# Dwell-time stability conditions for infinite dimensional impulsive systems

Sergey Dashkovskiy  Vitalii Slynko  Institute of Mathematics, University of
Wuerzburg, Emil-Fischer-Str. 40, 97074 Wuerzburg, Germany
email<EMAIL_ADDRESS>
###### Abstract

We consider nonlinear impulsive systems on Banach spaces subjected to
disturbances and look for dwell-time conditions guaranteeing the the ISS
property. In contrary to many existing results our conditions cover the case
where both continuous and discrete dynamics can be unstable simultaneously.
Lyapunov type methods are use for this purpose. The effectiveness of our
approach is illustrated on a rather nontrivial example, which is feedback
connection of an ODE and a PDE systems.

###### keywords:

Nonlinear systems, impulsive systems, input-to-state stability, Lyapunov
methods

††thanks: This work was supported by the German Research Foundation (DFG),
Grant number DA767/12-1. Corresponding author: Sergey Dashkovskiy.

##  1 Introduction

Hybrid systems accommodate continuous and discontinuous behavior, which allow
to model modern practical processes, where a combination of analog and digital
effects takes place as well as other processes where, for example, collisions
can change the systems state instantaneously. The general theory of hybrid
systems includes results on existence, uniqueness, continuous dependence on
initial data, stability and robustness of solutions, see [20, 19, 10].
Impulsive systems are a particular subclass of hybrid ones, see [15, 9], where
many stability results were developed, in particular by means of the linear
approximations and by the direct Lyapunov method as well as by means of the
corresponding comparison principle. Such systems can be seen as a combination
of continuous and discrete subsystems. Stability conditions for nonlinear
impulsive systems using continuously differential Lyapunov functions were
provided in [15]. The proofs are based on the comparison principle and require
that either of the continuous and discrete subsystems is stable. More general
stability conditions were developed in [9] on the base of Lyapunov functions
discontinuous in time. These conditions allow also to establish stability even
in the case, when both dynamics are unstable. Moreover in this class of
Lyapunov functions these conditions are necessary and sufficient [7].
Practical applications of these results lead to difficulties in contructions
of suitable Lyapunov functions. This problem was considered in [16, 2] for the
case of linear finite dimensional impulsive systems with constant
coefficients. Even in this relatively simple case these works need to use
lyapunov functions, which depend explicitly on time. In general, a
construction of Lyapunov function leads to a rather complex boundary value
problem for a systems of matrix differential equations [2].

In contrary to the above mentioned works, we will consider nonlinear infinite
dimensional impulsive systems. Our aim is to establish stability conditions by
means of smooth Lyapunov functions, which do not depend on time at least in
case when the right hand sides of the equations do not depend on time. As well
we would like to cover the case, where both continuous and discrete dynamics
can be unstable. Our class of systems is essentially more general, than in the
above mentioned literature and the existing methods cannot be applied directly
for our purposes.

Certain results in this direction exist: The work [11] uses second order and
the work [5] uses even higher order derivatives of Lyapunov functions, which
leads to additional restrictions on the equations constituting the impulsive
system. Different related approaches were developed in [17, 1], devoted to
linear impulsive systems on Banach spaces containing continuous operators
only. Also these results cannot be extended directly to the class of systems
considered in this paper.

Furthermore, we will consider systems with disturbances and hence we will work
in the ISS framework. This allows for potential applications in view of
studying stability of interconnected systems. Several related results in this
framework exist, see [4, 14, 3], for example. The work [4] provides stability
conditions of the dwell-time type for systems in Banach spaces, however it was
assumed that either discrete or continuous dynamics possesses the ISS
property. Interconnections of impulsive systems, such that not all of them are
ISS were studied in [14]. The work [3] studies ISS properties of impulsive
systems where jump mappings may depend on time and provides dwell-time
conditions establishing the ISS property. However only finite dimensional
systems were considered in the last two papers.

The main result of our work extends the results of [4] to the general case,
where we do not require that either continuous or discrete dynamics is ISS. To
this end we use completely different approach. We split the state space into
two subsets $X=G^{-}\cup G^{+}$ with $\text{int }G^{-}\cap G^{+}=\emptyset$
and such that $G^{+}$ is forward invariant. We use two auxiliary functions,
one is $V$ of the Lyapunov type the other one is $W$ of Chetaev type. The
latter one is used to establish the invariance property of $G^{+}$.

The Lyapunov function $V$ decays along the trajectories in $G^{-}$ but can
increase in $G^{+}$ and allows to estimate the change of solutions between the
jumps. It is assumed that the jumps from the state in $G^{+}$ are always
stabilizing, and there is no such restriction in $G^{-}$. Finally, the ISS
property is guaranteed by a dwell-time condition. In the particular case, when
both continuous and discrete dynamics are not stable this conditions restricts
the jumps frequency both from below and above.

We apply our result to a feedback connection of a linear ODE and a nonlinear
PDE of the parabolic type. This illustrates how our approach can be applied
and demonstrates its powerfulness. Let us note that this example cannot be
handled in view of stability by the existing results because of its
nonlinearity, possible instability of both dynamics as well as irregular time
instants of jumps.

##  2 Preliminaries

We will use the following classes of continuous functions, frequently called
comparison functions
$\mathcal{P}=\\{\gamma:\,\mathbb{R}_{+}\to\mathbb{R}_{+}|\,\gamma(r)=0\;\Leftrightarrow\;r=0\\},$
$\mathcal{K}=\\{\gamma\in\mathcal{P}|\,\gamma\text{ is strictly
increasing}\\},$
$\mathcal{K}_{\infty}=\\{\gamma\in\mathcal{K}|\,\gamma\text{ is
unbounded}\\},$
$\mathcal{K}_{\infty}^{2}=\\{\gamma\,:\,\mathbb{R}_{+}^{2}\to\mathbb{R}_{+}\,|\,\gamma(\cdot,s)\in\mathcal{K}_{\infty},\,\gamma(s,\cdot)\in\mathcal{K}_{\infty}\\},$
$\mathcal{L}=\\{\gamma:\mathbb{R}_{+}\to\mathbb{R}_{+}|\,\text{decreasing
with}\lim\limits_{t\to\infty}\gamma(t)=0\\},$
$\mathcal{K}\mathcal{L}=\\{\beta:\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}_{+}|\,\,\beta(\cdot,t)\in\mathcal{K},\,\beta(s,\cdot)\in\mathcal{L}\\}.$
For any $\varrho\in\mathcal{K}$ and any $a,b\in\mathbb{R}_{+}$ it follows by
the monotonicity that

$\varrho(a+b)\leq\varrho(2a)+\varrho(2b).$ (1)

$C[0,l]$ denotes the space of functions continuous on $[0,l]$ with values in
$\mathbb{R}$ and the norm $\|f\|_{C[0,l]}=\max\limits_{x\in[0,l]}|f(x)|$,
$C^{k}[0,l]$ stands for the space of $k$-times continuously differentiable
functions normed by
$\|f\|_{C^{k}[0,l]}=\max\limits_{p=0,\dots,k}\max\limits_{x\in[0,l]}|f^{(p)}(x)|$.
$H^{0}[0,l]=L^{2}[0,l]$ is the Hilbert space of measurable and square
integrable functions with the scalar product
$(f,g)_{L^{2}[0,l]}=\int\limits_{0}^{l}f(z)g(z)\,dz$.
$\mathfrak{L}(L^{2}[0,l])$ denotes the Banach algebra of linear bounded
operators on $L^{2}[0,l]$.

The Hilbert space $H^{k}[0,l]\subset L^{2}[0,l]$ is a subset of $L^{2}[0,l]$
of functions $f$ such that $f^{(p)}\in L^{2}[0,l]$ for , $p=0,\dots,k$ and the
scalar product defined by

$(f,g)_{H^{k}[0,l]}=\sum\limits_{p=0}^{k}\int\limits_{0}^{l}f^{(p)}(z)g^{(p)}(z)\,dz.$
(2)

Recall that $H^{k}[0,l]$ is the completion of $C^{k}[0,l]$ with respect to the
norm $\|f\|_{H^{k}[0,l]}=\sqrt{(f,f)_{H^{k}[0,l]}}$. $C_{0}^{\infty}[0,l]$ is
the space of infinitely smooth on $[0,l]$ functions vanishing in the vicinity
of $x=0$ and $x=l$. $C_{0}^{\infty}([0,T],C_{0}^{\infty}[0,l])$ is the set of
mappings $f\,:\,[0,T]\to C_{0}^{\infty}[0,l]$ vanishing in the vicinity of
$t=0$ and $t=T$. Completion of $C_{0}^{\infty}[0,l]$ with respect to the norm
$\eqref{0.1}$ is denoted by $H_{0}^{k}[0,l]$. In the space $H_{0}^{1}[0,l]$,
due to the inequality of Friedrichs, the norms $\|\cdot\|_{H^{1}[0,l]}$ and
$\|\cdot\|_{H^{1}_{0}[0,l]}$ defined by
$\|f\|_{H^{1}_{0}[0,l]}=\|f_{z}\|^{2}_{L^{2}[0,l]}$, are equivalent.
$L^{\infty}(\mathbb{R}_{+})$ is the space of measurable and essentially
bounded functions $f:\mathbb{R}_{+}\to\mathbb{R}$,

