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%--------------------------- article -----------------------
\begin{document}
%\baselineskip=15pt
\title[Evolution semigroups and stability]
{Evolution semigroups and robust stability\\ of evolution operators
on Banach spaces}
\author{Y.~Latushkin}
\address{ Department of Mathematics\\
University of Missouri\\ Columbia, MO 65211}
\email{mathyl@mizzou1.missouri.edu}
\author{S.~Montgomery-Smith}
\address{ Department of Mathematics\\
University of Missouri\\ Columbia, MO 65211}
\email{stephen@mont.cs.missouri.edu}
\author{T.~Randolph}
\address{ Department of Mathematics\\
University of Missouri\\ Rolla, MO 65409}
\email{randolph@umr.edu}
%\makeatother
\thanks{The first author was supported by the NSF grant DMS-9400518;
the second author was supported by the NSF grant DMS-9201357.}
\keywords{Exponential stability, stability radius, evolution semigroups,
spectral mapping theorems}
\subjclass {93D09, 47D99, 34G10, 34C35}
%\maketitle
\begin{abstract}
Using the theory of evolution semigroups, this paper investigates
the stability radius for a wide class of linear
nonautonomous systems in Banach spaces. A spectral-mapping theorem for
evolution semigroups acting on vector-valued functions on $[0,\infty)$ is
proven first. This allows the stability radius to be expressed in terms
of the spectrum of the generator of the evolution semigroup. Consequences
include explicit formulas for bounds on the stability radius for
Banach space systems; time-varying and constant perturbations are
considered.
\end{abstract}
\maketitle
%-------------------------- section 0 -----------------------
\section{Introduction}
\noindent The concept of the stability radius was introduced for
finite-dimensional autonomous systems by D.~Hinrichsen and
A.~J.~Pritchard in \cite{HP86} as a ``distance from instability," that is,
as the size of the largest operator, $\Delta$, for which the perturbation,
$x'=(A+\Delta)x$, of an exponentially stable system
\begin{equation}\lb{acp}
x'(t)=Ax(t),\quad t\ge0,
\end{equation}
remains stable.
More recently, these authors have developed a very
general theory on stability of control systems and have studied
the property of robustness of stability for
infinite-dimensional nonautonomous systems with unbounded
time-varying linear control operators (see \cite{HP94} and the
references therein). The in-depth works of B.~Jacob
\cite{Jacob} (see also \cite{JDP}) and F.~Wirth
\cite{FW} build on this study and consider more
general classes of perturbations, including nonlinear perturbations.
Our work also addresses infinite-dimensional nonautonomous systems
\begin{equation}\lb{nacp}
x'(t)=A(t)x(t),\qquad t\ge 0,
\end{equation}
where $A(t)$ is a (possibly unbounded) linear operator on a
Banach space $X$, but does so by using recent developments involving
so-called evolution semigroups and exploiting their role in determining
the asymptotic behavior of solutions for systems such as \eqref{nacp} (on
$\bbR$, see \cite{LMS1,LMS2,LMR,LR,RS1,RS2,Rau1,Rau2}). We describe, in
particular, a new way of viewing the stability radius and obtain new
bounds for it. For autonomous equations we generalize a known formula for
the stability radius in Hilbert space by obtaining upper and lower bounds
for it in the setting of Banach spaces.
Assume, for a moment, that \eqref{nacp}, for $t\in\bbR$, is well-posed in the
sense that there exists an evolution (solving) family of
operators $\U=\{U(t,s)\}_{t\ge s}$ which gives a differentiable solution.
This means that $x(\cdot):t\mapsto U(t,s)x(s)$, $t\ge s$ in $\bbR$, is
differentiable for any given initial conditions
$x(s)=x_s\in \calD(A(s))$ from the domain $\calD(A(s))$ of the operator $A(s)$,
$x(t)\in \calD(A(t))$, and \eqref{nacp} holds.
For a given evolution family $\U$ an ``evolution semigroup" $\{T^t\}_{t\ge
0}$ is defined on a ``super-space'', $F$, of functions $f:\bbR\to X$
by the rule
\begin{equation}\lb{evsg}
(T^tf)(\t)=U(\t,\t-t)f(\t-t),\quad \t\in\bbR,\; t\ge 0.
\end{equation}
For appropriate spaces $F$ (e.g., $F=\LpR$, $1\leq p<\infty$, or $\CoR$,
the space of continuous functions vanishing
at infinity), the semigroup $\{T^t\}_{t\ge 0}$ is strongly continuous
on $F$ with a generator, $\Gamma$.
A significant feature of the evolution semigroup is that it can be used to
overcome the fact that the asymptotic behavior of solutions to
\eqref{acp} or \eqref{nacp} is not, in general, determined by the
spectral properties of the operators $A$ or $A(t)$.
For example, there exist
strongly continuous semigroups $\{e^{tA}\}_{t\ge 0}$
on infinite-dimensional
Banach spaces that {\em are not} uniformly exponentially stable while
$\mbox{Re}\lambda\le s<0$ for all $\lambda\in\sigma(A)$, the spectrum of
$A$; see, e.g., \cite{Nagel}.
For nonautonomous equations the
situation is worse. Indeed, even for finite-dimensional $X$, the spectra
of the operators $A(t)$ do not determine the
asymptotic behavior of solutions to \eqref{nacp}. However, in all of
these cases {\em it is determined} by the spectrum of $T^t$ or $\Gamma$;
we refer to \cite{LMS1,LMS2,LMR,LR,RS2,Rau2} which study the exponential
dichotomy of solutions to \eqref{nacp} on $\bbR$.
%This is made possible by the following surprising properties: The
%spectrum $\sigma(\Gamma)$ is invariant under translations along
%$i\bbR$, the spectrum $\sigma(T^t)$ is invariant under rotations
%about origin, and the spectral mapping property holds
%$e^{t\sigma(\Gamma)}=\sigma(T^t)\setminus\{0\},\quad t>0.$
In this paper we consider \eqref{nacp} and prove
that the property of uniform exponential stability is completely
determined by the spectrum of an evolution
semigroup or its generator defined for ``super-spaces''
$F_+$ of $X$-valued functions on the half-line, $\bbR_+=[0,\infty)$.
More specifically, we extend a result of J.~van Neerven \cite{vanN1}
(concerning autonomous equations) in order to show that
exponential stability of solutions to
\eqref{nacp} on $\bbR_+$ can be characterized by the boundedness of a
particular convolution operator on $F_+$; this operator is, in fact, the
inverse of the generator of the evolution semigroup. This result is then
used to address questions concerning stability, including the robustness
of stability and the stability radius. In a sense, a nonautonomous problem
is reduced to an autonomous one by considering the
evolution semigroup. So for certain questions concerning stability,
complications arising from the dependence on time are removed.
The class of problems being addressed is very large since we allow the
operators $A(t)$ to be unbounded. Moreover, since the asymptotic
properties are completely determined by the evolution family,
we do not require solutions to be differentiable. Instead, we begin
with an exponentially stable evolution family, $\U$, and study
perturbations of the evolution semigroup. This allows us to consider
properties of a perturbed equation, $x'(t)=(A(t)+B(t))x(t)$, even if
there is no classical solution; that is, even when solutions exist only
in a ``mild" sense. We proceed by verifying that, if $\U$ corresponds to
solutions of \eqref{nacp} (possibly in a mild sense), and if $\Gamma$
denotes the generator of the induced evolution semigroup, then, under
certain assumptions, the corresponding perturbation, $\Gamma+\calB$, of
$\Gamma$ also generates an evolution semigroup. The stability of the
perturbed problem is then determined by the spectrum of this semigroup or
its generator. Robustness of the stability is an immediate consequence.
In order to prove that the exponential stability of solutions to
\eqref{nacp} is determined by the spectrum of the generator of the
evolution semigroup, we first prove a spectral mapping theorem for such a
semigroup defined on functions $f\colon\bbR_+\to X$ by the rule
\begin{equation}\lb{evsg+}
(T_+^tf)(s)=\cases
U(s,s-t)f(s-t),\quad s-t\ge 0;\\ 0,\qquad\mbox{otherwise}.\endcases
\end{equation}
Formally, this defines a semigroup of operators, $\{T_+^t\}_{t\ge0}$;
on appropriate Banach spaces of functions, $F_+$, such a semigroup is
strongly continuous (one can take, for example,
$F_+=L_p(\bbR_+,X)$ or $F_+=\{f\in C_0(\bbR_+,X):f(0)=0\}$).
Letting $\Gamma_+$ denote the generator for $\{T_+^t\}_{t\ge0}$, we begin
in Section~1 by showing that
\begin{equation}\label{essmt}
e^{t\sigma(\Gamma_+)}=\sigma(T_+^t)\setminus\{0\},\quad t>0.
\end{equation}
Spectral mapping theorems of this type have been proven for evolution
semigroups \eqref{evsg} on spaces of vector-valued functions on
$\bbR$ \cite{LMS2,LMR,RS1}. The analogous theorem for functions on
$\bbR_+$ is natural and of particular interest in the setting of
the present paper. For autonomous equations \eqref{acp} the solutions
are given by a strongly continuous semigroup $\{e^{tA}\}_{t\ge0}$ generated by
$A$.
In this case $U(t,s)=e^{(t-s)A}$. The question of whether the spectral
mapping theorem holds for $T_+^t$ was posed in \cite{vanN1} and later
proven for the autonomous case in \cite{vanN2}.
We stress that the spectral mapping theorem \eqref{essmt} holds for the
evolution semigroup $\{T^t_+\}_{t\ge 0}$ even if
$\sigma(e^{tA})\setminus\{0\}$ {\em is not} equal to $e^{t\sigma(A)}$.
For the nonautonomous case the spectral
mapping theorem \eqref{essmt} has
recently been proven by R.~Schnaubelt in \cite{Roland}. A different proof
is given here which is quite brief; it uses a simple change of variables and
exploits the previous result for functions on $\bbR$.
