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

\AtBeginDocument{{\noindent\small
Eighth Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
{\em Electronic Journal of Differential Equations},
Conf. 19 (2010),  pp. 37--44.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{37}
\title[\hfilneg EJDE-2010/Conf/19/\hfil Quasireversibility]
{Quasireversibility for inhomogeneous ill-posed problems
in Hilbert spaces}

\author[B. M. Campbell Hetrick \hfil EJDE/Conf/19 \hfilneg]
{Beth M. Campbell Hetrick}  % in alphabetical order

\address{Beth M. Campbell Hetrick \newline
Department of Mathematics, Gettysburg College, Gettysburg, PA
17325, USA}
\email{bcampbel@gettysburg.edu}

\thanks{Published September 25, 2010.}
\subjclass[2000]{47A52, 35R25, 35A35}
\keywords{Quasireversibility; ill-posed problems}

\begin{abstract}
 In a Hilbert space $\mathcal{H}$, the inhomogeneous ill-posed
 abstract Cauchy problem is given by
 $\frac{du}{dt} = Au(t) + h(t)$, $u(0) = \chi$,
 $0 \leq t < T$; where $A$ is a positive self-adjoint linear
 operator acting on $\mathcal{H}$, $\chi \in \mathcal{H}$,
 and $h: [0,T) \to \mathcal{H}$. Using semigroup theory,
 we obtain H\"older continuous dependence for the control
 problem generated by the method of quasireversibility.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

In a Hilbert space $\mathcal{H}$, we consider the problem
\begin{equation}
\begin{gathered}
 \frac{du}{dt} = Au(t) + h(t), \quad 0 \leq t < T, \\
 u(0) = \chi,
\end{gathered} \label{inhomCP_H}
\end{equation}
where $A$ is a positive self-adjoint linear operator,
$\chi \in \mathcal{H}$, and $h: [0,T) \to \mathcal{H}$.
Since $A$ is unbounded, the problem is ill-posed. Lattes and
Lions introduced the method of quasireversibility in \cite{LL}
in the 1960s as a way to generate approximate solutions to
ill-posed problems. As part of their technique, they perturb
the operator $A$ to construct an approximate problem.
We do the same, considering the approximate problem
\begin{equation}
\begin{gathered}
 \frac{dv}{dt} = f(A)v(t) + h(t), \\
 v(0) = \chi,
\end{gathered}\label{approx_inhomCP_H}
\end{equation}
where $f(\lambda)$ is a real-valued Borel function bounded above.
Since $f(A)$ is bounded above, the approximate problem
\eqref{approx_inhomCP_H} is well-posed with solution $v(t)$.
 Following work done by Ames and Hughes \cite{AH},
we have proved that the solution to the ill-posed problem,
if it exists, depends continuously on the solution to
the approximate problem. These results have been obtained
both in Hilbert space (including the nonlinear case)
\cite{Campbell_nonlin} and in Banach space \cite{Campbell}.

As mentioned above, Lattes and Lions perturb the operator to
define an approximate problem. However, they use the solution
$v(t)$ to this approximate problem to generate data $w(T)$
used to solve the problem
\begin{equation}
\begin{gathered}
 \frac{dw}{dt} = Aw(t), \quad 0 \leq t < T, \\
 w(T) = v(T).
\end{gathered} \label{fvp_homCP_H}
\end{equation}
As a final-value problem, this problem is well-posed.
Lattes and Lions show that
\[
\|u(0) - w(0) \| \leq \epsilon.
\]
Note that the method of Lattes and Lions does not give an
approximation for $u(t)$ where $t > 0$. In \cite{Miller},
Miller points out an additional concern: that the norm of
the operator $\mathrm{e}^{f(A)}$ is large for small $\epsilon$.
 Miller refines the method of quasireversibility, making additional
assumptions on $f$ in order to obtain a logarithmic convexity
result for the difference of the solutions $u(t)$ and $w(t)$.
He calls this approach a stabilized quasireversibility method.
We show here that we are able to obtain the same results as Miller
using our assumptions. In particular, we use the solution
to \eqref{approx_inhomCP_H} to create the final-value problem
\begin{equation}
\begin{gathered}
 \frac{dw}{dt} = Aw(t) + h(t), \quad 0 \leq t < T, \\
 w(T) = v(T).
\end{gathered} \label{fvp_inhomCP_H}
\end{equation}
Under the appropriate stabilizing conditions, we show that there
exist computable constants $C$ and $M$, independent of
$0 < \beta < 1$, such that
\[
\| u(t) - w(t)\| \leq C \beta^{1- \frac{t}{T}} M^{t/T},
\]
where $u(t)$ and $w(t)$ are solutions to \eqref{inhomCP_H}
and \eqref{fvp_inhomCP_H}, respectively, assuming a solution
to \eqref{inhomCP_H} exists.

