\LARGE \textcolor{red
Andrii Mironchenko$^{1
For bilinear infinite-dimensional dynamical systems, we show the equivalence between uniform global asymptotic stability and integral input-to-state stability. We provide two proofs of this fact. One applies to general systems over Banach spaces. The other is restricted to Hilbert spaces, but is more constructive and results in an explicit form of iISS Lyapunov functions.
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%\title{\LARGE \bf \textcolor{red}{Need to be changed!} 
%Integral input-to-state stability of bilinear infinite-dimensional systems* }
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%\author{Andrii Mironchenko$^{1}$ and Hiroshi Ito$^{2}$ % <-this % stops a space
%\thanks{*This work was supported by Japan Society for Promotion of Science (JSPS).}% <-this % stops a space
%\thanks{$^{1}$ Andrii Mironchenko is with Department of Systems Design and Informatics, Kyushu Institute of Technology, 
%680-4 Kawazu, Iizuka, Fukuoka 820-8502, Japan 
        %{\tt\small andrii.mironchenko@mathematik.uni-wuerzburg.de}}%
%\thanks{$^{2}$				
%Hiroshi Ito is with Department of Systems Design and Informatics, Kyushu Institute of Technology, 
%680-4 Kawazu, Iizuka, Fukuoka 820-8502, Japan 
        %{\tt\small hiroshi@ces.kyutech.ac.jp}}%
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%\begin{abstract}
%For bilinear infinite-dimensional dynamical systems, 
%we show the equivalence between uniform global asymptotic stability and integral input-to-state stability. We provide two proofs of this fact. One applies to general systems over Banach spaces. The other is restricted to Hilbert spaces, but is more constructive and results in an explicit form of iISS Lyapunov functions.  
%\end{abstract}
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%\textbf{Keywords}: bilinear systems, infinite-dimensional systems, integral input-to-state stability, Lyapunov methods.
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%%bilinear systems \and infinite-dimensional systems \and integral input-to-state stability \and Lyapunov methods 
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%% Place the running title of the paper with 40 letters or less in []
 %% and the full title of the paper in { }.
\title[\emph{i}ISS of bilinear systems] %Use the shortened version of the full title
      {Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions}

% Place all authors' names in [ ] shown as running head, Leave { } empty
% Please use `and' to connect the last two names if applicable
% Use FirstNameInitial.  MiddleNameInitial. LastName, or only last names of authors if there are too many authors
\author[Andrii Mironchenko and Hiroshi Ito]{}

% It is required to enter 2010 MSC.
\subjclass{Primary:  37C75, 93C25, 93D30 ; Secondary: 93C20, 93C10.}
% Please provide minimum  5 keywords.
 \keywords{Bilinear systems, infinite-dimensional systems, integral input-to-state stability, Lyapunov methods.}

% Email address of each of all authors is required.
% You may list email addresses of all other authors, separately.
 \email{andrii.mironchenko@mathematik.uni-wuerzburg.de}
 \email{hiroshi@ces.kyutech.ac.jp}

% Put your short thanks below. For long thanks/acknowlegements,
%please go to the last acknowlegments section.
\thanks{This work was supported by Japan Society for Promotion of Science (JSPS)}



\begin{document}
\maketitle

% Enter the first author's name and address:
\centerline{\scshape Andrii Mironchenko }
\medskip
{\footnotesize
% please put the address of the first author
 \centerline{Department of Systems Design and Informatics}
   \centerline{Kyushu Institute of Technology}
   \centerline{680-4 Kawazu, Iizuka, Fukuoka 820-8502, Japan}
} % Do not forget to end the {\footnotesize by the sign }

\medskip

\centerline{\scshape Hiroshi Ito}
\medskip
{\footnotesize
 % please put the address of the second  and third author
 \centerline{Department of Systems Design and Informatics}
   \centerline{Kyushu Institute of Technology}
   \centerline{680-4 Kawazu, Iizuka, Fukuoka 820-8502, Japan}
}

\bigskip

% The name of the associate editor will be entered by an editorial staff
% "Communicated by the associate editor name" is not needed for special issue.
 \centerline{(Communicated by the associate editor name)}


%The abstract of your paper
\begin{abstract}
For bilinear infinite-dimensional dynamical systems, 
we show the equivalence between uniform global asymptotic stability and integral input-to-state stability. We provide two proofs of this fact. One applies to general systems over Banach spaces. The other is restricted to Hilbert spaces, but is more constructive and results in an explicit form of iISS Lyapunov functions.  
\end{abstract}
















\section{Introduction}


Stability and robustness are fundamental for control systems and typically they have been addressed within two different concepts. One is Lyapunov stability characterizing behavior of dynamical systems without inputs near equilibrium points. The other is input-output stability which neglects information about the state of a system and studies response of a system to external inputs. 
In many cases it is not satisfactory to rely only on one of them. 
Input-to-state stability (ISS) was introduced in \cite{Son89} to unify these two concepts and provided powerful tools for systems of ordinary differential equations (ODEs) such as ISS Lyapunov functions \cite{SoW95}, design methods for nonlinear control systems \cite{KoA01,FrK08}, and the ISS small-gain theorem \cite{JTP94,JMW96} that has been extensively utilized in theory and application to establish stability and robustness of interconnections and networks of ISS systems \cite{DRW07,JiW08}.
%
%ISS plays an important role in constructive nonlinear control \cite{KoA01}, in particular in robust stabilization of nonlinear systems \cite{FrK08}, design of robust (w.r.t. errors in measurements and/or quantization) nonlinear observers \cite{Lib07}, stability of nonlinear networked control systems \cite{JMW96}, \cite{DRW07} etc. 
%nonlinear detectability \cite{SoW97}, \cite{KSW01},
%
The usefulness of ISS that was widely recognized first for ODE systems 
led to generalization, sophistication and abstractions to cover other types of control systems, such as time-delay systems, discrete-time, hybrid systems 
and trajectory-based systems (e.g. \cite{JiW01,PeJ06,InS02,KaJ11} to name a few).

The study of ISS of general infinite-dimensional systems and in particular 
of partial differential equations (PDEs) started relatively recently. 
In \cite{JLR08}, \cite{DaM13}, \cite{DaM13b}, \cite{Log13}, ISS of infinite-dimensional systems
\begin{equation}
\label{InfiniteDim}
\dot{x}(t)=Ax(t)+f(x(t),u(t)), \ x(t) \in X, u(t) \in U,
\end{equation}
has been addressed via methods of semigroup theory \cite{JaZ12}, \cite{CuZ95}. 
Here the state space $X$ and the space of input values $U$ are Banach spaces, $A:D(A) \to X$ is the generator of a $C_0$-semigroup over $X$ and $f:X \times U \to X$ is Lipschitz w.r.t. the first argument. Many classes of evolution PDEs, such as parabolic and hyperbolic PDEs are of this kind \cite{Hen81}, \cite{CaH98}.

In \cite{DaM13} sufficient conditions, a linearization method and a Lyapunov-based small-gain theorem for ISS of systems \eqref{InfiniteDim} have been developed, resulting in an efficient method to construct Lyapunov functions for interconnections of ISS systems and in this way to prove ISS of the interconnections.
On the basis of \cite{DaM13}, several results within ISS theory for impulsive infinite-dimensional systems have been proposed in \cite{DaM13b}.
In \cite{JLR08} and \cite{Log13}, systems \eqref{InfiniteDim} with linear function $f$ have been investigated via frequency-domain methods. 
%In \cite{JLR08}, relations between circle-criterion and ISS for \eqref{InfiniteDim} have been provided and in \cite{Log13} the problem of stabilization of infinite-dimensional systems via observer-based feedback has been addressed.
A substantial effort has been devoted to constructions of ISS Lyapunov functions for nonlinear parabolic systems over $L_2$ spaces in \cite{MaP11}.
In \cite{PrM12} the construction of ISS-Lyapunov functions for time-variant linear systems of hyperbolic equations (balance laws) has been provided and these results have been applied to design a stabilizing boundary feedback
control for the Saint--Venant--Exner equations.
In \cite{DaM10} ISS of some classes of monotone parabolic systems has been considered.

%A similar strategy has been adopted in \cite{AWP13}, \cite{APW13b} for robust stabilization of a diffusion equation with time-varying distributed coefficients and in  \cite{AWP12} for verification of ISS of diffusion equation w.r.t. boundary inputs. These results have been applied to the control of the poloidal magnetic flux profile in a tokamak plasma.
%
%Last but not least in \cite{KaJ11}, \cite{KaJ07} a general vector Lyapunov small-gain theorem for abstract control systems satisfying weak semigroup property has been proved. For this class of systems in \cite{KaJ12} the trajectory-based small-gain results have been obtained and applied to stabilization of a chemostat. Although in these papers the partial differential equations are not considered, the abstract level of exposition makes it possible under certain effort to include some classes of PDEs into this framework, and this emphasizes the general trend of "going infinite-dimensional" in the recent years. This framework formed a basis for a study of stability and stabilizability of nonlinear systems in a monograph \cite{KaJ11b}.

In spite of powerful tools developed within ISS theory for ODEs, 
ISS property is often too restrictive in practical systems, since in many cases boundedness of their trajectories is not guaranteed in the presence of inputs, or their trajectories grow to infinity for large enough inputs. Such a situation is usual in biochemical processes, population dynamics and traffic flows etc. due to saturation and limitations in actuators and processing rates. Such systems are never ISS, but many of them enjoy a weaker robustness property, called integral input-to-state stability (iISS) \cite{Son98}. In \cite{ASW00} a Lyapunov type necessary and sufficient condition for an ODE system to be iISS has been proved. In \cite{ITOTAC06,ItJ09,IJD13,ANGSGiISS,KaJ12} small-gain theorems for interconnections whose subsystems are not necessarily ISS have been developed. 
%, which required resolutions of several fundamental problems such as absence of ISS gain, incompatibility of signal spaces in trajectory-based approaches, and insufficiency of max-type Lyapunov functions popular in ISS Lyapunov-based approaches. The approach proposed in \cite{ITOTAC06,ItJ09} allows us to construct iISS Lyapunov functions of interconnections and to address external disturbances as in the ISS small-gain theorem.  

%In particular, within the approach developed in \cite{ItJ09,IJD13}, Lyapunov 
%functions for interconnected systems are explicitly constructed, which allows us to use 
%small-gain criteria and to address external disturbances 
%in exactly the same formulation as in the ISS small-gain theorem.  
%Although the practical issues of saturation and limitations are not 
%limited to finite-dimensional systems, 
%to the best of the authors knowledge, the iISS 
%has not been yet utilized for infinite-dimensional systems (excluding time-delay systems) in the 
%literature. 

It is well-known that for linear ODE systems the notions of ISS and iISS coincide.
In \cite{Son98} it was proved by a direct construction of an iISS Lyapunov function, that bilinear ODE systems with Hurwitz autonomous matrices are always iISS, although many of them are not ISS. Bilinear systems have allowed us to understand a basic class of pure iISS systems and provided clues to dealing with more complicated iISS systems \cite{Son98,Ito10}. 


In this work we are going to generalize the result from \cite{Son98} to bilinear infinite-dimensional systems \eqref{InfiniteDim}. On this way several difficulties arise. One of them is that the construction of Lyapunov functions, extending the original technique from \cite{Son98} directly works only for systems with a Hilbert state space. Therefore in order to prove the the equivalence between iISS and uniform global asymptotic stability for systems over Banach spaces, this paper develops a different method. Another difficulty is a need to use various density arguments, since the direct check of the properties of Lyapunov functions on the whole state space is often not possible. The work leads to an observation that non-uniformity over the spatial variables of a PDE system can make ISS fragile, while the system remains iISS. 