The spectral mapping theorem \eqref{essmt} allows one to prove that the
exponential stability of an evolution family $\U$ is equivalent to the
fact that $\Gamma_+$ is invertible. This
observation begins Section~2 and leads to a natural extension of a
theorem by J.~van Neerven \cite{vanN1} which characterizes the property of
uniform exponential stability of a semigroup, $\{e^{tA}\}_{t\ge0}$,
in terms of a convolution operator:
$f\mapsto\int_0^\cdot e^{tA}f(\cdot-t)\,dt$, on $F_+$. In our
consideration of nonautonomous equations, we define the analogue, $\hat
G$, of this operator by the rule
\begin{equation}\lb{hatG}
(\hat Gf)(s)=\int_0^s U(s,s-\t)f(s-\t)\,d\t,\quad f\in F_+, \quad s\ge 0.
\end{equation}
It turns out that this operator equals $-\Gamma_+^{-1}$ whenever either one
is a bounded operator on $F_+$ and, hence, it can be used to characterize
the property of exponential stability.
Our arguments in Section~2 are based entirely on the
insights given by J.~van Neerven in \cite{vanN1}
for the autonomous case.
In Section~3 the generator $\Gamma_+$ of the evolutionary semigroup
$\{T^t_+\}_{t\ge 0}$ again plays an important role in the study of
the stability radius. Assume, for now, that $\U$ is an evolution
family corresponding to a uniformly exponentially stable equation
\eqref{nacp}. If $\Delta(\cdot)\in C_b(\bbR_+,B_s(X))$ (the set of
strongly continuous $B(X)$-valued functions bounded on $\bbR_+$),
denote by $\calDelta$ the multiplication operator,
$\calDelta f(t)=\Delta(t)f(t)$, for $f\in F_+$. Then stability of a
perturbed equation
\begin{equation}\lb{nacp+Del}
x'(t)=(A(t)+\Delta(t))x(t), \quad t\ge0,
\end{equation}
%with initial condition $x(s)=x\in \calD(A(s))$, $s\le t$.
is determined by whether or not $\Gamma_+ +\calDelta$ is invertible.
Therefore, the stability radius, $\rstab(\U)$, of $\U$ with respect to a
perturbation in $C_b(\bbR_+,B_s(X))$ may be viewed as
the largest $\|\Delta(\cdot)\|_\infty$ such that
$\Gamma_+ +\calDelta$ is invertible. When viewed in this context, a
straightforward algebraic argument can be used to show that $\rstab(\U)$
satisfies
\[
\frac{1}{\|\Gamma_+^{-1}\|} \le \rstab(\U) \le
\frac{1}{r(\Gamma_+^{-1})};
\]
here $r(\cdot)$ denotes the spectral radius.
More generally, we consider ``structured" perturbations of the form
\begin{equation}\lb{nacp+DDeltaE}
x'(t)=(A(t)+D(t)\Delta(t)E(t))x(t), \quad t\in\bbR_+,
\end{equation}
where, for Banach spaces $U$ and $Y$, the operator-valued functions
\[D(\cdot)\in C_b(\bbR_+,B_s(U,X))\quad\mbox{ and }\quad
E(\cdot)\in C_b(\bbR_+,B_s(X,Y))\]
are fixed. In the context of control systems, the operators $D$ and $E$ are
given
``scaling operators" describing the structure of the perturbation, and
$\Delta(\cdot)\in C_b(\bbR_+,B_s(Y,U))$ represents an unknown
disturbance. Let $\calD$ denote the ``multiplication" operator
$\calD: L_p(\bbR_+,U)\to L_p(\bbR_+,X)$, $\calD u(t)=D(t)u(t)$. Let
$\calDelta$ and $\calE$ denote analogously defined multiplication operators.
In studying the stability of
\eqref{nacp+DDeltaE}, we again view the stability radius, $\rstab(\U,D,E)$,
of $\U$ with respect to a perturbation in $C_b(\bbR_+,B_s(Y,U))$, as the
largest $\|\Delta(\cdot)\|_\infty$ such that $\Gamma_+ +\calD\calDelta\calE$ is
invertible. It is well-known \cite{CP,HP94} that the stability radius
$\rstab(\U,D,E)$ can be characterized in terms of the so-called
``input-output" operator $\bbL_0:\LpRU\to\LpRY$,
\[
(\bbL_0u)(t)=E(t)\int_0^t U(t,\t)D(\t)u(\t)\, d\t,\quad u\in\LpRU.
\]
We observe that if $\U$ is exponentially stable,
then $\Gamma_+$ is invertible
and $\Gamma_+^{-1}=-\hat G$. This implies that the
``input-output" operator $\bbL_0$ can be expressed via the generator
$\Gamma_+$ of the evolution semigroup as follows:
\[\bbL_0=\calE\hat G\calD=-\calE\Gamma_+^{-1}\calD.\]
This allows for a new, relatively simple proof of the
following lower bound on
the stability radius (cf.~\cite{HIP,HP94}) for the Banach space setting:
\[
\frac{1}{\|\bbL_0\|}\le \rstab(\U,D,E).
\]
In the case where $A$, $D$, and $E$ are constant
and $\U$ is replaced by the semigroup $\{e^{tA}\}_{t\ge0}$
the following upper and lower
bounds can be obtained for the Banach space setting:
\begin{equation}\label{bounds}
\frac{1}{\|\bbL_0\|}\le \rstab(\{e^{tA}\},D,E)\le
\frac{1}{\sup_{s\in\bbR}\|E(A-is)^{-1}D\|}
\end{equation}
(see also \cite[Theorem 3.5]{HP94} concerning equality of these three
quantities when $X$, $U$ and $Y$ are Hilbert spaces).
Our proof of this inequality is quite elementary.
It uses a recent result of \cite{LMS1}
which extends a spectral mapping theorem of L.~Gearhart (see, e.g.,
\cite{Nagel})
for semigroups on Hilbert spaces to a theorem for Banach spaces.
Using the spectral mapping theorem, we give a simple proof
of the fact that all three quantities in \eqref{bounds} are equal
for the Hilbert space setting.
Also, we give an example where at least one inequality in
\eqref{bounds} is strict for the Banach space setting.
We obtain, in addition, in the Banach space setting,
the following formula for the lower bound
of $\rcstab(\{e^{tA}\},D,E)$,
the {\em constant} stability radius related to the time-independent
perturbations $\Delta(t)\equiv\Delta$:
\begin{multline*}
\rcstab(\{e^{tA}\},D,E)\ge\\
\left(
\sup_{\xi\in[0,1]}
\sup_{\{u_k\}}
\frac{\|\sum_kE(A-i\xi-ik)^{-1}Du_ke^{ik(\cdot)}\|_{L_p([0,2\pi],Y)}}
{\|\sum_k u_ke^{ik(\cdot)}\|_{L_p([0,2\pi],U)}}
\right)^{-1}.
\end{multline*}
Here the second $\sup$ is taken over
all finite sequences $\{u_k\}_{k=-N}^N$ in $U$.
In the development which leads to the bounds mentioned above we introduce the
new idea of a ``pointwise" stability radius,
\[
\rcstab^\lambda(e^{t_0 A},D,E):=\sup\{r>0: \|\Delta\|_{B(Y,U)}\le r
\Rightarrow \lambda\in\rho(e^{t_0(A+D\Delta E)})\},
\]
for a point $\lambda\in\rho(e^{t_0A})$, $t_0>0$ and obtain
upper and lower bounds for it.
In addition, for a {\it hyperbolic} strongly continuous semigroup
$\{e^{tA}\}_{t\ge 0}$ we study the constant {\it dichotomy radius}
which is the size of largest $\Delta$ such that the differential
equation $x'=(A+D\Delta E)x$ has exponential dichotomy.
For infinite-dimensional nonautonomous systems
\eqref{nacp+DDeltaE}, the stability radius as defined in \cite{HP94} allows
for unbounded perturbations of the form $D(t)\Delta(t)E(t)$.
We remark that the structure
of three Banach spaces, $\Xu\subseteq X\subseteq\Xo$ as described by
\cite{HP94}, would allow one to consider unbounded perturbation operators
$D(t)$ and $E(t)$ under the assumption that $E(\cdot)\in
C_b(\bbR_+,B_s(\Xu,Y))$ and
$D(\cdot)\in C_b(\bbR_+,B_s(U,\Xo))$. Many of the arguments follow much as in
the bounded case after making appropriate assumptions (and applying theorems of
J.~Voigt \cite{Voigt} and F.~R\"abiger et.~al \cite{RRS}) to ensure
that the unbounded perturbation, $\Gamma_+ +\calD\calDelta\calE$, of
$\Gamma_+$ generates an evolution semigroup. This, however, will not be
pursued in the present paper.
The following notation will be used throughout. If $A$ is a linear operator
on a Banach space, $X$, then $\calD(A)$ denotes the domain of $A$,
$\rho(A)$ denotes the resolvent set and
$\sigma(A)$ denotes the spectrum relative to $B(X)$, the algebra of bounded
linear operators on $X$. For $\lambda\in\rho(A)$, $R(\lambda,A)$ denotes
the resolvent
operator $(\lambda-A)^{-1}\in B(X)$. If $A$ is a bounded linear operator,
$r(A)$ denotes the spectral radius. If $A$ is the generator of a strongly
continuous semigroup, $\{e^{tA}\}_{t\ge0}$, on $X$, define the
growth bound of the semigroup by $\omega_0(\{e^{tA}\}):=
\inf\{\omega\in\bbR:\|e^{tA}\|\le Me^{t\omega}\mbox{ for some }M\ge0,\mbox{ all
}t\ge0\}$, and the spectral bound of $A$ by
$s(A):=\sup\{\mbox{Re}\lambda:\lambda\in\sigma(A)\}$.