For $M\subset\mathbb{R}$ and a Banach space $X$ by $L^{\infty}(M,X)$ we denote
the space of mappings $f\,:\,M\to X$ normed by $\|f\|_{L^{\infty}}=\sup_{m\in
M}\|f(m)\|_{X}$, where in the particular case $M=\mathbb{Z}_{+}$ we write
$L^{\infty}(\mathbb{Z}_{+},X)=l^{\infty}(X)$. $B_{r}(x)$ denotes the open ball
centered at $x\in X$ of radius $r>0$. For Banach spaces $X,Y$ and mappings
$f\,:\,\mathbb{R}\to X$, $g\,:\,\mathbb{R}\to Y$ the Cartesian product
$f\times g\,:\,\mathbb{R}\to X\times Y$ is defined by $(f\times
g)(t)=(f(t),g(t))$. For $\alpha\in(0,1]$ the space of locally Hölder
continuous functions $=f:\mathbb{R}_{+}\to X$ is denoted by
$H_{\alpha}(\mathbb{R}_{+},X)$ and let
$H_{loc}(\mathbb{R}_{+},X):=\cup_{\alpha\in(0,1]}H_{\alpha}(\mathbb{R}_{+},X)$.

For a linear bounded operator $A$ defined on a Bacnach space by $\sigma(A)$ we
denote its spectrum and by $r_{\sigma}(A)$ its spectral radius.

$\mathbb{R}^{n\times m}$ denotes the linear space of matrices of the size
$n\times m$, where in case $m=n$, the space $\mathbb{R}^{n\times n}$ is a
Banach algebra. For $A\in\mathbb{R}^{n\times n}$ we denote $\mathbb{R}^{n}$:
$\|A\|=\sup_{\|x\|=1}\|Ax\|=\lambda_{\max}^{1/2}(A^{\operatorname{T\,}}A)$.
$\mathbb{S}^{n}$ denotes the set of symmetric matreces of the size $n$. For
$P,Q\in\mathbb{S}^{n}$ we write $P\succ Q$ if the matrix $P-Q$ is positive
definite. For $A\in\mathbb{S}^{n}$ by $\lambda_{\min}(A)$ and
$\lambda_{\max}(A)$ we denote the smallest and the largest eigenvalues of $A$
respectively. The inequality $P\leq Q$ should be understood element-wise.

The following well-known inequalities will be used

$\begin{gathered}xy\leq\frac{x^{p_{1}}}{p_{1}}+\frac{y^{p_{2}}}{p_{2}},\quad
x\geq 0,\quad y\geq 0,\quad p_{1}\in(0,\infty),\quad\\\
\frac{1}{p_{1}}+\frac{1}{p_{2}}=1,\end{gathered}$ (3)

and for any $f\in L^{p_{1}}[0,l]$ and $g\in L^{p_{2}}[0,l]$ it holds that

$\begin{gathered}\Big{|}\int\limits_{0}^{l}f(t)g(t)\,dt\Big{|}\leq\Big{(}\int\limits_{0}^{l}|f(t)|^{p_{1}}\,dt\Big{)}^{1/p_{1}}\Big{(}\int\limits_{0}^{l}|g(t)|^{p_{2}}\,dt\Big{)}^{1/p_{2}},\\\
\quad
p_{1}\in(1,\infty),\quad\frac{1}{p_{1}}+\frac{1}{p_{2}}=1,\end{gathered}$ (4)

known as Young’s and Hölder’s inequalities, in the particular case
$p_{1}=p_{2}=2$ they are also known as Cauchy inequalities. For any $f\in
H_{0}^{1}(0,l)$ also holds

$\begin{gathered}\|f\|_{H^{1}_{0}(0,l)}^{2}\geq\frac{\pi^{2}}{l^{2}}\|f\|_{L^{2}[0,l]}^{2},\end{gathered}$
(5)

and if additionally $f\in H_{0}^{1}(0,l)\cap H^{2}(0,l)$, then

$\begin{gathered}\|\partial_{zz}f\|_{L^{2}[0,l]}^{2}\geq\frac{\pi^{2}}{l^{2}}\|\partial_{z}f\|_{L^{2}[0,l]}^{2},\end{gathered}$
(6)

##  3 Stability notions

Let us introduce dynamical systems that we will consider adapting definitions
from [18, 12, 13].

######  Definition 1.

Let $X$ be the state space with the norm $\|\cdot\|_{X}$ and
$\mathcal{U}_{1}\subset\\{f\,:\mathbb{R}\to U_{1}\\}$ be the space of input
signals normed by $\|\cdot\|_{\mathcal{U}_{1}}$ with values in a nonempty
subset $U_{1}$ of some linear normed space and invariant under the time
shifts, that is if $d_{1}\in\mathcal{U}_{1}$ and $\tau\in\mathbb{R}$, then
$\mathcal{S}_{\tau}d_{1}\in\mathcal{U}_{1}$, where
$\mathcal{S}_{s}\,:\,\mathcal{U}_{1}\to\mathcal{U}_{1}$, $s\in\mathbb{R}$ is
the linear operator defined by $\mathcal{S}_{s}u(t)=u(t+s)$ and satisfying
$\|\mathcal{S}_{s}\|\leq 1$. It is also assume that for all
$u_{1},u_{2}\in\mathcal{U}$ and any $t\geq t_{0}$ we have that
$u(\tau):=u_{1}(\tau)$ for $\tau\in[t_{0},t]$ and $u(\tau):=u_{2}(\tau)$ for
$\tau>t_{0}$ it holds that $u\in\mathcal{U}$.

The triple $\Sigma_{c}=(X,\mathcal{U}_{1},\phi_{c})$ is called dynamical
system with inputs if the mapping
$\phi_{c}\,\,:\,(t,t_{0},x,d_{1})\mapsto\phi_{c}(t,t_{0},x,d_{1})$ defined for
all
$(t,t_{0},x,d_{1})\in[t_{0},t_{0}+\epsilon_{t_{0},x,d_{1}})\times\mathbb{R}\times
X\times\mathcal{U}_{1}$ for some positive $\epsilon_{t_{0},x,d_{1}}$ and
satisfies the following axioms

($\Sigma_{c}1$) for $t_{0}\in\mathbb{R}$, $x\in X$, $d_{1}\in\mathcal{U}_{1}$,
$t\in[t_{0},t_{0}+\epsilon_{t_{0},x,d_{1}})$ the value of
$\phi_{c}(t,t_{0},x,d_{1})$ is well defined and the mapping
$t\mapsto\phi_{c}(t,t_{0},x,d_{1})$ is continuous on
$(t_{0},t_{0}+\epsilon_{t_{0},x,d_{1}})$ with $\lim_{t\to
t_{0}+}\phi_{c}(t,t_{0},x,d_{1})=x$;

($\Sigma_{c}2$) $\phi_{c}(t,t,x,d_{1})=x$ for any $(x,d_{1})\in
X\times\mathcal{U}_{1}$, $t\in\mathbb{R}$

($\Sigma_{c}3$) for any $t_{0}\in\mathbb{R}$,
$(t,x,d_{1})\in[t_{0},t_{0}+\epsilon_{t_{0},x,d_{1}})\times
X\times\mathcal{U}_{1}$ and $\widetilde{d}_{1}\in\mathcal{U}_{1}$ with
$d_{1}(s)=\widetilde{d}_{1}(s)$ for $s\in[t_{0},t]$ it holds that
$\phi_{c}(t,t_{0},x,d_{1})=\phi_{c}(t,t_{0},x,\widetilde{d}_{1})$;

($\Sigma_{c}4$) for any $(x,d_{1})\in X\times\mathcal{U}_{1}$ and
$t\geq\tau\geq t_{0}$ with $\tau\in[t_{0},t_{0}+\epsilon_{t_{0},x,d_{1}})$,
$t\in[\tau,\tau+\epsilon_{\tau,\phi(\tau,t_{0},x,d_{1}),d_{1}})\cap[t_{0},t_{0}+\epsilon_{t_{0},x,d_{1}})$
it holds that

$\phi_{c}(t,t_{0},x,d_{1})=\phi_{c}(t,\tau,\phi(\tau,t_{0},x,d_{1}),d_{1}),$

($\Sigma_{c}5$) for any $(x,d_{1})\in X\times\mathcal{U}_{1}$ and
$t\in[t_{0},t_{0}+\epsilon_{t_{0},x,d_{1}})$, it holds that

$\begin{gathered}\epsilon_{t_{0}+\tau,x,d_{1}}=\epsilon_{t_{0},x,\mathcal{S}_{\tau}d_{1}},\\\
\phi_{c}(t+\tau,t_{0}+\tau,x,d_{1})=\phi_{c}(t,t_{0},x,\mathcal{S}_{\tau}d_{1}).\end{gathered}$

Note that $(\Sigma_{c}5)$ implies that for all
$t\in[\tau,\tau+\epsilon_{\tau,x_{0},d_{1}})$, $\tau\leq t$

$\phi_{c}(t,\tau,x,d_{1})=\phi_{c}(t-\tau,0,x,\mathcal{S}_{\tau}d_{1}).$ (7)

Systems with impulsive actions are defined as follows

######  Definition 2.

Let $\mathcal{E}=\\{\tau_{k}\\}_{k=0}^{\infty},\tau_{k}\in\mathbb{R}$ be a
strictly increasing time sequence of impulsive actions with
$\lim\limits_{k\to\infty}\tau_{k}=\infty$. Let
$\mathcal{U}_{2}\subset\\{f\,:\,\mathbb{Z}_{+}\to\mathcal{U}_{2}\\}$ be the
space of input signals normed by $\|\cdot\|_{\mathcal{U}_{2}}$ and taking
values in a nonempty subset $U_{2}$ of some linear normed space. Let
$g\,:\,X\times U_{2}\to X$ be a mapping defining impulsive actions and the
mapping $\phi$ be defined for all
$(t,t_{0},x,d_{1},d_{2})\in\mathbb{R}\times\mathbb{R}\times
X\times\mathcal{U}_{1}\times\mathcal{U}_{2}$, $t\geq t_{0}$.