The structure of the paper is as follows. 
Having introduced basic stability notions in Section~\ref{Problem_Formulation}, we define in Section~\ref{LyapFunk} iISS Lyapunov functions and 
prove that the existence of an iISS Lyapunov function for a system \eqref{InfiniteDim} implies iISS of this system. ISS Lyapunov functions are also defined in a dissipative form so that they become a special case of iISS Lyapunov functions, while ISS Lyapunov functions are defined in an implicative form in \cite{DaM13}. In Section~\ref{LyapFunk}, we also show that both formulations of ISS Lyapunov functions coincide, provided that the operator $A$ generates an analytic semigroup and certain additional conditions on a nonlinearity hold. 
In Section~\ref{SectLinSys} we investigate iISS of bilinear systems. After a short discussion of infinite-dimensional linear systems, we prove that uniformly globally asymptotically stable bilinear systems are necessarily iISS. First we address this question for systems whose state space is an arbitrary Banach space. Next  we establish the abovementioned theorem for Hilbert spaces, which results in an explicit construction of an iISS Lyapunov function for bilinear systems. 
We illustrate our findings on an example of a parabolic system in Section~\ref{sec:Example} 
and conclude the paper in Section~\ref{sec:Conclusion}. In Appendix we prove two technical results.
%
The shortened preliminary version of this paper was submitted to 
the 53rd IEEE Conference on Decision and Control \cite{MiI14}.

We use the following notation throughout the paper. For linear normed spaces $X,Y$ let $L(X,Y)$ be the space of bounded linear operators from $X$ to $Y$ and $L(X):=L(X,X)$. A norm in these spaces we denote by $\| \cdot \|$. 
By $C(X,Y)$ we denote the space of continuous functions from $X$ to $Y$, $C(X):=C(X,X)$ and by 
$PC(X,Y)$ the space of piecewise right-continuous functions from $X$ to $Y$. Both are equipped with the standard $\sup$-norm.

We define $\R:=(-\infty,\infty)$ and $\R_+:=[0,\infty)$. Let $\N$ denote the set of natural numbers. Let $L_p(0,d)$, $p \geq 1$ be a space of $p$-th power integrable functions $f:(0,d) \to \R$ with the norm $\|f\|_{L_p(0,d)}=\left( \int_0^d{|f(x)|^p dx} \right)^{\frac{1}{p}}$.




\section{Problem formulation}
\label{Problem_Formulation}

Consider a system \eqref{InfiniteDim} and assume throughout the paper that $X$ and $U$ are Banach spaces and
$f(0,0)=0$, i.e., $x \equiv 0$ is an equilibrium point of \eqref{InfiniteDim}. 
Let $\phi(t,\phi_0,u)$ denote the state of a system 
\eqref{InfiniteDim} at moment $t\in\R_+$ associated with 
an initial condition $\phi_0\in X$ at $t=0$, and input $u\in U_c$, 
where $U_c$ is a linear normed space of admissible inputs equipped 
with a norm $\|\cdot\|_{U_c}$. 

%For the formulation of stability properties the following classes of functions are useful:
%\begin{equation*}
%\begin{array}{ll}
%{\K} &:= \left\{\gamma:\R_+ \to \R_+ \left|\ \right. \gamma\mbox{ is continuous and strictly increasing, }\gamma(0)=0\right\}\\
%{\K_{\infty}}&:=\left\{\gamma\in\K\left|\ \gamma\mbox{ is unbounded}\right.\right\}\\
%{\LL}&:=\left\{\gamma:\R_+ \to \R_+ \left|\ \gamma\mbox{ is continuous and decreasing with}\right.
 %\lim\limits_{t\rightarrow\infty}\gamma(t)=0 \right\}\\
%{\KL} &:= \left\{\beta: \R_+^2 \to \R_+ \left|\ \right. \beta(\cdot,t)\in{\K},\ \forall t \geq 0,\  \beta(r,\cdot)\in {\LL},\ \forall r >0\right\}
%\end{array}
%\end{equation*}
We use the following classes of comparison functions
\begin{equation*}
\begin{array}{ll}
{\PD} &:= \left\{\gamma:\R_+\rightarrow\R_+\left|\ \gamma\mbox{ is continuous, }  \gamma(0)=0 \mbox{ and } \gamma(r)>0 \mbox{ for } r>0 \right. \right\} \\
{\K} &:= \left\{\gamma \in \PD \left|\ \gamma \mbox{ is strictly increasing}  \right. \right\}\\
%{\K} &:= \left\{\gamma:\R_+\rightarrow\R_+\left|\ \gamma\mbox{ is continuous, }\gamma(0)=0\right.\right.\\
%&\phantom{aaaaaaaaaaaaaaaaaaa}\left.\text{and strictly increasing}\right\}\\
{\K_{\infty}}&:=\left\{\gamma\in\K\left|\ \gamma\mbox{ is unbounded}\right.\right\}\\
{\LL}&:=\{\gamma:\R_+\rightarrow\R_+\left|\ \gamma\mbox{ is continuous and strictly decreasing with } \lim\limits_{t\rightarrow\infty}\gamma(t)=0\right. \}\\
{\KL} &:= \left\{\beta:\R_+\times\R_+\rightarrow\R_+\left|\ \beta \mbox{ is continuous, } \beta(\cdot,t)\in{\K},\ \forall t \geq 0, \  \beta(r,\cdot)\in {\LL},\ \forall r > 0\right.\right\} \\
\end{array}
\end{equation*}
%
%\begin{equation*}
%\begin{array}{ll}
%{\PD} &:= \left\{\gamma:\R_+\rightarrow\R_+\left|\ \gamma\mbox{ is continuous, } \right. \right. \\
%&\phantom{aaaaaaaaaaaaa}\left. \gamma(0)=0 \mbox{ and } \gamma(r)>0 \mbox{ for } r>0 \right\} \\
%{\K} &:= \left\{\gamma \in \PD \left|\ \gamma \mbox{ is strictly increasing}  \right. \right\}\\
%%{\K} &:= \left\{\gamma:\R_+\rightarrow\R_+\left|\ \gamma\mbox{ is continuous, }\gamma(0)=0\right.\right.\\
%%&\phantom{aaaaaaaaaaaaaaaaaaa}\left.\text{and strictly increasing}\right\}\\
%{\K_{\infty}}&:=\left\{\gamma\in\K\left|\ \gamma\mbox{ is unbounded}\right.\right\}\\
%{\LL}&:=\left\{\gamma:\R_+\rightarrow\R_+\left|\ \gamma\mbox{ is continuous and strictly}\right.\right.\\
%&\phantom{aaaaaaaaaaaaaa} \text{decreasing with } \lim\limits_{t\rightarrow\infty}\gamma(t)=0\}\\
%{\KL} &:= \left\{\beta:\R_+\times\R_+\rightarrow\R_+\left|\ \beta \mbox{ is continuous,}\right.\right.\\
%&\phantom{aaaaaaa}\left.\beta(\cdot,t)\in{\K},\ \forall t \geq 0, \  \beta(r,\cdot)\in {\LL},\ \forall r > 0\right\} \\
%\end{array}
%\end{equation*}

Under solutions of \eqref{InfiniteDim} we understand solutions of an integral equation
\begin{equation}
\label{InfiniteDim_Integral_Form}
x(t)=T(t)x(0) + \int_0^t T(t-s)f(x(s),u(s))ds. %\ x(t) \in X, u(t) \in U.
\end{equation}
belonging to the class $C([0,\tau],X)$ for all $\tau>0$ (so-called weak solutions).

\begin{Def}
We call $f:X \times U \to X$ Lipschitz continuous on bounded subsets of $X$, uniformly w.r.t. the second argument if $\forall w>0 \; \exists L(w)>0$, such that $\forall x,y: \|x\|_X \leq w,\ \|y\|_X \leq w$, $\forall v \in U$
\begin{eqnarray}
\|f(y,v)-f(x,v)\|_X \leq L(w) \|y-x\|_X.
\label{eq:Lipschitz}
\end{eqnarray}
\end{Def}

We will use the following assumption concerning nonlinearity $f$ throughout the paper
\begin{Ass}
\label{Assumption1}
We assume that $f:X \times U \to X$ is Lipschitz continuous on bounded subsets of $X$, uniformly w.r.t. the second argument and that $f(x,\cdot)$ is continuous for all $x \in X$.
\end{Ass}
Assumption~\ref{Assumption1} ensures that the weak solution of \eqref{InfiniteDim} exists and is unique, according to a variation of a classical existence and uniqueness theorem \cite[Proposition 4.3.3]{CaH98}.

Next we introduce stability properties for the system \eqref{InfiniteDim}.
\begin{Def}
System \eqref{InfiniteDim} is {\it globally asymptotically
stable at zero uniformly with respect to state} (0-UGASs), if $\exists \beta \in \KL$, such that $\forall \phi_0 \in X$, $\forall t\geq 0$ it holds
\begin{equation}
\label{UniStabAbschaetzung}
\left\| \phi(t,\phi_0,0) \right\|_{X} \leq  \beta(\left\| \phi_0 \right\|_{X},t) .
\end{equation}
%
%If $\beta$ can be chosen as $\beta(r,t)=Me^{-at} r $ $\forall r,t \in \R_+$, for some $a,M>0$, then 
%$\Sigma$ is called exponentially 0-UGASs.
\end{Def}

%Another notion is widely used in stability theory:
%\begin{Def}
%\eqref{InfiniteDim} is {\it globally asymptotically
%stable at zero} (0-GAS), if it holds
%\begin{enumerate}
	%\item $\forall \varepsilon>0 \ \exists \delta>0: \|x\|_X < \delta,\ t\geq 0 \Rightarrow \|\phi(t,x,0)\|_X < \varepsilon$,
	%\item  $\forall x \in X$ $\| \phi(t,x,0)\|_X \to 0,\ t \to \infty$.
%\end{enumerate}
%\end{Def}
%
%In other words, $\Sigma$ is 0-GAS if it is locally stable and globally attractive (see, e.g. \cite{SoW96}).
%Note, that the 0-UGASs property is not equivalent to the 0-GAS in general.

%\begin{Def}
%Element of state space $\phi_0 \in X$ is called an equilibrium point of control system $\Sigma$ if $\phi(t,\phi_0,0) = \phi_0$, for all $t\geq 0$.
%\end{Def}

To study stability properties of \eqref{InfiniteDim} with respect to external inputs, we use the notion of input-to-state stability \cite{DaM13}:
\begin{Def}
\label{Def:ISS}
System \eqref{InfiniteDim} is called {\it input-to-state stable
(ISS) w.r.t. space of inputs $U_c$}, if there exist $\beta \in \KL$ and $\gamma \in \K$ 
such that the inequality
\begin {equation}
\label{iss_sum}
\begin {array} {lll}
\| \phi(t,\phi_0,u) \|_{X} \leq \beta(\| \phi_0 \|_{X},t) + \gamma( \|u\|_{U_c})
\end {array}
\end{equation}
holds $\forall \phi_0 \in X$, $\forall u\in U_c$ and $\forall t\geq 0$.

We emphasize that the above definition does not yet exactly 
correspond to ISS of finite dimensional systems \cite{Son08} since 
Definition \ref{Def:ISS} allows the flexibility of the choice $U_c$. 
A \textit{system \eqref{InfiniteDim} is said to be ISS}, without expressing the 
normed space of inputs explicitly, if it is ISS w.r.t. $U_c=C(\R_+,U)$ 
endowed with a usual supremum norm. 
This terminology follows that of ISS for finite dimensional systems. 