Also, we denote: $C_0(\bbR_+,X)$
--- the space of continuous functions vanishing at infinity;
$C_{00}(\bbR_+,X)$ --- its subspace of functions with $f(0)=0$;
$C^1_c$ --- differentiable functions with compact support.
%---------------------------- section 1 --------------------------
\section{Evolution semigroups; spectral mapping theorem}
\setcounter{equation}{0}
An {\it evolution family} (see, e.g., \cite{DK}) is a family of
bounded linear operators $\U=\{U(t,s)\}_{t\ge s}$
on $X$ with the following properties:
\begin{enumerate}
\item $U(t,s)=U(t,r)U(r,s)$ and $U(t,t)=I$ for all $t\geq r\geq s$;
\item for each $x\in X$ the function $(t,s)\mapsto U(t,s)x$
is continuous for $t\geq s$;
\item $\|U(t,s)\|\leq C e^{\gamma (t-s)}$, $t\geq s$, for some
constants $C>1$ and $\gamma$.
\end{enumerate}
An evolution family is said to be {\em exponentially stable} if
there exist constants $M>1,\,\beta >0$ such that
$\|U(t,s)\|\leq Me^{-\beta (t-s)}$ for all $t\geq s$.
An evolution family $\U$ is said to {\em solve} the nonautonomous Cauchy
problem \eqref{nacp}
with initial condition $x(s)=x_s\in \calD(A(s))$ ($t\ge s$) if
$x(\cdot)=U(\cdot,s)x_s$ is the unique classical solution of
\eqref{nacp} for every $x_s\in \calD(A(s))$; that is, $x(\cdot)$ is
differentiable, $x(t)\in \calD(A(t))$ for $t\ge s$, and \eqref{nacp} holds.
Since we assume that $\U$ is a {\it strongly}
continuous family, the operators $A(t)$ here might be unbounded. In
particular, for the autonomous case, if $A$ is a generator of a
strongly continuous semigroup, $\{e^{tA}\}_{t\ge0}$, on $X$ and $A(t)\equiv
A$, then
$U(t,s)=e^{(t-s)A}$.
Now let $\{e^{tA_0}\}_{t\ge0}$ be a strongly continuous semigroup
generated by $A_0$, let $A_1(t)\in B(X)$ for $t\ge0$, and define
$A(t)=A_0+A_1(t)$. Then even if $t\mapsto A_1(t)$
is continuous, equation \eqref{nacp} may not have
a differentiable solution for all initial conditions $x(0)=x\in
\calD(A)=\calD(A_0)$ (see, e.g.,\cite{Phillips}). Consequently, it is
important
to be able to consider solutions which may exist only in a ``mild sense''.
Let $\U$ be an evolution family of operators corresponding to
the solution of
\eqref{nacp}, and consider the nonautonomous inhomogeneous equation
\begin{equation}\lb{inhomog}
x'(t)=A(t)x(t)+f(t),\qquad t\ge0,
\end{equation}
where $f$ is a locally integrable $X$-valued function on $\bbR_+$.
Recall, that $x(\cdot)$ is a {\em mild} solution of \eqref{inhomog}
with initial value $x(s)=x \in \calD(A(s))$ if
\[
x(t)=U(t,s)x+\int_s^tU(t,\t)f(\t)\,d\t, \quad t\ge s.
\]
Now let $\U$ be an evolution family.
We define the {\em evolution semigroup}
$\{T_+^t\}_\t\ge0$ {\em induced by} $\U$ by the rule given in
\eqref{evsg+}. This defines a strongly continuous
semigroup on $F_+=\LpRp$, $1\le p<\infty$, and $F_+=\Coo$.
In either case, the evolution semigroup on
$F_+$ will be denoted by $\{T_+^t\}_t\ge0$, and its generator will be
denoted by $\Gamma_+$.
In order to study the asymptotic {\em stability} of solutions to
\eqref{nacp} on $\bbR_+$, we now prove the spectral mapping theorem for
the operators $T_+^t$ and $\Gamma_+$.
Of particular interest is the consequence that the growth bound
$\omega_0(\{T_+^t\})$ equals the spectral bound $s(\Gamma_+)$.
We begin by observing that the spectrum $\sigma(\Gamma_+)$ is invariant under
translations along $i\bbR$ and the spectrum $\sigma(T^t_+)$ is invariant
under rotations about origin. This spectral symmetry is a consequence of
the fact that for $\xi\in\bbR$,
\begin{equation}\lb{specsym}
T_+^t e^{i\xi\cdot}f=e^{i\xi\cdot} e^{-i\xi t}T_+^t f,
\quad\mbox{and}\quad
\Gamma_+ e^{i\xi\cdot}=e^{\xi\cdot}(\Gamma_+-i\xi).
\end{equation}
To prove the relationship
$e^{t\sigma(\Gamma_+)}=\sigma(T_+^t)\setminus\{0\},\quad t>0$, we employ a
``change-of-variables" technique as used in \cite[Theorem 3.1]{LMS2}. That is,
we exploit the known spectral relationships between a strongly continuous
semigroup and the
evolution semigroup it induces for functions on
$\bbR$. Specifically, let
$\{e^{tA}\}_{t\ge0}$ denote any strongly continuous semigroup generated by $A$
on a
Banach space $E$. Let $F=C_0(\bbR,E)$ or $L_p(\bbR,E)$. Let
$\{S^t\}_{t\ge0}$ be the induced evolution semigroup
$$(S^tf)(\t)=e^{tA}f(\t-t),\quad f\in F, $$
and denote the generator of $\{S^t\}_{t\ge0}$ by $B$. Then
Theorem 2.5 in \cite{LMS2} shows that the following are equivalent:
\newcounter{ii}
\begin{list}{\roman{ii})}{\usecounter{ii}}
\item $\sigma(e^{tA})\cap\bbT=\emptyset$ on $E$;
\item $\sigma(S^t)\cap\bbT=\emptyset$ on $F$;
\item $0\in\rho(B)$ on $F$.
\end{list}
\begin{thm}\lb{SMT+}
Let $F_+$ denote $\Coo$ or $\LpRp$. The spectrum $\sigma(\Gamma_+)$ is
invariant under translations along $i\bbR$, the spectrum $\sigma(T_+^t)$
is invariant under rotations about origin, and
\begin{equation}\lb{SMT+eq}
e^{t\sigma(\Gamma_+)}=\sigma(T_+^t)\setminus\{0\},\quad t>0.
\end{equation}
\end{thm}
\begin{proof}
The arguments for the two cases $F_+=\Coo$ and $F_+=\LpRp$ are similar, so
only the first one is considered here. The inclusion
$e^{t\sigma(\Gamma_+)}\subseteq
\sigma(T_+^t)\setminus\{0\}$ follows from the standard spectral
inclusion for strongly continuous semigroups \cite{Nagel}.
In view of the spectral
symmetry implied by \eqref{specsym}, it suffices to show that
$\sigma(T_+^t)\cap\bbT=\emptyset$ whenever $0\in\rho(\Gamma_+)$.
To this end, we replace the Banach space $E$ by $\Coo$ in the
discussion preceding the theorem and consider the evolution semigroup
$\{\scrS^t\}_{t\ge0}$ on $C_0(\bbR,\Coo)$ induced by $T_+^t$:
\begin{equation}\lb{scrS}
(\scrS^tF)(s)=T_+^t(F(s-t,\cdot)),\quad s\in\bbR,\;t\ge0.
\end{equation}
Denoting the generator of $\{\scrS^t\}_{t\ge0}$ by $\scrB$, we note that
the equivalence of (i)--(iii), above, holds in this setting with $T_+^t$,
$\scrS^t$ and $\scrB$ replacing $e^{tA}$, $S^t$ and $B$, respectively.
Next, consider the semigroup $\{\scrT^t\}_{t\ge0}$ on $C_0(\bbR,\Coo)$
defined by
\[
(\scrT^t F)(s)=T_+^tF(s,\cdot).
\]
The generator of $\{\scrT^t\}_{t\ge0}$ will be denoted by $\calG$.
Note that $\scrT^t$ is the operator of multiplication by $T_+^t$.
Similarly, the generator, $\calG$, of $\{\scrT^t\}_{t\ge0}$ satisfies
$$
(\calG F)(s)=\Gamma_+ F(s,\cdot),\qquad\mbox{where }
F(s,\cdot)\in\calD(\Gamma_+)\mbox{ for }s\in\bbR.
$$
In particular, if $0\in\rho(\Gamma_+)$, then
$\Gamma_+^{-1}(g(s))=(\calG^{-1}g)(s)$, for $g\in C_0(\bbR,\Coo)$, and so
$0\in\rho(\calG)$.
Now define an isometry $\calJ$ on $C_0(\bbR,\Coo)$ by
$(\calJ F)(s,\t)=F(s+\t,\t)$, for $s\in\bbR,\,\t\in\bbR_+$.
This gives the relationship
\begin{equation}\lb{TJ=JS}
(\scrT^t\calJ F)(s,\t)=(\calJ\scrS^tF)(s,\t),\quad
s\in\bbR,\,\t\in\bbR_+,
\end{equation}
and so $\calG\calJ F=\calJ\scrB F$ for $F\in\calD(\scrB)$, and
$\calJ^{-1}\calG F=\scrB\calJ^{-1}F$ for $F\in\calD(\calG)$.
Consequently, $\sigma(\calG)=\sigma(\scrB)$ on $C_0(\bbR,\Coo)$.
We conclude that if
$0\in\rho(\Gamma_+)$, then $0\in\rho(\calG)=\rho(\scrB)$.
The equivalence of (i)--(iii) above (with $T_+^t$, $\scrS^t$, and $\scrB$
on $C_0(\bbR,\Coo)$) implies that $\sigma(T_+^t)\cap\bbT=\emptyset$.
