[;H^2(\Omega ))\cap C^1(]0,T[; L^2(\Omega ))$
be a solution of the (ill posed) problem 
\begin{equation}\label{eq:sch1}
\begin{array}{c}
u_t+i\Delta  u=f\quad\mbox{in}\quad ]0, T[\times \Omega 
\\[1mm]
u=0\quad  \mbox{on}\quad [0, T]\times \partial\Omega \, .
\end{array}
\end{equation}
with $f\in L^2(]0,T[\times \Omega )$ and let $\phi$ be given as in
  $(\ref{d:psi})$ with $\psi$ satisfying $(\ref{A})$ and $(\ref{A'})$.
Then there exists $\lambda_0$ so that for each $\lambda >\lambda_0$ there
exists $\tau_\lambda$ so that
 for $\tau > \tau_\lambda$ the following estimate holds uniformly
in $\lambda,\tau$:
\begin{equation}\label{eq:cesch1}
C_2 \| e^{\tau \phi }\tilde\tau^{\frac 12} u\|_{1,\tilde\tau }^2\le 
\int_{[0,T]\times \partial\Omega } \tilde\tau e^{2\tau \phi} \partial_\nu \psi 
|\partial_\nu u|^2d\sigma +\| e^{\tau \phi} f\|^2\, ,  
\end{equation}
here $\tilde\tau =\lambda \tau g e^{\lambda \psi }$ and $C_2$ is a suitable
positive constant.
\end{Lemma} 
\paragraph{Proof.} 
Let us compute the conjugated operator to $P=\partial_t+i\Delta $. 
We have that 
\[
P_\tau =e^{\tau \phi }Pe^{-\tau\phi} =iD_t-\tau \phi_t-iD^2+i\tau^2
|\nabla \phi |^2 +\tau (D\cdot \nabla\phi+\nabla \phi \cdot D)\, .
\]
Clearly, we can split the operator $P_\tau $ into its symmetric and
antisymmetric parts as follows   
\[ 
P_\tau =P_\tau^s+P_\tau^a
\]
with 
\[
P_\tau^s= -\tau \phi_t +\tau (D\cdot \nabla \phi +\nabla \phi \cdot D)
\]
and 
\[
P_\tau^a=iD_t-iD^2+i\tau^2 |\nabla \phi |^2\, .
\]
Set $v=e^{\tau \phi}u$ and observe that 
\begin{equation}\label{eq:cd1} 
v(0,x)=v(T,x)=0\quad \quad \forall  x\in \Omega 
\end{equation}
and 
\begin{equation}\label{eq:cd2}
v(t,x)=0\quad \quad  \forall (t,x)\in [0,T]\times \partial\Omega .
\end{equation}  
So, estimate (\ref{eq:cesch1}) reduces to 
\begin{equation}\label{eq:wiwp}
C_2 \| \tilde\tau^{\frac 12} v\|^2_{1,\tilde\tau }\le 2\int_{[0,T]\times \partial\Omega } 
\tilde\tau \partial_\nu \psi | \partial_\nu v|^2\, d\sigma +\| e^{\tau \phi } f\|^2\, . 
\end{equation}
Clearly,  
\begin{equation}\label{eq:0}
\| P_\tau v\|^2 =\| P_\tau^s v\|^2+\| P_\tau^a v\|^2
+2\Re \langle P_\tau^s v,P_\tau^a v\rangle  
\end{equation}
and    
\[
2\Re \langle P_\tau^s v,P_\tau^a v\rangle  
=2\tau \Re \langle 
(-\phi_t +D\cdot \nabla \phi 
+\nabla \phi \cdot D )v, i(D_t-D^2+\tau^2 |\nabla \phi |^2 )v\rangle . 
\]
In order to complete the proof we must only to compute explicitly the 
terms of the above scalar product. 
It is immediate to see that  
\begin{equation}\label{eq:1}
\Re \langle \phi_t v, i\tau^2 |\nabla \phi |^2 v\rangle =0\, .
\end{equation}
Integrating by parts and using (\ref{eq:cd2}), we get  
\begin{eqnarray}
2\tau \Re \langle (D \cdot \nabla \phi  +\nabla \phi \cdot D ) v
,i\tau^2 |\nabla \phi |^2 v\rangle &=& 2\tau^3 
\langle v,\nabla \phi \cdot \nabla |\nabla \phi |^2  v\rangle 
\label{eq:2} \\
&=& 4\tau^3 \langle \nabla^2\phi \nabla \phi \cdot \nabla \phi \,  v,
v\rangle . \nonumber
\end{eqnarray}
Moreover, recalling (\ref{eq:cd1}), we have that   
\begin{equation}
2\tau \Re \langle -\phi_t v, iD_t v\rangle =\tau \Re \langle [-\phi_t , iD_t]v,v\rangle 
=\tau \langle \phi_{tt} v,v\rangle \, (=\mbox{lower order term}), 
\label{eq:3}
\end{equation} 
while (\ref{eq:cd1}) and (\ref{eq:cd2}) imply that 
\begin{eqnarray}
\lefteqn{2\tau \Re \langle (D \cdot \nabla \phi +\nabla \phi \cdot D ) v,
iD_t v\rangle }\nonumber    \\
&=&2\tau \Re (\langle \nabla \phi \cdot D v, iD_t v\rangle 
+\langle (D\cdot \nabla D_t \phi +D\cdot \nabla \phi D_t )v,iv\rangle )
\label{eq:4}\\
&=&2\tau \Re (\langle \nabla \phi \cdot D v, iD_t v\rangle 
+\langle D\cdot \nabla D_t \phi v , iv \rangle +
\langle D_t v,i\nabla \phi \cdot Dv\rangle )
\nonumber \\
&=&-2\tau \Re \langle \nabla \phi_t\cdot  Dv,v\rangle 
\, (=\mbox{lower order term}).  \nonumber
\end{eqnarray}
(Here and in the sequel, lower order terms means terms which can be   
absorbed in the LHS of (\ref{eq:wiwp}) taking $\tau$ large enough.)
 