\end{Def}

%\begin{remark}
%Note that the question, whether \eqref{InfiniteDim} is ISS or not in general depends on the input space $U_c$.
%% It is possible, that for some classes of inputs the system is ISS, and for another classes it is not.
%\end{remark}

The following notion is central in this paper
\begin{Def}
\label{Def:iISS}
System \eqref{InfiniteDim} is called integral input-to-state stable (iISS) if there exist $\theta \in \Kinf$, $\mu \in \K$ and $\beta \in \KL$ 
such that the inequality 
\begin{equation}
\label{iISS_Estimate}
\|\phi(t,\phi_0,u)\|_X \leq \beta(\|\phi_0\|_X,t) + 
\theta\left(\int_0^t \mu(\|u(s)\|_U)ds\right)
\end{equation}
holds $\forall \phi_0 \in X$, $\forall u\in U_c=C(\R_+,U)$ and $\forall t\geq 0$.
\end{Def}

In the next section we will provide a Lyapunov sufficient condition for iISS of systems \eqref{InfiniteDim}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





\section{Lyapunov characterization of {\rm i}ISS}
\label{LyapFunk}

\begin{Def}\label{def:iISSV}
A continuous function $V:X \to \R_+$ is called an \textit{iISS Lyapunov function},  if there exist
$\psi_1,\psi_2 \in \Kinf$, $\alpha \in \PD$ and $\sigma \in \K$ 
such that 
\begin{equation}
\label{LyapFunk_1Eig}
\psi_1(\|x\|_X) \leq V(x) \leq \psi_2(\|x\|_X), \quad \forall x \in X
\end{equation}
and system \eqref{InfiniteDim} satisfies 
\begin{equation}
\label{DissipationIneq}
\dot{V}_u(x) \leq -\alpha(\|x\|_X) + \sigma(\|u(0)\|_U)
\end{equation}
for all $x \in X$ and $u\in U_c$, 
where the Lie derivative of $V$ corresponding to the input $u$ is defined by
\begin{equation}
\label{LyapAbleitung}
\dot{V}_u(x)=\mathop{\overline{\lim}} \limits_{t \rightarrow +0} {\frac{1}{t}(V(\phi(t,x,u))-V(x)) }.
\end{equation}
Furthermore, if 
\begin{equation}
\label{eq:ISSalphsig}
\lim_{\tau\to\infty}\alpha(\tau)=\infty 
\,\mbox{ or }\, 
\liminf_{\tau\to\infty}\alpha(\tau)\geq
\lim_{\tau\to\infty}\sigma(\tau)
\end{equation}
holds, system $V$ is called an \textit{ISS Lyapunov function}. 
\end{Def}

The following Lyapunov-type theorem provides us with a way to prove iISS of \eqref{InfiniteDim}.

\begin{proposition}\label{PropSufiISS}
If there exist an iISS (resp. ISS) Lyapunov function for \eqref{InfiniteDim}, then \eqref{InfiniteDim} is iISS (resp. ISS).
%
%\begin{enumerate}
	%\item If there exist an iISS Lyapunov function for \eqref{InfiniteDim}, then \eqref{InfiniteDim} is iISS.
	%\item If there exist an ISS Lyapunov function for \eqref{InfiniteDim}, then \eqref{InfiniteDim} is ISS.
%\end{enumerate}
\end{proposition}

\begin{proof}
Let $\tau>0$ be such that $[0,\tau)$ is the maximal time interval 
over which a system \eqref{InfiniteDim} admits a solution. 
For a given initial condition $\phi_0\in X$ and 
a given input $u\in U_c$, let $y(t)=V(x(t))$, where $x(t)=\phi(t,\phi_0,u)$. 
With the help of \eqref{LyapFunk_1Eig}, 
definition \eqref{LyapAbleitung} and property 
\eqref{DissipationIneq} imply 
\begin{align*}
\dot{y}(t) \leq -\alpha(\psi_2^{-1}(y(t)))
+ \sigma(\|u(t)\|_U)  
\end{align*}
for each $t\in [0,\tau)$.  
From \cite[Corollary IV.3]{ASW00} it follows the existence of 
$\hat{\beta}\in\KL$ satisfying 
\begin{align*}
y(t) \leq \hat{\beta}(y(0),t)+\int_0^t
2\sigma(\|u(s)\|_U)ds . 
\end{align*}
Again, using \eqref{LyapFunk_1Eig} we have 
\begin{align}
\label{iISSprop_in_y}
\psi_1(\|x(t)\|_X) \leq \tilde{\beta}(\|x(0)\|_X,t)
+\int_0^t 2\sigma(\|u(s)\|_U)ds ,   
\end{align}
where $\tilde{\beta}=\hat{\beta}(\psi_2(\cdot),\cdot)\in\KL$. 
Since $\psi_1$ is of class $\K_\infty$ and satisfies 
$\psi_1^{-1}(a+b)\leq \psi_1^{-1}(2a)+\psi_1^{-1}(2b)$ for 
any $a,b\in\R_+$, 
we arrive at \eqref{iISS_Estimate} 
for all $t\in[0,\tau]$ 
with $\theta(s)=\psi_1^{-1}(2s)$ and $\mu(s)=2\sigma(s)$, 
and $\beta(s,r)=\psi_1^{-1}(2\tilde{\beta}(s,r))$. 

Although $\tau$ representing the endpoint of the maximal interval 
was assumed to be finite, property \eqref{iISSprop_in_y} 
implies that $\tau$ cannot be finite, because otherwise the solution  $x(\cdot)$ will be unbounded near $t=\tau$ (under Assumption~\ref{Assumption1}), due to a variation of \cite[Theorem 4.3.4]{CaH98}.
Therefore, 
property \eqref{iISS_Estimate} must hold for 
all $t\geq 0$, and thus \eqref{InfiniteDim} is iISS. 




Finally, we can prove ISS when \eqref{eq:ISSalphsig} holds as in the finite-dimensional case \cite{SoW95}, with the help of 
\cite[Theorem 1 and Proposition 5]{DaM13}.
 %
%In the case of $\lim_{\tau\to\infty}\alpha(\tau)=\infty$ 
%or $\lim_{\tau\to\infty}\alpha(\tau)>\lim_{\tau\to\infty}\sigma(\tau)$, 
%pick a constant $K>1$ such that 
%$\lim_{\tau\to\infty}\alpha(\tau)\geq K\lim_{\tau\to\infty}\sigma(\tau)$.  
%As in the finite-dimensional case \cite{SoW95}, applying 
%$\tilde{\chi}=\alpha^{-1}\circ(K \sigma)\in\K$ to \eqref{DissipationIneq} yields 
%\begin{equation*}
%\|x\|_X\geq\tilde{\chi}(\|u(0)\|_U)
%\ \Rightarrow \ 
%\dot{V}_u(x) \leq -\left(1-\frac{1}{K}\right)\alpha(\|x\|_X)
%\end{equation*}
%for all $x \in X$ and $u\in U_c$. From 
%\cite[Theorem 1 and Proposition 5]{DaM13} we obtain ISS of 
%\eqref{InfiniteDim}. 
%In the case of $\infty>\lim_{\tau\to\infty}\alpha(\tau)=
%\lim_{\tau\to\infty}\sigma(\tau)$, pick a continuous function 
%$\omega$ satisfying 
%\begin{align}
%& \label{eq:alpshigISS1}
%id+\omega\in\K_\infty 
%\\
%& \label{eq:alpshigISS2}
%\omega(\sigma(s))>0, \quad s\in(0,\infty)  
%\\
%& \label{eq:alpshigISS3}
%\lim_{\tau\to\infty}\omega(\sigma(\tau))=0 .
%\end{align}
%Then property \eqref{DissipationIneq} with 
%$\tilde{\chi}=\alpha^{-1}\circ(id+\omega)\circ\sigma$ yields 
%\begin{align*}
%&
%\|x\|_X\geq\tilde{\chi}(\|u(0)\|_U)
%\nonumber \\
%& \hspace{9ex} \Rightarrow \ 
%\dot{V}_u(x) \leq -\left(id-(id+\omega)^{-1}\right)\circ\alpha(\|x\|_X)
%\end{align*}
%for all $x \in X$ and $u\in U_c$. Here, 
%properties \eqref{eq:alpshigISS1}-\eqref{eq:alpshigISS3} guarantee 
%$\tilde{\chi}\in\Kinf$ and $(id-(id+\omega)^{-1})\circ\alpha\in\PD$. 
%Again \cite[Theorem 1 and Proposition 5]{DaM13} proves ISS of 
%\eqref{InfiniteDim}. 
\end{proof}

%In Definition \ref{def:iISSV} 
%the ISS Lyapunov function is introduced in a so-called dissipative form 
%in parallel to the iISS Lyapunov function. 
%In the preceding study \cite{DaM13} on ISS, however, a so-called implicative 
%form has been used  as follows:
%\begin{Def}
%A continuous function $V:X \to \R_+$ is called an 
%\textit{ISS Lyapunov function in an implicative form}, if there exist
%$\psi_1,\psi_2 \in \Kinf$, $\eta \in \Kinf$ and $\gamma \in \K$ 
%such that \eqref{LyapFunk_1Eig} holds and system \eqref{InfiniteDim} satisfies 
%\begin{equation}
%\label{GainImplikation}
%\|x\|_X \geq \gamma(\|u(0)\|_U) \Rightarrow   \dot{V}_u(x) \leq -\eta(\|x\|_X) 
%\end{equation}
%for all $x \in X$ and $u\in U_c$.
%\end{Def}
%
%The dissipative and implicative definitions coincide for finite-dimensional systems (see \cite[Remark 2.4, p. 353]{SoW95} and \cite{SoT95}). 
%For infinite-dimensional systems, it can be verified that they are equivalent under some additional assumptions on nonlinearity $f$.
%We will need
%\begin{Def}
%We say that $\Sigma$ depends continuously on inputs and on initial states, if
%$\forall x \in X, \forall u \in U_c, \forall \tau>0$ and $\forall \eps>0$ there exist $\delta=\delta(x,u,\tau,\eps)>0$, such that $\forall x' \in X: \|x-x'\|_X< \delta$ and 
%$\forall u' \in U_c: \|u-u'\|_{U_c}< \delta$ it holds
%\[
%\|\phi(t,x,u)-\phi(t,x',u')\|_X< \eps, \quad \forall t \in [0,\tau].
%\]
%\end{Def}
%Now we are able to formulate the following result:
%\begin{Satz}
%\label{thm:Dissipative_Implicative}
%Let
%\begin{enumerate}
	%\item $f$ be Lipschitz on bounded subsets of $X$ uniformly w.r.t. the second argument.
  %\item \eqref{InfiniteDim} depend continuously on initial states and inputs.
	%\item $f$ admit the existence of $q \in \Kinf$ satisfying, 
%for all $v \in U$, 
%\begin{eqnarray}
%\| f(0,v)\|_X \leq q(\| v\|_U) . 
%\label{eq:Boundedness_wrt_u}
%\end{eqnarray}
	%\item $V$ be Fr\'echet differentiable in $X$ and its derivative $\frac{\partial V}{\partial x}$ be bounded on bounded balls. 
	%\end{enumerate}
%Then $V$ is an ISS Lyapunov function in an implicative form iff it is an ISS Lyapunov function in a dissipative form.
%\end{Satz}
%The proof is given in Appendix~\ref{app:Implication_Dissipation}.
%
%In many cases it is hard to compute the derivative of a Lyapunov function for all $x \in X$  directly, but it is much more convenient to differentiate $V$ on some dense subspaces of $X$ and $U_c$, 
%and to verify dissipation inequality \eqref{DissipationIneq} on the 
%dense subspaces. This directly leads to iISS/ISS of a system 
%only on the dense subspaces. We will show next, that under a
%natural assumption, this already implies iISS of the system on 
%the original state and input spaces.  
%For this purpose, 
%let $\Sigma:=(X,U_c,\phi)$ denote a system \eqref{InfiniteDim} defined with 
%the transition map $\phi$ corresponding to the spaces $X$, $U_c$. 
%Let $\hat{\Sigma}:=(\hat{X},\hat{U}_c,\phi)$ be system \eqref{InfiniteDim} 
%defined with the same transition map $\phi$ 
%but restricted to the state space $\hat{X}$ and the input space $\hat{U}_c$
%which are dense linear normed subspaces of $X$ and $U_c$, respectively, 
%and endowed with the norms in original spaces $\|\cdot\|_X$ and 
%$\|\cdot\|_{U_c}$, respectively.
%The following result will be used frequently in the sequel. 
%
%\begin{proposition}
%\label{DensityArg}
%Let $\Sigma$ depend continuously on inputs and on initial states and let $\hat{\Sigma}$ be iISS. Then $\Sigma$ is also iISS with the same $\beta, \alpha$ and 
%$\mu$ in the estimate \eqref{iISS_Estimate}.
%\end{proposition}
%
%The proof is given in Appendix~\ref{app:Density Argument}.
%