\end{proof}
Evolution semigroups induced by an evolution family as in \eqref{evsg}
have been studied by several authors including R.~Schnaubelt \cite{Roland}
who has characterized such semigroups in terms of their generators on
general Banach function spaces of $X$-valued functions (e.g., $\LpR$ and
$\CoR$). Of particular interest to us is the following theorem of
R.~Schnaubelt \cite{Roland} (see also R\"abiger et al.~\cite{RRS}) which shows
exactly when a strongly continuous semigroup on a Banach function space arises
from a
strongly continuous evolution family on $X$. We state a version of the result
that will be used in Section~3; a more general version is proven in
\cite{Roland}.
\begin{thm}\lb{RRS-evsgThm}
Let $\{T_+^t\}_{t\ge0}$ be a strongly continuous semigroup on $F_+=\LpRp$
generated
by $\Gamma_+$. The following are equivalent:
\setcounter{ii}{0}
\begin{list}{\roman{ii})}{\usecounter{ii}}
\item $\{T_+^t\}_{t\ge0}$ is an evolution semigroup (induced by an
evolution family $\U$);
\item there exists a core, $\scrC$, of $\Gamma_+$ such that for all
$\vphi\in C_c^1(\bbR_+)$, and $f\in\scrC$, it follows that $\vphi f\in
\calD(\Gamma_+)$ and $\Gamma_+(\vphi f)=-\vphi'f+\vphi\Gamma_+ f$. Moreover,
there exists $\lambda\in\rho(\Gamma_+)$ such that
$R(\lambda,\Gamma_+): F_+\to \Coo$ is continuous with dense range.
\end{list}
\end{thm}
%\vfill\eject
%---------------------------- section 2 --------------------------
\section{Exponential Stability}
\setcounter{equation}{0}
\noindent To make use of Theorem \ref{SMT+} in this section, the
following connection between the spectral properties of the
evolution semigroup $\{T_+^t\}_{t\ge0}$ on $F_+$ and the evolution
family $\U$ on $X$ is needed,
$F_+=L_p(\bbR_+,X)$, $1\le p<\infty$, or $F_+=C_{00}(\bbR_+,X)$.
In the case of the whole line, $\bbR$,
the exponential dichotomy of an evolution family $\U$ has been
characterized by the property that the spectrum of the corresponding
evolution semigroup operators does not intersect the unit circle (see
the references in Section~1). For the property of exponential stability
on $\bbR_+$, the argument is straightforward:
\begin{thm}\lb{EST+}
The evolution family $\U$ is exponentially stable if and only if the
growth bound, $\omega_0(\{T_+^t\})$, of the the evolution semigroup
on $F_+$, is less than zero.
\end{thm}
\begin{proof} Let $F_+=\Coo$. If $\U$ is exponentially stable, then there
exist $M>1$,
$\beta>0$ such that $\|U(t,s)\|_{B(X)}\le Me^{-\beta(t-s)}$, $t\ge s$.
For $\t\ge0$ and $f\in\Coo$,
$$
\begin{aligned} \|T_+^\t f\|_{\Coo} = \sup_{t>0}\|T_+^\t f(t)\|_X
&=\sup_{t>\t}\|U(t,t-\t)f(t-\t)\|_X \\
&\le\sup_{t>\t}\|U(t,t-\t)\|_{B(X)} \|f(t-\t)\|_X \\
&\le Me^{-\beta \t} \|f\|_{\Coo}.
\end{aligned}
$$
Conversely, assume there exist $M>1$, $\alpha>0$ such that
$\|T_+^t\|\le Me^{-\alpha t}$, $t\ge 0$. Let $x\in X$,
$\|x\|=1$. For fixed $t> s > 0$, choose $f\in\Coo$ such that
$\|f\|_{\Coo}=1$ and $f(s)=x$. Then,
$$
\begin{aligned}
\|U(t,s)x\|_X=\|U(t,s)f(s)\|_X &=\|T_+^{(t-s)}f(t)\|_X \\
&\le\sup_{\t>0}\|T_+^{(t-s)}f(\t)\|_X\\
&=\|T_+^{(t-s)}f\|_{\Coo}\\
&\le Me^{-\alpha(t-s)}.
\end{aligned}
$$
A similar argument works for $F_+=\LpRp$.
\end{proof}
J.~van Neerven has characterized the exponential
stability of solutions to \eqref{acp} in terms of a convolution operator,
$\hat G$, induced by $\{e^{tA}\}_{t\ge0}$. In this autonomous setting,
$T_+^tf(s)=e^{tA}f(s-t)$, for $s\ge t$, and $0$ otherwise; the convolution
operator takes the form
\begin{equation}\lb{convop}
(\hat Gf)(s):=\int_0^s e^{\t A}f(s-\t)\,d\t
=\int_0^\infty (T_+^\t f)(s)\,d\t ,\quad s\ge 0,
\end{equation}
for $f\in F_+$ (see \cite{vanN1}, Theorem~1.3, and Corollary~2.2).
Specifically, if $\{e^{tA}\}_{t\ge0}$ is a strongly
continuous semigroup on $X$, and $1\le p<\infty$, then the
following are equivalent:
\setcounter{ii}{0}
\begin{list}{\roman{ii})}{\usecounter{ii}}
\item $\omega_0(\{e^{tA}\})<0$;
\item $\hat Gf\in\LpRp$ for all $f\in\LpRp$;
\item $\hat Gf\in\CoRp$ for all $f\in\CoRp$;
%\item $\displaystyle{\sup_{s>0}
% \left\|\int_0^s e^{tA}f(t)\,dt\right\|_X <\infty}$
% for all $f\in\CoRp$.
\end{list}
In this section this result will be extended so that it may be used to
describe exponential stability of the nonautonomous equation
\eqref{nacp}. We define the convolution operator $\hat G$
in a natural way as follows. If $\U$ is an evolution family and
$\{T_+^t\}_{t\ge0}$ is the evolution semigroup defined in \eqref{evsg+},
define $\hat G$ on $F_+$ by:
\begin{align}\hat G f(s):=&\int_0^\infty (T_+^\t f)(s)\,d\t
=\int_0^s U(s,s-\t)f(s-\t)\,d\t\nonumber\\
=&\int_0^s U(s,\t)f(\t)\,d\t,\qquad s\ge0. \label{defGhat}
\end{align}
By standard semigroups properties, $\hat{G}=-\Gamma_+^{-1}$
provided the semigroup $\{T^t_+\}_{t\ge 0}$ or the evolution
family $\U$ is uniformly exponentially stable.
With this setup in mind, the following theorem can be proven
using the technique of \cite[Theorem 1.3]{vanN1}. The
argument here is in fact shorter since, in view of Theorem \ref{SMT+},
we may consider the resolvent, $(\lambda-\Gamma_+)^{-1}$, for only
{\em real} $\lambda$.
\begin{thm}\lb{main} The following are equivalent for an evolution
family of operators $\U$ on $X$.
\setcounter{ii}{0}
\begin{list}{\roman{ii})}{\usecounter{ii}}
\item $\U$ is exponentially stable;
\item $\hat G$ is a bounded operator on $\LpRp$;
\item $\hat G$ is a bounded operator on $\CoRp$.
%\item $\displaystyle{\sup_{s>0}\left\|\int_0^s
%U(s,s-t)f(t)\,dt\right\|<\infty}$ for all $f\in\CoRp$.
\end{list}
\end{thm}
Before proceeding with the proof, note that $(ii)$ is equivalent to the
statement that $\hat Gf\in\LpRp$ for each $f\in\LpRp$. Indeed, an
application of the Closed-Graph Theorem shows (as in \cite{vanN1}) that
if the latter holds, then there exists a $M>0$ such that
$\|\hat G f\|_{\LpRp}\le M\|f\|_{\LpRp}$, for all $f\in\LpRp$.
Similarly, for any $\lambda>0$, the rescaled
semigroup $\{e^{-\lambda t}T_+^t\}_{t\ge0}$ induces a convolution
operator $\hat G_\lambda$, on $\LpRp$,
$$
\hat G_\lambda f(s)=\int_0^\infty (e^{-\lambda t}T_+^\t f)(s)\,d\t=
\int_0^s e^{-\lambda \t}U(s,s-\t)f(s-\t)\,d\t,
$$
with the property that there exists a $M_\lambda>0$ such that
$\|\hat G_\lambda f\|_{\LpRp}\le
M_\lambda\|f\|_{\LpRp}$, for all $f\in\LpRp$.
As for the proof of $(ii)\Rightarrow(i)$,
the idea is to prove the existence of the
operator $(\lambda-\Gamma_+)^{-1}$ for any $\lambda\ge 0$. This shows
that $s(\Gamma_+)<0$ and hence by Theorem \ref{SMT+},
$\omega_0(\{T_+^t\})<0$; Theorem \ref{EST+} then shows that $\U$ is
exponentially stable.
\begin{proof} Beginning with the proof of $(ii)\Rightarrow(i)$, we make
the following definitions: For $T>0$ and $f\in\LpRp$, set
\[
\FTf(\lambda)=\int_0^T e^{-\lambda \t}T_+^\t f\,d\t, \qquad
\lambda\in\bbR_+.
\]
This defines a continuous $\LpRp$-valued function on $\bbR_+$.
For $T_0>0$, and $g\in\LpRp$, let $\piTo(g)$ denote the restriction,
$g|_{[0,T_0]}$. Then for $T\ge T_0$, set
\[
\FTTf=\piTo(\FTf).
\]
For each $\lambda\in\bbR_+$, the map $f\mapsto\FTTfl$ is a bounded
operator from $\LpRp$ to $\LpTo$.
Now consider the growth bound $\omega_0(\{T_+^t\})$. A standard property of
strongly continuous semigroups is that for $\lambda>\omega_0(\{T_+^t\})$,
the resolvent $(\lambda-\Gamma_+)^{-1}$ exists and can be expressed as
\[
(\lambda-\Gamma_+)^{-1}f=\int_0^\infty e^{-\lambda \t}T_+^\t f\,d\t,
\qquad f\in\LpRp.