The following data $\Sigma=(X,\Sigma_{c},\mathcal{U}_{2},g,\phi,\mathcal{E})$
defines a (forward complete) impulsive system if

$(\Sigma_{1})$ for all $(k,x,d_{1})\in\mathbb{Z}_{+}\times
X\times\mathcal{U}_{1}$ the system $\Sigma_{c}$ satisfies

$\tau_{p(t_{0})}-t_{0}<\epsilon_{t_{0},x,d_{1}},\quad
T_{k}:=\tau_{k+1}-\tau_{k}<\epsilon_{\tau_{k},x,d_{1}}$

where we denote
$p(t_{0}):=\min\\{k\in\mathbb{Z}_{+}\,:\tau_{k}\in\mathcal{E}_{t_{0}}\\}$ with
$\mathcal{E}_{t_{0}}=[t_{0},\infty)\cap\mathcal{E}$; and

$(\Sigma_{2})$ the mapping $\phi$ satisfies

$\begin{gathered}\phi(t,t_{0},x,d_{1},d_{2})=\phi_{c}(t,t_{0},x,d_{1}),\quad\text{
for all }\quad t\in[t_{0},\tau_{p(t_{0})}],\\\
\phi(t,t_{0},x,d_{1},d_{2})=\phi_{c}(t,\tau_{k},g(\phi(\tau_{k},t_{0},x,d_{1},d_{2}),d_{2}(k)),d_{1})\\\
\quad\text{ for all }\quad t\in(\tau_{k},\tau_{k+1}],\quad
k\in\mathbb{Z}_{+},k\geq p(t_{0}).\end{gathered}$

We will denote for short

$\begin{gathered}\phi(\tau_{k}^{+},t_{0},x,d_{1},d_{2})=g(\phi(\tau_{k},t_{0},x,d_{1},d_{2}),d_{2}(k)),\\\
k\geq p(t_{0}),\;\tau_{k}\geq t_{0}\end{gathered}$

The conditions $(\Sigma_{c}1)$ and $(\Sigma_{2})$ imply

$\begin{gathered}\lim\limits_{t\to\tau_{k}+}\phi(t,t_{0},x,d_{1},d_{2})=\phi(\tau_{k}^{+},t_{0},x,d_{1},d_{2}),\\\
\lim\limits_{t\to\tau_{k}-}\phi(t,t_{0},x,d_{1},d_{2})=\phi(\tau_{k},t_{0},x,d_{1},d_{2});\end{gathered}$

and $(\Sigma_{c}4)$, $(\Sigma_{c}5)$, $(\Sigma_{2})$ imply that for
$t\geq\tau\geq t_{0}$, $(x,d_{1},d_{2})\in
X\times\mathcal{U}_{1}\times\mathcal{U}_{2}$ the following holds

$\begin{gathered}\phi(t,t_{0},x,d_{1},d_{2})=\phi(t,\tau,\phi(\tau,t_{0},d_{1},d_{2}),d_{1},d_{2}).\end{gathered}$
(8)

The system $\Sigma_{c}$ describes the continuous dynamics of the impulsive
system $\Sigma$. One can also consider its discrete dynamics separately as a
system $\Sigma_{d}$ defined next

######  Definition 3.

A discrete dynamical system with input
$\Sigma_{d}=(X,g,\phi_{d},\mathcal{U}_{2})$ is defined by a normed state space
$(X,\|\cdot\|_{X})$; a space of input signals
$\mathcal{U}_{2}\subset\\{f\,:\,\mathbb{Z}_{+}\to U_{2}\\}$ with norm
$\|\cdot\|_{\mathcal{U}_{2}}$ and values in a nonempty subset $U_{2}$ of a
linear normed space; a mapping $g\,:\,X\times U_{2}\to X$; and a mapping
$\phi_{d}\,\,:(k,l,x,d_{2})\mapsto\phi_{d}(k,l,x,d_{2})$, for
$(k,l,x,d_{2})\in\mathbb{Z}_{+}\times\mathbb{Z}_{+}\times
X\times\mathcal{U}_{2}$, $k\geq l$ such that

$(\Sigma_{d}1)$ $d(k,k,x,d_{2})=x$ and
$\phi_{d}(k+1,l,x,d_{2})=g(\phi_{d}(k,l,x,d_{2}),d_{2}(k))$ for all $k\geq l$.

We assume that $\Sigma$ satisfies the following

######  Assumption 1.

There exist $\xi,\,\xi_{\tau}\in\mathcal{K}_{\infty}$, $\tau\in\mathbb{R}_{+}$
and $\eta_{\tau},\eta\in\mathcal{K}_{\infty}$ such that

$\begin{gathered}\|\phi_{c}(t,0,x,d_{1})\|\leq\xi_{\tau}(\|x\|)+\eta_{\tau}(\|d_{1}\|_{\mathcal{U}_{1}}),\quad
t\in[0,\tau],\end{gathered}$ (9)

where $(x,d_{1})\in X\times\mathcal{U}_{1}$ and

$\begin{gathered}\|g(x,d_{2})\|\leq\xi(\|x\|)+\eta(\|d_{2}\|_{\mathcal{U}_{2}}),\end{gathered}$
(10)

where $(x,d_{2})\in X\times\mathcal{U}_{2}$.

Now we define the main stability property of this paper.

######  Definition 4.

For a fixed time sequence $\mathcal{E}$ of impulsive actions the system
$\Sigma$ is called input-to-state stable (ISS) if there exist functions
$\beta_{t_{0}}\in\mathcal{K}\mathcal{L}$,
$\gamma_{t_{0}}\in\mathcal{K}_{\infty}$, such that for all $x\in X$ and all
$(d_{1},d_{2})\in\mathcal{U}_{1}\times\mathcal{U}_{2}$ it holds that

$\begin{gathered}\|\phi(t,t_{0},x,d_{1},d_{2})\|_{X}\leq\beta_{t_{0}}(\|x\|_{X},t)+\gamma_{t_{0}}(d),\quad
t\geq t_{0}\end{gathered}$ (11)

where $d:=\max\\{\|d_{1}\|_{\mathcal{U}_{1}},\|d_{2}\|_{\mathcal{U}_{2}}\\}$.

######  Definition 5.

The Lie derivative of a function $V\,:\,X\to\mathbb{R}$ is defined by

$\begin{gathered}\dot{V}(x,\xi)=\lim_{t\to
0+}\frac{1}{t}(V(\phi_{c}(t,0,x,\xi))-V(x)),\quad(x,\xi)\in X\times
U_{1}\end{gathered}$

In many practical particular cases there are simpler expressions for
calculation of the Lie derivative possible, see Remark 2.15 in [13].

The aim of our work is to establish conditions guaranteeing the ISS property
for nonlinear impulsive systems. Next we define the class of functions that we
will use as Lyapunov functions for studying the ISS property.