In Definition \ref{def:iISSV} 
the ISS Lyapunov function is introduced in a so-called dissipative form 
in parallel to the iISS Lyapunov function. 
In the preceding study \cite{DaM13} on ISS, however, a so-called implicative 
form has been used  as follows:
\begin{Def}
A continuous function $V:X \to \R_+$ is called an 
\textit{ISS Lyapunov function in an implicative form}, if there exist
$\psi_1,\psi_2 \in \Kinf$, $\eta \in \Kinf$ and $\gamma \in \K$ 
such that \eqref{LyapFunk_1Eig} holds and system \eqref{InfiniteDim} satisfies 
\begin{equation}
\label{GainImplikation}
\|x\|_X \geq \gamma(\|u(0)\|_U) \Rightarrow   \dot{V}_u(x) \leq -\eta(\|x\|_X) 
\end{equation}
for all $x \in X$ and $u\in U_c$.
\end{Def}
The dissipative and implicative definitions coincide for finite-dimensional systems (see \cite[Remark 2.4, p. 353]{SoW95} and \cite{SoT95}). 
For infinite-dimensional systems, it can be verified that they are equivalent under some additional assumptions on nonlinearity $f$.
\begin{Satz}
\label{thm:Dissipative_Implicative}
Let
\begin{enumerate}
	\item $f$ be Lipschitz on bounded subsets of $X$ uniformly w.r.t. the second argument.
  \item $A$ generates an analytic semigroup.
	\item $V$ be Fr\'echet differentiable in $X$ and its derivative $\frac{\partial V}{\partial x}$ be bounded on bounded balls. 
	\item $f$ and $V$ admit the existence of $p \in \K$ and $q \in \Kinf$ satisfying
\begin{align}
\left\| \frac{\partial V}{\partial x}f(0,v)\right\|_X \hspace{-1.3ex}\leq p(\| x\|_X)+q(\| v\|_U) 
, \ \forall x \in X, \ v \in U . 
\label{eq:Boundedness_wrt_u}
\end{align}
	\end{enumerate}
Then $V$ is an ISS Lyapunov function in an implicative form iff it is an ISS Lyapunov function in a dissipative form.
\end{Satz}
The proof is given in Appendix~\ref{app:Implication_Dissipation}. 

\begin{remark}
Importantly, for infinite-dimensional systems, posing 
\eqref{eq:Boundedness_wrt_u} is fundamentally less restrictive than 
$\|f(0,v)\|_X\leq q(\| v\|_U)$ since 
norms are in general not equivalent, as opposed to 
finite-dimensional systems.
\end{remark}

In many cases it is hard to compute the derivative of a Lyapunov function for all $x \in X$  directly, but it is much more convenient to differentiate $V$ on some dense subspaces of $X$ and $U_c$, 
and to verify dissipation inequality \eqref{DissipationIneq} on the 
dense subspaces. This directly leads to iISS/ISS of a system 
only on the dense subspaces. We will show next, that under a
natural assumption, this already implies iISS of the system on 
the original state and input spaces. 
For this purpose, 
let $\Sigma:=(X,U_c,\phi)$ denote system \eqref{InfiniteDim} defined with 
the transition map $\phi$ corresponding to the spaces $X$, $U_c$. 
Let $\hat{\Sigma}:=(\hat{X},\hat{U}_c,\phi)$ be system \eqref{InfiniteDim} 
defined with the same transition map $\phi$ 
but restricted to the state space $\hat{X}$ and the input space $\hat{U}_c$
which are dense linear normed subspaces of $X$ and $U_c$, respectively, 
and endowed with the norms in original spaces $\|\cdot\|_X$ and 
$\|\cdot\|_{U_c}$, respectively.

When considering the approximations of dynamical systems, the following property is useful
\begin{Def}
We say that $\Sigma$ depends continuously on inputs and on initial states, if
$\forall x \in X, \forall u \in U_c, \forall \tau>0$ and $\forall \eps>0$ there exist $\delta=\delta(x,u,\tau,\eps)>0$, such that $\forall x' \in X: \|x-x'\|_X< \delta$ and 
$\forall u' \in U_c: \|u-u'\|_{U_c}< \delta$ it holds
\[
\|\phi(t,x,u)-\phi(t,x',u')\|_X< \eps, \quad \forall t \in [0,\tau].
\]
If under the same assumptions 
\[
\|\phi(t,x,u)-\phi(t,x',u)\|_X< \eps, \quad \forall t \in [0,\tau].
\]
holds, $\Sigma$ is called continuously dependent on initial states.
\end{Def}


The following result will be used frequently in the sequel. 
\begin{proposition}
\label{DensityArg}
Let $\Sigma$ depend continuously on inputs and on initial states and let $\hat{\Sigma}$ be iISS. Then $\Sigma$ is also iISS with the same $\beta, \theta$ and 
$\mu$ in the estimate \eqref{iISS_Estimate}.
\end{proposition}
The proof is given in Appendix~\ref{app:Density Argument}.

Often only approximations of the state space are needed, therefore we state another proposition whose proof is analogous to the proof of Proposition~\ref{DensityArg}.
\begin{proposition}
\label{DensityArg_StateAppr}
Let $\Sigma$ depend continuously on initial states, $\hat{\Sigma}$ be iISS and $\hat{U}=U$. Then $\Sigma$ is also iISS with the same $\beta, \theta$ and $\mu$ in the estimate \eqref{iISS_Estimate}.
\end{proposition}
This result will be used several times in this paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{ {\rm i}ISS and ISS of bilinear systems}
\label{SectLinSys}

Throughout this section, 
let $\T=\{T(t),\ t \geq 0\}$ be a $C_0$-semigroup on a Banach space $X$
with an infinitesimal generator $A:D(A) \to X$, $Ax =\lim\limits_{t \rightarrow +0}{\frac{1}{t}(T(t)x-x)}$, whose domain of definition $D(A)$ consists of those $x \in X$, for which this limit exists.




\subsection{Linear systems}

We will begin with a class of linear systems. 
Consider system \eqref{InfiniteDim} with $f(x(t),u(t))=Bu(t)$:
\begin{equation}
\label{LinSys}
\begin{array}{l}
{\dot{x}(t)=Ax(t)+Bu(t),} \\
x(0)=\phi_0,
\end{array}
\end{equation}
where $x:\R_+ \to X$, $u:\R_+ \to U$, and $B:U \to X$ is a linear operator. 

It is well-known that for linear finite-dimensional systems 0-GAS (local stability + global attractivity), 0-UGASs, ISS and iISS are identical properties \cite{Son89,Son98}.
For infinite-dimensional systems, 0-GAS is not equivalent to 0-UGASs in general.
In \cite[Proposition 3]{DaM13} it was shown that 0-UGASs is equivalent to ISS provided that $B$ is a bounded operator. 
Moreover, in \cite[p. 8]{DaM13} and \cite[p. 247]{MaP11} 
examples of 0-GAS infinite-dimensional systems are provided whose 
solutions go to infinity even for inputs of arbitrarily small magnitude.
Next we investigate relations between ISS and iISS for systems \eqref{LinSys}.

\begin{proposition}
\label{ISS_Lp_spaces}
Let $B \in L(U,X)$. Then,  
\eqref{LinSys} is 0-UGASs $\Iff$ \eqref{LinSys} is ISS $\Iff$ \eqref{LinSys} is ISS w.r.t. $L_p(\R_+,U)$ for some $p \geq 1$. 
\end{proposition}

\begin{proof}
The fact that \eqref{LinSys} is 0-UGASs $\Iff$ \eqref{LinSys} is ISS has been proved in \cite[Proposition 3]{DaM13}.
Next, by definition, it is obvious that 
if \eqref{LinSys} is ISS w.r.t. $L_p(\R_+,U)$ for a real number $p \geq 1$, 
then it is 0-UGASs. 
To prove the converse, let \eqref{LinSys} be 0-UGASs. Then $T$ is an exponentially stable semigroup, see e.g. \cite[Lemma 1]{DaM13}. Thus
$\exists M,\lambda>0: \|T(t)\| \leq Me^{-\lambda t}$.

We estimate the solution $x(t)=\phi(t,\phi_0,u)$ of \eqref{LinSys}:
\begin{align*}
\|x(t)\|_X  = & \|T(t)\phi_0 + \int_0^t{T(t-r)B u(r)dr}\|_X \\
						\leq & \|T(t)\|\|\phi_0\|_X + \int_0^t{\|T(t-r)\|\|B\| \|u(r)\|_U dr}, \\
						\leq & M e^{-\lambda t} \|\phi_0\|_X + M \|B\| \int_0^t{e^{-\lambda (t-r)} \|u(r)\|_U dr}.
\end{align*}
Now, estimating $e^{-\lambda (t-r)}  \leq 1$, $r \leq t$ we obtain that \eqref{LinSys} is ISS w.r.t. $L_1(\R_+,U)$. 

To prove the claim for $p>1$, pick any $q \geq 1$ so that $\frac{1}{p}+ \frac{1}{q} = 1$. 
We continue the above estimates, using the H\"older's inequality
\begin{align*}
\|x(t)\|_X  \leq & M e^{-\lambda t} \|\phi_0\|_X  + M \|B\| \int_0^t{e^{-\frac{\lambda}{2} (t-r)} e^{-\frac{\lambda}{2} (t-r)} \|u(r)\|_U dr}  \\
\leq & M e^{-\lambda t} \|\phi_0\|_X + M \|B\|  \Big(\int_0^t{e^{- \frac{q \lambda}{2} (t-r)}dr} \Big)^{\frac{1}{q}} \cdot \Big(\int_0^t{e^{-\frac{p \lambda}{2} (t-r)}\|u(r)\|^p_U dr} \Big)^{\frac{1}{p}} \\
 \leq & M e^{-\lambda t} \|\phi_0\|_X + M \|B\|  \Big( \frac{2}{q\lambda} \Big)^{\frac{1}{q}}  \Big(\int_0^t{\|u(r)\|^p_U dr} \Big)^{\frac{1}{p}} \\
 = & M e^{-\lambda t} \|\phi_0\|_X + w \|u\|_{L_p(\R_+,U)},
\end{align*}
where $w:=M \|B\|  \Big( \frac{2}{q\lambda} \Big)^{\frac{1}{q}}$.
%
This means that \eqref{LinSys} is ISS w.r.t. $L_p(\R_+,U)$, $p\geq 2$. 
%\hfill $\blacksquare$
\end{proof}

The following statement is a consequence of 
Proposition~\ref{ISS_Lp_spaces} 
(by taking $\theta:= {\rm id}$ and $\mu := c\cdot {\rm id}$ for large enough $c>0$ in Definition \ref{Def:iISS}):
\begin{corollary}
System \eqref{LinSys} is ISS iff it is iISS.
\end{corollary}

 
For finite-dimensional systems, in the presence of nonlinearities which are locally Lipschitz w.r.t. state, 
0-GAS implies local ISS \cite[Lemma I.1]{SoW96}, 
i.e., the ISS property for initial states and inputs with a sufficiently small norm. 
In contrast to this finite-dimensional fact, we next show 
an infinite-dimensional linear system illustrating that 
\textit{for unbounded operator $B$, 0-UGASs implies neither ISS nor iISS, even if 
the initial state and the input is restricted to 
sufficiently small neighborhoods of the origin. 
}


\begin{examp}\label{examp:Linear_Unbounded_InpOpe_nonISS}
Consider the following ODE ensemble defined on 
the interval $(0,\pi/2)$ of the spatial variable $l$: 
\begin{eqnarray}
\dot{x}(l,t)=-x(l,t)+(\tan l)^{\frac{1}{8}} u(l,t), \quad l \in (0,\pi/2) . 
\label{eq:Example_Unbounded_Input_operator}
\end{eqnarray}
Let $X=C(0,\pi/2)$ be a space of bounded continuous 
functions on $(0,\pi/2)$. The functions $x(l,t)$ and $u(l,t)$ are scalar-valued. 
The input operator $B:D(B) \to X$ for \eqref{eq:Example_Unbounded_Input_operator}
is defined by $(Bv)(l)=(\tan l)^{\frac{1}{8}} v(l)$ which 
is unbounded with a domain of definition
\[
D(B)=\{ v \in C(0,\pi/2): \sup_{l \in (0,\pi/2)}|(\tan l)^{\frac{1}{8}} v(l)| < \infty \}.
\]
Since $x(\cdot,t)=e^{-t}x(\cdot,0)$ holds for 
$u(\cdot,t)=0$, $\forall t\geq 0$,
system \eqref{eq:Example_Unbounded_Input_operator} is 
0-UGASs. But it is neither ISS nor iISS for $U=D(B)$. To verify this fact, consider an input $u(l,t)=\hat{u}(l)$ given by 
\[
\hat{u}(l)=\left\{ \begin{array}{ll}
b &,  \ l< \arctan{(c^8)}  \\ 
bc(\tan l)^{-\tfrac{1}{8}} &, \ l \geq \arctan{(c^8)}
\end{array}\right.
\]
for real $b, c>0$. 
It is easy to see that $\hat{u}\in D(B)$ and $\|\hat{u}\|_U=b$ from 
$\|B\hat{u}\|_X=\sup_{l \in (0,\pi/2)}|\hat{u}(l)(\tan l)^{\frac{1}{8}}|=bc$ 
and the definition of $\hat{u}$. 
The solution of \eqref{eq:Example_Unbounded_Input_operator} for 
$\phi_0=0$ is computed as 
$\phi(t,0,u)=\int_0^t e^{-(t-r)}\hat{u}(l)(\tan l)^{\frac{1}{8}}  dr=(1-e^{-t})\hat{u}(l)(\tan l)^{\frac{1}{8}}$. Thus, by definition, the solution satisfies 
\[
%\label{Ex_Unb_Inpoper_solu}
\sup_{l \in (0,\pi/2)}\phi(t,0,u)=bc(1-e^{-t}). 
\]
Now, assume that system \eqref{eq:Example_Unbounded_Input_operator} is iISS.  
From Definition \ref{Def:iISS} it follows that there exist 
$\alpha$, $\mu \in \Kinf$ satisfying 
$\|\phi(t,0,u)\|_X \leq \alpha^{-1} (t \mu(b))$ for $t \geq 0$.
Clearly, for any given $\alpha$, $\mu \in \Kinf$ and any $t>0$, 
one can find $c>0$ so that $bc(1-e^{-t}) > \alpha^{-1} (t \mu(b))$. 
Since $\|\hat{u}\|_U=b$ is satisfied for any $c>0$, 
system \eqref{eq:Example_Unbounded_Input_operator} is not iISS. 