\]
We now show that this holds for all $\lambda\ge 0$. To this end, fix
$\omega>\omega_0(\{T_+^t\})$. Let $T\ge T_0\ge0$, and consider the operators
$f\mapsto\FTTfl$ for $0\le\lambda\le\omega$. Then for $f\in\Coo\cap\LpRp$
and $0\le s\le T_0$,
\begin{align*}
[\FTf(\lambda)](s)
&=\left(\int_0^T e^{-\lambda \t}T^\t_+f\,d\t\right)(s)\\
&=\int_0^s e^{-\lambda \t}U(s,s-\t)f(s-\t)\,d\t\\
&=(\hat G_\lambda f)(s).
\end{align*}
Therefore,
\[
\begin{aligned}
\|\FTTfl\|_{\LpTo}&=
\left\|{\piTo}\left(\int_0^{(\cdot)}
e^{-\lambda \t}T_+^\t f\,d\t\right)\right\|_{\LpRp}\\
&\le\|\piTo\|\,\|\hat G_\lambda f\|_\LpRp \\
&\le M_\lambda\|f\|_\LpRp
\end{aligned}
\]
for some $M_\lambda>0$ (see the remark preceding the proof).
On the interval $[0,\omega]$, the function $\FTTfl$ is a uniformly
continuous function of $\lambda$, so there exists $K>0$ such that
$\|\FTTfl\|_\LpTo\le K\|f\|_\LpRp$ for all $\lambda\in [0,\omega]$.
Since $\Coo\cap\LpRp$ is dense in $\LpRp$, this holds for all $f\in\LpRp$.
As a consequence, the limit
$$
F_{\infty,T_0,f}(\lambda):=\lim_{T\to\infty}\FTTfl=
\piTo\left(\int_0^\infty e^{-\lambda \t}T_+^\t f\,d\t\right)
$$
exists for all $\lambda\in[0,\omega]$.
Finally, the limit
\[
F_{\infty,\infty,f}(\lambda):=
\lim_{T_0\to\infty}F_{\infty,T_0,f}(\lambda)
=\int_0^\infty e^{-\lambda t}T_+^\t f\,d\t
\]
exists for all $\lambda\ge0$ since for each $T_0$, $\LpTo$ is a closed
subspace of $\LpRp$. Consequently, if
$\lambda\ge 0$, then
$\lambda\in\rho(\Gamma_+)$ and $(\lambda-\Gamma_+)^{-1}=\Fiifl$
\cite[p.~6]{Nagel}. This proves $(ii)\Rightarrow(i)$.
The above argument can be modified, as in
\cite{vanN1}, and used to prove $(iii)\Rightarrow(i)$.
First note that if, as above, $\hat G_\lambda$ denotes the convolution
operator on $\CoRp$ induced by $\{e^{-\lambda t}T^t_+\}_{t\ge0}$
($\lambda>0$), then there exists $M_\lambda>0$ such
that $\|\hat G_\lambda f\|_\CoRp\le M_\lambda\|f\|_\CoRp$ for
$f\in\CoRp$ (this follows from the uniform boundedness principle).
Next, applying the truncation, $\piTo$, to a function in $\Coo$
may not result in a function in $\Coo$, so it is necessary to replace the
operator $\piTo$ by a ``continuous" truncation: for $T_0\ge 1$, redefine
$\piTo$ by the rule
$\piTo f(t)=\alpha_{T_0}(t)f(t)$, for $0\le t\le T_0$, where
\[
\alpha_{T_0}=\begin{cases}1,\quad &0\le t\le T_0-1\\
T_0-t,\quad&T_0-1\le t\le T_0\\ 0, \quad &t\ge T_0.
\end{cases}
\]
Then $\piTo$ is an operator from $\Coo$ to $C_{00}([0,T_0],X)$,
the subspace of functions vanishing on $[T_0,\infty)$; it satisfies
$\|\piTo f\|_{C_{00}([0,T_0],X)}\le\|f\|_\Coo$, for $f\in\Coo$. The
previous argument can now be used to prove $(iii)\Rightarrow(i)$.
Finally, assume that $(i)$ holds.
Then there exist $M>0$, $\beta>0$ such that
$\|U(t,s)\|\le Me^{-\beta(t-s)}$. Therefore,
\[
\|\hat Gf(s)\|=\|\int_0^sU(s,s-\t)f(s-\t)\,d\t\|
\le\int_0^sMe^{-\beta\t}\|f(s-\t)\|\,d\t.
\]
This coincides with the inequality that occurs in the autonomous case;
that is, the above inequality holds if $\|e^{tA}\|\le Me^{-\beta t}$
for some $M,\beta>0$.
Therefore, the proofs of $(i)\Rightarrow(ii)$ and $(i)\Rightarrow(iii)$ will
follow exactly as in \cite{vanN1}.
\end{proof}
This theorem makes explicit, in the case of
the half line $\bbR_+$, the relationship between the stability of an
evolution family $\U$ and the generator, $\Gamma_+$, of the
corresponding evolution semigroup \eqref{evsg+}. Indeed, as shown
above, stability is equivalent to the boundedness of $\hat G$, in which
case $\hat G=-\Gamma_+^{-1}$. Combining Theorems \ref{EST+} and
\ref{main} yields:
\begin{cor}\lb{vanNCor}
Let $\U$ be an exponentially bounded evolution family and let $\Gamma_+$
denote the generator of the induced evolution semigroup
on $F_+=L_p(\bbR_+,X)$, $1\le p<\infty$
or $F_+=C_{00}(\bbR_+,X)$.
The following are
equivalent:
\setcounter{ii}{0}
\begin{list}{\roman{ii})}{\usecounter{ii}}
\item $\U$ is exponentially stable;
\item $\Gamma_+$ is invertible;
\item $s(\Gamma_+)<0$.
\end{list}
\end{cor}
%\vfill\eject
%---------------------------- section 3 --------------------------
\section{Perturbations: the stability radius}
\setcounter{equation}{0}
\noindent In this section we consider
perturbations of \eqref{nacp} of the form
\begin{equation}\lb{nacp+B}
x'(t)=(A(t)+B(t))x(t), \quad t\in\bbR_+,
\end{equation}
with initial condition $x(s)=x\in \calD(A(s))$, $s\le t$.
As mentioned in the introduction, we do not
restrict attention to differentiable solutions, and so we do not assume
that either \eqref{nacp} or \eqref{nacp+B} are well-posed.
Instead we begin with an evolution family $\U$ and, under appropriate
hypotheses, prove the existence of a perturbed
evolution {\em semigroup} from which the stability properties of
the corresponding perturbed evolution family, $\Uone$, are
determined.
%Our approach parallels, yet differs slightly from, that of
%\cite{HP94} where the question of existence and uniqueness
%(of a mild solution) is treated
%jointly with the problem of exponential stability.
If $X$ and $Y$ are Banach spaces, let $B_s(X,Y)$ denote the set of
bounded operators from $X$ to $Y$ endowed with the strong operator
topology, and let $C_b(\bbR_+,B_s(X,Y))$ denote the set of strongly
continuous functions, $B(\cdot)\colon\bbR_+\to B(X,Y)$, which are bounded
on $\bbR_+$; the norm is defined by
$\|B(\cdot)\|_\infty=\sup_{t\ge0}\|B(t)\|_{B(X,Y)}$. Given a function
$B(\cdot)\in C_b(\bbR_+,B_s(X,Y))$, a calligraphic letter will be used to
denote the multiplication operator $\calB\colon\LpRp\to\LpRY$
defined by $\calB f(t)=B(t)f(t)$. The map $B(\cdot)\mapsto\calB$ is an
isometric embedding from
\[C_b(\bbR_+,B_s(X,Y))\quad\mbox{ into }\quad B(\LpRp,\LpRY).\]
We begin in Subsection 3.1 with some observations concerning bound\-ed
perturbations of \eqref{nacp} of the form \eqref{nacp+B} where
$B(\cdot)\in C_b(\bbR_+,B_s(X))$. For such perturbations,
\eqref{nacp+B} has a
mild solution in the sense that there exists a unique evolution
family $\Uone=\{U_1(t,s)\}_{t\ge s}$ satisfying
\begin{equation}\lb{varpar}
U_1(t,s)x=U(t,s)x+\int_s^tU(t,\t)B(\t)U_1(\t,s)x\,d\t.
\end{equation}
for all $x\in X$. A byproduct of our approach is immediate information
concerning stability and its robustness for such solutions, and a
straightforward algebraic argument which identifies bounds for the
stability radius.
Assume, for a moment, that $\U$ is an exponentially stable evolution
family corresponding to solutions of \eqref{nacp}.
Let $X$, $U$ and
$Y$ denote Banach spaces.
Let $\Delta(\cdot)\in C_b(\bbR_+,B(Y,U))$
denote a ``disturbance" operator. Let
\[ E(\cdot)\in C_b(\bbR_+,B(X,Y))\quad\mbox{ and }\quad
D(\cdot)\in C_b(\bbR_+,B(U,X))\]
denote given operators describing the ``structure" of the perturbation
of \eqref{nacp}. Then \eqref{nacp+B} takes the form
\begin{equation}\lb{nacp+DDelE}
x'(t)=[A(t)+D(t)\Delta(t)E(t)]x(t),\quad t\ge0.
\end{equation}
The stability radius measures the size of the largest
disturbance, $\Delta(\cdot)$, such that solutions (possibly,
in the ``mild sense'') to equation
\eqref{nacp+DDelE} remain stable. Our next goal is
to obtain upper and lower bounds for this quantity. For the
nonautonomous situation some general facts are obtained in
Subsection~3.1. More specific results for autonomous problems
are considered in Subsection~3.2.
