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\title[Stability estimates in inverse problem]{Stability estimates in inverse problem
 \\
 for potentials in system of two \\ Schr\"{o}dinger equations with \\
Dirichlet and Neumann conditions}


\author{Atef Saci}

\address{Atef Saci,
\newline\hphantom{iii} Department of Mathematics,
\newline\hphantom{iii} University Center of Barika, Algeria,
\newline\hphantom{iii} Laboratory of Science for mathematics,
\newline\hphantom{iii}  computer science and
engineering applications}
\email{atef.saci@cu-barika.dz}


\thanks{\sc A. Saci, Stability estimates in inverse problem
 for potentials in system of two  Schr\"{o}dinger equations with
Dirichlet and Neumann conditions}
\thanks{\copyright \ Saci A. 2026}
\thanks{\it Submitted January 27, 2025.}

\maketitle {\small
\begin{quote}
\noindent{\bf Abstract.} We study the inverse problem on identifying two unknown potential coefficients in a system of coupled Schr\"{o}dinger equations in a
bounded domain in $\mathds{R}^{n}$  subject to non--homogeneous Dirichlet and Neumann boundary conditions by
using Neumann and Dirichlet boundary measurements. Under specific convexity
assumptions on the geometry of  domain and minimal regularity conditions
on the data, we establish the Lipschitz stability of this inverse problem by
employing uniqueness results as well as the observability inequality.

\medskip

\noindent{\bf Keywords:} inverse problems, uniqueness, stability, coupled Schr\"{o}dinger equations, observability inequality.

\medskip
\noindent{\bf Mathematics Subject Classification:}  35R30, 35L10, 35Q40, 49K20

\end{quote}
}

\vskip30pt

\section{Introduction}

Inverse problems for partial differential equations (PDEs) play a central
role in many scientific and engineering applications. Unlike direct
problems, where the solution of a PDE is determined from known inputs such
as boundary conditions, coefficients, and source terms, inverse problems aim
to deduce these unknown inputs from observed data. These problems are
inherently more complex, as they are often ill-posed in the sense of
Hadamard, meaning that solutions may not exist, may not be unique, or may
not depend continuously on the data. Solving inverse problems typically
involves developing robust mathematical models, regularization techniques to
deal with ill-posedness, and computational algorithms for implementation.
Analytical approaches such as uniqueness theorems and stability estimates
are also crucial for understanding the theoretical aspects of inverse
problems. Historically, the Carleman estimation method was introduced in the
field of inverse problems by Bukhgeim and Klibanov, who first presented it
in   \cite{b5}. This method is referred to as the Bukhgeim~---~Klibanov
method. The applications of such inequalities are numerous. The major use of
these inequalities is in demonstrating uniqueness results and stability
inequalities in inverse problems. In   \cite{b26} and \cite{b25}, local
Carleman inequalities and their roles in uniqueness results and H\"{o}lder-
type stability estimates were discussed. They were also used in \cite{8}
for the numerical resolution of a class of inverse problems on
determining the coefficients of certain partial differential equations from
boundary measurements of the solution or a function of the solution. Our
work   follows the same approach as that   in   \cite{b5}.

There is a large literature dealing with inverse problems for partial
differential equations, see for example \cite{3}, \cite{4}, \cite{7}, \cite{8}, \cite{16} and the references therein. Inverse problems for
the Schr\"{o}dinger equation were studied, for instance, in \cite{14}, \cite{1}, \cite{12}, \cite{2}, \cite{15}, \cite{6}, \cite{18}. In \cite{14}, the authors employed Carleman inequalities, which are
powerful tools in the analysis of partial differential equations, with
specific focus on degenerate weights.



The main objective of the study is to
establish stability estimates for the inverse problem of recovering
potential coefficients in the Schr\"{o}dinger equation from partial boundary
measurements. In \cite{1}, the authors studied    the inverse problem related to
the Schr\"{o}dinger equation aimed on
determining unknown potential functions in the Schr\"{o}dinger equation
based on observed measurements. The authors explored the conditions, under
which this inverse problem has a unique solution, as well as the stability of these
solutions  under data perturbations. The paper also provides
theoretical results concerning the well-posedness of the problem, offering
insights into both uniqueness and stability. In \cite{2},
an inverse problem for the magnetic Schr\"{o}dinger equation was treated. The authors
focused on estimating the stability of the problem, specifically addressing
how the magnetic potential can be determined from the Dirichlet--to--Neumann
map. The Dirichlet--to--Neumann map is a mathematical object that connects the
values of a solution to a partial differential equation on the boundary of a
domain with the values of its normal derivative on the boundary. The main
contribution of the paper is a  stability estimate, which
quantifies the extent to which the solution to the inverse problem can be
influenced by small errors or perturbations in the data. The results in the
paper have significant implications for understanding the uniqueness and
stability of inverse problems in quantum mechanics, especially in the
context of magnetic fields.   Nakamura, Sun, and Uhlmann \cite{15}
studied on the global identifiability of an inverse problem for the
Schr\"{o}dinger equation in the presence of a magnetic field. The inverse
problem refers to determining unknown parameters (such as the magnetic
potential) of a system by observing its behavior. In this work, the authors
explored the conditions, under which these parameters can be uniquely
identified from the solution of the Schr\"{o}dinger equation. They provided
mathematical results related to the global identifiability of these
parameters, contributing to the understanding of inverse problems in quantum
mechanics. The study is significant for applications in areas like quantum
physics and medical imaging. Triggiani and Zhang \cite{18}
investigated the inverse problem on determining the electric potential
coefficient in Schr\"{o}dinger equations on Riemannian manifolds. The focused
on the global uniqueness and stability of the solution to this inverse
problem. The authors provided mathematical proofs and established conditions,
under which the potential coefficient can be uniquely determined from
boundary measurements.   Dou and Yamamoto \cite{6}
addressed an inverse problem related to coupled Schr\"{o}dinger equations.
The focus was on the stability of the inverse problem, specifically
logarithmic stability. This means the authors investigated how small changes
in the data can affect the accuracy of the solution to the inverse problem,
which is crucial for practical applications, where data may not be perfectly
accurate. The study provides mathematical analysis and results that help to
understand the behavior and limitations of solutions to this class of
inverse problems. In \cite{bp46}, the authors employed Carleman estimates
for the Schr\"{o}dinger equations as established by Lasiecka et al. \cite{11},
along with the associated observability inequality, to demonstrate
uniqueness and Lipschitz stability for the inverse problem of recovering the
unknown potential coefficient of the Schr\"{o}dinger equation in a bounded
domain of $\mathds{R}^{n}$. In \cite{12} the inverse problem of recovering unknown potential parameters for a more general coupled system of Schr\"odinger equations with magnetic potential terms was studied with  Neumann boundary conditions. A uniqueness result was established by using Carleman inequalities.

The system we study in this work is a particular case of the general model in \cite{12}, but with mixed Dirichlet and Neumann boundary conditions. To the best of our knowledge, the stability of this inverse problem has not been studied in the literature. The aim of the present paper is to fill this gap. More specifically, we  prove a stability inequality for the simultaneous determination of two unknown potential coefficients in a coupled system of two Schr\"odinger equations with non--homogeneous Dirichlet and Neumann boundary data from observations of the Neumann and Dirichlet traces.