Next, suppose that system \eqref{eq:Example_Unbounded_Input_operator} is ISS.  
Definition \ref{Def:ISS} implies the existence of 
$\gamma \in \Kinf$ satisfying 
$\|\phi(t,0,u)\|_X \leq \gamma(b)$ for $t \geq 0$ 
when $u=\hat{u}$ is applied to \eqref{eq:Example_Unbounded_Input_operator}. 
For any given $\gamma \in \Kinf$ and any $t>0$, 
there exists $c>0$ such that $bc(1-e^{-t}) > \gamma(b)$. 
Hence, system \eqref{eq:Example_Unbounded_Input_operator} is not ISS either. 
Since we can take $\phi_0=0$ and $b>0$ is arbitrary, the system 
\eqref{eq:Example_Unbounded_Input_operator} is 
neither iISS nor ISS even if the initial state and the input 
are restricted to arbitrarily small neighborhoods of the origin. 
\qed
\end{examp}

\begin{remark}
\label{LISS_ISS_LinSys}
The construction of a control $\hat{u}$ in Example~\ref{examp:Linear_Unbounded_InpOpe_nonISS} indicates that the notions of global and local ISS coincide for systems \eqref{LinSys}. More precisely, system \eqref{LinSys} is locally ISS, i.e. there exist $\delta>0$ s.t. the inequality \eqref{iss_sum} holds only for $\phi_0 \in X: \; \|\phi_0\|_X  \leq \delta$ and $u \in U_c: \; \|u\|_{U_c}  \leq \delta$ iff \eqref{LinSys} is ISS.
\end{remark}

%\begin{proof}
%It is not hard to see (see e.g. \cite[Lemma 1]{DaM13}), that if \eqref{LinSys} is locally ISS for some $\beta \in \KL$  and $\gamma \in K$, then there exist
%$M,\lambda,C,\delta>0$ so that 
%\begin {equation*}
%\begin {array} {lll}
%\| \phi(t,\phi_0,u) \|_{X} \leq M e^{-\lambda t} \| \phi_0 \|_{X} + C \|u\|_{U_c}
%\end {array}
%\end{equation*}
%for all $\phi_0 \in X: \; \|\phi_0\|_X  \leq \delta$ and $u \in U_c: \; \|u\|_{U_c}  \leq \delta$.
%Since for any $\tilde{\phi_0} \in X, \tilde{u} \in U_c$ there exists $K\leq 1$: $\tilde{\phi_0}=K\phi_0$ and $\tilde{u}=Ku$ for some $\phi_0,u$: $\|\phi_0\|_X  \leq \delta$ and $\|u\|_{U_c}  \leq \delta$.
%Since $\phi(t,\tilde{\phi_0},\tilde{u})=K\phi(t,\phi_0,u)$ for solutions of \eqref{LinSys}, the claim of the lemma follows.
%\end{proof}

In the following example the same equation 
\eqref{eq:Example_Unbounded_Input_operator} is studied 
as in Example \ref{examp:Linear_Unbounded_InpOpe_nonISS}. 
It highlights the dependency of ISS and iISS on the choice of spaces. 

\begin{examp}\label{examp:Linear_Unbounded_InpOpe_ISS}
Consider the system \eqref{eq:Example_Unbounded_Input_operator} again. 
The system is ISS if we choose $X=L_2(0,\pi/2)$, $U=L_4(0,\pi/2)$. 
%To see this, let the input operator $B:U \to X$ be defined as in 
%Example \ref{examp:Linear_Unbounded_InpOpe_nonISS} but with 
%the domain of definition
%\[
%\tilde D(B)=\{ v \in U: 
%\int_0^{\pi/2}{|(\tan l)^{\frac{1}{8}} v(l)|^2dl}<\infty\}.
%\]

Choose 
\[
V(x)=\int_0^{\pi/2}{x^2(l)dl} = \|x\|^2_{L_2(0,\pi/2)}
\]
For the solutions $x(\cdot,t)=\phi(t,\phi_0,u)$ of 
\eqref{eq:Example_Unbounded_Input_operator} we obtain 
\begin{align*}
\frac{d}{dt}V(x) &= 2 \int_0^{\pi/2}{x(l,t) \Big(-x(l,t) + u(l,t)(\tan l)^{\frac{1}{8}} \Big)dl} \\
&\leq -2 V(x) + w V(x) + \frac{1}{w}\int_0^{\pi/2}{u(l,t)^2(\tan l)^{\frac{1}{4}} dl}\\
%&\leq -2 V(x) + K(w V(x) + \frac{1}{w} \|u(\cdot,t)\|^2_{L_2(0,d)}) \\
&\leq (-2 +w) V(x) + \frac{K}{w} \|u(\cdot,t)\|^2_{L_4(0,d)},
\end{align*}
for any $w>0$ (between lines 1 and 2 Young's inequality has been used). Here, it is important to notice that 
$K:=\int_0^{\pi/2}{(\tan l)^{\frac{1}{2}}dl}< \infty$. Hence, 
taking $w<2$, Proposition \ref{PropSufiISS} proves that 
system \eqref{eq:Example_Unbounded_Input_operator} is ISS 
for $X=L_2(0,\pi/2)$ and $U=L_4(0,\pi/2)$. 
\qed
\end{examp}


\subsection{iISS of generalized bilinear systems}\label{sec:Binlin}

While for linear infinite-dimensional systems with bounded input operators the properties of ISS and iISS coincide, 
the difference between these two properties arises for 
bilinear systems which is one of 
the simplest classes of nonlinear systems. 
For finite-dimensional bilinear systems, 
Sontag \cite{Son98} demonstrated that 0-GAS systems  are seldom ISS (for example, one-dimensional bilinear systems are never ISS), and that a system is 0-GAS if and only if it is iISS. 
To generalize this result to infinite-dimensional systems, 
consider 
\begin{equation}
\label{BiLinSys}
\begin{array}{l}
{\dot{x}(t)=Ax(t)+ Bu(t) + C(x(t),u(t)),} \\
x(0)=\phi_0,
\end{array}
\end{equation}
where $B \in L(U,X)$, and $C: X \times U \to X$ is such that there exist $K>0$ and $\xi \in \K$ so that 
\begin{align}
\|C(x,u)\|_X \leq K \|x\|_X \xi(\|u\|_U).
\label{eq:BilinOperator}
\end{align}
for all $x \in X$ and all $u \in U$. 

\begin{remark}
This class of includes systems with $C$ linear in both variables and bounded in the sense that $\|C\|:=\sup_{\|x\|_X =1,\|u\|_U=1}\|C(x,u)\|_X < \infty$  (then $K=\|C\|$ and $\xi(r)=r$ for all $r \in \R_+$). 
\end{remark}
In this class, there are systems which are not ISS. The following is a simple example of such a system. 

\begin{examp}\label{examp:Bilinear_nonISS_diffusion}
Consider the following parabolic PDE:
\begin{align}
\begin{array}{l} 
\dfrac{\partial x}{\partial t} = \dfrac{\partial^2 x}{\partial l^2}(l,t) + x(l,t)u(l,t),\  (l,t) \in (0,1)\times(0,\infty) \\[1.5ex]
x(0,t) = x(1,t)=0.
\end{array}
\label{eq:Example_BilinearPDE_nonISS}
\end{align}
with a state space $X=C_0[0,1]$ (functions, continuous on $[0,1]$ and zero on the boundary) and input space $U=C(0,1)$.
A closure $\bar{A}_c$ of an operator $A_c=\dfrac{\partial^2} {\partial l^2} + cI$
with a domain of definition  $D(A)=C^2(0,1) \cap C_0[0,1]$
generates an analytic semigroup and the system \eqref{eq:Example_BilinearPDE_nonISS} is bilinear in the above sense. 

Operator $A_c$ has a spectrum $\{-(\pi n)^2 + c | \ n \geq 1 \}$. Taking $u \equiv c>\pi^2$, we obtain a system whose spectral bound is positive, and thus there are solutions of \eqref{eq:Example_BilinearPDE_nonISS}, which grow exponentially to infinity. Thus \eqref{eq:Example_BilinearPDE_nonISS} is not ISS.
\qed
\end{examp}

Next we prove the equivalence between 0-UGASs and iISS for the general system \eqref{BiLinSys} in Banach spaces. 
%
For infinite-dimensions, we employ the notion of 0-UGASs instead of 0-GAS. 
%
To establish iISS from 0-GAS for finite-dimensional systems, 
Sontag \cite{Son98} constructed Lyapunov functions for systems 
with Hurwitz $A$ by means of the Lyapunov equation. 
To the best of authors' knowledge, there is no generalization of the method for construction of Lyapunov functions by means of the Lyapunov equation for linear systems on Banach spaces, although for systems on Hilbert spaces such a construction exists.
%To the best of authors' knowledge, this method cannot be used for the general system \eqref{BiLinSys} on Banach spaces, since the construction of Lyapunov functions via Lyapunov equation for exponentially stable semigroups over infinite-dimensional spaces exists only if $X$ is a Hilbert space. 
Thus, to allow for Banach spaces, we employ another method to prove that iISS is equivalent to 0-UGASs. 