We will drop the subscript ``+" when referring to
the space $F=\LpRp$ and when referring to
the evolution semigroup \eqref{evsg+} or its generator. Hence, if $\U$
is an evolution family on $X$, the evolution semigroup \eqref{evsg+} and
its generator on $F=\LpRp$ will be denoted simply by $\{T^t\}_{t\ge0}$ and
$\Gamma$.
%--------------------------------------
\subsection{Nonautonomous case}
Let $\U$ be an evolution family on $X$ and let $\Gamma$ be the generator
of the corresponding evolution semigroup.
If $B(\cdot)\in C_b(\bbR_+,B_s(X))$, then the multiplication
operator $\calB$ is a bounded operator on $F$ and so the
operator $\Gamma_1=\Gamma+\calB$ generates a strongly continuous
semigroup, $\{T_1^t\}_{t\ge0}$ on $F$ (see, e.g., \cite{Pazy}).
Moreover, the following simple fact holds.
\begin{prop}\lb{GammaRobustStab} Let $\U=\{U(t,s)\}_{t\ge s}$ be an
evolution family, and $B(\cdot)\in C_b(\bbR_+,B_s(X))$. Then there
exists a unique evolution family $\Uone=\{U_1(t,s)\}_{t\ge s}$ which
solves the integral equation \eqref{varpar}. Moreover, $\Uone$ is
exponentially stable if and only if $\Gamma+\calB$ is invertible.
\end{prop}
\begin{proof} As already observed,
$\Gamma_1:=\Gamma+\calB$ generates a strongly continuous semigroup,
$\{T_1^t\}_{t\ge0}$ on $F$. To see that this is, in fact, an
{\em evolution semigroup},
recall Theorem \ref{RRS-evsgThm} and note that for
$\lambda\in\rho(\Gamma)\cap\rho(\Gamma_1)$,
\[\Range(R(\lambda,\Gamma))=D(\Gamma)=D(\Gamma+\calB)
=\Range(R(\lambda,\Gamma_1))\] is dense in $\CoRp$.
Also, if $\scrC$
is a core for $\Gamma$, then it is a core for $\Gamma_1$, and so
for $\vphi\in C_c^1(\bbR)$, $f\in\scrC$,
\[
\Gamma_1(\vphi f)=\Gamma(\vphi f)+\calB(\vphi f)=
-\vphi'f+\vphi\Gamma f+\vphi \calB f
=-\vphi'f+\vphi(\Gamma_A+\calB)f.
\]
Consequently, $\{T_1^t\}_{t\ge0}$ corresponds to an
evolutionary family, $\Uone=\{U_1(t,s)\}_{t\ge s}$, and
$x(t)=U_1(t,s)x(s)$ is
seen to define a mild solution to \eqref{nacp+B}. Indeed,
\begin{equation}\lb{varparF}
T_1^tf=T_0^tf+\int_0^tT_0^{(t-\t)}\calB T_1^\t f \,d\t,
\end{equation}
holds for all $f\in F$. In particular, for $x\in X$, and any
$\vphi\in C_c^1(\bbR)$, setting $f=\vphi\otimes x$ in \eqref{varparF}
%(where $\vphi\otimes x(t)=\vphi(t)x$)
and using a change of variables leads to
\[
\vphi(s)U_1(t,s)x=\vphi(s)U(t,s)x+
\vphi(s)\int_s^tU(t,\t)B(\t)U_1(\t,s)x\,d\t.
\]
Therefore, \eqref{varpar} holds for all $x\in X$.
Finally, Theorem \ref{EST+} shows that $\Uone$ is exponentially stable if
and only if $\Gamma_1$ is invertible.
\end{proof}
The existence of mild solutions under bounded perturbations of this type
is well known (see, e.g., \cite{CP}), but an immediate consequence of the
approach given here is the property of robustness for the stability of
$\U$. Indeed, by continuity properties of the spectrum of an operator
$\Gamma$, there exists $\epsilon>0$ such that $\Gamma_1$ is invertible
whenever $\|\Gamma_1-\Gamma\|<\epsilon$; that is, $\Uone$ is exponentially
stable whenever $\|B(\cdot)\|_\infty<\epsilon$. Moreover, this approach
provides insight into the concept of the stability radius of a controlled
system.
Let $\U$ be an evolution family on $X$.
Fix $D(\cdot)\in C_b(\bbR_+,B_s(U,X))$, and
$E(\cdot)\in C_b(\bbR_+,B_s(X,Y))$.
For any $\Delta(\cdot)\in C_b(\bbR_+,B_s(Y,U))$, we consider the
perturbation $B(t)=D(t)\Delta(t)E(t)$.
The {\em stability radius} of an exponentially
stable evolution family $\U$ with respect to the perturbation structure
$D(\cdot)$ and $E(\cdot)$ is defined as
\begin{equation*}
\begin{aligned}
\rstab(\U,D,E)=\sup\{r>0&: \Delta(\cdot)\in C_b(\bbR_+,B_s(Y,U)),\ \
\|\Delta(\cdot)\|_\infty\le r \\
&\mbox{ implies } \Uone \mbox{ is exponentially stable}\}.
\end{aligned}
\end{equation*}
Here, $\Uone=\{U_1(t,s)\}_{t\ge s}$ is the perturbed evolution family as in
Proposition \ref{GammaRobustStab} with $B(t)=D(t)\Delta(t)E(t)$.
In view of this proposition, we have
\begin{equation*}
\begin{aligned}
\rstab(\U,D,E)=\sup\{r>0&: \Delta(\cdot)\in
C_b(\bbR_+,B_s(Y,U)),\ \ \|\Delta(\cdot)\|_\infty\le r \\
&\mbox{ implies } \Gamma+\calD\calDelta\calE \mbox{ is invertible}\}.
\end{aligned}
\end{equation*}
We will also need to consider
the {\em constant stability radius}, denoted by
$\rcstab(\{e^{tA}\},D,E)$, which is
defined in the same way as $\rstab(\{e^{tA}\},D,E)$ except that $\Delta$
does not depend on $t$.
For an exponentially stable evolution family $\U$ consider the
``input-output" operator $\bbL_0:\LpRU\to\LpRY$ defined as:
\[
(\bbL_0u)(t)=E(t)\int_0^t U(t,\t)D(\t)u(\t)\, d\t,\quad u\in\LpRU.
\]
As shown by the next theorem, the norm of this operator provides a
lower bound for $\rstab(\U,D,E)$. (The inequality
\eqref{L0Bound}, below, is proven in \cite[Theorem 3.2]{HP94}
using a completely different approach.)
\begin{thm}\lb{L0BoundThm}
Let $\U$ be an exponentially stable evolution family.
Let $D(\cdot)\in C_b(\bbR_+,B_s(U,X))$ and
$E(\cdot)\in C_b(\bbR_+,B_s(X,Y))$.
Then $$\bbL_0=\calE\hat G\calD=-\calE\Gamma^{-1}\calD$$ and
\begin{equation}\lb{L0Bound}
\frac{1}{\|\bbL_0\|}\le \rstab(\U,D,E).
\end{equation}
In the ``unstructured" case, where $U=Y=X$ and $D=E=I$, one has
\[\bbL_0=-\Gamma^{-1},\quad\mbox{ and }\quad
\frac{1}{\|\Gamma^{-1}\|} \le \rstab(\U,I,I) \le
\frac{1}{r(\Gamma^{-1})},
\]
where $r(\cdot)$ denotes the spectral radius.
\end{thm}
\begin{proof} Since $\U$ is exponentially stable,
$\Gamma$ is invertible and $\Gamma^{-1}=\hat{G}$.
The required formula for $\bbL_0$ follows from \eqref{defGhat}.
Set $\calH:=\Gamma^{-1}\calD\calDelta$. To prove \eqref{L0Bound},
let $\Delta(\cdot)\in C_b(\bbR_+,B_s(Y,U))$ and
suppose that $\|\Delta(\cdot)\|< 1/\|\bbL_0\|$. Then
$\|\bbL_0\calDelta\|<1$, and hence
$I-\bbL_0\calDelta=I+\calE\Gamma^{-1}\calD\calDelta$ is invertible
on $\LpRY$. That is,
$I+\calE\calH$ is invertible on $\LpRY$, and hence
$I+\calH\calE$ is invertible on $\LpRX$ (with inverse
$(I-\calH(I+\calE\calH)^{-1}\calE)$). Now,
\[
\Gamma+\calD\calDelta\calE
=\Gamma(I+\Gamma^{-1}\calD\calDelta\calE)
=\Gamma(I+\calH\calE)
\]
and so $\Gamma+\calD\calDelta\calE$ is
invertible. Since $\Gamma+\calD\calDelta\calE$ generates the
evolutionary semigroup corresponding to $U_1(\cdot,\cdot)$, it follows
that $1/\|\bbL_0\|\le \rstab(\U,D,E)$.
For the last assertion, suppose that
$\rstab(\U,I,I)>1/r(\Gamma^{-1})$. Then there exists $\lambda$ such
that $|\lambda|=r(\Gamma^{-1})$ and
$\lambda+\Gamma^{-1}$ is not invertible.
But then setting $\calDelta\equiv\frac{1}{\lambda}$ gives
$\|\calDelta\|=\frac{1}{|\lambda|}<\rstab(\U,I,I)$, and so
$\Gamma+\calDelta=\calDelta(\lambda+\Gamma^{-1})\Gamma$ is invertible,
a contradiction.
\end{proof}
The nonautonomous scalar equation given by Example 4.4 of \cite{HIP} shows
that, in general, the inequality in \eqref{L0Bound} is strict.
%-----------------------------------------
\subsection{Autonomous case}
The development in this subsection is motivated by considering
the autonomous equation \eqref{acp} and constant perturbation
operators $D\in B(U,X)$ and $E\in B(X,Y)$:
\[
x'(t)=(A+D\Delta(t) E)x(t),\quad t\ge0.