\section{Theory}
In Hilbert space, the linear inhomogeneous ill-posed problem is
given by
\begin{equation}
\begin{gathered}
 \frac{du}{dt} = Au(t) + h(t), \\
 u(0) = \chi ,
\end{gathered} \label{inhomCauchyprob}
\end{equation}
for $0 \leq t < T$, where $A$ is a positive self-adjoint operator
on a Hilbert space $\mathcal{H}$, $\chi \in \mathcal{H}$, and
$h: [0, T) \to \mathcal{H}$. We assume that $h$
is differentiable on $(0, T)$ and that
$h' \in L^1((0, T); \mathcal{H})$. The following theorem
states conditions under which a solution exists.

\begin{theorem}[{\cite[Corollary 4.2.10]{Pazy}}] \label{solution}
Let $X$ be a Banach space and let $A$ be the infinitesimal
generator of a $C_0$ semigroup $T(t)$ on $X$.
If $h: [0, T) \to X$ is differentiable almost
everywhere on $[0, T]$ and $h' \in L^1((0, T);X)$, then for
 every $\chi \in \operatorname{Dom}(A)$ the initial value
problem \eqref{inhomCauchyprob} has a unique strong solution
$u$ on $[0,T]$ given by
\begin{equation} \label{solution_u(t)}
u(t) = T(t)\chi + \int_0^t T(t-s) h(s) ds.
\end{equation}
\end{theorem}

 Recall that self-adjoint operators bounded above generate
$C_0$ semigroups. This yields the following corollary:

\begin{corollary} \label{bddabove_solution}
Let $\mathcal{H}$ be a Hilbert space, and let
$A: \mathcal{H} \to \mathcal{H}$ be a closed, densely-defined
linear operator. If $A$ is self-adjoint and bounded above
and $h: [0, T) \to \mathcal{H}$ is differentiable on $(0, T)$
with $h' \in L^1(0, T)$, then for every $\chi \in \operatorname{Dom}(A)$
the initial value problem \eqref{inhomCauchyprob} has a
unique solution $u$ on $[0,T]$ given by \eqref{solution_u(t)}.
\end{corollary}

We approximate the inhomogeneous ill-posed problem with
\begin{gather*}
 \frac{dv}{dt} = f(A)v(t) + h(t), \\
 v(0) = \chi ,
\end{gather*}
where $f$ is a real-valued Borel function bounded above that
approximates $A$ in a suitable sense. Take
$f(A) = A - \epsilon A^2$, following Lattes and Lions \cite{LL},
Miller \cite{Miller1}, and Ames \cite{Ames}, or
$f(A) = A(I + \epsilon A)^{-1}$, following Showalter \cite{Showalter}.
Since $f(A)$ is bounded above, by Corollary \ref{bddabove_solution}
the approximate problem is well-posed with solution
\[
v(t) = \mathrm{e}^{tf(A)} \chi + \int_0^t \mathrm{e}^{(t-s) f(A)}
h(s) \, ds.
\]
Now consider the \emph{final}-value problem given by
\begin{equation}
\begin{gathered}
 \frac{dw}{dt} = Aw(t) + h(t), \\
 w(T) = v(T) = \mathrm{e}^{Tf(A)} \chi
 + \int_0^T \mathrm{e}^{(T-s) f(A)} h(s) \, ds.
\end{gathered} \label{FVP}
\end{equation}
This problem is well-posed with solution
\[
w(t) = \mathrm{e}^{(t-T)A} \Big( \mathrm{e}^{Tf(A)} \chi
+ \int_0^T \mathrm{e}^{(T-s) f(A)} h(s) \, ds \Big)
- \int_t^T \mathrm{e}^{(t-s)A} h(s) \, ds,
\]
where $\chi \in \operatorname{Dom}(f(A))$.
Under certain stabilizing conditions, we prove that
\[
\| u(t) - w(t)\| \leq C \beta^{1- \frac{t}{T}} M^{t/T},
\]
where $0 < \beta < 1$ and $C$ and $M$ are computable constants
independent of $\beta$.