\begin{Satz}
System \eqref{BiLinSys} is iISS $\Iff$ \eqref{BiLinSys} is 0-UGASs.
\end{Satz}

\begin{proof}
Clearly, iISS implies 0-UGASs. To prove the converse, assume that 
\eqref{BiLinSys} be 0-UGASs, that is 
let $T$ be an exponentially stable semigroup.
%
Integrating \eqref{BiLinSys}, we obtain
\begin{equation*}
\label{iISS_Syst_IntEq}
x(t)=T(t)x(0) + \int_0^t{T(t-r) \big( Bu(r)+ C(x(r),u(r))\big)dr} .
\end{equation*}
From $B \in L(U,X)$, inequality \eqref{eq:BilinOperator} 
and since $T$ is an exponentially stable semigroup it follows for some $K,M,\lambda>0$, that
\begin{align*}
\label{iISS_Syst_IntEq_Estim1}
\|x(t)\|_X & \leq  \|T(t)\|\|x(0)\|_X + \int_0^t \|T(t-r)\| \big( \|B\|\|u(r)\|_U+ \|C(x(r),u(r))\|_X\big)dr \\
& \leq  Me^{-\lambda t} \|x(0)\|_X + \int_0^t Me^{-\lambda (t-r)} \big( \|B\|\|u(r)\|_U + K \|x(r)\|_X \xi(\|u(r)\|_U)\big)dr . 
\end{align*}
We multiply both sides of the inequality by $e^{\lambda t}$ and define $z(t) = x(t) e^{\lambda t}$. From $\lambda>0$ we obtain
\begin{align*}
\|z(t)\|_X \leq & M \Big( \|z(0)\|_X + \|B\| \int_0^t e^{\lambda r} \|u(r)\|_U dr\Big)  + \int_0^t{MK \|z(r)\|_X \xi(\|u(r)\|_U)dr}.
\end{align*}
Since $q: t \mapsto M\big( \|z(0)\|_X + \|B\| \int_0^t e^{\lambda r} \|u(r)\|_U dr \big)$ is a non-decreasing function, Gronwall's inequality (see e.g. \cite[Lemma 2.7, p.42]{Tes12}) yields 
\begin{align*}
\|z(t)\|_X \leq& M\Big( \|z(0)\|_X + \|B\| \int_0^t e^{\lambda r} \|u(r)\|_U dr \Big)  e^{\int_0^t{MK \xi(\|u(r)\|_U)dr}}.
\end{align*}
Coming back to original variables and using $\lambda>0$, we have 
\begin{align*}
\|x(t)\|_X 
%\leq& Me^{-\lambda t}\Big( \|x(0)\|_X + \|B\| \int_0^t e^{\lambda r} \|u(r)\|_U dr\Big) \\
%&& \cdot e^{\int_0^t{K \|u(r)\|_Udr}} \\
\leq & M\Big( e^{-\lambda t}\|x(0)\|_X + \|B\| \int_0^t e^{-\lambda (t-r)} \|u(r)\|_U dr\Big)  e^{\int_0^t{MK \xi(\|u(r)\|_U)dr}} \\
\leq & M\Big( e^{-\lambda t}\|x(0)\|_X + \|B\| \int_0^t \|u(r)\|_U dr\Big)  e^{\int_0^t{MK \xi(\|u(r)\|_U)dr}}.
\end{align*}
Applying to the both sides the function $\alpha \in \Kinf$ defined for all $r \geq 0$ as $\alpha(r)=\ln(1+r)$ results in 
\begin{align*}
\alpha(\|x(t)\|_X ) \leq& \ln\Big(1+ M\big( e^{-\lambda t}\|x(0)\|_X +\|B\| \int_0^t \|u(r)\|_U dr\big) e^{\int_0^t{MK \xi(\|u(r)\|_U)dr}}\Big).
\end{align*}
Now since for all $a,b \in \R_+$ it holds that
\[
\ln(1+ae^b) \leq \ln((1+a)e^b)=\ln(1+a) + b , 
\]
we obtain
\begin{align*}
\alpha(\|x(t)\|_X ) \leq& \ln\Big( 1+ M\big( e^{-\lambda t}\|x(0)\|_X  +\|B\| \int_0^t \|u(r)\|_U dr\big) \Big) \\
 & \quad\quad + \int_0^t{MK \xi(\|u(r)\|_U)dr}.
\end{align*}
Moreover, for all $a,b \in \R_+$ it holds that
\[
\ln(1+a + b) \leq \ln((1+a)(1+b))=\ln(1+a) + \ln(1+b),
\]
which implies
\begin{align*}
\alpha(\|x(t)\|_X ) \leq& \ln\big(1+ M e^{-\lambda t}\|x(0)\|_X\big)\\
& + \ln\Big(1+ M\|B\| \int_0^t \|u(r)\|_U dr\Big)  + \int_0^t{MK \xi(\|u(r)\|_U)dr}.
\end{align*}
Since $\beta:(s,t) \mapsto \ln(1+ M e^{-\lambda t}s)$ is a $\KL$-function, with the help of 
$\alpha^{-1}(a+b)\leq \alpha^{-1}(2a)+\alpha^{-1}(2b)$ holding for 
any $a, b\in\R_+$, 
the above estimate shows us that \eqref{BiLinSys} is iISS 
as defined in Definition \ref{Def:ISS}.  
\end{proof}


\subsection{Lyapunov functions for generalized bilinear systems}\label{sec:Lyap_Binlin}

This section develops a method to construct an 
iISS Lyapunov function for the infinite-dimensional system 
\eqref{BiLinSys} 
analogous to the the finite-dimensional case \cite{Son98}. 
For this purpose, in this section, let $X$ be a Hilbert space with a 
scalar product $\lel \cdot, \cdot \rir$. Note that 
if \eqref{BiLinSys} is 0-UGASs, the operator $A$ generates exponentially stable semigroup \cite[Lemma 1]{DaM13}. 
Since $X$ is a Hilbert space, the exponential stability of this semigroup is equivalent to existence of a positive self-adjoint operator $P\in L(X)$ satisfying the Lyapunov equation
\begin{equation}
\label{OperatorBedingung}
\lel Ax,Px \rir + \lel Px,Ax \rir=-\|x\|^2_X, \quad \forall x \in D(A)
\end{equation}
see \cite[Theorem 5.1.3, p. 217]{CuZ95}.
%
Recall that a self-adjoint operator $P \in L(X)$ is said to be positive if $\lel P x,x \rir > 0$ holds for all $x \in X\setminus\{0\}$. A positive operator $P \in L(X)$ is called coercive if there exists $k>0$ such that
\[
\lel Px,x \rir \geq k \|x\|^2_X \quad \forall x \in D(P).
\]

\begin{Satz}
\label{Thm:LF_Constructions_for_bilinear_systems}
Consider a system \eqref{BiLinSys} over a Hilbert space $X$. 
Let Assumption~\ref{Assumption1} hold and let 
there exists a coercive positive self-adjoint operator $P \in L(X)$ satisfying \eqref{OperatorBedingung}. 
If $A$ generates an analytic semigroup, then a system \eqref{BiLinSys} is iISS and its iISS Lyapunov function can be constructed as
\begin{align}
W(x)=\ln\Big(1+ \lel Px,x \rir \Big).
\label{eq:iISS_LF}
\end{align}
If $A$ doesn't necessarily generate an analytic semigroup, but all the trajectories, emanating from $D(A)$ under arbitrary input $u \in U_c$ remain in $D(A)$, then \eqref{BiLinSys} is still iISS and $W$ is its iISS Lyapunov function on $D(A)$.
\end{Satz}

\begin{proof}
Let assumptions of the theorem hold. Consider a function $V: x \mapsto \lel Px,x \rir $. 
%
Since $P$ is bounded and coercive, for some $k>0$ it holds 
\[
k \|x\|^2_X \leq V(x) \leq \|P\| \|x\|^2_X, \quad \forall x \in X,
\]
and property \eqref{LyapFunk_1Eig} is verified.
%
Let us compute the Lie derivative of $V$ with respect to system \eqref{InfiniteDim}. 
For $x \in D(A)$ we have
\begin{align*}
\dot{V}(x) =& \lel P\dot{x},x \rir +  \lel Px,\dot{x} \rir \\
 =&\lel P(Ax +Bu + C(x,u)),x \rir \\
&+ \lel Px,Ax +Bu + C(x,u) \rir  \\
=& \lel P(Ax),x \rir +  \lel Px,Ax \rir \\
& + \lel P(Bu + C(x,u)),x \rir +  \lel Px,Bu + C(x,u) \rir.
\end{align*}
From $\lel P(Ax),x \rir = \lel Ax,Px \rir$, 
\eqref{eq:BilinOperator} and \eqref{OperatorBedingung} 
with the help of Cauchy-Schwarz inequality, we obtain 
\begin{align*}
\dot{V}(x) \leq& -\|x\|^2_X + \|P(Bu + C(x,u))\|_X  \|x\|_X\\
& + \|Px\|_X \|Bu + C(x,u)\|_X \\ 
\leq& -\|x\|^2_X + \|P\| \|Bu + C(x,u)\|_X  \|x\|_X \\
& + \|P\| \|x\|_X \|Bu + C(x,u)\|_X \\ 
\leq& -\|x\|^2_X + 2 \|P\| \|x\|_X (\|B\| \|u\|_U + K \|x\|_X \xi(\|u\|_U)) \\
\leq& -\|x\|^2_X \!\!+\! 2 K \|P\| \|x\|^2_X \xi(\|u\|_U)  \!\!+\! 2 \|P\| \|B\| \|x\|_X \|u\|_U.
\end{align*}
Let $\eps>0$. 
Using Young's inequality 
\begin{align*}
2 \|x\|_X \|u\|_U \leq \eps \|x\|^2_X + \dfrac{1}{\eps} \|u\|^2_U , 
\end{align*}
we can continue the above estimates as 
\begin{align*}
\dot{V}(x) \leq 
& -\Big(1 - \eps \|P\|\|B\|\Big) \|x\|^2_X + 2 K \|P\| \|x\|^2_X \xi(\|u\|_U) \\
& + \dfrac{\|P\|\|B\|}{\eps} \|u\|^2_U . 
\end{align*}
Defining $W$ as in \eqref{eq:iISS_LF} yields 
\begin{align*}
\dot{W}(x) \leq& \frac{1}{1+V(x)}
\Bigl[-\left(1 - \eps \|P\|\|B\|\right) \|x\|^2_X
\\
& + 2 K \|P\| \|x\|^2_X \xi(\|u\|_U) 
+ \dfrac{\|P\|\|B\|}{\eps} \|u\|^2_U \Bigr] \\
\leq & 
-\left(1 - \eps \|P\|\|B\|\right)\frac{\|x\|^2_X}{1+V(x)}
\\
& + \frac{2 K \|P\|\|x\|^2_X}{1+k\|x\|^2_X}\xi(\|u\|_U) 
+ \dfrac{\|P\|\|B\|}{\eps} \|u\|^2_U. 
\end{align*}
which finally leads to
\begin{align}
\dot{W}(x) \leq& 
-\left(1 - \eps \|P\|\|B\|\right)\frac{\|x\|_X^2}{1+\|P\|\|x\|_X^2}
\nonumber \\
& + \frac{2 K \|P\|}{k}\xi(\|u\|_U) 
+ \dfrac{\|P\|\|B\|}{\eps} \|u\|^2_U. 
\label{LyapEstimate}
\end{align}
These derivations hold for $x \in D(A) \subset X$. 
If $x \notin D(A)$, then for all admissible $u$ the solution $x(t) \in D(A)$ and $t \to W(x(t))$ is a continuously differentiable function for all $t >0$ (these properties follow from the properties of solutions $x(t)$, see Theorem 3.3.3 in \cite{Hen81}). 
%
Therefore, by the mean-value theorem, $\forall t>0$ $\exists t_* \in (0,t)$
\[
\frac{1}{t}(W(x(t))-W(x))=\dot{W}(x(t_*)),
\]
where $x=x(0)$. 
Taking the limit when $t \to +0$ we obtain that \eqref{LyapEstimate} holds for all $x \in X$.
%
Pick $\eps>0$ such that $\eps<{1}/(\|P\|\|B\|)$. 
According to Proposition \ref{PropSufiISS}, system \eqref{BiLinSys} 
is iISS and $W$ is an iISS Lyapunov function. 

Let now all the trajectories, emanating from $D(A)$ under arbitrary input $u \in U_c$ remain in $D(A)$. This means, that \eqref{BiLinSys} generates a control system on $D(A)$ with inputs $U_c$. According to above argument $W$ is an iISS Lyapunov function for this restricted system, which shows its iISS. 
Similarly to \cite[Proposition 4.3.7]{CaH98} one can prove, that Assumption~\ref{Assumption1} implies that \eqref{BiLinSys} depends continuously on initial data. Thus, due to Proposition~\ref{DensityArg_StateAppr} iISS of \eqref{BiLinSys} for state from $D(A)$ and inputs belonging to $U$ implies iISS of \eqref{BiLinSys} on all spaces $X,U$. 
\end{proof}

As we see the proof of the previous theorem consists basically of two parts: the verification that $W$ is an iISS Lyapunov function for $x \in D(A)$ and then use of a density argument for analytic semigroups.
We will see that this strategy will be useful also in the proof of Theorem~\ref{thm:Dissipative_Implicative}.


\begin{remark}
Note, that in the case when $A$ generates a nonanalytic semigroup we do not claim that $W$ is an iISS Lyapunov function on the whole space $X$, since we do not know the Lie derivative $\dot{W}_u$ for $x \notin D(A)$.
\end{remark}
%
%{
%\color{red}
%Actually it is not completely clear, whether this already proves iISS of \eqref{BiLinSys} on the space $D(A)$.
%If the trajectories, starting in $D(A)$ can leave $D(A)$ (while staying in $X$), then it will not be iISS.
%And density argument cannot be used to prove iISS on whole $X$ (since there we compare forward-complete trajectories, starting in $X\backslash D(A)$ with forward-complete trajectories emanating from $D(A)$).
%}
%
%
%\begin{remark}
%Actually, in a linearization theorem for ISS systems \cite[Theorem 3]{DaM13} in a similar situation another density argument has been employed, which uses smoothification property of analytical semigroups. This resulted in an additional assumption in the statement of the theorem that $A$ must be a generator of an analytic semigroup. Using a simple density argument Proposition~\ref{DensityArg_StateAppr} this requirement is no more needed, and it is enough to assume that $A$ generates a strongly continuous semigroup.
%\end{remark}





\section{An example}
\label{sec:Example}

In this concluding section we give an example of an iISS system. 
Let $c>0$ and $L>0$. 
Consider the following reaction-diffusion system
\begin{equation}
\label{GekoppelteNonLinSyst}
\left
\{
\begin{array}{l} 
{\dfrac{\partial x}{\partial t}(l,t) = c \dfrac{\partial^2 x}{\partial l^2}(l,t) + \dfrac{x(l,t)}{1+|l-1|x(l,t)^2}u(l,t),} \\[1.5ex]
{x(0,t) = x(L,t)=0;} 
\end{array}
\right.	
\end{equation}
on the region $(l,t) \in (0,L) \times (0,\infty)$ of 
the $\R$-valued functions $x(l,t)$ and $u(l,t)$. 