Finally, integrating by parts once and recalling (\ref{eq:cd2}), we get 
\begin{equation}
2\tau \Re \langle -\phi_t v, -iD^2 v\rangle = 
-2\tau \Re \langle \nabla \phi_t\cdot  Dv,v\rangle 
\, (=\mbox{lower order term}). 
\label{eq:5}
\end{equation}
Now, it remains to compute the higher order terms. 
Using once more integrations by parts and (\ref{eq:cd2}) we obtain that 
\begin{eqnarray}
\lefteqn{ -2\tau \Re \langle (D \cdot \nabla \phi  
+\nabla \phi \cdot D)v,iD^2 v\rangle }  \nonumber\\
&=&-2\tau \Re \sum_{k=1}^n \langle D_k ( (D\cdot \nabla \phi +\nabla
\phi \cdot D)v), iD_kv\rangle 
-4\tau \int_{ [0,T]\times \partial\Omega } \partial_\nu \phi
|\partial_\nu v|^2\, d\sigma  \nonumber\\
&=&-2\tau \Re \sum_{k=1}^n \langle (D\cdot \nabla (D_k\phi )+\nabla (D_k
\phi )\cdot D)v), iD_kv\rangle \nonumber \\ 
&&-2\tau \int_{ [0,T]\times \partial\Omega } \partial_\nu \phi
|\partial_\nu v|^2\, d\sigma \label{eq:6} \\
&=&{2\tau \Re \sum_{k=1}^n \langle (D\cdot \nabla \phi_k +\nabla 
\phi_k \cdot D)v), D_kv\rangle              -2\tau \int_{ [0,T]\times \partial\Omega } \partial_\nu \phi
|\partial_\nu v|^2\, d\sigma }\nonumber \\
&=&{2\tau \Re \Big ( \int_{]0,T[\times \Omega } 2\nabla^2 \phi Dv
\cdot \overline{Dv} +\frac 1i \, v \nabla (\Delta \phi )\cdot
\overline{Dv} \, dt\, dx \Big )}   \nonumber \\ % \label{eq:7}
& & -{ 2\tau \int_{ [0,T]\times \partial\Omega } \partial_\nu \phi
|\partial_\nu v|^2\, d\sigma } \nonumber
\end{eqnarray}
Substituting (\ref{eq:1})--(\ref{eq:6}) in (\ref{eq:0}), we get 
\begin{eqnarray} 
\lefteqn{ \| P_\tau v \|^2 }\nonumber\\ 
&\geq &{4 \tau^3 \langle \nabla^2 \phi \nabla \phi \cdot
\nabla
\phi \, v,v\rangle +\tau \langle \phi_{tt}v,v\rangle }-4\tau \Re \langle
\nabla \phi_t \cdot Dv,v\rangle  \nonumber \\ 
&&+{2\tau \sum_{j=1}^n \Re \langle \Delta (D_j\phi ) D_j\,  v,v \rangle }- { 
2\tau \int_{ [0,T]\times \partial\Omega } \partial_\nu \phi |\partial_\nu v|^2\, d\sigma 
+4\tau \sum_{j,k=1}^n \langle \phi_{jk} D_kv\, ,\, D_j
v\rangle }\nonumber \\
&\geq &{\Re \langle (4\tau^3 \nabla^2\phi \nabla \phi \cdot
\nabla \phi +\tau \phi_{tt} -(4\tau | \nabla \phi_t |)^2 -(2\tau |\nabla (\Delta
\phi )|)^2)v,v \rangle }\nonumber \\
&&+ {\int_{]0,T[\times \Omega } 4\tau \nabla^2 \phi Dv\cdot
\overline{Dv}-2|Dv|^2\, dt\, dx}-{2\tau \int_{ [0,T]\times \partial\Omega } \partial_\nu \phi |\partial_\nu v|^2\, d\sigma .}
\nonumber
\end{eqnarray}
Moreover, it is not difficult to verify that 
\[
4\tau^3 \nabla^2 \phi \nabla \phi \cdot \nabla \phi +\tau \phi_{tt} -(4
\tau
|\nabla \phi_t |)^2 -(2\tau | \nabla (\Delta \phi )|)^2\approx \lambda \tilde\tau^3 
\]
and  
\[
4\tau \nabla^2 \phi -2 \mbox{I} \approx \tilde\tau \mbox{I} 
\]
provided that $\lambda $ and $\tau $ are sufficiently large. 
So, 
\[
\| P_\tau v\|^2 +2\int_{[0,T]\times \partial\Omega } \tilde\tau \partial_\nu \psi | \partial_\nu v|^2\,
d\sigma \geq C_2 \| \tilde\tau^{\frac 12} v\|_{_{1,\tilde\tau }}^2,  
\]
provided that $\lambda $ and $\tau$ are
sufficiently large.\hfill$\diamondsuit$