\begin{definition}[{\cite[Definition 1]{AH}}]
\label{Cond_A} \rm
Let $A$ be a positive self-adjoint operator on a Hilbert
space $\mathcal{H}$. Let $f:[0, \infty) \to \mathbb{R}$
 be a Borel function, and assume that there exists
$\omega \in \mathbb{R}$ such that $f(\lambda) \leq \omega$
for all $\lambda \in [0, \infty)$. Then $f$ is said to
satisfy \emph{Condition ($\mathcal{A}$)} if there exist positive
constants $\beta$, $\delta$, with $0 < \beta < 1$, for which
$\operatorname{Dom}(A^{1+\delta}) \subseteq \operatorname{Dom}(f(A))$,
and
\begin{equation}
\label{ConditionA}
\|(-A + f(A)) \psi \| \leq \beta \|A^{1+\delta} \psi \|,
\end{equation}
for all $\psi \in \operatorname{Dom}(A^{1+\delta})$.
\end{definition}
Note that $f(A)$ and $A^{1+ \delta}$ are defined by the functional
calculus for self-adjoint operators that follows from the
Spectral Theorem. Set
\begin{equation} \label{g}
g(\lambda) = - \lambda + f(\lambda).
\end{equation}

\begin{lemma}[{\cite[Lemma 1]{AH}}] \label{AHLemma1}
For all $t \geq 0$,
\[
\mathrm{e}^{tg(A)} = \mathrm{e}^{-tA}\mathrm{e}^{tf(A)}.
 \]
\end{lemma}

We will use this repeatedly in our proofs, together with the fact
that this relationship holds for all $\alpha \in \mathbb{C}$:
\[
\mathrm{e}^{\alpha g(A)} = \mathrm{e}^{-\alpha A}\mathrm{e}
^{\alpha f(A)}.
\]
Recall that $A$ is unbounded and thus not defined everywhere.
We need to regularize our data so that it is in the domain of
these operators. In Hilbert space, we use the resolution of
the identity for this regularization. As mentioned above,
we also rely on the functional calculus for unbounded operators
that follows from the Spectral Theorem. Before stating our result
and its proof, we review these ideas.

\begin{theorem} {\rm Spectral Theorem for Unbounded Self-Adjoint
Operators \cite[Theorem~XII.2.3]{DS2}}
Let $T$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$.
Then its spectrum is real and there is a uniquely determined regular
countably additive self-adjoint spectral measure $E$
defined on the Borel sets of the plane, vanishing on the complement
of the spectrum, and related to $T$ by the equations
\begin{itemize}
\item[(a)] $\operatorname{Dom}(T) = \{x \in \mathcal{H}:
 \int_{\sigma (T)} \lambda ^2 d(E(\lambda ) x, x) < \infty \}$, and

\item[(b)] $Tx = \lim_{ n \to \infty} \int_{-n}^n
\lambda dE(\lambda ) x$, where $x \in \operatorname{Dom}(T)$.
\end{itemize}
\end{theorem}

\begin{definition}[{\cite[Definition~XII.2.4]{DS2}}] \rm
The unique spectral measure associated with a self-adjoint
operator $T$ as in the above theorem is called the
\emph{resolution of the identity} for $T$.
\end{definition}

\section{Results}

We assume that there exists a strong solution $u(t)$ to the
ill-posed inhomogeneous problem given in \eqref{inhomCauchyprob}.