Let $X=L_2(0,L)$ and $U=C(0,L)$.
It is easy to see that the above system is bilinear since its nonlinearity satisfies inequality \eqref{eq:BilinOperator}. Clearly, this system is 0-UGASs, therefore it is iISS for any $L>0$. Below we give an explicit construction of an iISS Lyapunov function for this system. Afterwards we will prove that this system is ISS for $L<1$.

Define 
\[
W(x)=\int_0^L{x^2(l)dl} = \|x\|^2_{L_2(0,L)} . 
\]
Since $1+|l-1|x(l,t)^2\geq 1$, we obtain 
\begin{align*}
\dot{W}(x) =& 2 \int_0^L{x(l)\Big( c \dfrac{\partial^2 x}{\partial l^2}(l,t) + \dfrac{x(l,t)}{1+|l-1|x(l,t)^2}u(l,t)\Big)}dl \\
\leq& -2c  \int_0^L \left( \dfrac{\partial x}{\partial l}(l,t) \right)^2 dl + 2 \int_0^L{x^2(l,t)|u(l,t)|dl}.
\end{align*}
Using the Friedrich's inequality (see e.g. \cite[p. 67]{MPF91}) in the first term, we continue estimates:
\begin{align*}
\dot{W}(x) \leq  -2c \left( \frac{\pi}{L} \right)^2 W(x) + 2 W(x) \|u\|_{C(0,L)}
\end{align*}
Choosing  
\begin{align}
V(x)=\ln\left(1+W(x)\right)  
\label{eq:LF_1st_Sys}
\end{align}
yields 
\begin{align}
\dot{V}(x) \leq& 
-2c \!\left( \frac{\pi}{L} \right)^2\!\frac{ W(x)}{1\!+\!W(x)}
\!+2 \frac{ W(x)}{1\!+\!W(x)} \|u\|_{C(0,L)}    \nonumber\\
	%\leq& 
%-2c \!\left( \frac{\pi}{L} \right)^2\!\frac{ W(x)}{1\!+\!W(x)}
%+   2 \|u\|_{C(0,L)}. \nonumber \\
\leq &  -2c \!\left( \frac{\pi}{L} \right)^2\!\frac{\|x\|_{L_2(0,L)}^2}{1\!+\!\|x\|_{L_2(0,L)}^2}
+   2 \|u\|_{C(0,L)},
\label{Examp_Conclusing_iISS_estim} \\
=&
-\alpha(\|x\|_{L_2(0,L)}) + \sigma(\|u\|_{C(0,L)}) , 
\nonumber 
\end{align}
where 
\begin{align}
\alpha(s)=
2c \!\left( \frac{\pi}{L} \right)^2\!\frac{s^2}{1\!+\!s^2}
, \quad 
\sigma(s)=2s . 
\label{Examp_Conclusing_iISS_estimb}
\end{align}
Thus, Proposition \ref{PropSufiISS}
establishes iISS of \eqref{GekoppelteNonLinSyst} irrespective of a value of $L$.


\begin{remark}
Since $x(\cdot,t) \in L_2(0,L)$, the spatial derivative of $x$ above may not exist. However, the above derivations hold for smooth enough functions $x$, and the general result for all $x(\cdot,t) \in L_2(0,L)$ will follow due to Proposition~\ref{DensityArg}.
\end{remark}

Interestingly, when $L<1$, the system \eqref{GekoppelteNonLinSyst} is ISS for 
the input space $U=C(0,L)$ as well as $U=L_2(0,L)$. 
To verify this, we first note that 
\begin{align}
\sup_{s\in\R}
\left|\dfrac{s}{1+|l-1|s^2}\right|=
\dfrac{1}{2\sqrt{1-l}}  
\end{align}
holds for $l<1$. 
Assume $L<1$.  Using the same Lyapunov function $W$ we obtain 
\begin{align}
\dot{W}(x) \leq & 
2 \int_0^L x(l,t)c \frac{\partial^2 x}{\partial l^2}(l,t)dl 
+ 2 \int_0^L \dfrac{1}{2\sqrt{1-l}}|x(l,t)u(l,t)|dl
\nonumber \\
\leq& 
-2c \left( \frac{\pi}{L} \right)^2 \|x\|_{L_2(0,L)}^2
+\dfrac{1}{\sqrt{1-L}}\|x\|_{L_2(0,L)}\|u\|_{L_2(0,L)}
\nonumber \\
\leq& 
-\left(2c \left( \frac{\pi}{L} \right)^2-w\right) \|x\|_{L_2(0,L)}^2
+\dfrac{1}{4(1-L)w}\|u\|_{L_2(0,L)}^2
\label{Example_Estim}
\end{align}
for $0<w<2c({\pi}/{L})^2$.  Recall that 
$\|u\|_{L_2(0,L)}^2\leq L\|u\|_{C(0,L)}^2$. 
Thus, by virtue of Proposition \ref{PropSufiISS}, 
system \eqref{GekoppelteNonLinSyst} is ISS whenever $L<1$. It is 
stressed that the coefficient of $\|u\|_{L_2(0,L)}^2$ 
in \eqref{Example_Estim} goes to $\infty$ as $L$ tends to $1$ from below. Hence, the ISS estimate \eqref{Example_Estim} is valid only if $L<1$. 

For the choice of input space $U=L_p(0,1)$ with $p \geq 1$, the case of 
$L \geq 1$ does not allow us to have an ISS estimate like 
\eqref{Example_Estim}. 
In fact, if $L \geq 1$ and $U=L_p(0,1)$ for any $p \geq 1$, the right hand side of \eqref{GekoppelteNonLinSyst} system is undefined. To see this take $u:l \mapsto |l-1|^{-\tfrac{1}{2p}} \in L_p(0,L)$ and 
$x:l \mapsto |l-1|^{-\tfrac{1}{2}+\tfrac{1}{2p}} \in L_2(0,L)$.
Then $f(x,u):l \mapsto \dfrac{x(l,t)}{1+|l-1|x(l,t)^2}u(l,t) \notin L_2(0,L)$.
Thus, according to our formulation of \eqref{InfiniteDim}, the system \eqref{GekoppelteNonLinSyst} is not well-defined for $U=L_p(0,L)$  for any real $p \geq 1$.

For the choice of input space $U=C(0,L)$, we expect that the
system \eqref{GekoppelteNonLinSyst} is not ISS for $L \geq 1$, 
but we have not proved it at this time. 
The blow-up of the $\sigma$-term in \eqref{Example_Estim} corresponding to 
the dissipation inequality \eqref{GainImplikation} for $V=W$ 
suggests the absence of 
ISS for the system \eqref{GekoppelteNonLinSyst} provided $L \geq 1$. 
The notion of iISS describes the absence of ISS with 
a bounded decay rate $\alpha$ satisfying 
$\lim_{s\to\infty}\alpha(s)<\lim_{s\to\infty}\sigma(s)$, and 
it allows the dissipation inequality \eqref{GainImplikation} to be 
well-defined for all $u\in U$ and all $x\in X$ as in 
the iISS estimate \eqref{Examp_Conclusing_iISS_estim}, i.e., 
\eqref{Examp_Conclusing_iISS_estimb} 
that is valid uniformly over all $L>0$. 
Being able to uniformly characterize iISS irrespectively of whether 
systems are ISS or not 
should be advantageous in many applications. For instance, 
ISS of subsystems is not always necessary for stability of 
their interconnections, and there are examples of 
UGAS interconnections involving iISS systems which are not ISS 
\cite{ITOTAC06,ItJ09,ANGSGiISS,KaJ12}. 




\section{Conclusion}
\label{sec:Conclusion}

We have proved that infinite-dimensional bilinear systems described by differential equations in Banach spaces are integral input-to-state stable provided they are uniformly globally asymptotically stable. For the systems whose state space is Hilbert we have obtained under some additional restrictions, another proof of this result, which leads to a construction of an iISS Lyapunov function for the system. 

The possible directions for future research are investigation of iISS of more general nonlinear control systems and development of novel methods for construction of iISS Lyapunov functions for such systems. Another challenging problem is a study of interconnected infinite-dimensional systems, whose subsystems are iISS or ISS.


\section{Appendix}



\subsection{Proof of Proposition~\ref{DensityArg}}
\label{app:Density Argument}

%Often it is easier to verify first an iISS of a distributed parameter system on certain dense subspaces of a state space and a space of inputs. We will show in this section, that under natural additional assumption this will imply iISS of the system on the original state and input spaces.  
%
%Let $\Sigma:=(X,U_c,\phi)$ be a control system. Let $\hat{X}$, $\hat{U}_c$ be dense linear normed subspaces of $X$ and $U_c$ respectively, and let $\hat{\Sigma}:=(\hat{X},\hat{U}_c,\phi)$ be the system, generated by the same as in $\Sigma$ transition map $\phi$, but restricted to the state space $\hat{X}$ and space of admissible inputs $\hat{U}_c$, endowed with the norms in original spaces $\|\cdot\|_X$ and $\|\cdot\|_{U_c}$ respectively.
%

%In this section we give a proof of Proposition~\ref{DensityArg}.

Since $\hat{\Sigma}$ is iISS, there exist $\beta \in \KL$ and $\mu\in\K$, $\theta \in \Kinf$, such that $\forall \hat{x} \in \hat{X},\; \forall \hat{u} \in \hat{U}_c$ and $\forall t\geq 0$ it holds that 
\begin {equation}
\label{DichteArg_1}
\| \phi(t,\hat{x},\hat{u}) \|_{X} \leq \beta(\| \hat{x} \|_{X},t) + \theta \Big(\int_0^t \mu(\|\hat{u}(s)\|_{U})ds\Big).
\end{equation}
Let $\Sigma$ be not iISS with the same $\beta,\mu,\theta$. Then there exist $t^*>0$, $x\in X$, $u \in U_c$:
\begin {equation}
\label{DichteArg_2}
\| \phi(t^*,x,u) \|_{X} = \beta(\| x \|_{X},t^*) + \theta \Big(\int_0^{t^*} \mu(\|u(s)\|_U)ds\Big) + r,
\end{equation}
where $r=r(t^*,x,u)>0$.
%
From \eqref{DichteArg_1} and \eqref{DichteArg_2} we obtain
%\begin {equation}
%\label{DichteArg_3}
%\| \phi(t^*,x,u) \|_{X} - \| \phi(T,\hat{x},\hat{u}) \|_{X} \geq (\beta(\| x \|_{X},T) - \beta(\| \hat{x} \|_{X},T) ) +  (\gamma( \|u\|_{U_c}) - \gamma( \|\hat{u}\|_{U_c}) ) + r.
%\end{equation}
\begin{align}
\| \phi(t^*,x,u) \|_{X} - & \| \phi(t^*,\hat{x},\hat{u}) \|_{X} \geq 
\beta(\| x \|_{X},t^*) - \beta(\| \hat{x} \|_{X},t^*) 
\nonumber \\
&\quad + \theta \Big(\int_0^{t^*}\hspace{-1.2ex}\mu(\|u(s)\|_U)ds\Big) - \theta \Big(\int_0^{t^*}\hspace{-1.2ex}\mu(\|\hat{u}(s)\|_{U})ds\Big) + r.
\label{DichteArg_3}
\end{align}
Since $\hat{X}$ and $\hat{U}_c$ are dense in $X$ and $U_c$ respectively, 
and since operator $u \mapsto \theta \Big(\int_0^{t^*} \mu(\|u(s)\|_U)ds\Big)$ is continuous, we can find sequences $\{\hat{x}_i \} \subset \hat{X}$: $\|x-\hat{x}_i \|_X \to 0$ and $\{\hat{u}_i \} \subset \hat{U}_c$: $\|u-\hat{u}_i \|_{U_c} \to 0$.
From \eqref{DichteArg_3} it follows that for each arbitrary $\eps>0$, 
there exist $\hat{x}_i$ and $\hat{u}_i$ such that 
\begin{equation*}
\| \phi(t^*,x,u) - \phi(t^*,\hat{x}_i,\hat{u}_i) \|_{X}  \geq
\left| \| \phi(t^*,x,u) \|_{X} - \| \phi(t^*,\hat{x}_i,\hat{u}_i) \|_{X}  \right| \\
\geq r - 2\eps.
\end{equation*}
This contradicts to the assumption of continuous dependence of $\Sigma$ on initial states and inputs.
Thus, $\Sigma$ is iISS with the same $\theta$, $\beta$ and $\mu$ 
in \eqref{iISS_Estimate}. 