\section{Problem statement}

Let $T>0$ and let $\Omega \subset \mathds{R}^{n}$, $n=2,3$ be an open bounded domain with boundary of class $C^{2}$. Throughout this paper, we use the notation
\begin{equation*}
\Gamma= \partial\Omega = \overline{\Gamma_0 \cup \Gamma_1},\quad \Gamma_0 \cap \Gamma_1 = \emptyset,\qquad \nabla \upsilon = \left(\frac{\partial \upsilon}{\partial x_1},\dots,\frac{\partial \upsilon}{\partial x_n}\right),\qquad
\Delta \upsilon = \sum\limits_{i=1}^{n}\frac{\partial^{2}\upsilon}{\partial x_i^{2}},
\end{equation*}
by  $nu= (\nu_1,\dots,\nu_n)$ we denote the unit outward normal vector to $\Gamma = \partial\Omega,$ and
\begin{equation*}
\frac{\partial \upsilon}{\partial \nu}=\nabla \upsilon \cdot \nu
\end{equation*}
is the normal derivative.
Following \cite{10}, \cite{11}, \cite{17}, \cite{12}, we make the following assumptions.

\begin{assumption}\label{as2.1}%[H]
There exists a non--negative function $d:\overline{\Omega}\to\mathds{R}_+$ of class $C^{3}$ such that
\begin{enumerate}

\vskip5pt

\item[(i)] The normal derivative of $d$ is {red}{non--negative} on $\Gamma_0$,
 %i.e., if we denote $\nabla d$ by $h$, then
\begin{equation}
\nu(x)\nabla d(x){\geqslant} 0 \quad \text{on}\quad \Gamma_0. \label{11}
\end{equation}

\vskip5pt

\item[(ii)] The function $d$ is strictly convex in $\overline{\Omega},$ namely, there exists $\rho>0$ such that for all $x\in\overline{\Omega}$ and all $\xi\in\mathds{R}^n$,
\begin{equation}
\mathcal{H}_d(x)\xi\cdot\xi \ge \rho \vert\xi\vert^{2}, \label{12}
\end{equation}
where $\mathcal{H}_d(x)$ denotes the Hessian matrix of $d(x)$.

\vskip5pt

\item[(iii)] The function $d$ has no critical point on $\overline{\Omega}$
\begin{equation}
\inf_{x\in\overline{\Omega}} \Vert\nabla d(x)\Vert = s>0.\label{13}
\end{equation}
\end{enumerate}
\end{assumption}

\vskip5pt

Choosing the strictly convex potential function $d(x)$ satisfying Assumption~\ref{as2.1} and $d(x)\ge d_0>0$, we next introduce the pseudo--convex function $\varphi(x,t)$
\begin{equation}
\varphi(x,t) = d(x) - c\left(t-\frac{T}{2}\right)^{2}, \qquad x\in\Omega,\quad t\in[0,T], \label{31}
\end{equation}
where $T>0$ is arbitrary.

We  study the  system of two Schr\"odinger equations for  the unknowns $w=w(x,t)$, $z=z(x,t)$:
\begin{align}
& iw_t + \Delta w = n(x)w + q(x)z\hphantom{.,t)} \quad \text{in}\quad Q, \label{14a} \\
& iz_t + \Delta z = m(x)z + p(x)w\hphantom{;,t)} \quad \text{in}\quad Q, \label{14b} \\
& w\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) = w_0(x)\hphantom{z+ p(x)w .ht)}\quad \text{in}\quad \Omega, \label{14c} \\
& z\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) = z_0(x)  \hphantom{)z+ p(x)w Jlt)}\quad   \text{in} \quad\Omega, \label{14d} \\
& w = g_1(x,t),\qquad \frac{\partial z}{\partial\nu}=g_2(x,t) \quad \text{on}\quad \Sigma, \label{14e}
\end{align}
where $\Sigma = \Gamma\times[0,T]$, $Q = \Omega\times[0,T]$.
Here $w_0(x)$, $z_0(x)$ are the given initial conditions and $g_1(x,t)$, $g_2(x,t)$ are  given Dirichlet and Neumann boundary conditions. Instead, the potentials $q(x)$, $p(x)$ are time--independent unknown coefficients.


This paper treats two kinds of inverse problems. We first note that the map $\{q,p\}|_{\Omega} \to \{\frac{\partial w}{\partial\nu}, z\}|_{\Gamma_1\times[0,T]}$ is nonlinear and hence we consider the following nonlinear inverse problem.

\vskip10pt

\textsl{1. Nonlinear inverse problem.}
Let $\{w=w(q,p),\; z=z(q,p)\}$ be a solution to system \eqref{14a}--\eqref{14e}.  We are interesting in following two  problems.

\textsl{Uniqueness in the nonlinear inverse problem.}
Under Assumption~\ref{as2.1}, does the implication holds?
\begin{equation}
\left\{
\begin{aligned}
&\frac{\partial w(q_1,p_1)}{\partial\nu}\Big|_{\Gamma_1\times[0,T]} = \frac{\partial w(q_2,p_2)}{\partial\nu}\Big|_{\Gamma_1\times[0,T]}\\
&z(q_1,p_1)\big|_{\Gamma_1\times[0,T]} = z(q_2,p_2)\big|_{\Gamma_1\times[0,T]}
\end{aligned}
\right\}
\implies
\left\{
\begin{aligned}
&q_1(x)=q_2(x)\\
&p_1(x)=p_2(x)
\end{aligned}
\right\}\quad
\text{on}\quad \Omega? \label{16}
\end{equation}

\textsl{Stability in the nonlinear inverse problem.}
Under Assumption~\ref{as2.1}, is it possible to estimate $q_1-q_2|_{\Omega}$ and $p_1-p_2|_{\Omega}$ by
\begin{equation*}
\Bigl(\frac{\partial w(q_1,p_1)}{\partial\nu} - \frac{\partial w(q_2,p_2)}{\partial\nu}\Bigr)\Big|_{\Gamma_1\times[0,T]}
\quad\text{and}\quad
\bigl(z(q_1,p_1)-z(q_2,p_2)\bigr)\big|_{\Gamma_1\times[0,T]}
\end{equation*}
in suitable norms?

\vskip10pt

As usual, the nonlinear inverse problem is converted into a linear inverse problem for an auxiliary, corresponding problem. Let
\begin{align}
&f(x) = q_1(x)-q_2(x), \qquad\hphantom{g(x} g(x)=p_1(x)-p_2(x), \nonumber \\
&R_1(x,t) = z(q_2,p_2)(x,t), \qquad R_2(x,t) = w(q_2,p_2)(x,t), \nonumber \\
&u(x,t) = w(q_1,p_1)(x,t)-w(q_2,p_2)(x,t), \nonumber
 \\
&\upsilon(x,t)= z(q_1,p_1)(x,t)-z(q_2,p_2)(x,t). \label{17}
\end{align}
Subtracting system \eqref{14a}--\eqref{14e} with coefficients $(q_1,p_1)$ from the same system with coefficients $(q_2,p_2)$, we obtain
\begin{align}
& iu_t + \Delta u = n(x)u + q(x)\upsilon + f(x)R_1(x,t) \qquad \text{in}\quad Q, \label{18a} \\
& i\upsilon_t + \Delta\upsilon = m(x)\upsilon + p(x)u + g(x)R_2(x,t) \qquad \text{in}\quad Q, \label{18b} \\
& u\bigl(\,\cdot\,,\tfrac{T}{2}\bigr)=0 \qquad \text{in}\quad \Omega, \label{18c} \\
& \upsilon\bigl(\,\cdot\,,\tfrac{T}{2}\bigr)=0 \qquad \text{in}\quad \Omega, \label{18d} \\
& u=0,\quad \frac{\partial\upsilon}{\partial\nu}=0 \qquad \text{on}\quad \Sigma. \label{18e}
\end{align}
 Here $q$ in \eqref{18a} is actually $q_1$ in the notation of \eqref{17}, while $p$ in \eqref{18b} is actually $p_1$ in \eqref{17}.  The terms $f,g$ are unknown time--independent coefficients.  The $\{u,\upsilon\}$-problem has the advantage over the original $\{w,z\}$-problem in \eqref{14a}--\eqref{14e} that the map $\{f,g\}|_{\Omega} \to \{\frac{\partial u}{\partial\nu}, \upsilon\}|_{\Gamma_1\times[0,T]}$ is linear.

\vskip10pt



\textsl{2. Linear inverse problem.}  We are interesting in following two  problems.
%Similarly in problem \eqref{18a}--\eqref{18e}, we will more precisely answer the %following questions.