\begin{theorem} \label{thm2}
Let $A$ be a positive self-adjoint operator acting on a Hilbert
space $\mathcal{H}$ and let $f$ satisfy Condition ($\mathcal{A}$).
Assume that $h(t): [0, T) \to \mathcal{H}$ is continuously
differentiable with $h'(t) \in L^1(0, T)$ and
$h(t) \in \operatorname{Dom}(\mathrm{e}^{TA})$ for all
$t \in [0,T)$. Also, assume that there exists a constant
$\gamma$, independent of $\beta$ and $\omega$, such that
$(g(A) \psi, \psi) \leq \gamma (\psi, \psi)$, for all
 $\psi \in \operatorname{Dom}(g(A))$. Further, suppose
that $\chi \in \operatorname{Dom}(\mathrm{e}^{TA})$ and
$\| e^{TA} \chi\| \leq L$, $\| e^{TA} h(t) \| \leq N$ for all
$t \in [0, T)$. Then there exist constants $C$ and $M$,
independent of $\beta$, such that for $0 \leq t < T$,
\[
\| u(t) - w(t) \| \leq C\beta^{1 - \frac{t}{T}}M^{t/T}.
\]
\end{theorem}

As discussed above, since $A$ is unbounded we need to regularize
our data so that it is in the domain of the operators with
which we are working. We use the resolution of the identity
for this regularization. Let $\{E (\cdot)\}$ represent the
resolution of the identity for the linear operator $A$.
Set $e_n = \{ \lambda \in [0, \infty) : |g(\lambda)| \leq n \}$.
 Using the definition of $g$ given above in \eqref{g},
where $f$ satisfies Condition ($\mathcal{A}$), we see
that $e_n$ is a bounded set since
\begin{align*}
 e_n &= \{ \lambda \in [0, \infty) : |g(\lambda)| \leq n \} \\
 &\subseteq \{ \lambda : 0 \leq \lambda \leq n + \omega \}.
\end{align*}
Let $E_n = E(e_n)$. The following lemma is used repeatedly
throughout this work.

\begin{lemma} \label{E_n_lemma}
Let $A$ be a self-adjoint operator with $E$ the resolution
of the identity for $A$ and
$e_n = \{ \lambda \in [0, \infty) : |g(\lambda)| \leq n \}$.
Let $\tau \in \mathcal{H}$. Then
$E_n \tau \in \operatorname{Dom}(f(A))$, where $f$ is a
complex Borel function defined $E$-almost everywhere on the
real axis and bounded on bounded sets.
\end{lemma}

Our proof begins with approximations $u_n(t)$ and $w_n(t)$.
Set $\chi_n = E_n \chi$ and $h_n(s) = E_n h(s)$.
Note that $f(\lambda) = \mathrm{e}^{t\lambda}$ is a
Borel function bounded on bounded sets, so by
Lemma~\ref{E_n_lemma} $\chi_n$,
$h_n \in \operatorname{Dom}(A) \cap \operatorname{Dom}(\mathrm{e}^{tA})$.
 Define
\begin{gather*}
 u_n(t) = E_n u(t), \\
 w_n(t) = E_n w(t).
\end{gather*}

\begin{lemma}[{\cite[Lemma~9]{Campbell_nonlin}}] \label{u_n(t)}
\[
u_n(t) = \mathrm{e}^{tA} \chi_n + \int_0^t \mathrm{e}^{(t-s)A}
h_n (s) ds.
\]
\end{lemma}

We have an analogous result for $w_n$:
\begin{lemma} \label{w_n(t)}
\[
w_n(t) = \mathrm{e}^{(t-T)A} \Big( \mathrm{e}^{Tf(A)} \chi_n
+ \int_0^T \mathrm{e}^{(T-s) f(A)} h_n(s) \, ds \Big)
- \int_t^T \mathrm{e}^{(t-s)A} h_n(s) \, ds.
\]
\end{lemma}