\subsection{Proof of Theorem~\ref{thm:Dissipative_Implicative}}
\label{app:Implication_Dissipation}

%We have introduced the ISS Lyapunov function in a so-called dissipative form.
%In the paper \cite{DaM13}, however, so-called implicative form has been used.
%
%\begin{Def}
%A continuous function $V:X \to \R_+$ is called an \textit{ISS Lyapunov function},  if there exist
%$\psi_1,\psi_2 \in \Kinf$, $\alpha \in \Kinf$ and $\gamma \in \K$ 
%such that  \eqref{LyapFunk_1Eig} holds and system \eqref{InfiniteDim} satisfies 
%\begin{equation}
%\label{GainImplikation}
%V(x) \geq \gamma(\|u(0)\|_U) \Rightarrow   \dot{V}_u(x) \leq -\alpha(\|x\|_X) 
%\end{equation}
%for all $x \in X$ and $u\in U_c$.
%\end{Def}
%
%For finite-dimensional systems these two definitions coincide, see \cite{SoW95}. We are going to investigate whether the equivalence holds for infinite-dimensional systems.
%
%\begin{Satz}
%Let the following holds
%\begin{enumerate}
	%\item $f$ be Lipschitz on bounded subsets of $X$ uniformly w.r.t. the second argument.
  %\item \eqref{InfiniteDim} depend continuously on initial states and inputs.
	%\item There exist $q \in \Kinf$ s.t. for all $v \in U$ it holds 
%\begin{eqnarray}
%\| f(0,v)\|_X \leq q(\| v\|_U)
%\label{eq:Boundedness_wrt_u}
%\end{eqnarray}
	%\item $V$ be Fr\'echet differentiable in $X$ and its derivative $\frac{\partial V}{\partial x}$ be bounded on bounded balls. 
	%\end{enumerate}
%Then $V$ is an ISS Lyapunov function in an implicative form iff it is an ISS Lyapunov function in a dissipative form.
%\end{Satz}


First, assume that $V$ is an ISS Lyapunov function in an 
implicative formulation for \eqref{InfiniteDim}. 
Pick any $x \in X$ and $u \in U_c$ s.t. $\|x\|_X \leq \gamma(\|u(0)\|_U)$.
%
Since $V$ is Fr\'echet differentiable in $X$ and since for $x \in D(A)$ the trajectory $\phi(\cdot,x,u)$ is differentiable, 
 $V(x(t))$ is also differentiable (see \cite[par. 2.2]{ATF87}) and can be computed as
\begin{align}
\dot{V}_u(x) &=  \frac{\partial V}{\partial x}(x) \big(Ax + f(x,u(0))\big) 
\nonumber \\
&=  \frac{\partial V}{\partial x}(x) (Ax + f(x,0)) 
+ \frac{\partial V}{\partial x}(x)( f(x,u(0)) - f(x,0)) 
\nonumber \\
&\leq  -\eta(\|x\|_X) + \Big\| \frac{\partial V}{\partial x}(x)\left(f(x,u(0)) - f(x,0)\right)\Big\|_X
\label{TempEstimate-1}
\end{align}
Here $\frac{\partial V}{\partial x}(x)$ denotes a Fr\'echet derivative of $V$ at point $x \in X$ (which is a bounded linear operator from $X$ to $\R$ with an operator norm $\|\cdot\|$). 
Since $\frac{\partial V}{\partial x}$ is bounded on bounded balls, there exists $q_2 \in \K$ such that 
\begin{eqnarray*}
\Big\| \frac{\partial V}{\partial x}(x) \Big\| \leq q_2(\|x\|_X) .
\end{eqnarray*}
Moreover, since $f$ is Lipschitz w.r.t. $x$ on bounded subsets of $X$, we have
\begin{eqnarray*}
\|f(x,u(0)) - f(0,u(0))\|_X \leq w(\|x\|_X)\|x\|_X
\end{eqnarray*}
for some continuous non-decreasing function $w: \R_+\to\R_+$. 
Thus, we have 
\begin{eqnarray*}
\left\| \frac{\partial V}{\partial x}(x)\left(f(x,u(0)) - f(0,u(0))\right)\right\|_X \leq \hat{w}(\|x\|_X)\|x\|_X
\end{eqnarray*}
for some $\hat{w}\in\K$. 
This implies
\begin{align*}
 \left\| \frac{\partial V}{\partial x}(x)\left(f(x,u(0)) - f(x,0)\right)\right\|_X
\leq& \|\tfrac{\partial V}{\partial x}(x)(f(x,u(0)) - f(0,u(0)))\|_X \\
&\hspace{10ex}+ \|\tfrac{\partial V}{\partial x}(x)(f(0,u(0)) - f(x,0))\|_X  \\
\leq& \hat{w}(\|x\|_X)\|x\|_X + \|\tfrac{\partial V}{\partial x}(x)f(0,u(0))\|_X \\
&\hspace{12ex}+ \|\tfrac{\partial V}{\partial x}(x)f(x,0)\|_X \\
\leq& 2 \hat{w}(\|x\|_X)\|x\|_X + \|\tfrac{\partial V}{\partial x}(x)f(0,u(0))\|_X.  
\end{align*}
Due to \eqref{eq:Boundedness_wrt_u} we proceed from \eqref{TempEstimate-1} to
\begin{align*}
\dot{V}_u(x) \leq &-\eta(\|x\|_X) + 2 \hat{w}(\|x\|_X)\|x\|_X + p(\|x\|_X)+q(\|u(0)\|_U).
\end{align*} 
And for $\|x\|_X \leq \gamma(\|u(0)\|_U)$
we obtain finally 
\begin{eqnarray}
\label{AG-like_estimate}
\dot{V}_u(x) \leq -\eta(\|x\|_X)+\sigma(\| u(0)\|_U)
\end{eqnarray} 
with  $\sigma(r):= 2 \hat{w}(\gamma(r))\gamma(r) + p(\gamma(r))+q(r)$ 
which is of class $\K_\infty$. 
Pick $\alpha=\eta$, which is of class $K_\infty$ due to Definition 6. 
Combining \eqref{AG-like_estimate} with the implication \eqref{GainImplikation} we obtain that
for all $x \in D(A)$ and all $u \in U_c$
\begin{eqnarray*}
\dot{V}_u(x) \leq  -\alpha(\|x\|_X)  + \sigma(\| u(0)\|_U),
\end{eqnarray*}
i.e. $V$ is an ISS Lyapunov function in a dissipative form 
\eqref{DissipationIneq} for states from $D(A)$.

Since $A$ generated an analytic semigroup, we can apply a density argument for analytic semigroups as in the proof of Theorem~\ref{Thm:LF_Constructions_for_bilinear_systems} to prove that $V$ is an ISS Lyapunov function in a dissipative form for \eqref{InfiniteDim} on the whole $X$.

%Since \eqref{InfiniteDim} depends continuously on initial states and inputs, $V$ is an ISS Lyapunov function for \eqref{InfiniteDim} 
%on all $X$ since $D(A)$ is dense in $X$. 
%
%{
%\color{red}
%
%The density argument here raises questions as in the theorem about construction of iISS-LFs.
%First of all, question about invariance of $D(A)$ (as described in that theorem) remains.
%
%Secondly, even if $D(A)$ is invariant w.r.t. flow, the density argument will prove only that the system  \eqref{InfiniteDim} will be iISS, and will not prove that $V$ will be an iISS LF in a dissipative form for \eqref{InfiniteDim} on all $X$.
%
%The density argument for the analytic semigroups solves these questions.
%}


Now let us prove the converse. Basically we can follow 
the argument for finite-dimensional systems. 
Suppose that $V$ is an ISS Lyapunov function in a dissipative 
formulation, i.e., \eqref{DissipationIneq}. 
Due to $\sigma\in\K$, property \eqref{eq:ISSalphsig} ensures 
the existence of $\hat{\alpha}\in\K$ such that 
\begin{align*}
&
\hat{\alpha}(s)\leq\alpha(s), \quad \forall s\in\R_+
\nonumber \\
&
\lim_{\tau\to\infty}\hat{\alpha}(\tau)\geq
\lim_{\tau\to\infty}\sigma(\tau) . 
\end{align*}
If either $\lim_{\tau\to\infty}\hat{\alpha}(\tau)=\infty$
or $\lim_{\tau\to\infty}\hat{\alpha}(\tau)>\lim_{\tau\to\infty}\sigma(\tau)$ holds, we can pick a constant $K>1$ such that 
$\lim_{\tau\to\infty}\hat{\alpha}(\tau)\geq K
\lim_{\tau\to\infty}\sigma(\tau)$.  
Then $V$ achieves \eqref{GainImplikation} 
with $\gamma=\hat{\alpha}^{-1}\circ K\sigma\in\K$ 
and $\eta=(1-1/K)\hat{\alpha}\in\K$.  
%
In the case of $\infty>\lim_{\tau\to\infty}\hat{\alpha}(\tau)=
\lim_{\tau\to\infty}\sigma(\tau)$, there exists a continuous function 
$\omega: \R_+\to\R_+$ satisfying ${\rm id}+\omega\in\K_\infty$ and  
\begin{align*}
& 
\omega(\sigma(s))>0, \quad s\in(0,\infty)  
\\
& 
\lim_{\tau\to\infty}\omega(\sigma(\tau))=0 .
\end{align*}
Then property \eqref{DissipationIneq} with 
$\gamma=\hat{\alpha}^{-1}\circ({\rm id}+\omega)\circ\sigma\in\K_\infty$ yields 
\eqref{GainImplikation} with 
$\eta=\left({\rm id}-({\rm id}+\omega)^{-1}\right)\circ\hat{\alpha}
\in\PD$. 
The function $\eta$ obtained in the above two cases is 
only guaranteed to satisfy either $\eta\in\K$ or $\eta\in\PD$. 
The function $V$ can be transformed into another continuous function 
$\hat{V}:X \to \R_+$ by applying the technique in \cite{SoT95} 
to obtain $\eta\in\K_\infty$ in \eqref{GainImplikation}. 
The transformed $\hat{V}$ is an ISS Lyapunov function of 
\eqref{InfiniteDim} in an implicative form. 



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%\appendix
%
%\subsection{A variation of Wirtinger's inequality}
%
%For $L>0$, let $W^{2,1}(0,L)$ denote a Sobolev space
%of functions $x\in L_2(0,L)$ which have the first 
%order weak derivatives, all of which belong to 
%$L_2(0,L)$. 
%We used the following in subsection \ref{sec:parablic}. 
%
%\begin{Satz}[A variation of Wirtinger's inequality]\label{theorem:Wirtinger}
%For every $x \in W^{2,1}(0,L)$ it holds that
%\begin{align}
%\label{Wirtinger_Variation_Ineq}
%L x^2(0) +\frac{4 L^2}{\pi^2} \int_0^L{\left( \frac{\partial x(l)}{\partial l} \right)^2 dl} \geq \int_0^L {x^2(l)}dl
%\end{align}
%\end{Satz}

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iisspde_AM_2014_06_09_MCRF_Format.bbl
\begin{thebibliography}{10}

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