\begin{Theorem}\label{c:1}
Let $u$,  $\phi$ and $\psi$ be as in Lemma~\ref{l:1}. 
Then there exists $\lambda_0$ so that for each $\lambda >\lambda_0$ there
exists $\tau_\lambda$
so that for $\tau >\tau_\lambda $ the following estimate holds uniformly in $\lambda ,\tau $:
\begin{equation}\label{eq:cesf1} 
 C_3 \| e^{\tau \phi } u\|^2_{1,\tilde\tau } \le \int_{[0,T]\times \partial\Omega } 
e^{2\tau \phi } \partial_\nu \psi |\partial_\nu u|^2\, d\sigma +\| e^{\tau \phi }
\tilde\tau^{-\frac 12} f\|^2 \, , 
\end{equation}
here $\tilde\tau =\lambda \tau g e^{\tau \psi }$ and $C_3$ is a suitable positive
constant.  
\end{Theorem} 

\paragraph{Proof.} 
Set $P=\partial_t +i\Delta .$ 
Applying Lemma~\ref{l:1} to $\tilde\tau^{-\frac 12}u$ we obtain 
\[
C_2 \| e^{\tau \phi }  u\|_{1,\tilde\tau }^2\le 
\int_{[0,T]\times \partial\Omega }  e^{2\tau \phi} \partial_\nu \psi 
|\partial_\nu u|^2d\sigma +\| e^{\tau \phi}([P ,\tilde\tau^{-\frac 12}
]+\tilde\tau^{-\frac 12}P) u\|^2\, ,  
\] 
for large enough $\tau $ the commutator is small compared to the LHS
therefore we get 
\[
C_3 \| e^{\tau \phi } u\|_{1,\tilde\tau }^2\le 
\int_{[0,T]\times \partial\Omega } e^{2\tau \phi} \partial_\nu \psi 
|\partial_\nu u|^2d\sigma +\| e^{\tau \phi} \tilde\tau^{-\frac 12}f\|^2\, ,  
\]
and the conclusion follows.\hfill$\diamondsuit$ 