######  Definition 6.

A continuous function $V:\,X\to\mathbb{R}_{+}$ is called ISS-Lyapunov function
if for some $\alpha_{1}$, $\alpha_{2}\in\mathcal{K}_{\infty}$ it holds that

$\begin{gathered}\alpha_{1}(\|x\|_{X})\leq V(x)\leq\alpha_{2}(\|x\|_{X}),\quad
x\in X,\end{gathered}$ (12)

and there exists a function $W\in C(X,\mathbb{R})$, $W(0)=0$ such that the Lie
derivatives $\dot{V}(x,\xi)$ and $\dot{W}(x,\xi)$ exist for all $(x,\xi)\in
X\times U_{1}$, the sets

$\begin{gathered}G^{+}=\\{x\in X\,:\,W(x)\geq 0\\},\\\ G^{-}=\\{x\in
X\,:\,W(x)\leq 0\\},\\\ \end{gathered}$ (13)

are not empty and for some $\chi\in\mathcal{K}_{\infty}$ and
$\varphi_{i}\in\mathcal{P}$, $\psi_{i}\in\mathcal{P}$, $i=1,2$ so that

$\begin{gathered}x\in
G^{-},\quad\|x\|_{X}\geq\max\\{\chi(\|\xi\|_{U_{1}}),\chi(\|\mu\|_{U_{2}})\\}\\\
\Rightarrow\begin{cases}\dot{V}(x,\xi)\leq-\varphi_{1}(V(x)),\\\
V(g(x,\mu))\leq\psi_{1}(V(x))\end{cases}\end{gathered}$ (14)
$\begin{gathered}x\in
G^{+},\quad\|x\|_{X}\geq\max\\{\chi(\|\xi\|_{U_{1}}),\chi(\|\mu\|_{U_{2}})\\}\\\
\Rightarrow\begin{cases}\dot{V}(x,\xi)\leq\varphi_{2}(V(x)),\\\
V(g(x,\mu))\leq\psi_{2}(V(x))\end{cases}\end{gathered}$ (15)

and

$\begin{gathered}W(x)=0,\;\xi\neq
0,\;\|x\|\geq\chi(\|\xi\|)\Rightarrow\dot{W}(x,\xi)>0.\end{gathered}$ (16)

This definition defers from the known ones, as for example in [4], due to the
auxiliary function $W$ which is of Chetaev type.

##  4 Main result

Our main results establish conditions guaranteeing the ISS property of
$\Sigma$.

######  Theorem 1.

Let the impulsive system $\Sigma$ satisfy the Assumption 1 and possesses an
ISS-Lyapunov function $V$, satisfying (12)–(15) such that for some constants
$\theta_{1}$ and $\theta_{2}$ ($\theta_{1}\leq\theta_{2}$) and $\delta>0$ for
all $a>0$ holds

$\begin{gathered}\int\limits_{a}^{\psi_{1}(a)}\frac{ds}{\varphi_{1}(s)}\leq\theta_{1}-\delta,\end{gathered}$
(17)
$\begin{gathered}\int\limits_{\psi_{2}(a)}^{a}\frac{ds}{\varphi_{2}(s)}\geq\theta_{2}+\delta.\end{gathered}$
(18)

Then for any $\mathcal{E}$ such that the dwell-time
$T_{k}=\tau_{k+1}-\tau_{k}$, $k\in\mathbb{Z}_{+}$ satisfies $\theta_{1}\leq
T_{k}\leq\theta_{2}$ the system $\Sigma$ is ISS.

The proof of this theorem is split into several steps. Without loss of
generality we assume $t_{0}\leq\tau_{0}$, $p(t_{0})=0$.

######  Proposition 1.

Let $\Sigma$ satisfy the Assumtion 1 and for some
$\beta_{\tau_{0}^{+}}\in\mathcal{K}\mathcal{L}$,
$\gamma_{\tau_{0}^{+}}\in\mathcal{K}_{\infty}$ its solutions satisfy for all
$t>\tau_{0}$ the inequality

$\begin{gathered}\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\\\
\leq\beta_{\tau_{0}^{+}}(\|\phi(\tau_{0}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X},t)+\gamma_{\tau_{0}^{+}}(d),\end{gathered}$
(19)

Then $\Sigma$ is ISS.

Proof. Let us fix any initial state $\phi_{0}$ and disturbance $d_{1},d_{2}$.
From $\Sigma_{2}$, (7) and (9) it follows that for all $t\in[t_{0},\tau_{0}]$
the next estimate holds

$\begin{gathered}\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}=\|\phi_{c}(t-t_{0},0,\phi_{0},\mathcal{S}_{t_{0}}d_{1})\|_{X}\\\
\leq\xi_{\tau_{0}-t_{0}}(\|\phi_{0}\|_{X})+\eta_{\tau_{0}-t_{0}}(d).\end{gathered}$
(20)

Also from (10) and (1) we have the next estimate
$\|\phi(\tau_{0}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\leq\xi(\|\phi(\tau_{0},t_{0},\phi_{0},d_{1},d_{2})\|_{X})+\eta(d)$
$\leq\xi(\xi_{\tau_{0}-t_{0}}(\|\phi_{0}\|_{X})+\eta_{\tau_{0}-t_{0}}(d))+\eta(d)\\\
\leq\widehat{\xi}_{\tau_{0}-t_{0}}(\|\phi_{0}\|_{X})+\widehat{\eta}_{\tau_{0}-t_{0}}(d),$
for some fixed $\widehat{\xi}_{\tau_{0}-t_{0}}\in\mathcal{K}_{\infty}$,
$\widehat{\eta}_{\tau_{0}-t_{0}}\in\mathcal{K}_{\infty}$.

Finally, from (19) and (1) we have

$\begin{gathered}\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|\leq\beta_{\tau_{0}^{+}}(\|\phi(\tau_{0}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X},t)\\\
+\gamma_{\tau_{0}^{+}}(d)\leq\beta_{\tau_{0}^{+}}(\widehat{\xi}_{\tau_{0}-t_{0}}(\|\phi_{0}\|_{X})+\widehat{\eta}_{\tau_{0}-t_{0}}(d),t)+\gamma_{\tau_{0}^{+}}(d)\\\
\leq\widehat{\beta}_{\tau_{0}^{+}}(\|\phi_{0}\|_{X},t)+\widetilde{\beta}_{\tau_{0}^{+}}(\widetilde{\eta}_{\tau_{0}-t_{0}}(d),\tau_{0}^{+})+\gamma_{\tau_{0}^{+}}(d)\\\
\leq\widehat{\beta}_{\tau_{0}^{+}}(\|\phi_{0}\|_{X},t)+\widehat{\gamma}_{\tau_{0}^{+}}(d),\quad
t>\tau_{0}\end{gathered}$ (21)

for some $\widehat{\beta}_{\tau_{0}^{+}}$,
$\widetilde{\beta}_{\tau_{0}^{+}}\in\mathcal{K}\mathcal{L}$,
$\widehat{\gamma}_{\tau_{0}^{+}}\in\mathcal{K}_{\infty}$. Let us define for
$s\geq 0$ and $t\geq t_{0}$

$\begin{gathered}b_{t_{0}}(s,t):=\begin{cases}\xi_{\tau_{0}-t_{0}}(s),\quad
t\in[t_{0},\tau_{0}],\\\ \xi_{\tau_{0}-t_{0}}(s)e^{-t+\tau_{0}},\quad
t>\tau_{0};\end{cases}\\\
\beta_{t_{0}}(s,t):=\max\\{b_{t_{0}}(s,t),\widehat{\beta}_{\tau_{0}^{+}}(s,t)\\};\end{gathered}$
$\begin{gathered}\gamma_{t_{0}}(s):=\max\\{\widehat{\gamma}_{\tau_{0}^{+}}(s),\eta_{\tau_{0}-t_{0}}(s)\\}.\end{gathered}$

From these definitions follows $\beta_{t_{0}}\in\mathcal{K}\mathcal{L}$,
$\gamma_{t_{0}}\in\mathcal{K}_{\infty}$ and (20) with (21) imply the ISS
property for the impulsive system $\Sigma$, which proves the proposition.

######  Lemma 1.

Let $V$ be an ISS-Lyapunov function of $\Sigma$ satisfying (12)–(15) with
$\varphi_{1},\varphi_{2}$ satisfying (17)-(18). Fix any $\phi_{0}\in X$,
$(d_{1},d_{2})\in\mathcal{U}_{1}\times\mathcal{U}_{2}$ and $r>\chi(d)$. If
$\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\geq r$ for all
$t\in(\tau_{p},\tau_{m}]$ with some $m>p$, then

$\begin{gathered}F(v(\tau_{l}^{+}),v(\tau_{p}^{+}))\geq\delta(l-p),\quad p\leq
l\leq m\end{gathered}$ (22)

where $v(t):=V(\phi(t,t_{0},\phi_{0},d_{1},d_{2}))$ and
for any $s>0,\;q>0$

$\begin{gathered}F(s,q):=\int\limits_{s}^{q}\frac{ds}{\widehat{\varphi}(s)},\quad\widehat{\varphi}(s):=\min\\{\varphi_{1}(s),\varphi_{2}(s),s\\},\end{gathered}$

so that $F(s,q)\to\infty$ for $s\to 0+$ for any fixed $q>0$.

Proof. For $l=p$ the inequality (22) is trivially satisfied. For $l=p+1$ we
consider the solution $\phi(t,t_{0},\phi_{0},d_{1},d_{2})$ of $\Sigma$ for
$t\in(\tau_{p},\tau_{p+1}]$ and either of two possible cases 1) $\phi_{0}\in
G^{+}$ and 2) $\phi_{0}\in\operatorname{int\,}(G^{-})$, recalling that
$G^{+}\cup G^{-}=X$ and $G^{+}\cap\operatorname{int\,}(G^{-})=\emptyset$ by
definition.