\vskip5pt

\textsl{Uniqueness in the linear inverse problem}
Let $\{u=u(f,g),\; \upsilon=\upsilon(f,g)\}$ be the solution to system \eqref{18a}--\eqref{18e}. Under Assumption~\ref{as2.1}, does the implication holds
\begin{equation}
\left\{
\begin{aligned}
&\frac{\partial u(f,g)}{\partial\nu}\Big|_{\Gamma_1\times[0,T]} = 0\\
&\upsilon(f,g)\big|_{\Gamma_1\times[0,T]} = 0
\end{aligned}
\right\}
\implies
\left\{
\begin{aligned}
&f(x)=0\\
&g(x)=0
\end{aligned}
\right\}\quad
\text{on}\quad \Omega ? \label{110}
\end{equation}

\vskip5pt

\textsl{Stability in the linear inverse problem.}
Under Assumption~\ref{as2.1}, is it possible to estimate $f|_{\Omega}$ and $g|_{\Omega}$ by $\frac{\partial u(f,g)}{\partial\nu}\big|_{\Gamma_1\times[0,T]}$ and $\upsilon(f,g)\big|_{\Gamma_1\times[0,T]}$ in suitable norms?

\vskip10pt

The goal of the present paper is to give an affirmative and quantitative answer to the above stability questions for the linear and nonlinear inverse problems.
\section{Important previous results}

In this section we  present the most important theorems that play a key
role in proving our main results.

\subsection{Uniqueness in  linear case}

\begin{theorem}\label{th21}
Under Assumption~\ref{as2.1} let
\begin{equation}
T>0 \label{21}
\end{equation}
and in  $\{u,\upsilon\}$-system \eqref{18a}--\eqref{18e} we suppose
\begin{align}
&
\begin{aligned}
& n,\,m\in W^{1,\infty}(\Omega),\quad q,\,p\in W^{1,\infty}(\Omega),\quad f,\,g\in W^{1,\infty}(\Omega),\qquad f|_{\Gamma}=0,\\
& R_i,\,R_{it},\,R_{itt}\in C\bigl([0,T],H^{1}(\Omega)\bigr),\\
& R_{ix_j}(x,\tfrac{T}{2}),\, R_{ix_jx_k}(x,\tfrac{T}{2}),\, R_{ix_jx_kx_l}\big(x,\tfrac{T}{2}\big)\in L^{\infty}(\Omega),\qquad i=1,2,\quad 1\le j,k,l\le n,
\end{aligned}
\label{22a}
\\
&
  fR_1,\,fR_{1t},\,fR_{1tt}\in C\bigl([0,T],H_0^{1}(\Omega)\bigr),\qquad
 gR_2,\,gR_{2t},\,gR_{2tt}\in C\bigl([0,T],H^{1}(\Omega)\bigr),
\label{22b}
\\
&
|R_1(x,\tfrac{T}{2})|\ge r_1>0,\qquad |R_2(x,\tfrac{T}{2})|\ge r_2>0,\qquad x\in\overline{\Omega}, \label{23}
\end{align}
with some positive constants $r_1$, $r_2$. Assume that the solution $\{u=u(f,g),\upsilon=\upsilon(f,g)\}$ to problem \eqref{18a}--\eqref{18e} possesses the smoothness
\begin{equation}
u,u_t,u_{tt}\in C\bigl([0,T],H_0^{1}(\Omega)\bigr),\quad
\upsilon,\upsilon_t,\upsilon_{tt}\in C\bigl([0,T],H^{1}(\Omega)\bigr).\label{24}
\end{equation}
If
\begin{equation}
 \frac{\partial u(f,g)}{\partial\nu}(x,t)=0,\qquad
\upsilon(f,g)(x,t)=0,\qquad (x,t)\in\Gamma_1\times[0,T], \label{25}
\end{equation}
then
\begin{equation}
f(x)=0,\qquad
g(x)=0,\qquad x\in\Omega. \label{26}
\end{equation}
\end{theorem}

\begin{proof}
The methodology for proving the uniqueness result \eqref{25}, \eqref{26} is analogous to that employed in \cite[Sect. 4]{12}, where  a more general coupled Schr\"odinger system
\begin{align*}
iu_t + \Delta u &= a(x)\cdot\nabla u + n(x)u + \beta(x)\cdot\nabla\upsilon + q(x)\upsilon + f(x)R_1,\\
i\upsilon_t + \Delta\upsilon &= b(x)\cdot\nabla\upsilon + m(x)\upsilon + \gamma(x)\cdot\nabla u + p(x)u + g(x)R_2,
\end{align*}
was considered
with Neumann boundary conditions for both equations. Despite the presence of additional first--order terms and the difference in boundary conditions (our system \eqref{18a}--\eqref{18e} involves a Dirichlet condition for $u$ and a Neumann condition for $\upsilon$ on $\Sigma$), the same essential steps  in \cite[Sect. 4]{12} can be adapted to our setting. Consequently, the uniqueness result follows exactly as in \cite{12}.  The proof is complete.
%To avoid redundancy, we omit the detailed calculations here.
\end{proof}

\vskip10pt

\subsection{Uniqueness in the nonlinear case}

\begin{theorem}\label{th22}
Let  Assumption~\ref{as2.1} holds. Consider problem \eqref{14a}--\eqref{14e} on $[0,T]$, with $T$ as in \eqref{21} and with potential coefficients $q_1,p_1\in W^{1,\infty}(\Omega)$, and   potential coefficients $q_2,p_2\in W^{1,\infty}(\Omega)$, and suppose that
\begin{align}
&n,m\in W^{1,\infty}(\Omega),\quad q_1,p_1,q_2,p_2\in W^{1,\infty}(\Omega),\quad \{q_1,q_2\}|_{\Gamma}=0, \label{27}\\
&w,w_t,w_{tt}\in C\bigl([0,T],H_0^{1}(\Omega)\bigr),\quad
z,z_t,z_{tt}\in C\bigl([0,T],H^{1}(\Omega)\bigr), \label{28a}\\
&w_{0x_j},\,w_{0x_jx_k},\,w_{0x_jx_kx_l},\,
z_{0x_j},\,z_{0x_jx_k},\,z_{0x_jx_kx_l}\in L^{\infty}(\Omega),\quad 1\le j,k,l\le n. \label{28b}
\\
&|w_0(x)|\ge w_0>0,\qquad |z_0(x)|\ge z_0>0,\quad x\in\overline{\Omega}. \label{29}
\end{align}
Then
\begin{equation}
%\left\{
%\begin{aligned}
%&
\frac{\partial w(q_1,p_1)}{\partial\nu}%\bigg|_{\Gamma_1\times[0,T]}
= \frac{\partial w(q_2,p_2)}{\partial\nu},%\bigg|_{\Gamma_1\times[0,T]},
\qquad
%&
z(q_1,p_1)%(x,t)
= z(q_2,p_2)
%(x,t),
%\end{aligned}
%\right.
\quad\text{on}\quad\Gamma_1\times[0,T], \label{210}
\end{equation}
implies
\begin{equation}
q_1(x)=q_2(x),\qquad p_1(x)=p_2(x),\quad x\in\Omega. \label{211}
\end{equation}
\end{theorem}

\begin{proof}
The result is a direct consequence of Theorem \ref{th21}. Define \(f,g,R_1,R_2,u,\upsilon\) as in \eqref{17}.
By \eqref{27}--\eqref{28b} we obtain that \(n,m,q,p\) (with \(f=q_1-q_2\), \(g=p_1-p_2\)) belong to \(W^{1,\infty}(\Omega)\), that \(f,g\in W^{1,\infty}_0(\Omega)\), and that \(R_1,R_2\) satisfy  required regularity  \eqref{22a}, \eqref{22b}. Condition \eqref{29} implies \eqref{23} with \(r_1=z_0\), \(r_2=w_0\). The regularity assumptions on \(w,z\) given in \eqref{28a} guarantee \eqref{24} for \(u,\upsilon\).
Moreover, condition \eqref{210} becomes exactly \eqref{25}.
 Theorem \ref{th21} yields \(f\equiv0\) and \(g\equiv0\) in \(\Omega\), that is,
\(q_1\equiv q_2\) and \(p_1\equiv p_2\).  The proof is complete.
\end{proof}