The next result follows arguing as in Lemma~\ref{l:1} and in
Theorem~\ref{c:1}.


\begin{Theorem}\label{c:2} 
Let $u\in C(]0,T[;H^4(\Omega ))\cap C^1(]0,T[; H^2(\Omega ))\cap 
C^2(]0,T[\times L^2(\Omega ))$ be a solution of the
(ill posed) problem 
\begin{equation}\label{eq:sch2}
\begin{array}{c}
u_t-i\Delta  u=f\quad \mbox{in}\quad  ]0, T[\times \Omega
\\[1mm]
u=\partial_\nu u=0\quad  \mbox{on}\quad  [0, T]\times \partial\Omega
\end{array} 
\end{equation}
and let $\phi$ be given as in $(\ref{d:psi})$ with $\psi$ satisfying $(\ref{A})$ and $(\ref{A'})$.
Then there exists $\lambda_0$ so that for each $\lambda >\lambda_0$
there exists $\tau_\lambda$ so that
 for $\tau > \tau_\lambda$ the following estimate holds uniformly
in $\lambda,\tau$: 
\begin{equation}\label{eq:cesf2} 
C_4 \| e^{\tau \phi } u\|^2_{1,\tilde\tau } \le 
\| e^{\tau \phi } \tilde\tau^{-\frac 12} f\|^2 \, , 
\end{equation}
here $\tilde\tau =\lambda \tau g e^{\tau \psi }$ and $C_4$ is a suitable positive
constant.  
\end{Theorem} 

Now, the main result of this section can be easily proved.  
\begin{Theorem}\label{t:1}
Let $w$  be the solution of the  problem $(\ref{eq:plate1})$ 
and let $\phi$ be given as in
  $(\ref{d:psi})$ with $\psi$ satisfying $(\ref{A})$ and $(\ref{A'}).$
Then there exists $\lambda_0$ so that for each $\lambda > \lambda_0$
there exists $\tau_\lambda >0$ so that
 for $\tau > \tau_\lambda$ the following estimate holds uniformly
in $\lambda,\tau$: 
\begin{equation}\label{eq:ce}
 C \| e^{\tau \phi }w\|_{2,\tilde\tau }^2\le  \int_{[0,T]\times \partial\Omega } e^{2\tau \phi} \partial_\nu \psi |\partial_\nu \Delta w|^2 \, d\sigma +\| e^{\tau \phi}\tilde\tau^{-\frac 12}  f\|^2\, , 
\end{equation} 
here $\tilde\tau =\lambda \tau g e^{\lambda \psi }$ and $C$ is a suitable
positive constant. 
\end{Theorem}
{\bf Proof } --
Clearly, problem (\ref{eq:plate1}) can be recast as follows 
\begin{equation}\label{eq:sp1}
\begin{array}{c}
w_t-i\Delta  w=u\quad \mbox{in}\quad  ]0, T[\times \Omega
\\[1mm]
w=\partial_\nu w=0\quad  \mbox{on}\quad  [0, T]\times \partial\Omega 
\end{array}
\end{equation}
and 
\begin{equation}\label{eq:sp2}
\begin{array}{c}
u_t+i\Delta  u=f\quad \mbox{in}\quad  ]0, T[\times \Omega
\\[1mm]
u=0\quad  \mbox{on}\quad  [0, T]\times \partial\Omega \, .
\end{array}
\end{equation}
Now, applying Theorem~\ref{c:2} to the solution of problem (\ref{eq:sp1}) 
we get 
\begin{equation}\label{eq:i1} 
C_4 \| e^{\tau \phi }  w\|^2_{1, \tilde\tau }\le \| e^{\tau \phi }\tilde\tau^{-\frac 12}
u \|^2
\end{equation}
whilst using Theorem~\ref{c:1}, we deduce that the solution of  
(\ref{eq:sp2}) satisfies 
\begin{equation}\label{eq:i2} 
C_3 \| e^{\tau \phi }  u\|^2_{1,\tilde\tau }\le 
\int_{ [0,T]\times \partial\Omega } e^{2\tau \phi } \partial_\nu \psi | \partial_\nu \Delta
w|^2\, d\sigma +\| e^{\tau \phi }\tilde\tau^{-\frac 12}f\|^2  
\end{equation}
for sufficiently large $\lambda $ and $\tau .$ 
Now, the conclusion follows putting together (\ref{eq:i1}) and
(\ref{eq:i2}).\hfill$\diamondsuit$ 