1) For $\phi_{0}\in G^{+}$ we will show that
$\phi(t,t_{0},\phi_{0},d_{1},d_{2})\in G^{+}$ for $t\in(\tau_{p},\tau_{p+1}]$.
Assume, this is not the case, that is there exists

$\begin{gathered}\widetilde{t}=\sup\\{t\in(\tau_{p},\tau_{p+1}]\,:\,\phi(t,t_{0},\phi_{0},d_{1},d_{2})\in
G^{+}\\}\in(\tau_{p},\tau_{p+1})\end{gathered}$

such that $\phi(t,t_{0},\phi_{0},d_{1},d_{2})\in G^{+}$,
$t\in(\tau_{p},\widetilde{t}]$ and
$\phi(\widetilde{t},t_{0},\phi_{0},d_{1},d_{2})\in\partial G^{+}$. Let us
denote for short $w(t):=W(\phi(t,t_{0},\phi_{0},d_{1},d_{2}))$, where
$t\mapsto W(\phi(t,t_{0},\phi_{0},d_{1},d_{2}))$ is absolutely continuous.
Hence $w(\widetilde{t})=0$. By assumptions of the lemma we have
$\|\phi(\widetilde{t},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\geq r>\chi(d)$, hence
due to (16) and (4.8) from [8] follows $\dot{w}(t)>0$. This means that for
some $\epsilon>0$ for $t\in(\widetilde{t},\widetilde{t}+\epsilon)$ it holds
that $w(\widetilde{t})\geq 0$, that is
$\phi(\widetilde{t},t_{0},\phi_{0},d_{1},d_{2})\in G^{+}$ contradicting the
choice of $\widetilde{t}$.

This means that $\phi(t,t_{0},\phi_{0},d_{1},d_{2})\in G^{+}$ and
$\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\geq r>\chi(d)$ for
$t\in(\tau_{p},\tau_{p+1}]$, hence (15) imples

$\begin{gathered}\dot{v}(t)\leq\varphi_{2}(v(t)),\quad
t\in(\tau_{p},\tau_{p+1}].\end{gathered}$

We calculate

$\begin{gathered}\int\limits_{v(\tau_{p}^{+})}^{v(\tau_{p+1})}\frac{ds}{\varphi_{2}(s)}=\int\limits_{\tau_{p}}^{\tau_{p+1}}\frac{dv(s)}{\varphi_{2}(v(s))}\leq\int\limits_{\tau_{p}}^{\tau_{p+1}}ds=\tau_{p+1}-\tau_{p}=T_{p}\leq\theta_{2}.\end{gathered}$

Setting $a=v(\tau_{p+1})$ in (18) we obtain

$\begin{gathered}\int\limits_{\psi_{2}(v(\tau_{p+1}))}^{v(\tau_{p+1})}\frac{ds}{\varphi_{2}(s)}\geq\theta_{2}+\delta\geq\int\limits_{v(\tau_{p}^{+})}^{v(\tau_{p+1})}\frac{ds}{\varphi_{2}(s)}+\delta.\end{gathered}$

which implies

$\begin{gathered}\int\limits_{\psi_{2}(v(\tau_{p+1}))}^{v(\tau_{p}^{+})}\frac{ds}{\varphi_{2}(s)}\geq\delta.\end{gathered}$

Due to $\|\phi(\tau_{p+1},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\geq r>\chi(d)$
from (18) follows $v(\tau_{p+1}^{+})\leq\psi_{2}(v(\tau_{p+1}))$ which implies

$\begin{gathered}\int\limits_{v(\tau_{p+1}^{+})}^{v(\tau_{p}^{+})}\frac{ds}{\widehat{\varphi}(s)}\geq\int\limits_{v(\tau_{p+1}^{+})}^{v(\tau_{p}^{+})}\frac{ds}{\varphi_{2}(s)}\geq\delta\end{gathered}$

or equivalently

$\begin{gathered}F(v(\tau_{p+1}^{+}),v(\tau_{p}^{+}))\geq\delta.\end{gathered}$
(23)

2) Now let $\phi_{0}\in\operatorname{int\,}G^{-}$, then either

(i) $\phi(t,t_{0},\phi_{0},d_{1},d_{2})\in\operatorname{int\,}G^{-}$ for
$t\in(\tau_{p},\tau_{p+1}]$ or

(ii) $\phi(\widetilde{t}_{1},t_{0},\phi_{0},d_{1},d_{2})\in G^{+}$ for some
$\widetilde{t}_{1}\in(\tau_{p},\tau_{p+1}]$.

In case (i) from $\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\geq r>\chi(d)$
and (14) it follows that

$\begin{gathered}\dot{v}(t)\leq-\varphi_{1}(v(t)),\quad
t\in(\tau_{p},\tau_{p+1}].\end{gathered}$

which means

$\begin{gathered}\int\limits_{v(\tau_{p}^{+})}^{v(\tau_{p+1})}\frac{ds}{\varphi_{1}(s)}=\int\limits_{\tau_{p}}^{\tau_{p+1}}\frac{dv(s)}{\varphi_{1}(v(s))}\leq-(\tau_{p+1}-\tau_{p}),\end{gathered}$

and hence

$\begin{gathered}\theta_{1}\leq
T_{p}=\tau_{p+1}-\tau_{p}\leq\int\limits_{v(\tau_{p+1})}^{v(\tau_{p}^{+})}\frac{ds}{\varphi_{1}(s)}.\end{gathered}$

Setting $a=v(\tau_{p+1})$ in (17) we obtain

$\begin{gathered}\theta_{1}\geq\int\limits_{v(\tau_{p+1})}^{\psi_{1}(v(\tau_{p+1}))}\frac{ds}{\varphi_{1}(s)}+\delta.\end{gathered}$

That is

$\begin{gathered}\int\limits_{v(\tau_{p+1})}^{v(\tau_{p}^{+})}\frac{ds}{\varphi_{1}(s)}\geq\int\limits_{v(\tau_{p+1})}^{\psi_{1}(v(\tau_{p+1})}\frac{ds}{\varphi_{1}(s)}+\delta,\end{gathered}$

and

$\begin{gathered}\int\limits_{\psi_{1}(v(\tau_{p+1}))}^{v(\tau_{p}^{+})}\frac{ds}{\varphi_{1}(s)}\geq\delta.\end{gathered}$

From $\phi(\tau_{p+1},t_{0},\phi_{0},d_{1},d_{2})\in G^{-}$ and
$\|\phi(\tau_{p+1},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\geq r>\chi(d)$, follows
$v(\tau_{p+1}^{+})\leq\psi_{1}(v(\tau_{p+1}))$, and hence

$\begin{gathered}\int\limits_{v(\tau_{p+1}^{+})}^{v(\tau_{p}^{+})}\frac{ds}{\widehat{\varphi}(s)}\geq\int\limits_{v(\tau_{p+1}^{+})}^{v(\tau_{p}^{+})}\frac{ds}{\varphi_{1}(s)}\geq\int\limits_{\psi_{1}(v(\tau_{p+1}))}^{v(\tau_{p}^{+})}\frac{ds}{\varphi_{1}(s)}\geq\delta\end{gathered}$

or in other words

$\begin{gathered}F(v(\tau_{p+1}^{+}),v(\tau_{p}^{+}))\geq\delta.\end{gathered}$
(24)

In case (ii) we define

$\begin{gathered}\widehat{t}=\inf\\{t\in(\tau_{p},\tau_{p+1}]\,:\,\phi(t,\tau_{p}^{+},\phi_{0},d_{1},d_{2})\in\operatorname{int\,}G^{+}\\},\end{gathered}$

so that $\phi(\widehat{t},t_{0},\phi_{0},d_{1},d_{2})\in\partial G^{+}\subset
G^{+}$ and $\widehat{t}>\tau_{p}$. From the properties of $W$ it follows that
$\phi(t,t_{0},\phi_{0},d_{1},d_{2})\in\operatorname{int\,}G^{-}$ for
$t\in(\tau_{p},\widehat{t})$ and $\phi(t,t_{0},\phi_{0},d_{1},d_{2})\in G^{+}$
for $t\in[\widehat{t},\tau_{p+1}]$ (similarly to the case 1) above).