\vskip10pt

\subsection{Observability inequality}

Our keys for the proofs of Theorem \ref{th23} are an observability
inequality and a compactness--uniqueness argument. In this section, we state
an observability inequality for a system of   Schr\"odinger
equations with homogeneous Dirichlet and Neumann boundary conditions
\begin{align}
&iu_t(x,t) + \Delta u(x,t) = n(x) u(x,t) + q(x) \upsilon(x,t) \qquad \text{in}\quad Q, \label{61a} \\
&i\upsilon_t(x,t) + \Delta \upsilon(x,t) = m(x) \upsilon(x,t) + p(x) u(x,t) \quad  \hphantom{w} \text{in}\quad Q, \label{61b} \\
&u\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) = u_0(x) \hphantom{at) = n(x) u(x,t) + q(x) v(x,t)}\qquad \text{in}\quad \Omega, \label{61c}
\\
&\upsilon\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) = \upsilon_0(x) \hphantom{at) = n(x) u(x,t) + q(x) v(x,t)}\qquad \text{in}\quad \Omega, \label{61d} \\
&u = 0,\quad \frac{\partial \upsilon}{\partial \nu} = 0 \hphantom{it) = n(x) u(x,t) + q(x) v(x,t)}\qquad \text{on}\quad \Sigma,\label{61e}
\end{align}
where
\begin{equation}
\{u_0,\upsilon_0\} \in H_0^1(\Omega) \times H^1(\Omega), \qquad  n,m \in W^{1,\infty}(\Omega),\qquad q,p \in W^{1,\infty}(\Omega). \label{62}
\end{equation}
Then
\begin{equation}
\{u,\upsilon\} \in C\bigl([0,T], H_0^1(\Omega) \times H^1(\Omega)\bigr). \label{63}
\end{equation}

The next theorem states the observability inequality.

\begin{theorem}\label{th61}
Let Assumption~\ref{as2.1} hold as well as \eqref{62}, and $T>0$. Let $\{u,\upsilon\}$ be solutions of problem \eqref{61a}--\eqref{61e}. Then there exists a constant $C=C(\Omega,T,q,p)>0$ such that
\begin{equation}
C_T E(0) \le \int\limits_0^T \int\limits_{\Gamma_1} \left( \Bigl|\frac{\partial u}{\partial\nu}\Bigr|^2 + |\upsilon|^2 + |\upsilon_t|^2 \right) d\Gamma_1 dt, \label{64}
\end{equation}
where
\begin{equation}
E(t) = E_u(t) + E_\upsilon(t), \label{65}
\end{equation}
and
\begin{equation*}
E_w(t) = \int\limits_\Omega \bigl( |\nabla w(x,t)|^2 + |w(x,t)|^2 \bigr) d\Omega.
\end{equation*}
\end{theorem}

\begin{proof}
The observability result \eqref{64} is the same as the observability result in  \cite[Thm. 1.4, Case 3]{17},  where  a Carleman estimate  different from the Carleman estimate in \cite[Thm. 3.1]{12} was employed. However,  the approach from \cite[Sect. 3]{17} still works and allows one  to demonstrate estimate \eqref{64} by using Carleman estimate \cite[Thm. 3.1]{12}. The proof is compplete.
\end{proof}

\vskip20pt

\section{Main Results}

Our main results are the following theorems.

\begin{theorem}[Stability of the Linear Inverse Problem]\label{th23} Consider problem \eqref{18a}--\eqref{18e} and we suppose that Assumption~\ref{as2.1}, \eqref{21}, \eqref{22a}, \eqref{22b}, and \eqref{23} hold. Then there exists a constant
\begin{equation*}
C = C(\Omega,T,\Gamma_1,\varphi,q,p,R_1,R_2) > 0,
\end{equation*}
 depending on the data of problem \eqref{18a}--\eqref{18e}, but not on the unknown coefficients $f$ and $g$, such that
\begin{equation}
\Vert f\Vert_{H_0^1(\Omega)} + \Vert g\Vert_{H^1(\Omega)}
\le C \left(
\Bigl\Vert \frac{\partial u(f,g)}{\partial\nu} \Bigr\Vert_{H^1(0,T;L^2(\Gamma_1))}
+ \bigl\Vert \upsilon_t(f,g) \bigr\Vert_{H^1(0,T;L^2(\Gamma_1))}
\right) \label{212}
\end{equation}
for all $f,g\in W^{1,\infty}(\Omega)$.
\end{theorem}

\begin{theorem}[Stability of Nonlinear Inverse Problem]\label{th24}
Suppose   Assumption~\ref{as2.1}, \eqref{27}, \eqref{28a}, \eqref{28b}, and \eqref{29} hold. Let $\{w(q_1,p_1),z(q_1,p_1)\}$, $\{w(q_2,p_2),z(q_2,p_2)\}$ be respectively solutions of problem \eqref{14a}--\eqref{14e} on $[0,T]$, with $T$ as in \eqref{21} and with potential coefficients $q_1,p_1\in W^{1,\infty}(\Omega)$ and  $q_2,p_2\in W^{1,\infty}(\Omega).$ Then there exists a constant
\begin{equation*}
C = C(\Omega,T,\Gamma_1,\varphi,M,w_0,w_1,z_0,z_1,g_1,g_2) > 0,
\end{equation*}
such that
\begin{equation}
\begin{aligned}
\Vert q_1-q_2\Vert_{H_0^1(\Omega)} + \Vert p_1-p_2\Vert_{H^1(\Omega)} \le C \bigg\{&
\Bigl\Vert \frac{\partial w(q_1,p_1)}{\partial\nu} - \frac{\partial w(q_2,p_2)}{\partial\nu} \Bigr\Vert_{H^1(0,T;L^2(\Gamma_1))}
 \\
& + \Bigl\Vert z_t(q_1,p_1) - z_t(q_2,p_2) \Bigr\Vert_{H^1(0,T;L^2(\Gamma_1))} \bigg\}.
\end{aligned}\label{213}
\end{equation}
\end{theorem}

\vskip20pt

\section{Stability in the linear case}

In this section we prove Theorem~\ref{th23}. Before providing the proof, we establish an auxiliary statement.


\begin{lemma}\label{l71}
Consider the system
\begin{align}
&i(\tilde{u}_t)_t + \Delta(\tilde{u}_t) = n(x) \tilde{u}_t + q(x) \tilde{\upsilon}_t + f(x) R_{1t}(x,t) \hspace{25.6pt} \text{in}\quad Q, \label{755a} \\
&i(\tilde{\upsilon}_t)_t + \Delta(\tilde{\upsilon}_t) = m(x) \tilde{\upsilon}_t + p(x) \tilde{u}_t + g(x) R_{2t}(x,t) \qquad \text{in}\quad Q, \label{755b} \\
&\tilde{u}_t\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) = 0 \hspace{206.3pt} \text{in}\quad \Omega, \label{755c} \\
&\tilde{\upsilon}_t\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) = 0 \hspace{206.7pt} \text{in}\quad \Omega, \label{755d} \\
&\tilde{u}_t = 0,\quad \frac{\partial \tilde{\upsilon}_t}{\partial \nu} = 0 \hspace{181pt} \text{on}\quad \Sigma. \label{755e}
\end{align}
 with data
\begin{equation}
\begin{aligned}
& n,\,m\in W^{1,\infty}(\Omega),\qquad q,\,p\in W^{1,\infty}(\Omega),\qquad f,\,g\in W^{1,\infty}(\Omega),\qquad f|_{\Gamma}=0,\\
& R_{it},\,R_{itt}\in C\bigl([0,T],H^1(\Omega)\bigr),\qquad i=1,2.
\end{aligned} \label{78}
\end{equation}
Then the  operators
\begin{equation}
\begin{aligned}
& \bigl(K\{f,g\}\bigr)(x,t)=\frac{\partial\tilde{u}_t}{\partial\nu}\,:\,W^{1,\infty}(\Omega)\times W^{1,\infty}(\Omega)\longrightarrow L^2(\Gamma_1\times[0,T]),\\[2mm]
& \bigl(L\{f,g\}\bigr)(x,t)=\tilde{\upsilon}_t(x,t)\,:\,W^{1,\infty}(\Omega)\times W^{1,\infty}(\Omega)\longrightarrow L^2(\Gamma_1\times[0,T]),\\[2mm]
& \bigl(L_1\{f,g\}\bigr)(x,t)=\tilde{\upsilon}_{tt}(x,t)\,:\,W^{1,\infty}(\Omega)\times W^{1,\infty}(\Omega)\longrightarrow L^2(\Gamma_1\times[0,T])
\end{aligned} \label{79}
\end{equation}
are compact.
\end{lemma}