\begin{Remark}{\rm 
We note that a stronger estimate could be proved. In fact, the
interior (microlocal) Carleman estimate for the plate operator  shows that
one can  replace the $H^2$  (anisotropic) norm  in  the LHS of (\ref{eq:ce})  with the $H^3$ (anisotropic) norm  (see e.g. \cite{T2}).    
}\end{Remark}
 
\section{The observability estimate}

In this section we prove the observability result for the system
(\ref{e:state}). 

\begin{Theorem}
Suppose that $(w,\theta )$ is a solution of the system $(\ref{e:state})$
and let $\phi$ be given as in
  $(\ref{d:psi})$ with $\psi$ satisfying $(\ref{A})$ and $(\ref{A'})$.
Then there exists $\lambda_0$ so that for each $\lambda >\lambda_0$ there exists
$\tau_\lambda>0$ so that for $\tau > \tau_\lambda$ the following estimate holds
uniformly in $\lambda,\tau$  
\begin{eqnarray}\label{CarlemanHW}
\| e^{\tau \phi }w\|_{_{2,\tilde\tau }}^2+\| e^{\tau \phi }\tilde\tau^{-\frac
12}\theta
\|_{_{2,\tilde\tau }}^2 &\leq&
C\Big (\int_{[0,T ]\times \partial\Omega }  e^{2\tau \phi }\partial_\nu \psi  |\partial_\nu
\Delta w|^2\, 
d\sigma \,\nonumber \\ 
&& +\int_{[0,T]\times \partial\Omega } \tilde\tau e^{2\tau \phi }\partial_\nu \psi |\partial_\nu
\theta|^2\, d\sigma \Big )\, ,
\end{eqnarray}  
for a suitable positive constant $C.$ 

\label{obs}\end{Theorem}

\begin{Remark}{\rm 
This estimate shows that  solutions to (\ref{e:state}) can be 
reconstructed  in a stable manner if one observes their normal derivative
on the boundary. As one can see from the above estimate, this observation
needs not be on the entire boundary; it suffices to observe  $(w,\theta )$  in the region $\Gamma$,
given by
\[
\Gamma  =[0,T]\times  \{ x \in \partial\Omega;\
\partial_\nu \psi (x)> 0\}\, .
\]
}\end{Remark} 
{\bf Proof of Theorem~\ref{obs}:} 
Applying Theorem~\ref{t:1} to $w$ we deduce that  
\begin{equation}\label{123}
 C \| e^{\tau \phi }w\|_{_{2,\tilde\tau }}^2\leq 
\int_{[0,T]\times \partial\Omega }   e^{2\tau \phi }\partial_\nu \psi |\partial_\nu \Delta w|^2\, 
d\sigma +\| e^{\tau \phi } \tilde\tau^{-\frac 12} \alpha \Delta \theta \|^2. 
\end{equation}
On the other hand, Theorem~\ref{hpseudo} implies that  
\begin{equation}\label{1234}
\lambda C_1 \| e^{\tau \phi }\tilde\tau^{-\frac 12}\theta \|_{_{2,\tilde\tau}}^2
\le \| e^{\tau \phi }\beta \Delta w\|^2\, 
+ \int_{[0,T]\times \partial\Omega } \tilde\tau e^{2\tau \phi } 
\partial_\nu \psi |\partial_\nu \theta|^2\, d\sigma \, .
\end{equation}
Then, the conclusion follows by adding $(\ref{123})$ and
(\ref{1234}) provided that $\lambda$ is sufficiently large.\hfill$\diamondsuit$ 