\begin{proof}
 Note that we may write the final value problem given in
\eqref{FVP} as an initial value problem by replacing $t$ with $T-t$.
Then the differential equation becomes
\[
\frac{dw}{dt} = - Aw(t) + h(t).
\]
Since $-A$ is bounded above, $-A$ generates a $C_0$ semigroup.
Then by Corollary~\ref{bddabove_solution}, the problem given
in \eqref{FVP} has a unique solution given by
\[
w(t) = \mathrm{e}^{(t-T)A} \Big( \mathrm{e}^{Tf(A)} \chi
+ \int_0^T \mathrm{e}^{(T-s) f(A)} h(s) \, ds \Big)
- \int_t^T \mathrm{e}^{(t-s)A} h(s) .
\]
Thus
\begin{align*}
 E_n w(t) &= E_n \Big[\mathrm{e}^{(t-T)A}
\Big( \mathrm{e}^{Tf(A)} \chi + \int_0^T \mathrm{e}^{(T-s) f(A)} h(s)
\, ds \Big) - \int_t^T \mathrm{e}^{(t-s)A} h(s) \Big] \\
 &= \mathrm{e}^{(t-T)A} \Big( \mathrm{e}^{Tf(A)} \chi_n
+ \int_0^T \mathrm{e}^{(T-s) f(A)} h_n(s) \, ds \Big)
- \int_t^T \mathrm{e}^{(t-s)A} h_n(s) ,
\end{align*}
so
\[ w_n(t) = \mathrm{e}^{(t-T)A} \Big( \mathrm{e}^{Tf(A)} \chi_n
+ \int_0^T \mathrm{e}^{(T-s) f(A)} h_n(s) \, ds \Big)
- \int_t^T \mathrm{e}^{(t-s)A} h_n(s).
\]
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm2}]
To obtain our result we use the Three Lines Theorem
(cf. \cite[p. 33]{RS2}), which requires us to extend $u_n$
and $w_n$ into the complex strip
$\{ \alpha = t + \mathrm{i} \eta : 0 \leq t \leq T, \eta \in \mathbb{R}
\}$. To do so, set
\begin{gather*}
 u_n (\alpha ) = \mathrm{e}^{\mathrm{i} \eta A} u_n(t) \\
 w_n (\alpha ) = \mathrm{e}^{\mathrm{i} \eta A} w_n(t),
\end{gather*}
Note that the complex-valued function $u_n - w_n$ is analytic
and continuous on the strip. Define
\[
\phi_n (\alpha) = (u_n(\alpha) - w_n (\alpha), \tau),
\]
where $(\cdot, \cdot)$ is the inner product in $\mathcal{H}$
and $\tau$ is an arbitrary element in $\mathcal{H}$.
To use the Three Lines Theorem, we must show that
$\phi_n (\alpha)$ is bounded in the strip.
For $\alpha = t + \mathrm{i} \eta$ we have
\newpage
\begin{align*}
& \| \phi_n (\alpha) \|\\
&\leq \| u_n (\alpha) - w_n (\alpha) \| \|\tau \| \\
&=  \| \mathrm{e}^{\mathrm{i} \eta A} \| \| u_n (t) - w_n (t) \| \|\tau \| \\
&=  \left \| \Big(\mathrm{e}^{tA}\chi_n
 + \int_0^t \mathrm{e}^{(t-s)A} h_n(s) ds \Big)
 - \Big( \mathrm{e}^{tA} \mathrm{e}^{T(-A +f(A))} \chi_n \right. \\
&\quad \left. +  \mathrm{e}^{(t-T)A} \int_0^T \mathrm{e}^{(T-s)f(A)}
 h_n(s) ds - \int_t^T \mathrm{e}^{(t-s)A} h_n(s) ds \Big) \right \|\,
  \|\tau \| \\
&\leq \left \| \big(I - \mathrm{e}^{T (-A + f(A))} \big)
 \mathrm{e}^{tA} \chi_n + \int_0^T \left ( \mathrm{e}^{(t-s)A}
 - \mathrm{e}^{(t-T)A} \mathrm{e}^{(T-s)f(A)} \right ) h_n(s)
 \, ds \right \|\, \|\tau \| \\
&\leq \left ( \| \big (I - \mathrm{e}^{T g(A)} \big)