From (14) follows

$\begin{gathered}v(\widehat{t})\leq v(\tau_{p}^{+}).\end{gathered}$ (25)

Since $\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\geq r>\chi(d)$ and
$\phi(t,t_{0},\phi_{0},d_{1},d_{2})\in G^{+}$ for
$t\in[\widehat{t},\tau_{p+1}]$, then from (15) follows

$\begin{gathered}\dot{v}(t)\leq\varphi_{2}(v(t)),\quad
t\in[\widehat{t},\tau_{p+1}].\end{gathered}$

This allows to calculate

$\begin{gathered}\int\limits_{v(\widehat{t})}^{v(\tau_{p+1})}\frac{ds}{\varphi_{2}(s)}=\int\limits_{\widehat{t}}^{\tau_{p+1}}\frac{dv(s)}{\varphi_{2}(v(s))}\leq\tau_{p+1}-\widehat{t}\leq\tau_{p+1}-\tau_{p}=T_{p}\leq\theta_{2},\end{gathered}$

so that (25) implies

$\begin{gathered}\int\limits_{v(\tau_{p}^{+})}^{v(\tau_{p+1})}\frac{ds}{\varphi_{2}(s)}\leq\int\limits_{v(\widehat{t})}^{v(\tau_{p+1})}\frac{ds}{\varphi_{2}(s)}\leq\theta_{2}.\end{gathered}$

Now we set $a=v(\tau_{p})$ into (18) and obtain

$\begin{gathered}\int\limits_{v(\tau_{p}^{+})}^{v(\tau_{p+1})}\frac{ds}{\varphi_{2}(s)}\leq\theta_{2}\leq\int\limits_{\psi_{2}(v(\tau_{p+1}))}^{v(\tau_{p+1})}\frac{ds}{\varphi_{2}(s)}-\delta\end{gathered}$

or

$\begin{gathered}\int\limits_{v(\tau_{p}^{+})}^{\psi_{2}(v(\tau_{p+1}))}\frac{ds}{\varphi_{2}(s)}\leq-\delta.\end{gathered}$

Having $\|\phi(\tau_{p+1},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\geq r>\chi(d)$,
$\phi(\tau_{p+1},t_{0},\phi_{0},d_{1},d_{2})\in G^{+}$ and (18) we conclude
that $v(\tau_{p+1}^{+})\leq\psi_{2}(v(\tau_{p+1}))$. Hence

$\begin{gathered}\int\limits_{v(\tau_{p}^{+})}^{v(\tau_{p+1}^{+})}\frac{ds}{\widehat{\varphi}(s)}\leq\int\limits_{v(\tau_{p}^{+})}^{v(\tau_{p+1}^{+})}\frac{ds}{\varphi_{2}(s)}\leq\int\limits_{v(\tau_{p}^{+})}^{\psi_{2}(v(\tau_{p+1}))}\frac{ds}{\varphi_{2}(s)}\leq-\delta.\end{gathered}$

In other words

$\begin{gathered}F(v(\tau_{p+1}^{+}),v(\tau_{p}^{+}))\geq\delta.\end{gathered}$
(26)

From (23)—(26) we conclude that (22) is true for $l=p+1$. Considering the
solution between the next two consequent jumps we obtain

$\begin{gathered}F(v(\tau_{p+1}^{+}),v(\tau_{l}^{+}))\geq\delta,\quad
l=p,p+1,\dots,m-1.\end{gathered}$ (27)

From the definition of $F$ for any $s\geq z\geq q>0$ we have
$F(s,z)+F(z,q)=F(s,q)$. Since the interval $(\tau_{p},\tau_{l})$ is split into
$l-p$ subintervals by the time instants of the impulsive actions we finally
obtain from (27) that (22) is proved.

######  Remark 1.

Let $F^{-1}(s,\cdot)$ be the inverse function to $F(\cdot,s)$,
$s\in\mathbb{R}_{+}$. From $F(\tau,s)\to\infty$ for $\tau\to 0+$ follows
$F^{-1}(s,\tau)\to 0$ for $\tau\to+\infty$. Also note that $F^{-1}(s,\cdot)$
is strictly decreasing whereas $F^{-1}(\cdot,s)$ is strictly increasing for
$s>0$.

######  Lemma 2.

Under the conditions of Theorem 1 let $r$ be such that $r>\chi(d)$, then there
exists $\tau_{k_{0}}\geq\tau_{0}$ such that
$\phi(\tau_{k_{0}}^{+},t_{0},\phi_{0},d_{1},d_{2})\in B_{r}(0)$.

Proof. Assume by contradiction that for all $k\in\mathbb{Z}_{+}$ we have
$\|\phi(\tau_{k},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\geq r$. Lemma 1 implies
that the sequence $\\{v(\tau_{k}^{+})\\}_{k=0}^{\infty}$ is decreasing. Being
bounded from below it has a limit $v^{*}\geq 0$. From (22) we have

$\begin{gathered}F(v(\tau_{m}^{+}),v(\tau_{0}^{+}))\geq\delta
m.\end{gathered}$

If $v^{*}\neq 0$ this inequality leads to a contradiction letting
$m\to\infty$. Hence, $v^{*}=0$. Due to (12) we have

$\begin{gathered}0\leq\alpha_{1}(r)\leq\alpha_{1}(\|\phi(\tau_{m}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X})\leq\\\
V(\phi(\tau_{m}^{+},t_{0},\phi_{0},d_{1},d_{2}))=v(\tau_{m}^{+})\end{gathered}$

and taking the limit for $m\to\infty$ we arrive to $\alpha_{1}(r)=0$ which
implies $r=0$ contradicting $r>\chi(d)$. This finishes the proof of the lemma.

######  Lemma 3.

Under the conditions of Theorem 1 the solution $\phi$ of $\Sigma$ satisfy
$\phi(\tau_{k_{0}}^{+},t_{0},\phi_{0},d_{1},d_{2})\in B_{r}(0)$ for some
$\tau_{k_{0}}\in{\mathcal{E}}$ and $r>\chi(d)$, then there exists
$R\in\mathcal{K}^{2}_{\infty}$ such that

$\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|\leq R(r,d),\quad t>\tau_{k_{0}}.$

Proof. We define for $s\geq 0,\;q\geq 0$

$\begin{gathered}R(s,q):=\max\\{R_{1}(s,q),R_{4}(s,q),R_{6}(s,q),s\\},\end{gathered}$

where we use the following combinations of functions from (9)-(10)

$\begin{gathered}R_{1}(s,q)=\xi_{\theta_{2}}(s)+\eta_{\theta_{2}}(q),\quad
R_{2}(s,q)=\xi(s)+\eta(q),\\\
R_{3}(s,q)=\max\\{\xi(R_{1}(s,q))+\eta(q),R_{2}(s,q)\\},\\\
R_{4}(s,q)=\xi_{\theta_{2}}(R_{3}(s,q))+\eta_{\theta_{2}}(q),\\\
R_{5}(s,q)=(\alpha_{1}^{-1}\circ\alpha_{2})(R_{3}(s,q)),\\\
R_{6}(s,q)=\xi_{\theta_{2}}(R_{5}(s,q))+\eta_{\theta_{2}}(q).\end{gathered}$

Let $\widehat{t}_{1}>\tau_{k_{0}}$ be such that
$\|\phi(\widehat{t}_{1},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\leq r$ and
$\|\phi(\widehat{t}_{1}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\geq r$ (if such
$\widehat{t}_{1}$ does not exist, then the result is proved). Define

$\begin{gathered}\widehat{t}_{2}:=\sup\\{t>\widehat{t}_{1}:\,\|\phi(s,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\geq
r\\\ \text{for}\quad
s\in[\widehat{t}_{1},t]\\}\in[\widehat{t}_{1},\infty],\end{gathered}$

so that

$\begin{gathered}\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\geq r,\quad
t\in[\widehat{t}_{1},\widehat{t}_{2}].\end{gathered}$

It is enough to show that $\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\leq
R(r,d)$ for $t\in[\widehat{t}_{1},\widehat{t}_{2}]$.

If
$\mathcal{E}_{[\widehat{t}_{1},\widehat{t}_{2}]}:=[\widehat{t}_{1},\widehat{t}_{2}]\cap\mathcal{E}=\emptyset$,
then by the properties (8), $\Sigma_{2}$, (7) for all
$t\in[\widehat{t}_{1},\widehat{t}_{2}]$ we conclude

$\begin{gathered}\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\\\
=\|\phi(t,\widehat{t}_{1},\phi(\widehat{t}_{1},t_{0},\phi_{0},d_{1},d_{2}),d_{1},d_{2})\|_{X}\\\
=\|\phi_{c}(t,\widehat{t}_{1},\phi(\widehat{t}_{1},t_{0},\phi_{0},d_{1},d_{2}),d_{1})\|_{X}\\\
=\|\phi_{c}(t-\widehat{t}_{1},0,\phi(\widehat{t}_{1},t_{0},\phi_{0},d_{1},d_{2}),\mathcal{S}_{\widehat{t}_{1}}d_{1})\|_{X}\\\
\leq\xi_{\theta_{2}}(r)+\eta_{\theta_{2}}(d)=R_{1}(r,d).\end{gathered}$

If otherwise $\mathcal{E}_{[\widehat{t}_{1},\widehat{t}_{2}]}\neq\emptyset$ we
denote its minimal element by $\tau_{p}\geq\widehat{t}_{1}$ and consider two
possible cases (i) $\widehat{t}_{1}=\tau_{p}$ and (ii)
$\tau_{p}>\widehat{t}_{1}$ separately.