\begin{proof}
\textsl{Step (i).} By assumptions \eqref{78}  we have the following regularity for solution to    system \eqref{755a}--\eqref{755e}:
\begin{equation}
\tilde{u}_t\in C\bigl([0,T],H_0^1(\Omega)\bigr),\qquad
\tilde{\upsilon}_t\in C\bigl([0,T],H^1(\Omega)\bigr). \label{611}
\end{equation}
By differentiating in $t$ we
%After that, we derive the $\{\tilde{u}_t,\tilde{\upsilon}_t\}$-system %\eqref{755a}--\eqref{755e} in time, then we find the %$\{\tilde{u}_{tt},\tilde{\upsilon}_{tt}\}$-system
get
\begin{align}
& i(\tilde{u}_{tt})_t + \Delta(\tilde{u}_{tt}) = n(x)\tilde{u}_{tt} + q(x)\tilde{\upsilon}_{tt} + f(x)R_{1tt}(x,t) \hspace{25.2 true pt} \text{in}\quad Q, \label{612a}\\
& i(\tilde{\upsilon}_{tt})_t + \Delta(\tilde{\upsilon}_{tt}) = m(x)\tilde{\upsilon}_{tt} + p(x)\tilde{u}_{tt} + g(x)R_{2tt}(x,t) \qquad \text{in}\quad Q, \label{612b}\\
& \tilde{u}_{tt}\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) = -i f(x) R_{1t}\bigl(x,\tfrac{T}{2}\bigr) \in H_0^1(\Omega) \hspace{92.1 true pt} \text{in}\quad \Omega, \label{612c}\\
& \tilde{\upsilon}_{tt}\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) = -i g(x) R_{2t}\bigl(x,\tfrac{T}{2}\bigr) \in H^1(\Omega)  \hspace{93.5 true pt}  \text{in}\quad \Omega, \label{612d}\\
& \tilde{u}_{tt}=0,\quad \frac{\partial\tilde{\upsilon}_{tt}}{\partial\nu}=0 \hspace{190 true pt} \text{on}\quad \Sigma. \label{612e}
\end{align}
Under the  assumptions \eqref{78}, we obtain also the following regularity for  solution to     system \eqref{612a}--\eqref{612e}
\begin{equation}
\tilde{u}_{tt}\in C\bigl([0,T],H_0^1(\Omega)\bigr),\qquad
\tilde{\upsilon}_{tt}\in C\bigl([0,T],H^1(\Omega)\bigr). \label{613}
\end{equation}

\vskip5pt

\textsl{Step (ii).} Under conditions \eqref{78}, \eqref{611} problem \eqref{755a}--\eqref{755e}, for $\{\tilde{u}_t,\tilde{\upsilon}_t\}$  yields that the map
\begin{align*}
W^{1,\infty}(\Omega)\times W^{1,\infty}(\Omega) &\longrightarrow C\bigl([0,T],H^1(\Omega)\bigr)\qquad \{f,g\} \longmapsto \{\tilde{\upsilon}_t\},
\end{align*}
is continuous. Hence, by the trace theory, the map
\begin{align}
W^{1,\infty}(\Omega)\times W^{1,\infty}(\Omega)  \longrightarrow C\bigl([0,T],H^{1/2}(\Gamma)\bigr) \qquad
\Upsilon_1\{f,g\}  \longmapsto \Upsilon\{f,g\} = \bigl\{ \tilde{\upsilon}_t|_{\Sigma} \bigr\} \label{615}
\end{align}
is continuous. Similarly, under conditions \eqref{78}, \eqref{613}, problem  yields that the map
\begin{align*}
W^{1,\infty}(\Omega)\times W^{1,\infty}(\Omega)  \longrightarrow C\bigl([0,T],H^1(\Omega)\bigr)\qquad
\{f,g\}  \longmapsto \{\tilde{\upsilon}_{tt}\},
\end{align*}
continuous and by the trace theory, the map
\begin{align}
W^{1,\infty}(\Omega)\times W^{1,\infty}(\Omega)  \longrightarrow C\bigl([0,T],H^{1/2}(\Gamma)\bigr) \qquad
\Upsilon_2\{f,g\}  \longmapsto \Upsilon\{f,g\} = \bigl\{ \tilde{\upsilon}_{tt}|_{\Sigma} \bigr\}  \label{617}
\end{align}
is continuous. Thus, by \eqref{78} for $R_{it},R_{itt}$, $i=1,2$ and \eqref{22b}, it follows from \eqref{615} and \eqref{617} that the mappings
\begin{equation*}
f,g\in W^{1,\infty}(\Omega) \;\longmapsto\; \tilde{\upsilon}_t(f,g)|_{\Sigma} \in C\bigl([0,T],H^{1/2}(\Gamma)\bigr),\quad
\tilde{\upsilon}_{tt}(f,g)|_{\Sigma} \in C\bigl([0,T],H^{1/2}(\Gamma)\bigr)
\end{equation*}
are continuously, i.e.,
\begin{align}
&\bigl\|\tilde{\upsilon}_t(f,g)\bigr\|_{C([0,T],H^{1/2}(\Gamma))} \le C_{R_{1t},R_{2t}}\bigl( \|f\|_{H^1(\Omega)}+\|g\|_{H^1(\Omega)}\bigr), \label{618a}\\
&\bigl\|\tilde{\upsilon}_{tt}(f,g)\bigr\|_{C([0,T],H^{1/2}(\Gamma))} \le C_{R_{1t},R_{2t},R_{1tt},R_{2tt}}\bigl( \|f\|_{H^1(\Omega)}+\|g\|_{H^1(\Omega)}\bigr). \label{618b}
\end{align}

\vskip10pt

\textsl{Compactness of $L$, $L_1$.} By \eqref{618a}, \eqref{618b} we see that
\begin{align*}
& \bigl(L\{f,g\}\bigr)(x,t)=\tilde{\upsilon}_t(x,t):W^{1,\infty}(\Omega)\times W^{1,\infty}(\Omega)\longrightarrow L^2(\Gamma_1\times[0,T]),\\
& \bigl(L_1\{f,g\}\bigr)(x,t)=\tilde{\upsilon}_{tt}(x,t):W^{1,\infty}(\Omega)\times W^{1,\infty}(\Omega)\longrightarrow L^2(\Gamma_1\times[0,T])
\end{align*}
are compact operators because the embedding $H^{\frac{1}{2}}(\Gamma)\hookrightarrow L^2(\Gamma)$ is compact.