Combining the straightforward energy estimates\footnote{Here the energy is
defined by $E(t)=2^{-1}\int_{\Omega } |\nabla \theta
(t,x)|^2+w_t^2(t,x)+|\Delta
w(t,x)|^2\, dx .$} with (\ref{CarlemanHW}) we obtain the following
consequence: 
\begin{Theorem}\label{c:3} 
For all solutions  $(w,\theta )$ of the system $(\ref{e:state})$ we have
\[
\|\nabla \theta(T)\|^2_{L^2(\Omega)}+\|w_t(T)\|^2_{L^2(\Omega)} 
+ \|\Delta w(T)\|^2_{L^2(\Omega)}  
\leq C (\|\partial_\nu \Delta w\|^2_{L^2(\Gamma )} + \|\partial_\nu
\theta\|^2_{L^2(\Gamma
)})
\] 
for a suitable positive constant $C$. 
\end{Theorem}


The next example shows in the case of $\Omega $ a ball of ${\mathbb R}^n$ how one can
choose the function $\phi .$  
\begin{Example}{\rm 
Let $\Omega =\{ x\in {\mathbb R}^n\, :\, |x|< 1\} $
and fix $\overline{x}\in {\mathbb R}^n\setminus \Omega .$ Then, define 
\[
\psi (x)\, =\, |x-\overline{x} |^2 
\]
and 
\[
\phi (t,x)=\frac 1{t(T-t)} (e^{\lambda \psi (x)}-2e^{\lambda \sup_{y\in \Omega}
\psi (y) }). 
\] 
It is immediate to verify that $\phi $ satisfies all the assumption of
Theorem~\ref{obs}. Moreover, if $(w,\theta )$ is a solution of equation (\ref{e:state}) and $\partial_\nu \Delta w $ and  $\partial_\nu \theta $ are equal to  $0$ on
\[
\Gamma =\{ (t,x)\in [0,T]\times  \partial\Omega \, :\, x \cdot (x-\overline{x}) >0\} ,
\]   
then $w$ and $\theta$ are equal to  $0$ in $]0,T[\times \Omega .$ 
}\end{Example} 

\appendix

\section{Proof of Theorem~\ref{hpseudo}} 

Set
\[
Q(D)=\partial_t-\Delta . 
\]
As usual, with the substitution $v=e^{\tau \phi}\theta $,  
the estimate (\ref{H}) reduces to
\begin{equation}\label{goalh}
C_1 \lambda \|\tilde\tau ^{-\frac 12}v\|^2_{_{2,\tilde\tau }}\le 
 \| Q_\tau v\|^2+\int_{[0,T]\times \partial\Omega } 
\tilde\tau \partial_\nu \psi  |\partial_\nu v|^2  \, d\sigma \, , 
\end{equation}
where $Q_\tau$ is the conjugated operator defined as 
\[
Q_\tau (t,x,D):=e^{\tau \phi}Q(D)e^{-\tau \phi} .
\]
Notice that 
\[
v(0,x)=v(T,x)=0\quad \quad \forall x\in \Omega 
\]
and 
\[
v(t,x)=0\quad \quad \forall (t,x)\in [0,T]\times \partial\Omega .
\]
Split $Q_\tau $ into
$$
Q_\tau (t,x,D)=Q_\tau ^a(t,x,D)+Q_\tau ^s(t,x,D)
$$
where
$$
Q_\tau^a(t,x,D)= \partial_t+\, \tau \, ( \nabla \cdot \nabla \phi  \,
+\nabla \phi  \cdot \nabla )\
$$
is the skew-symmetric part of $Q_\tau $ while
$$
Q_\tau ^s(t,x,D)v=-\Delta -\tau^2 | \nabla \phi |^2 \, -\tau \phi_t
$$
is the symmetric part of $Q_\tau .$ Since
\begin{equation}\label{Q}
\| Q_\tau v\|^2=\| Q_\tau^sv\|^2+\| Q_\tau^av\|^2+ 2     
\langle Q_\tau^s v, Q_\tau^a v\rangle \, ,
\end{equation}
the crucial step of the proof will be to estimate from below 
\[
2\langle Q_\tau ^s v,Q_\tau ^a v\rangle =
-2\langle (\Delta +\tau^2 |\nabla \phi |^2+\tau \phi_t )v,
(\partial_t +\tau (\nabla \cdot \nabla \phi +\nabla \phi \cdot  \nabla ))
v\rangle 
\] 
integrating by parts.  It is easy
to
see that $\langle \partial_t  v\, ,\, Q_\tau^s v\rangle $ is a lower order term
compared to the left hand side in (\ref{goalh}) since
\begin{eqnarray*}
2 \langle \partial_t  v\, ,\, Q_\tau^s v\rangle &=& 
\langle [Q_\tau^s,\partial_t] v,v\rangle \\ 
&=& + \langle (\partial_t(\tau^2 | \nabla \phi |^2 
 \, +\tau \phi_t) v,v\rangle
\end{eqnarray*}
therefore
\[
|\langle \partial_t v\, ,\, Q_\tau^s v\rangle| \leq 
c_\lambda \tau^2 \| g^{3/2} v\|^2\, ,
\]
for a suitable positive constant  $c_\lambda .$
Similarly, for some $c_\lambda >0$,  
\[
|\langle \, \tau  (\nabla \cdot \nabla \phi  
+ \nabla \phi \cdot \nabla ) v , \tau \phi_t v \rangle |
\leq  c_\lambda \tau^2 \| g^{3/2} v\|^2\, .
\]