\]
The evolution family $\U$ is now replaced by a semigroup
$\{e^{tA}\}_{t\ge0}$ generated by $A$.
The primary focus is on properties of the constant stability
radius, in which case $\Delta$ is independent of $t$.
When $U$ and $Y$ are Hilbert spaces it is known
\cite[Theorem 3.5]{HP94} that
\begin{equation}\lb{rcstabHilb}
\frac{1}{\|\bbL_0\|}=\rcstab(\{e^{tA}\},D,E)=
\frac{1}{\sup_{s\in\bbR}\|E(A-is)^{-1}D\|}.
\end{equation}
In this section we obtain explicit formulas for upper and lower bounds on
the constant
stability radius when $U$ and $Y$ are Banach spaces. The development
uses a generalization of the spectral mapping theorem of L.~Gearhart.
Gearhart's theorem (see \cite[p.~95]{Nagel}) says: if $A$ generates a $C_0$
semigroup $\{e^{tA}\}_{t\ge0}$ on a Hilbert space, then
{\it $1\in \rho(e^{2\pi A})$ if and only if $i\bbZ \subset\rho (A)$
and $\sup_{k\in \bbZ} \| (A-ik)^{-1}\|< \infty$\/}.
A generalization (see \cite{LMS2}) of this result for Banach spaces is stated
next. Here,
$\Fper$ denotes the Banach space $L_p([0,2\pi],X)$, $1\le
p<\infty$.
If $\{e^{tA}\}_{t\ge0}$ is a strongly continuous
semigroup on $X$, $\{T_{per}^t\}_{t\ge0}$
will denote the evolution semigroup defined on $\Fper$ by the rule
$T^t_{per}f(s)=e^{tA}f([s-t](\mod 2\pi))$; its generator will be
denoted by $\Gammaper$. The symbol
$\Lambda$ will be used to denote the set of all finite sequences
$\{v_k\}_{k=-N}^N$ in $X$, or $\{u_k\}_{k=-N}^N$ in $U$.
\begin{thm}\lb{LMS-Gearhart+DDeltaE}
Let $A$ generate a $C_0$ semigroup $\{e^{tA}\}_{t\ge0}$ on
$X$. Let $D$ and $E$ be as above, and $\Delta\in B(Y,U)$. Let
$\{e^{t(A+D\Delta E)}\}_{t\ge0}$ be the strongly continuous
semigroup generated by
$A+D\Delta
E$. Then the following are equivalent:
\setcounter{ii}{0}
\begin{list}{\roman{ii})}{\usecounter{ii}}
\item $1\in \rho (e^{2\pi (A+D\Delta E)})$;
\item $i\bbZ \subset \rho (A+D\Delta E)$ \quad and
\[\disp{ \sup_{\{v_k\}\in\Lambda}
\frac{\|\sum_k(A-ik+D\Delta E)^{-1}
v_ke^{ik(\cdot)}\|_\Fper}{\|\sum_k v_k
e^{ik(\cdot)}\|_\Fper} <\infty;}\]
\item $i\bbZ \subset \rho (A+D\Delta E)$ \quad and
\[\disp{\inf_{\{v_k\}\in\Lambda}
\frac{\|\sum_k(A-ik+D\Delta E)v_ke^{ik(\cdot)}\|_\Fper}{\|\sum_k v_k
e^{ik(\cdot)}\|_\Fper} >0.}\]
\end{list}
Further, if $\Gammaper$ denotes the generator of the evolution semigroup on
$\Fper$, as above, and if $1\in\rho(e^{2\pi A})$, then $\Gammaper$ is
invertible and
\begin{equation}\lb{NormEGD}
\|\calE\Gammaper^{-1}\calD\|=
\sup_{\{u_k\}\in\Lambda}
\frac{\|\sum_k E(A-ik)^{-1}Du_ke^{ik(\cdot)}\|_{L_p(0,2\pi],Y)}}
{\|\sum_k u_ke^{ik(\cdot)}\|_{L_p([0,2\pi],U)}}
\end{equation}
where $\calE\Gammaper^{-1}\calD\in B(L_p([0,2\pi],U),L_p([0,2\pi],Y))$.
\end{thm}
\begin{proof} The equivalence of {\em (i)--(iii)} follows as in
Theorem 2.3 of \cite{LMS2}. For the last statement, let $\{u_k\}$ be a
finite set in $U$ and consider functions
$f$ and $g$ of the form
\[f(s)=\sum_k(A-ik)^{-1}Du_ke^{iks},\quad\mbox{and}\quad
g(s)=\sum_kDu_ke^{iks}.\]
Then $f=\Gammaper^{-1}g$. For,
\begin{align*}
(\Gammaper f)(s)&=\left.\frac{d}{dt}\right|_{t=0}e^{tA}f([s-t]\mbox{mod}2\pi)\\
&= \sum_k [A(A-ik)^{-1}Du_ke^{iks}-ik(A-ik)^{-1}Du_ke^{iks}]=g(s).
\end{align*}
For functions of the form $h(s)=\sum_k u_ke^{iks}$, where
$\{u_k\}_k$ is a finite set in $U$, we have
$\calE\Gamma^{-1}\calD h=\sum_kE(A-ik)^{-1}Du_ke^{ik(\cdot)}$. Taking the
supremum over all such functions gives:
\begin{align*}
\|\calE\Gammaper^{-1}\calD\|&=
\sup_{h}\frac{\|\calE\Gammaper^{-1}\calD h\|}{\|h\|}\\
&=\sup_{\{u_k\}\in\Lambda}
\frac{\|\sum_k E(A-ik)^{-1}Du_ke^{ik(\cdot)}\|_{L_p([0,2\pi],Y)}}
{\|\sum_k u_ke^{ik(\cdot)}\|_{L_p([0,2\pi],U)}}.
\end{align*}
\end{proof}
In view of these facts we introduce a ``pointwise" variant of the
constant stability radius: for $t_0>0$ and $\lambda\in\rho(e^{t_0 A})$,
define
\[
\rcstab^\lambda(e^{t_0 A},D,E):=\sup\{r>0: \|\Delta\|_{B(Y,U)}\le r
\Rightarrow \lambda\in\rho(e^{t_0(A+D\Delta E)})\}.
\]
By rescaling,
the study of this quantity can be reduced to the case of $\lambda=1$ and
$t_0=2\pi$. Indeed,
\[\rcstab^\lambda(e^{t_0 A},D,E)=
\frac{2\pi}{t_0}\rcstab^\lambda(e^{2\pi C},D,E),\quad\mbox{
where }\quad C=\frac{t_0}{2\pi}A.\]
Also, after writing
$\lambda=|\lambda|e^{i\theta}$
($\theta\in\bbR$), note that
\[\rcstab^\lambda(e^{2\pi A},D,E)=\rcstab^1(e^{2\pi B},D,E)\quad
\mbox{ for }\quad
B=A-\frac{1}{2\pi}(\ln|\lambda|+i\theta).\]
Therefore,
\[
\rcstab^\lambda(e^{t_0 A},D,E)=\frac{2\pi}{t_0}\rcstab^1(e^{2\pi B},D,E),
\]
where
\[
B=\frac{1}{2\pi}(t_0A-\ln|\lambda|-i\theta)).
\]
In the following theorem we estimate
$\rcstab^1(e^{2\pi A},D,E)$.
\begin{thm}\lb{PtwiseBounds}
Let $\{e^{tA}\}_{t\ge0}$ be a strongly continuous semigroup generated
by $A$ on $X$, and assume $1\in\rho(e^{2\pi A})$. Let $\Gammaper$ denote the
generator of the induced evolution semigroup on $\Fper$.
Let $D\in B(U,X)$, and $E\in
B(X,Y)$. Then
\begin{equation}\lb{rstab1Bounds}
\frac{1}{\|\calE\Gammaper^{-1}\calD\|}
\le \rcstab^1(e^{2\pi A},D,E)
\le \frac{1}{\sup_{k\in\bbZ}\|E(A-ik)^{-1}D\|}.
\end{equation}
If $U$ and $Y$ are a Hilbert spaces and $p=2$, then equalities hold in
\eqref{rstab1Bounds}.
\end{thm}
\begin{proof} The first inequality follows from an argument as in Theorem
\ref{L0BoundThm}. For the second inequality, let $\epsilon>0$, and
choose $\bar u\in U$ with $\|\bar u\|=1$ and $k_0\in\bbZ$ such that
\[
\|E(A-ik_0)^{-1}D\bar{u}\|_Y\ge
\sup_{k\in\bbZ}\|E(A-ik)^{-1}D\|-\epsilon>0.
\]
Choose $y^*\in Y^*$ with $\|y^*\|\le 1$ such that
\[
\left\langle y^*,\frac{E(A-ik_0)^{-1}D\bar{u}}
{\|E(A-ik_0)^{-1}D\bar{u}\|_Y}\right\rangle=1.
\]
Define $\Delta\in B(Y,U)$ by
\[
\Delta y =
-\frac{\langle y^*,y\rangle}{\|E(A-ik_0)^{-1}D\bar u\|_Y}\bar{u},
\quad y\in Y.
\]
We note that
\begin{equation}\lb{eqnA}
\Delta E(A-ik_0)^{-1}D\bar u=-\frac{\langle y^*,E(A-ik_0)^{-1}D\bar u\rangle}
{\|E(A-ik_0)^{-1}D\bar{u}\|_Y}\bar u=-\bar{u},
\end{equation}
and
\begin{equation}\lb{eqnB}
\|\Delta\|\le\frac{1}{\|E(A-ik_0)^{-1}D\bar{u}\|_Y}
\le\frac{1}{\sup_{k\in\bbZ}\|E(A-ik_0)^{-1}D\bar{u}\|_Y-\epsilon}\, .