\vskip10pt

\textsl{Step (iii).} We apply the regularity property \cite[Lm. 2]{19} to \eqref{755a}--\eqref{755e} and \eqref{612a}--\eqref{612e} and we obtain
\begin{align}
\bigl\|\tilde{u}_t\bigr\|_{C(0,T;H_0^1(\Omega))} &\le C\bigl( \|f\|_{H_0^1(\Omega)}+\|g\|_{H^1(\Omega)}\bigr), \label{712a}\\
\bigl\|\tilde{\upsilon}_t\bigr\|_{C(0,T;H^1(\Omega))} &\le C\bigl( \|f\|_{H_0^1(\Omega)}+\|g\|_{H^1(\Omega)}\bigr), \label{712b}
\end{align}
and
\begin{align}
\bigl\|\tilde{u}_{tt}\bigr\|_{C(0,T;H_0^1(\Omega))} &\le C\bigl( \|f\|_{H_0^1(\Omega)}+\|g\|_{H^1(\Omega)}\bigr), \label{713a}\\
\bigl\|\tilde{\upsilon}_{tt}\bigr\|_{C(0,T;H^1(\Omega))} &\le C\bigl( \|f\|_{H_0^1(\Omega)}+\|g\|_{H^1(\Omega)}\bigr). \label{713b}
\end{align}
By \eqref{755a}, \eqref{755b} we find
\begin{equation}
\left\{
\begin{aligned}
\Delta \tilde{u}_t &= -i(\tilde{u}_t)_t + n(x)\tilde{u}_t + q(x)\tilde{\upsilon}_t + f(x)R_{1t}(x,t),\\
\Delta \tilde{\upsilon}_t &= -i(\tilde{\upsilon}_t)_t + m(x)\tilde{\upsilon}_t + p(x)\tilde{u}_t + g(x)R_{2t}(x,t),
\end{aligned}
\right. \label{714}
\end{equation}
so that, by  \eqref{712a}, \eqref{712b}, \eqref{713a},
\begin{equation}
\|\Delta\tilde{u}_t\|_{C(0,T;H^1(\Omega))} \le C\bigl( \|f\|_{H_0^1(\Omega)}+\|g\|_{H^1(\Omega)}\bigr), \label{715a}
\end{equation}
then
\begin{equation}
\|\Delta\tilde{u}_t\|_{C(0,T;L^2(\Omega))} \le C\bigl( \|f\|_{H_0^1(\Omega)}+\|g\|_{H^1(\Omega)}\bigr). \label{715c}
\end{equation}
On the other hand, we apply the regularity property \cite[Lm. 1]{19} to \eqref{755a}--\eqref{755e}
\begin{equation}
\|\tilde{u}_t\|_{C(0,T;L^2(\Omega))} \le C\bigl( \|f\|_{L^2(\Omega)}+\|g\|_{L^2(\Omega)}\bigr), \label{716a}
\end{equation}
hence
\begin{equation}
\|\tilde{u}_t\|_{C(0,T;L^2(\Omega))} \le C\bigl( \|f\|_{H_0^1(\Omega)}+\|g\|_{H^1(\Omega)}\bigr). \label{716c}
\end{equation}
Combining \eqref{715c} with \eqref{716c}, by  \cite[Ch. 5, Cor. 4]{24} we obtain
\begin{equation}
\|\tilde{u}_t\|_{C(0,T;H^2(\Omega))} \le C\bigl( \|f\|_{H_0^1(\Omega)}+\|g\|_{H^1(\Omega)}\bigr). \label{717a}
\end{equation}
Therefore, by the trace theorem,
\begin{equation}
\Bigl\|\frac{\partial\tilde{u}_t}{\partial\nu}\Bigr\|_{C(0,T;H^{1/2}(\Gamma))} \le C\bigl( \|f\|_{H_0^1(\Omega)}+\|g\|_{H^1(\Omega)}\bigr). \label{718a}
\end{equation}

\vskip10pt

\textsl{Compactness of $K$.} Similarly, by  \eqref{718a} we see that
\begin{equation*}
\bigl(K\{f,g\}\bigr)(x,t)=\frac{\partial\tilde{u}_t}{\partial\nu} : W^{1,\infty}(\Omega)\times W^{1,\infty}(\Omega) \longrightarrow L^2(\Gamma_1\times[0,T])
\end{equation*}
is a compact operator, because the embedding $H^{\frac{1}{2}}(\Gamma)\hookrightarrow L^2(\Gamma)$ is compact. The proof is complete.
\end{proof}

\vskip10pt

\begin{proof}[Proof of  Theorem~\ref{th23}]


\textit{Step 1.} We differentiate  system \eqref{18a}--\eqref{18e} in $t$
\begin{align}
&i(u_t)_t + \Delta(u_t) = n(x) u_t + q(x) \upsilon_t + f(x) R_{1t}(x,t) \qquad \text{in}\quad Q, \label{721a} \\
&i(\upsilon_t)_t + \Delta(\upsilon_t) = m(x) \upsilon_t + p(x) u_t + g(x) R_{2t}(x,t) \hspace{21.7 true pt} \text{in}\quad Q, \label{721b} \\
&u_t\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) = -i f(x) R_1\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) \in H_0^1(\Omega) \hspace{80.2 true pt} \text{in}\quad \Omega, \label{721c} \\
&\upsilon_t\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) = -i g(x) R_2\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) \in H^1(\Omega)  \hspace{81.7 true pt}  \text{in}\quad \Omega, \label{721d} \\
&u_t = 0,\quad \frac{\partial \upsilon_t}{\partial \nu} = 0  \hspace{179 true pt}  \text{on}\quad \Sigma. \label{721e}
\end{align}
We let
\begin{equation}
u_t = \bar{u}_t + \tilde{u}_t,\qquad \upsilon_t = \bar{\upsilon}_t + \tilde{\upsilon}_t. \label{73}
\end{equation}
Then, by the linearity of \eqref{721a}--\eqref{721e}, we have
\begin{align}
&i(\bar{u}_t)_t + \Delta(\bar{u}_t) = n(x) \bar{u}_t + q(x) \bar{\upsilon}_t \hspace{44.5pt} \text{in}\quad Q, \label{74a} \\
&i(\bar{\upsilon}_t)_t + \Delta(\bar{\upsilon}_t) = m(x) \bar{\upsilon}_t + p(x) \bar{u}_t \hspace{41.8pt}\text{in}\quad Q, \label{74b} \\
&\bar{u}_t\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) = -i f(x) R_1\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) \in H_0^1(\Omega) \hspace{22pt} \text{in}\quad \Omega, \label{74c} \\
&\bar{\upsilon}_t\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) = -i g(x) R_2\bigl(\,\cdot\,,\tfrac{T}{2}\bigr) \in H^1(\Omega) \qquad \text{in}\quad \Omega, \label{74d} \\
&\bar{u}_t = 0,\qquad \frac{\partial \bar{\upsilon}_t}{\partial \nu} = 0  \hspace{109pt}
\text{on}\quad \Sigma. \label{74e}
\end{align}
and system (\ref{755a})--(\ref{755e}).