 \mathrm{e}^{tA} \chi_n \| + \left \| \int_0^T \left ( \mathrm{e}^{(t-s)A}
 - \mathrm{e}^{(t-s+s-T)A} \mathrm{e}^{(T-s)f(A)} \right )
 h_n(s) \, ds \right \| \right) \|\tau \| \\
 &\leq \left ( \| \big (I - \mathrm{e}^{T g(A)} \big)
 \mathrm{e}^{tA} \chi_n \| + \left \| \int_0^T \mathrm{e}^{(t-s)A}
 \left (I - \mathrm{e}^{(T-s)(-A + f(A))} \right ) h_n(s) \, ds
 \right \| \right) \|\tau \| \\
&\leq \left ( \| \big (I - \mathrm{e}^{T g(A)} \big)
 \mathrm{e}^{tA} \chi_n \| + \left \| \int_0^T
 \left (I - \mathrm{e}^{(T-s) g(A)} \right ) \mathrm{e}^{(t-s)A}
  h_n(s) \, ds \right \| \right ) \|\tau \|.
\end{align*}
Recall that $(g(A) \psi, \psi) \leq \gamma (\psi, \psi)$
for all $\psi \in \operatorname{Dom}(g(A))$, and so $g(A)$
is the generator of a strongly continuous semigroup
$\{\mathrm{e}^{tg(A)} \}_{t \geq 0}$ of bounded operators
with $\| \mathrm{e}^{tg(A)} \| \leq \mathrm{e}^{\gamma t}$.
Thus $\| \left ( I - \mathrm{e}^{t g(A)} \right )
\psi \| \leq K \| \psi \|$ for all
$t \in [0, T]$, where $K$ is a constant. Using our assumptions
that $ \| \mathrm{e}^{TA} \chi \| \leq L$ and
$\| \mathrm{e}^{TA} h(t) \| \leq N$, we have
\begin{align}
\nonumber \| \phi_n (\alpha) \|
&\leq \left ( \| \big (I - \mathrm{e}^{T g(A)} \big)
  \mathrm{e}^{tA} \chi_n \| + \left \| \int_0^T
  \left (I - \mathrm{e}^{(T-s) g(A)} \right )
  \mathrm{e}^{(t-s)A} h_n(s) \, ds \right \| \right) \|\tau \| \\
\nonumber &\leq \left (K \| \mathrm{e}^{tA} \chi_n \| + K \int_0^T \|
 \mathrm{e}^{(t-s)A} h_n(s) \| \, ds \right ) \|\tau \| \\
&\leq K \left ( \| \mathrm{e}^{TA} \chi_n \| + \int_0^T \|
 \mathrm{e}^{TA} h_n(s) \| \, ds \right ) \|\tau \| \label{phi_bound} \\ 
\nonumber &\leq K ( L + TN) \| \tau \|,
\end{align} 
and thus $\phi_n$ is bounded. Hence we may apply the Three
Lines Theorem to this inner product. By the Three Lines Theorem,
\[
|(\phi_n(t), \tau) | \leq M(0)^{1 - \frac{t}{T}} M(T)^{t/T}
\]
for $0 \leq t \leq T$, where
\[
M(t) = \max_{\alpha = t + \mathrm{i} \eta, \eta
\in \mathbb{R}} | (\phi_n(\alpha), \tau) |.
\]
Using properties of semigroups and Condition ($\mathcal{A}$),
we have
\begin{align*}
 M(0) &\leq \left ( \| \big (I - \mathrm{e}^{T g(A)} \big)
 \chi_n \| + \left \| \int_0^T \left (I - \mathrm{e}^{(T-s) g(A)} \right )
  \mathrm{e}^{(-s)A} h_n(s) \, ds \right \| \right) \|\tau \| \\
&\leq \left ( \left \| \int_0^T \mathrm{e}^{\sigma g(A)} g(A)
 \chi_n \, d\sigma \right \| + \int_0^T \| \left ( I - \mathrm{e}^{(T-s)
 g(A)} \right ) \mathrm{e}^{(-s)A} h_n(s) \| \, ds \right) \|\tau \| \\
&\leq \left ( \mathrm{e}^{\gamma T} \int_0^T \| g(A)
 \chi_n \| \, d\sigma + \int_0^T \left \| \int_0^{T-s}