In case of (i) from (10) follows

$\begin{gathered}\|\phi(\widehat{t}_{1}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\leq\xi(\|\phi(\widehat{t}_{1},t_{0},\phi_{0},d_{1},d_{2})\|_{X})\\\
+\eta(\|d_{2}\|_{\mathcal{U}_{2}})\leq\xi(r)+\eta(d)=R_{2}(r,d).\end{gathered}$

In case of (ii) by means of (9) and (10) we obtain

$\begin{gathered}\|\phi(\tau_{p}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\leq\xi(\|\phi(\tau_{p},t_{0},\phi_{0},d_{1},d_{2})\|_{X})+\eta(\|d_{2}\|_{\mathcal{U}_{2}})\end{gathered}$

and with help of (8), $\Sigma_{2}$, (7) we get

$\begin{gathered}\|\phi(\tau_{p},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\\\
=\|\phi(\tau_{p},\widehat{t}_{1},\phi(\widehat{t}_{1},t_{0},\phi_{0},d_{1},d_{2}),d_{1},d_{2})\|_{X}\\\
=\|\phi_{c}(\tau_{p},\widehat{t}_{1},\phi(\widehat{t}_{1},t_{0},\phi_{0},d_{1},d_{2}),d_{1})\|_{X}\\\
=\|\phi_{c}(\tau_{p}-\widehat{t}_{1},0,\phi(\widehat{t}_{1},t_{0},\phi_{0},d_{1},d_{2}),\mathcal{S}_{\widehat{t}_{1}}d_{1})\|_{X}\\\
\leq\xi_{\theta_{2}}(\|\phi(\widehat{t}_{1},t_{0},\phi_{0},d_{1},d_{2})\|_{X})+\eta_{\theta_{2}}(\|d_{1}\|_{\mathcal{U}_{2}})\\\
\leq\xi_{\theta_{2}}(r)+\eta_{\theta_{2}}(d)=R_{1}(r,d).\end{gathered}$

Hence,

$\begin{gathered}\|\phi(\tau_{p}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\leq\xi(R_{1}(r,d))+\eta(d).\end{gathered}$

In bothe cases (i) and (ii) we see that
$\|\phi(\tau_{p}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\leq R_{3}(r,d)$ holds.

If $\sharp\mathcal{E}_{[\widehat{t}_{1},\widehat{t}_{2}]}=1$, then by means of
$\Sigma_{2}$, (7), (9) and (10) we obtain for $t\in(\tau_{p},\widehat{t}_{2}]$

$\begin{gathered}\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\\\
=\|\phi_{c}(t-\tau_{p},0,\phi(\tau_{p}^{+},t_{0},\phi_{0},d_{1},d_{2}),\mathcal{S}_{\tau_{p}}d_{1})\|_{X}\\\
\leq\xi_{\theta_{2}}(\|\phi(\tau_{p}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X})+\eta_{\theta_{2}}(\|d_{1}\|_{\mathcal{U}_{1}})\\\
\leq\xi_{\theta_{2}}(R_{3}(r,d))+\eta_{\theta_{2}}(d)=R_{4}(r,d).\end{gathered}$

If $\sharp\mathcal{E}_{[\widehat{t}_{1},\widehat{t}_{2}]}\geq 2$, then by
Lemma 1 and observing that $F(s,q)>0\;\Leftrightarrow\;s<q$ we obtain for
$l\geq p$ such that
$\tau_{l}\in\mathcal{E}_{[\widehat{t}_{1},\widehat{t}_{2}]}$

$\begin{gathered}\alpha_{1}(\|\phi(\tau_{l}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X})\leq
V(\phi(\tau_{l}^{+},t_{0},\phi_{0},d_{1},d_{2}))\\\ =v(\tau_{l}^{+})\leq
v(\tau_{p}^{+})=V(\phi(\tau_{p}^{+},t_{0},\phi_{0},d_{1},d_{2}))\\\
\leq\alpha_{2}(\|\phi(\tau_{p}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X})\leq\alpha_{2}(R_{3}(r,d)).\end{gathered}$

This implies
$\|\phi(\tau_{l}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\leq(\alpha_{1}^{-1}\circ\alpha_{2})(R_{3}(r,d))\\\
=R_{5}(r,d)$. Hence for all $t\in(\tau_{l},\tau_{l+1}]$, from (9) and
properties $\Sigma_{2}$, (7), (8) we obtain

$\begin{gathered}\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\\\
=\|\phi_{c}(t,\tau_{l},\phi(\tau_{l}^{+},t_{0},\phi_{0},d_{1},d_{2}),d_{1})\|_{X}\\\
=\|\phi_{c}(t-\tau_{l},0,\phi(\tau_{l}^{+},t_{0},\phi_{0},d_{1},d_{2}),\mathcal{S}_{\tau_{l}}d_{1})\|_{X}\\\
\leq\xi_{\theta_{2}}(\|\phi(\tau_{l}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X})+\eta_{\theta_{2}}(\|d_{1}\|_{\mathcal{U}_{1}})\\\
\leq\xi_{\theta_{2}}(R_{5}(r,d))+\eta_{\theta_{2}}(d)=R_{6}(r,d).\end{gathered}$

This implies the estimate

$\begin{gathered}\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\leq
R(r,d)\quad\text{for all}\quad t\in[\tau_{p},\tau_{m+1}].\end{gathered}$

If $t\in[\widehat{t}_{1},\tau_{p}]$, then from $\Sigma_{2}$, (7), (8) and (9)
we get

$\begin{gathered}\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\\\
=\|\phi(t,\widehat{t}_{1},\phi(\widehat{t}_{1},t_{0},\phi_{0},d_{1},d_{2}),d_{1},d_{2})\|_{X}\\\
=\|\phi_{c}(t,\widehat{t}_{1},\phi(\widehat{t}_{1},t_{0},\phi_{0},d_{1},d_{2}),d_{1})\|_{X}\\\
=\|\phi_{c}(t-\widehat{t}_{1},0,\phi(\widehat{t}_{1},t_{0},\phi_{0},d_{1},d_{2}),\mathcal{S}_{\widehat{t}_{1}}d_{1})\|_{X}\\\
\leq\xi_{\theta_{2}}(\|\phi(\widehat{t}_{1},t_{0},\phi_{0},d_{1},d_{2})\|_{X})+\eta_{\theta_{2}}(d)\\\
=\xi_{\theta_{2}}(r)+\eta_{\theta_{2}}(d)=R_{1}(r,d)\leq
R(r,d).\end{gathered}$

Since $[\widehat{t}_{1},\tau_{m+1}]\supseteq[\widehat{t}_{1},\widehat{t}_{2}]$
holds, the lemma is proved.

Proof of Theorem 1 Take $r=(1+\varepsilon)\chi(d)$ for some $\varepsilon>0$
and denote by $k_{0}$ the smallest integer for which
$\phi(\tau_{k_{0}}^{+},t_{0},\phi_{0},d_{1},d_{2})\in B_{r}(0)$ holds, that is
$\|\phi(\tau_{k}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\geq r$ for all $0\leq
k\leq k_{0}-1$. From Lemma 1 follows
$F(v(\tau_{k}^{+}),v(\tau_{0}^{+}))\geq\delta k$, hence (see Remark 1) we have

$\begin{gathered}v(\tau_{k}^{+})\leq
F^{-1}(v(\tau_{0}^{+}),k\delta).\end{gathered}$

From (9) and properties $\Sigma_{2}$, (8) and (7) it follows that for
$t\in(\tau_{k},\tau_{k+1}]$, $k=0,\dots,k_{0}-1$ the next inequality holds

$\begin{gathered}\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}=\|\phi_{c}(t,\tau_{k},\phi(\tau_{k}^{+},t_{0},\phi_{0},d_{1},d_{2}),d_{1},d_{2})\|_{X}\\\
=\|\phi_{c}(t-\tau_{k},0,\phi(\tau_{k}^{+},t_{0},\phi_{0},d_{1},d_{2}),\mathcal{S}_{\tau_{k}}d_{1})\|_{X}\\\
\leq\xi_{\theta_{2}}(\|\phi(\tau_{k}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X})+\eta_{\theta_{2}}(d).\end{gathered}$

Now (12) implies

$\begin{gathered}\alpha_{1}(\|\phi(\tau_{k}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X})\leq
V(\phi(\tau_{k}^{+},t_{0},\phi_{0},d_{1},d_{2}))\\\ =v(\tau_{k}^{+})\leq
F^{-1}(v(\tau_{0}^{+}),\delta k),\\\
v(\tau_{0}^{+})=V(\phi(\tau^{+}_{0},t_{0},\phi_{0},d_{1},d_{2}))\leq\alpha_{2}(\|\phi(\tau^{+}_{0},t_{0},\phi_{0},d_{1},d_{2})\|_{X}).\end{gathered}$

Defining $\beta_{k}(s):=F^{-1}(\alpha_{2}(s),\delta k),\;s>0$ and
$\beta_{k}(0):=0$ we can write

$\begin{gathered}\alpha_{1}(\|\phi(\tau_{k}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X})\\\
\leq
F^{-1}(\alpha_{2}(\|\phi(\tau^{+}_{0},t_{0},\phi_{0},d_{1},d_{2})\|_{X}),\delta
k)\\\
=\beta_{k}(\|\phi(\tau^{+}_{0},t_{0},\phi_{0},d_{1},d_{2})\|_{X}).\end{gathered}$

Which means

$\|\phi(\tau_{k}^{+},t_{0},\phi_{0},d_{1},d_{2})\|_{X}\leq(\alpha_{1}^{-1}\circ\beta_{k})(\|\phi(\tau^{+}_{0},t_{0},\phi_{0},d_{1},d_{2})\|_{X}).$

By the properties of $F^{-1}$ (see Remark 1) it follows that
$\beta_{k}\in\mathcal{K}$, $\beta_{k+1}(s)<\beta_{k}(s)$ and
$\lim_{k\to\infty}\beta_{k}(s)=0$ for $s>0$.