\vskip10pt

\textit{Step 2.} In view of condition \eqref{62} of Theorem \ref{th61}, we apply the continuous observability inequality \eqref{64} to $\{\bar{u}_t,\bar{\upsilon}_t\}$-problem \eqref{74a}--\eqref{74e}
\begin{equation}\label{76}
\begin{aligned}
\bigl\Vert -i f(x) R_1(\,\cdot\,,\tfrac{T}{2}) \bigr\Vert_{H_0^1(\Omega)}^2 &+ \bigl\Vert -i g(x) R_2(\,\cdot\,,\tfrac{T}{2}) \bigr\Vert_{H^1(\Omega)}^2
\\
&\le C_{T,q,p}^2 \int\limits_0^T \int\limits_{\Gamma_1} \left( \Bigl|\frac{\partial \bar{u}_t}{\partial\nu}\Bigr|^2 + |\bar{\upsilon}_t|^2 + |\bar{\upsilon}_{tt}|^2 \right) d\Gamma_1 dt,
\end{aligned}
\end{equation}
Using \eqref{23} and \eqref{73}, we obtain
\begin{equation}
\begin{aligned}
\Vert f\Vert_{H_0^1(\Omega)} + \Vert g\Vert_{H^1(\Omega)}
\le& C \left(
\Bigl\Vert \frac{\partial \bar{u}_t}{\partial\nu} \Bigr\Vert_{L^2(\Gamma_1\times[0,T])}
+ \Vert \bar{\upsilon}_t \Vert_{L^2(\Gamma_1\times[0,T])}
+ \Vert \bar{\upsilon}_{tt} \Vert_{L^2(\Gamma_1\times[0,T])}
\right) \\
= &C \Bigg(
\Bigl\Vert \frac{\partial (u_t-\tilde{u}_t)}{\partial\nu} \Bigr\Vert_{L^2(\Gamma_1\times[0,T])}
+ \Vert \upsilon_t-\tilde{\upsilon}_t \Vert_{L^2(\Gamma_1\times[0,T])}
\\
&\hphantom{C \Bigg(}+ \Vert \upsilon_{tt}-\tilde{\upsilon}_{tt} \Vert_{L^2(\Gamma_1\times[0,T])}
\Bigg)
\\
\le &C \left(
\Bigl\Vert \frac{\partial u}{\partial\nu} \Bigr\Vert_{H^1(0,T;L^2(\Gamma_1))}
+ \Vert \upsilon_t \Vert_{H^1(0,T;L^2(\Gamma_1))}
\right)
\\
& + C \left(
\Bigl\Vert \frac{\partial \tilde{u}_t}{\partial\nu} \Bigr\Vert_{L^2(\Gamma_1\times[0,T])}
+ \Vert \tilde{\upsilon}_t \Vert_{L^2(\Gamma_1\times[0,T])}
+ \Vert \tilde{\upsilon}_{tt} \Vert_{L^2(\Gamma_1\times[0,T])}
\right).
\end{aligned}\label{77}
\end{equation}
Once we eliminate the term
\begin{equation*}
\Bigl\Vert \frac{\partial \tilde{u}_t}{\partial\nu} \Bigr\Vert_{L^2(\Gamma_1\times[0,T])}
+ \Vert \tilde{\upsilon}_t \Vert_{L^2(\Gamma_1\times[0,T])}
+ \Vert \tilde{\upsilon}_{tt} \Vert_{L^2(\Gamma_1\times[0,T])}
\end{equation*}
in \eqref{77}, we can complete the proof of Theorem \ref{th23}.

\vskip10pt

\textsl{Step 3.} We drop the term
\begin{equation*}
\Bigl\|\frac{\partial \tilde{u}_t}{\partial\nu}\Bigr\|_{L^2(\Gamma_1\times[0,T])}
+ \|\tilde{\upsilon}_t\|_{L^2(\Gamma_1\times[0,T])}
+ \|\tilde{\upsilon}_{tt}\|_{L^2(\Gamma_1\times[0,T])},
\end{equation*}
using a compactness--uniqueness argument. %We need the following lemma.

\vskip10pt

\textsl{Step 4.} We use Lemma~\ref{l71} to delete the terms
\begin{equation*}
\bigl\|K\{f,g\}\bigr\|_{L^2(\Sigma_1)} = \Bigl\|\frac{\partial\tilde{u}_t}{\partial\nu}\Bigr\|_{L^2(\Sigma_1)},\quad
\|L\{f,g\}\|_{L^2(\Sigma_1)} = \|\tilde{\upsilon}_t\|_{L^2(\Sigma_1)},\quad
\|L_1\{f,g\}\|_{L^2(\Sigma_1)} = \|\tilde{\upsilon}_{tt}\|_{L^2(\Sigma_1)},
\end{equation*}
in estimate \eqref{77}, via the compactness--uniqueness argument.

Assume that  inequality in \eqref{212} fails. Then there exist sequences $f_n\in H_0^1(\Omega)$, $g_n\in H^1(\Omega)$, $n\ge 1$, such that
\begin{align}
&\|f_n\|_{H_0^1(\Omega)} = \|g_n\|_{H^1(\Omega)} = 1, \qquad n\ge 1, \label{622a}
\\
&\lim_{n\to+\infty} \Bigl( \Bigl\|\frac{\partial u(f_n,g_n)}{\partial\nu}\Bigr\|_{H^1(0,T;L^2(\Gamma_1))} + \|\upsilon_t(f_n,g_n)\|_{H^1(0,T;L^2(\Gamma_1))} \Bigr) = 0. \label{622b}
\end{align}
By \eqref{622a}, there exist subsequences, still denoted by $f_n$ and $g_n$, such that
\begin{align}
&\{f_n\} \text{ converges weakly in } H_0^1(\Omega) \text{ to some } f_0\in H_0^1(\Omega), \label{623a}
\\
&\{g_n\} \text{ converges weakly in } H^1(\Omega) \text{ to some } g_0\in H^1(\Omega). \label{623b}
\end{align}
Then Lemma~\ref{l71} yields
\begin{align}
&\lim_{m,n\to+\infty} \bigl\|K\{f_n,g_n\}-K\{f_m,g_m\}\bigr\|_{L^2(\Sigma_1)} = 0, \label{624a}
\\
&
\begin{aligned}
\lim_{m,n\to+\infty} \bigl\|L\{f_n,g_n\}-L\{f_m,g_m\}\bigr\|_{L^2(\Sigma_1)}
&= \lim_{m,n\to+\infty} \bigl\|L_1\{f_n,g_n\}-L_1\{f_m,g_m\}\bigr\|_{L^2(\Sigma_1)}
\\
&= 0.
\end{aligned}
\label{624b}
\end{align}
It follows from \eqref{77} and \eqref{79} that
\begin{equation}\label{6250}
\begin{aligned}
\|f_n-f_m\|_{H_0^1(\Omega)} &+ \|g_n-g_m\|_{H^1(\Omega)}
\\
\le &C\Bigl( \Bigl\|\frac{\partial u(f_n,g_n)}{\partial\nu} - \frac{\partial u(f_m,g_m)}{\partial\nu}\Bigr\|_{H^1(0,T;L^2(\Gamma_1))}
\\
&\hphantom{\le C\Bigl( }+ \|\upsilon_t(f_n,g_n)-\upsilon_t(f_m,g_m)\|_{H^1(0,T;L^2(\Gamma_1))} \Bigr)
 \\
& + C\Bigl( \|K\{f_n,g_n\}-K\{f_m,g_m\}\|_{L^2(\Sigma_1)}
+ \|L\{f_n,g_n\}-L\{f_m,g_m\}\|_{L^2(\Sigma_1)}
\\
&\hphantom{ + C\Bigl(} + \|L_1\{f_n,g_n\}-L_1\{f_m,g_m\}\|_{L^2(\Sigma_1)} \Bigr) \\
\le& C\Bigl(\Bigl\|\frac{\partial u(f_n,g_n)}{\partial\nu}\Bigr\|_{H^1(0,T;L^2(\Gamma_1))}
+ \Bigl\|\frac{\partial u(f_m,g_m)}{\partial\nu}\Bigr\|_{H^1(0,T;L^2(\Gamma_1))}
\\
& \hphantom{C\Bigl(}+ \|\upsilon_t(f_n,g_n)\|_{H^1(0,T;L^2(\Gamma_1))}
+ \|\upsilon_t(f_m,g_m)\|_{H^1(0,T;L^2(\Gamma_1))} \Bigr)
\\
& + C\Bigl( \|K\{f_n,g_n\}-K\{f_m,g_m\}\|_{L^2(\Sigma_1)}
+ \|L\{f_n,g_n\}-L\{f_m,g_m\}\|_{L^2(\Sigma_1)}
\\
& \hphantom{+ C\Bigl(} + \|L_1\{f_n,g_n\}-L_1\{f_m,g_m\}\|_{L^2(\Sigma_1)} \Bigr).
\end{aligned}
\end{equation}
Then, by \eqref{622b} and \eqref{624a}, \eqref{624b}, we find
\begin{equation}
\lim_{n,m\to+\infty} \|f_n-f_m\|_{H_0^1(\Omega)} = 0,\qquad
\lim_{n,m\to+\infty} \|g_n-g_m\|_{H^1(\Omega)} = 0. \label{62700}
\end{equation}
By uniqueness of the limit and  \eqref{623a}, \eqref{623b}  we obtain
\begin{equation}
\lim_{n\to+\infty} \|f_n-f_0\|_{H_0^1(\Omega)} = 0,\qquad
\lim_{n\to+\infty} \|g_n-g_0\|_{H^1(\Omega)} = 0. \label{627}
\end{equation}
Thus, in view of \eqref{622a}, it follows from \eqref{627} that
\begin{equation}
\|f_0\|_{H_0^1(\Omega)} = \|g_0\|_{H^1(\Omega)} = 1. \label{628}
\end{equation}