 \mathrm{e}^{\sigma g(A)} g(A) \mathrm{e}^{(-s)A} h_n(s) \, d\sigma
 \right \| ds \right) \|\tau \| \\
&\leq \Big( \beta T \mathrm{e}^{\gamma T} \|A^{1 + \delta}
  \chi_n \| + \int_0^T \int_0^{T-s} \mathrm{e}^{\gamma \sigma}
  \| g(A) \mathrm{e}^{(-s)A} h_n(s) \| \, d\sigma ds \Big) \|\tau \| \\
&\leq \Big( \beta T \mathrm{e}^{\gamma T} \|A^{1 + \delta}
  \chi_n \| + \int_0^T \mathrm{e}^{\gamma (T-s)} \int_0^{T-s}
  \beta \| A^{1 + \delta} \mathrm{e}^{(-s)A} h_n(s) \| \, d\sigma
  ds \Big) \|\tau \| \\
&\leq \Big( \beta T \mathrm{e}^{\gamma T} \|A^{1 + \delta}
  \chi_n \| + \int_0^T \mathrm{e}^{\gamma (T-s)} \beta (T-s) \|
  A^{1 + \delta} \mathrm{e}^{(-s)A} h_n(s) \| \, ds \Big)
  \|\tau \| \\
&\leq \beta T \mathrm{e}^{\gamma T}
 \Big( \|A^{1 + \delta} \chi_n \| + \int_0^T \| A^{1 + \delta}
 \mathrm{e}^{(-s)A} h_n(s) \| \, ds \Big) \|\tau \| \\
&\leq \beta T \mathrm{e}^{\gamma T}
 \Big( \|A^{1 + \delta} \chi_n \| + \int_0^T \|
 A^{1 + \delta} h_n(s) \| \, ds \Big) \|\tau \| \\
&\leq k \beta T \mathrm{e}^{\gamma T}
  \Big( \|\mathrm{e}^{TA} \chi_n \| + \int_0^T \|\mathrm{e}^{TA}
 h_n(s) \| \, ds \Big) \|\tau \|,
\end{align*}
where $k$ is a positive constant. Also, from \eqref{phi_bound}
we have
\begin{align*}
 M(T) &\leq \left (\| \big (I - \mathrm{e}^{T g(A)} \big)
\mathrm{e}^{TA} \chi_n \| + \left \| \int_0^T
\left (I - \mathrm{e}^{(T-s) g(A)} \right )
\mathrm{e}^{(T-s)A} h_n(s) \, ds \right \| \right) \|\tau \| \\
 &\leq K \Big( \| \mathrm{e}^{TA} \chi_n \|
 + \int_0^T \| \mathrm{e}^{TA} h_n(s) \| \, ds \Big) \|\tau \|.
\end{align*}
Thus by the Three Lines Theorem,
\begin{align*}
|\phi_n(t)|
&\leq \Big\{ k \beta T \mathrm{e}^{\gamma T}
 \Big( \|\mathrm{e}^{TA} \chi_n \|
 + \int_0^T \|\mathrm{e}^{TA} h_n(s) \| \, ds \Big)
 \Big\}^{1-\frac{t}{T}} \\
&\quad\times \Big\{K \Big( \| \mathrm{e}^{TA} \chi_n \|
 + \int_0^T \| \mathrm{e}^{TA} h_n(s) \| \, ds \Big)
 \Big\}^{t/T} \|\tau \|.
\end{align*}
Letting $n \to \infty$ and using our stability assumptions, we have
\[
|\phi(t)|
\leq \{ k \beta T \mathrm{e}^{\gamma T} ( L + TN ) \}^{1-\frac{t}{T}}
\left \{ K \left ( L + TN \right ) \right \}^{t/T} \|\tau \| \\
\leq C \beta^{1- \frac{t}{T}} M^{t/T},
\]
where $C$ and $M$ are computable constants independent of $\beta$ and
\[
\phi (t) = (u(t) - w(t), \tau).
 \]
Taking the supremum over all $\tau \in \mathcal{H}$ with
$\|\tau \| \leq 1$, we have
\[
\|u(t) - w(t)\| \leq C \beta^{1- \frac{t}{T}} M^{t/T},
\]
our desired result.
\end{proof}

\subsection*{Acknowledgments}
The author gratefully acknowledges the support and guidance
of Professor Rhonda Hughes.

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