\end{equation}
Now set $\bar{v}:= (A-ik_0)^{-1}D\bar u$ in $X$. By \eqref{eqnA},
$\Delta E\bar{v}=-\bar{u}$, and so
\[ (A-ik_0+D\Delta E)\bar{v}
=(A-ik_0)\bar{v}+D\Delta E\bar{v}=D\bar{u}- D\bar{u}=0.\]
Therefore,
\begin{multline*}
\inf_{\{v_k\}\in\Lambda}\frac{\|\sum_k(A-ik+D\Delta E)
v_ke^{ik(\cdot)}\|_\Fper}{\|\sum_ku_ke^{ik(\cdot)}\|_\Fper}\\
\le \frac{\|(A-ik_0+D\Delta E)
\bar{v}e^{ik_0(\cdot)}\|_\Fper}{\|\bar{v}e^{ik_0(\cdot)}\|_\Fper} =0.
\end{multline*}
By Theorem \ref{LMS-Gearhart+DDeltaE}, $1\notin\rho(e^{2\pi(A+D\Delta
E)})$. This shows that \[\rcstab^1(e^{2\pi A},D,E)\le\|\Delta\|.\]
To finish the proof, suppose that
\[
\rcstab^1(e^{2\pi A},D,E)>
\frac{1}{\sup_{k\in\bbZ}\|E(A-ik)^{-1}D\|}.
\]
Then $\epsilon>0$ can be chosen sufficiently small so that for
\[
r:=\frac{1}{\sup_{k\in\bbZ}\|E(A-ik_0)^{-1}D\bar{u}\|_Y-\epsilon}\, ,
\]
one has
\[ \frac{1}{\sup_{k\in\bbZ}\|E(A-ik_0)^{-1}D\bar{u}\|_Y}< r <
\rcstab^1(e^{2\pi A},D,E).
\]
But then by \eqref{eqnB}, $\|\Delta\|\le r < \rcstab^1(e^{2\pi A},D,E)$,
which is a contradiction.
For the last statement of the theorem, note that Parseval's formula applied to
\eqref{NormEGD} gives
\[
\|\calE\Gammaper^{-1}\calD\|=\sup_{\{u_k\}\in\Lambda}
\frac{\left(\sum_k\| E(A-ik)^{-1}Du_k\|^2_Y\right)^{1/2}}
{\left(\sum_k \|u_k\|^2_U\right)^{1/2}}
\le \sup_{k\in\bbZ}\|E(A-ik)^{-1}D\|.
\]
Therefore,
\[
\frac{1}{\|\calE\Gammaper^{-1}\calD\|}\ge
\frac{1}{\sup_{k\in\bbZ}\|E(A-ik)^{-1}D\|}
\]
and hence equalities hold in \eqref{rstab1Bounds}.
\end{proof}
%% inserted "dichotomy radius" def.
Next we consider the following ``hyperbolic'' variant of the constant
stability radius. Recall, that a strongly continuous semigroup
$\{e^{tA}\}_{t\ge 0}$ on $X$ is called {\it hyperbolic} if
\[
\sigma(e^{tA})\cap\bbT=\emptyset,\quad\mbox{ where }
\bbT=\{z\in\bbC: |z|=1\},
\]
for some (and, hence, for all) $t>0$ (see, e.g., \cite{vanN2}).
The hyperbolic semigroups are those for which the differential
equation $x'=Ax$ has exponential dichotomy (see, e.g., \cite{BAG,DK})
with the dichotomy projection $P$ being the Riesz projection for
the operator, say, $e^A$ that corresponds to the part of its spectrum
contained in the open unit disc.
For a given hyperbolic semigroup $\{e^{tA}\}_{t\ge 0}$ and
operators $D$, $E$ we define
the {\it constant dichotomy radius} as follows:
\begin{equation*}
\begin{aligned}
rc_{dich}(\{e^{tA}\},D,E):= \sup\{
r>0&: \|\Delta\|_{B(Y,U)}\le r \mbox{ implies }\\
&\sigma(e^{t(A+D\Delta E)})\cap\bbT=\emptyset
\mbox{ for all } t>0\}.
\end{aligned}
\end{equation*}
The dichotomy radius measures the size of the smallest $\Delta$
for which the perturbed equation $x'=[A+D\Delta E]x$ looses the
exponential dichotomy.
Now for any $\xi\in[0,1]$, consider the rescaled semigroup generated by
$A_\xi:=A-i\xi$ consisting of operators $e^{tA_\xi}=e^{-i\xi t}e^{tA}$,
$t\ge0$. The pointwise stability radius can be related to the
dichotomy radius as follows.
\begin{lem}\label{dichrad}
Let $\{e^{tA}\}_{t\ge0}$ be a hyperbolic semigroup.
Then
\[
rc_{dich}(\{e^{tA}\},D,E)=\inf_{\xi\in[0,1]}\rcstab^1(e^{2\pi A_\xi},D,E).
\]
\end{lem}
\begin{proof} Denote the left-hand side by $\alpha$ and the right-hand
side by $\beta$. First fix $r<\beta$. Let $\xi\in[0,1]$. If
$\|\Delta\|\le r$, then $1\in\rho(e^{2\pi(A_\xi+D\Delta E)})$ and so
$e^{i\xi 2\pi}\in\rho(e^{2\pi(A+D\Delta E)})$
for all $\xi\in[0,1]$. That is, $e^{is}\in\rho(e^{2\pi(A+D\Delta E)})$
for all $s\in\bbR$, and so $\sigma(e^{2\pi(A+D\Delta E)})\cap\bbT
=\emptyset$. This shows that $r\le\alpha$, and so
$\beta\le\alpha$.
Now suppose $r<\alpha$. If $\|\Delta\|\le r$, then
$\sigma(\{e^{t(A+D\Delta E)}\})\cap\bbT=\emptyset$, and so
$e^{i\xi t}\in\rho(e^{t(A+D\Delta E)})$ for all $\xi\in[0,1], t\in\bbR$.
That is, $1\in\rho(e^{t(A_\xi+D\Delta E)})$.
This says $r\le \beta$ and so $\alpha\le \beta$.
\end{proof}
%% inserted new proposition
Under the additional assumption that the semigroup
$\{e^{tA}\}_{t\ge 0}$ is exponentially stable
(that is, hyperbolic with a trivial dichotomy projection $P=I$),
Lemma~\ref{dichrad} gives, in fact, a formula for the constant {\it
stability} radius. Indeed, the following simple proposition holds.
\begin{prop}\label{dichst}
Let $\{e^{tA}\}_{t\ge0}$ be an exponentially stable semigroup.
Then
\[
rc_{dich}(\{e^{tA}\},D,E)= rc_{stab}(\{e^{tA}\},D,E).
\]
\end{prop}
\begin{proof} Denote the left-hand side by $\alpha$
and the right-hand side by $\beta$. Take $r<\beta$ and any $\Delta$
with $\|\Delta\|\le r$. By definition of the constant stability
radius, $\omega_0(\{e^{t(A+D\Delta E)}\})<0$. In particular,
$\sigma(e^{t(A+D\Delta E)})\cap\bbT=\emptyset$, and $r\le \alpha$ shows
that $\beta\le\alpha$.
Suppose that $\beta\omega\}\subset\rho(A)$ and
$\sup\{\|R(\lambda,A)\|:{\mbox{Re}{\lambda}>\omega}\}<\infty$.
If $s(A)$ and $\omega_0(A)$ denote the spectral bound
and growth bound of the semigroup, respectively,
then it is always the case that $s(A)\le s_0(A)\le \omega_0(A)$.
%In the case that $A$ generates a positive semigroup,
%then $s(A)\le s_0(A)$ (see \cite{vanN2}).
An example due to W.~Arendt (see \cite{vanN2}, Example 1.4.5) exhibits a
(positive) strongly continuous
semigroup with the property that $s_0(A)<\omega_0(A)<0$. Now,
for $\alpha$ such that $0\le\alpha\le-\omega_0(A)$, consider a rescaled
semigroup generated by $A+\alpha$, and denote by $\Gamma_{A+\alpha}$ the
generator of the induced evolution semigroup on $\LpRp$.
The following relationships hold:
\noindent for $0\le\alpha<-\omega_0(A)$,
\[
s_0(A+\alpha)=s_0(A)+\alpha<\omega_0(A)+\alpha=\omega_0(A+\alpha)<0;
\]
for $\alpha_0:=-\omega_0(A)$,
\[
s_0(A+\alpha_0)<\omega_0(A+\alpha_0)=0.
\]
This says that $s_0(A+\alpha)<0$ for all $\alpha\in[0,\alpha_0]$ and hence
\[
M:=\sup_{\alpha\in[0,\alpha_0]}\sup_{s\in\bbR}\|(A+\alpha-is)^{-1}\|<\infty.
\]
Now note that $\omega_0(A+\alpha)<0$ if and only if
$\|\Gamma_{A+\alpha}^{-1}\|<\infty$, and $\|\Gamma_{A+\alpha}^{-1}\|$ is a
continuous function of $\alpha$ on $[0,\alpha_0)$. Therefore, there exists
$\alpha_1 \in [0,\alpha_0)$ such that $\|\Gamma_{A+\alpha_1}^{-1}\|>M$, and
so at least one of the following inequalities is strict:
\[
\frac{1}{\|\bbL_0\|}=
\frac{1}{\|\Gamma_{A+\alpha_1}^{-1}\|}\le\rcstab(\{e^{t(A+\alpha_1)}\},I,I)
\le\frac{1}{\sup_{s\in\bbR}\|(A+\alpha_1-is)^{-1}\|}.
\]
\end{exmp}
{\scshape Acknowledgments:} The authors would like to thank Professor
S.~Clark,
University of Missouri-Rolla, for several helpful discussions.
%%----------------------------bibliography------------------------
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\end{document}
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