  Then it remains to estimate
\[
- 2\tau \,\langle (\nabla \cdot  \nabla \phi + \nabla \phi \cdot 
\nabla \,) v ,\, (\Delta +\tau^2 |\nabla \phi |^2  ) v\rangle \, .
\]
Compute first the leading zero order terms. Integrating by parts, 
we get 
\[
- 2 \tau \, \langle (\nabla \cdot  \nabla \phi + \nabla \phi \cdot  \nabla
) v\, ,\, 
\tau^2 | \nabla \phi |^2 v\rangle =  
2\tau^3 \, \langle v,\, \nabla \phi \cdot \nabla  |\nabla \phi |^2)
v\rangle \, .
\]
Since $\nabla \phi = \lambda g e^{\lambda \psi }\nabla \psi $, the
highest order
terms occur when the derivative falls on the exponential. Hence we get
\begin{equation}
-2  \tau \, \langle (\nabla \cdot \nabla \phi   
+ \nabla \phi \cdot \nabla ) v\, ,\, 
\tau^2 | \nabla \phi |^2 v\rangle =  
4\lambda \langle \tilde\tau^3 |\nabla \psi |^4 v\, , \, 
v\rangle +R
\label{e:2}
\end{equation}
where $R$ stands for lower order terms,
\[
R=O(\|\tilde\tau ^{-1/2}v\|^2_{_{2,\tilde\tau }}).
\]
Now compute the leading first order terms integrating by parts
the expression
\[
-2 \tau \,\langle (\nabla \cdot \nabla \phi + \nabla \phi \cdot \nabla  )
v\, 
,\, \Delta v \rangle \, .
\]
Since one operator is symmetric and the other is skew-symmetric,
we need to compute their commutator.
In fact, we have that
\begin{eqnarray*}
\lefteqn{ -2 \tau \,\langle \Delta v,\, (\nabla \cdot  \nabla \phi +
\nabla\phi\cdot \nabla )v\rangle  }\\
&=&\tau \,\langle (\nabla \cdot  \nabla \phi + \nabla \phi \cdot \nabla )
\Delta v,\, v\rangle 
- \tau \,\langle \Delta v,\, (\nabla \cdot  \nabla \phi + \nabla
\phi \cdot \nabla ) v\rangle .
\end{eqnarray*}
Hence,  using the equality  
\begin{eqnarray*}
\tau \,\langle \Delta v,\, (\nabla \cdot  \nabla \phi + \nabla
\phi \cdot \nabla ) v\rangle &=& -{\tau \int_{]0,T[\times \Omega } }\nabla v\cdot \nabla ( (\nabla
\cdot \nabla
\phi +\nabla \phi \cdot \nabla )v) \, dt\, dx \\
&&+2\tau {\int_{[0,T]\times \partial\Omega } \partial_\nu \phi |\partial_\nu v|^2 \,
d\sigma } 
\end{eqnarray*} 
we deduce that 
\begin{eqnarray*}
\lefteqn{ -2 \tau \,\langle \Delta v,\, (\nabla \cdot  \nabla \phi +
\nabla\phi\cdot \nabla )v\rangle }\\
&=&-\tau \langle [\Delta , (\nabla \cdot \nabla \phi +\nabla \phi \cdot \nabla
)] v , v\rangle 
- 2\tau \int_{[0,T]\times \partial\Omega } \partial_\nu \phi |\partial_\nu v|^2 \, d\sigma .
\end{eqnarray*}
Then, 
\begin{eqnarray*}
\lefteqn{\tau \langle [\Delta , (\nabla \cdot \nabla \phi +\nabla \phi \cdot \nabla
)] v , v\rangle =2\tau \sum_{k=1}^n \langle (\nabla \cdot \nabla \phi_k +\nabla \phi_k