For problem \eqref{18a}--\eqref{18e} under assumptions \eqref{22a} we see that the map
\begin{align}
W^{1,\infty}(\Omega)\times W^{1,\infty}(\Omega) \longrightarrow C\bigl([0,T],H^1(\Omega)\bigr),\qquad
\{f,g\} \longmapsto \{\upsilon(f,g)\} \label{442}
\end{align}
is continuous; by the trace theory, the map
\begin{align}
\Upsilon : W^{1,\infty}(\Omega)\times W^{1,\infty}(\Omega) &\longrightarrow C\bigl([0,T],H^{1/2}(\Gamma)\bigr), \qquad
\{f,g\} &\longmapsto \Upsilon\{f,g\} = \bigl\{\upsilon(f,g)|_{\Sigma}\bigr\} \label{443}
\end{align}
is continuous. The map
\begin{equation}
\{f,g\} \longmapsto \Bigl\{ \frac{\partial u(f,g)}{\partial\nu}\big|_{\Sigma} \Bigr\} \label{444}
\end{equation}
is also continuous, see \cite[Lm. 7]{1}.

Since for the $\{u,\upsilon\}$-problem \eqref{18a}--\eqref{18e} the map $\{f,g\}\mapsto \{u(f,g)|_{\Sigma},\upsilon(f,g)|_{\Sigma}\}$ is linear, it follows from \eqref{443} and \eqref{444} that
\begin{align}
&\Bigl\|\frac{\partial u(f_n,g_n)}{\partial\nu} - \frac{\partial u(f_0,g_0)}{\partial\nu}\Bigr\|_{C([0,T];L^2(\Gamma_1))}
\le C_{R_1,R_2}\bigl( \|f_n-f_0\|_{H_0^1(\Omega)} + \|g_n-g_0\|_{H^1(\Omega)}\bigr), \nonumber\\
&\|\upsilon(f_n,g_n)-\upsilon(f_0,g_0)\|_{C([0,T];L^2(\Gamma_1))}
\le C_{R_1,R_2}\bigl( \|f_n-f_0\|_{H_0^1(\Omega)} + \|g_n-g_0\|_{H^1(\Omega)}\bigr). \label{633}
\end{align}
Replacing \eqref{627} on the right‑hand side of \eqref{633}, we find
\begin{align}
&\lim_{n\to+\infty} \Bigl\|\frac{\partial u(f_n,g_n)}{\partial\nu} - \frac{\partial u(f_0,g_0)}{\partial\nu}\Bigr\|_{C([0,T];L^2(\Gamma_1))}= 0, \label{634a}\\
&\lim_{n\to+\infty} \bigl\|\upsilon(f_n,g_n)|_{\Sigma_1} - \upsilon(f_0,g_0)|_{\Sigma_1}\bigr\|_{C([0,T];L^2(\Gamma_1))}= 0. \label{634b}
\end{align}
Then, by virtue of \eqref{622b} together with \eqref{634a}, for $t\in[0,T]$ we obtain
\begin{equation}
\frac{\partial u(f_0,g_0)}{\partial\nu}(x,t)=0,\qquad (x,t)\in\Sigma_1. \label{634c}
\end{equation}
Similarly, for   $\{u_t,\upsilon_t\}$-problem \eqref{721a}--\eqref{721e},
\begin{equation}
\lim_{n\to+\infty} \bigl\|\upsilon_t(f_n,g_n)|_{\Sigma_1} - \upsilon_t(f_0,g_0)|_{\Sigma_1}\bigr\|_{C([0,T];L^2(\Gamma_1))}=0, \label{635}
\end{equation}
hence, by \eqref{622b} and \eqref{635},
\begin{equation}
\upsilon_t(f_0,g_0)|_{\Sigma_1}=0. \label{636}
\end{equation}
Returning to problem \eqref{18a}--\eqref{18e} with $f=f_n\in W^{1,\infty}(\Omega)$, $g=g_n\in W^{1,\infty}(\Omega)$, the initial conditions give
\begin{equation*}
\upsilon(f_n,g_n)\bigl(\,\cdot\,,\tfrac{T}{2}\bigr)=0,\quad x\in\overline{\Omega},
\end{equation*}
therefore
\begin{equation}
\upsilon(f_n,g_n)\bigl(\,\cdot\,,\tfrac{T}{2}\bigr)=0,\quad x\in\Gamma_1, \label{638}
\end{equation}
in the sense of trace in $H^{\frac{1}{2}}(\Gamma_1)$. Thus, in view of \eqref{638}, it follows from \eqref{634b} that
\begin{equation*}
\upsilon(f_0,g_0)\bigl(\,\cdot\,,\tfrac{T}{2}\bigr)=0,\quad x\in\Gamma_1,
\end{equation*}
and next, by \eqref{636},
\begin{equation}
\upsilon(f_0,g_0)=0,\qquad x\in\Gamma_1,\quad 0<t<T. \label{639}
\end{equation}
Finally, by virtue of \eqref{634c} and \eqref{639}, we apply Theorem \ref{th21}
\begin{equation}
f_0(x)=g_0(x)=0,\qquad x\in\Omega. \label{641}
\end{equation}
Then \eqref{641} contradicts \eqref{628}. This completes the proof.
\end{proof}


\section{Stability in  nonlinear case}

Here we prove  Theorem \ref{th24}, which is in fact a direct consequence of Theorem \ref{th23}. By \eqref{17}  we have
\begin{align}
&f(x) = q_1(x)-q_2(x), \qquad g(x) = p_1(x)-p_2(x), \notag\\
&R_1(x,t) = z(q_2,p_2)(x,t), \qquad R_2(x,t) = w(q_2,p_2)(x,t), \notag\\
&u(x,t) = w(q_1,p_1)(x,t) - w(q_2,p_2)(x,t), \notag\\
&\upsilon(x,t) = z(q_1,p_1)(x,t) - z(q_2,p_2)(x,t). \label{81}
\end{align}

Since
\begin{align*}
&q_1(x),q_2(x),p_1(x),p_2(x) \in W^{1,\infty}(\Omega),\qquad \{q_1,q_2\}|_{\Gamma}=0,
\\
& w(q_2,p_2)(x,t) \in C([0,T],H_0^1(\Omega)),\qquad z(q_2,p_2)(x,t) \in C([0,T],H^1(\Omega)),
\end{align*}
 it follows from \eqref{81} that
 \begin{equation*}
 f(\cdot)\in W^{1,\infty}(\Omega),\qquad f|_{\Gamma}=0,\qquad
g(\cdot)\in W^{1,\infty}(\Omega),\qquad
R_1(\cdot,\cdot),R_2(\cdot,\cdot)\in C([0,T],H^1(\Omega))
\end{equation*}
as assumed in Theorem~\ref{th23}.
Thus, we can apply Theorem \ref{th23} to the variables
$\{u = w(q_1,p_1)-w(q_2,p_2),$ $\upsilon = z(q_1,p_1)-z(q_2,p_2)\}$,
solve problem \eqref{18a}--\eqref{18e}, and we then obtain the desired stability estimate \eqref{213}, i.e.
\begin{align*}
 \|q_1-q_2\|_{H_0^1(\Omega)} + \|p_1-p_2\|_{H^1(\Omega)}  \le C \bigg(&
\Bigl\|\frac{\partial w(q_1,p_1)}{\partial\nu} - \frac{\partial w(q_2,p_2)}{\partial\nu}\Bigr\|_{H^1(0,T;L^2(\Gamma_1))}
\\
&+ \Bigl\|z_t(q_1,p_1) - z_t(q_2,p_2)\Bigr\|_{H^1(0,T;L^2(\Gamma_1))}
\bigg).
\end{align*}

\vskip30pt

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\end{document}