\cdot \nabla )v_k , v \rangle }\\
&=&-2 \tau 
{ \int_{]0,T[\times \Omega }} 
\nabla^2 \phi \nabla v\cdot \nabla v 
+ \sum_{k=1}^n v_k  \nabla \cdot (\nabla \phi_k v)\,  dt\, dx \\
&=&-4 \tau 
{ \int_{]0,T[\times \Omega }} 
\nabla^2 \phi \nabla v\cdot \nabla v\,  dt\, dx 
+\tau \langle (\Delta^2 \phi ) v , v\rangle 
\end{eqnarray*}
and 
\begin{eqnarray}
\lefteqn{ -2 \tau \,\langle \Delta v,\, (\nabla \cdot  \nabla \phi +
\nabla\phi\cdot \nabla )v\rangle } \nonumber\\
&=& 4 { \int_{]0,T[\times \Omega }} \tilde\tau \lambda |\nabla \psi \cdot
\nabla v|^2 dt\, dx+R-2{ \int_{[0,T]\times \partial\Omega }} \tilde\tau 
\partial_\nu\psi |\partial_\nu v|^2 \, d\sigma . 
\label{e:3} 
\end{eqnarray}
Hence, if we put together (\ref{e:2}) and (\ref{e:3}) then we get
\begin{eqnarray*}
4 \lambda \int_{]0,T[\times \Omega } \tilde\tau^3 |v|^2 + \tilde\tau | \nabla \psi \cdot \nabla v
|^2\,dt\,dx 
&\leq& 2  \langle Q_\tau ^sv,Q_\tau ^a v\rangle\\ 
&&+ 2  \int_{[0,T]\times \partial\Omega } \tilde\tau \partial_\nu \psi  
|\partial_\nu \theta |^2\, d\sigma  + R  \, . 
\end{eqnarray*}
Substituting this in (\ref{Q}) we obtain
\begin{eqnarray*}
4 \lambda \int_{]0,T[\times \Omega } \tilde\tau^3 |v|^2 
+ \tilde\tau | \nabla \psi \cdot \nabla v|^2
\,dt\,dx + \| Q_\tau ^s v\|^2 +\| Q_\tau ^a v\|^2  \leq \|Q_\tau v\|^2 \\
+  2 \int_{[0,T]\times \partial\Omega }  \tilde\tau \partial_\nu \psi 
|\partial_\nu \theta |^2\, 
d\sigma + R \, .
\end{eqnarray*}
In the first term on the left we already control the appropriate
weighted $L^2$ norm of $v$.  The corresponding norm of $\partial_t v$ is easily
obtained from the second and the fourth term, while the weighted $L^2$
norms of $\nabla  v$, respectively $\nabla^2  v$ are obtained in an elliptic
fashion from the first and the third term to obtain
\[
C\lambda \|\tilde\tau ^{-\frac 12}v\|^2_{_{2,\tilde\tau }}  \le 
 \| Q_\tau v\|^2+2
\int_{[0,T]\times \partial\Omega }  \tilde\tau \partial_\nu \psi |\partial_\nu \theta |^2\,  d\sigma +R\, .
\] 
Now the lower order terms in $R$ are negligible (i.e. much smaller than the
left hand side) for sufficiently large
$\lambda$, $\tau$ and we obtain (\ref{goalh}).\hfill$\diamondsuit$ 

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\end{thebibliography}

\noindent {\sc Paolo Albano} \\
Dipartimento di Matematica, Universit\`a di Roma ``Tor Vergata"\\
Via della Ricerca Scientifica, 00133 Roma, Italy\\
e-mail: albano@axp.mat.uniroma2.it  


\end{document}