For $t\in(\tau_{k},\tau_{k+1}]$ the following estimate holds

$\begin{gathered}\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\\\
\leq(\xi_{\theta_{2}}\circ\alpha_{1}^{-1}\circ\beta_{k})(\|\phi(\tau^{+}_{0},t_{0},\phi_{0},d_{1},d_{2})\|_{X})+\eta_{\theta_{2}}(d).\end{gathered}$

Defining
$\widehat{\beta}_{k}(s):=(\xi_{\theta_{2}}\circ\alpha_{1}^{-1}\circ\beta_{k})(s)$
for $k\in\mathbb{Z}_{+}$ and $s\geq 0$ we see that
$\widehat{\beta}_{k}\in\mathcal{K}$,
$\widehat{\beta}_{k+1}(s)<\widehat{\beta}_{k}(s)$ and
$\lim_{k\to\infty}\widehat{\beta}_{k}(s)=0$ for any $s>0$. Further, we define

$\begin{gathered}\beta_{\tau_{0}^{+}}(s,t)=\widehat{\beta}_{k}(s)+\frac{t-\tau_{k}}{\tau_{k+1}-\tau_{k}}(\widehat{\beta}_{k+1}(s)-\widehat{\beta}_{k}(s)),\\\
t\in(\tau_{k},\tau_{k+1}],\;s\geq 0\end{gathered}$

so that $\beta_{\tau_{0}^{+}}\in\mathcal{K}\mathcal{L}$ and for
$t\in(\tau_{0},\tau_{k_{0}}]$ the following estimate holds

$\begin{gathered}\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\leq\beta_{\tau_{0}^{+}}(\|\phi(\tau^{+}_{0},t_{0},\phi_{0},d_{1},d_{2})\|_{X},t)\\\
+\eta_{\theta_{2}}(d).\end{gathered}$ (28)

From Lemma 3 follows that for any $t>\tau_{k_{0}}$ we have

$\begin{gathered}\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\leq\sigma(d),\end{gathered}$
(29)

where we denote $\sigma(s):=R((1+\varepsilon)s,s)$, $\varepsilon>0$. Since
$\sigma\in\mathcal{K}$ then collecting the estimates (28) and (29) we obtain

$\begin{gathered}\|\phi(t,t_{0},\phi_{0},d_{1},d_{2})\|_{X}\leq\beta_{\tau_{0}^{+}}(\|\phi(\tau^{+}_{0},t_{0},\phi_{0},d_{1},d_{2})\|_{X},t)\\\
+\gamma_{\tau_{0}^{+}}(d),\quad t>\tau_{0},\end{gathered}$

where $\gamma_{\tau_{0}^{+}}(s):=\sigma(s)+\eta_{\theta_{2}}(s)$. The
statement of the theorem follows then from the Proposition 1.

######  Remark 2.

In the definition of an ISS-Lyapunov function $V$ we have assumed that the
sets $G^{+}$ and $G^{-}$ are not empty. Observe however that in case
$G^{+}=\emptyset$, $G^{-}=X$ Theorem 1 reduces to Theorem 1 from [4] and if
$G^{-}=\emptyset$, $G^{+}=X$ then Theorem 1 reduces to Theorem 3 from [4].

##  5 Nonlinear interconnection of an ODE and a PDE

Let $\widehat{X}=L^{2}[0,l]\times\mathbb{R}$,
$X=H_{0}^{1}(0,l)\times\mathbb{R}$,
$U_{1}=U_{2}=H_{0}^{1}(0,l)\times\mathbb{R}$ and the spaces of input signals
be

$\begin{gathered}\mathcal{U}_{1}=L^{\infty}(U_{1})\cap(H_{\operatorname{loc\,}}(\mathbb{R}_{+},H_{0}^{1}(0,l))\times
H_{\operatorname{loc\,}}(\mathbb{R}_{+},\mathbb{R})),\\\
\mathcal{U}_{2}=l^{\infty}(\mathbb{Z}_{+},U_{2}).\end{gathered}$

Consider the following nonlinear impulsive system

$\begin{gathered}\partial_{t}x(z,t)=a^{2}\partial_{zz}x(z,t)+\Phi(x(z,t))\\\
+B(z)y(t)+d_{11}(z,t),\quad t\neq\tau_{k},\\\
\dot{y}(t)=c^{2}y(t)+\int\limits_{0}^{l}D(z)x(z,t)\,dz+d_{12}(t),\quad
t\neq\tau_{k},\\\ x(z,t^{+})=\alpha(z)x(z,t)+\beta(z)y(t)+d_{21}(z,k),\quad
t=\tau_{k},\\\ y(t^{+})=\int\limits_{0}^{l}\gamma(z)x(z,t)\,dz+\delta
y(t)+d_{22}(k),\quad t=\tau_{k},\end{gathered}$ (30)

with initial and boundary conditions

$\begin{gathered}x(z,0)=x_{0}(z)\in H_{0}^{1}(0,l),\quad z\in[0,l],\\\ \quad
y(0)=y_{0}\in\mathbb{R},\\\ x(0,t)=x(l,t)=0,\quad t\in\mathbb{R}_{+},\\\
x_{0}(0)=x_{0}(l)=0.\end{gathered}$ (31)

Here $a$, $c$, $l$ are given positive constants, $\alpha\in C^{2}[0,l]$,
$\,\beta\in H^{1}_{0}(0,l)$, $\gamma\in L^{2}[0,l]$, $D\in L^{2}[0,l]$, $B\in
H_{0}^{1}(0,l)$ are given functions and
$d_{1}(t)=(d_{11}(\cdot,t),d_{12}(t))\in U_{1}$,
$d_{2}(k)=(d_{21}(\cdot,k),d_{22}(k))\in U_{2}$ are unknown disturbances.

Assume that $\,\Phi\,:\,\mathbb{R}\to\mathbb{R}$ satisfies:

(i) $\Phi\in C^{1}(\mathbb{R})$ with locally Lipshitz $\Phi^{\prime}$ that is
for any $s_{0}\in\mathbb{R}$ and $\varrho_{0}>0$ there exists
$L=L(s_{0},\varrho_{0})>0$ such that for all $s\in\mathbb{R}$ with
$|s-s_{0}|\leq\varrho_{0}$ it holds that
$|\Phi^{\prime}(s)-\Phi^{\prime}(s_{0})|\leq L|s-s_{0}|$.

(ii) It holds that $\Phi^{\prime}(s)\leq 0$, $s\Phi(s)\leq 0$ for all
$s\in\mathbb{R}$.

The problem (30)–(31) can be written in the following form

$\begin{gathered}\frac{d}{dt}\begin{pmatrix}x(\cdot,t)\\\
y(t)\end{pmatrix}+\mathcal{A}\begin{pmatrix}x(\cdot,t)\\\
y(t)\end{pmatrix}=f(t,x,y,d_{1}),\quad t\neq\tau_{k}\\\
\begin{pmatrix}x(\cdot,t^{+})\\\
y(t^{+})\end{pmatrix}=\mathcal{B}\begin{pmatrix}x(\cdot,t)\\\
y(t)\end{pmatrix}+d_{2}(k),\quad t=\tau_{k},\end{gathered}$ (32)

where $\mathcal{A}$ is the linear operator on $\widehat{X}$ defined by

$\begin{gathered}\mathcal{A}\begin{pmatrix}x\\\
y\end{pmatrix}=\begin{pmatrix}-a^{2}\partial_{zz}x(z)\\\
-c^{2}y\end{pmatrix}\end{gathered}$

with domain $\mathcal{D}(\mathcal{A})=(H_{0}^{1}(0,l)\cap
H^{2}(0,l))\times\mathbb{R}$, $\mathcal{B}$ is the linear operator defined by

$\begin{gathered}\mathcal{B}\begin{pmatrix}x\\\
y\end{pmatrix}=\begin{pmatrix}\alpha(z)x(z)+\beta(z)y\\\
\int\limits_{0}^{l}\gamma(z)x(z)\,dz+\delta y\end{pmatrix}\end{gathered}$

and $d_{2}(k)=(d_{21}(\cdot,k),d_{22}(k))$.

The operator $\mathcal{A}$ being a direct product of two sectorial operators
is sectorial and hence it generates on $\widehat{X}$ an analytic semi-group
[6]. The mapping $f\,:\,\mathbb{R}\times H_{0}^{1}(0,l)\times\mathbb{R}\to
L^{2}[0,l]\times\mathbb{R}$ in (32) is defined by

$\begin{gathered}f(t,x,y,d_{1})=\begin{pmatrix}\Phi(x(z))+B(z)y+d_{11}(z,t)\\\
\int\limits_{0}^{l}D(z)x(z)\,dz+d_{12}(t)\end{pmatrix}\end{gathered}$

We consider classical solutions of (30)—(31) defined as in Definition 3.3.1 of
[6].

######  Remark 3.

For any initial state $(x_{0},y_{0})\in H_{0}^{1}(0,l)\times\mathbb{R}$ and
input $(d_{1},d_{2})\in\mathcal{U}_{1}\times\mathcal{U}_{2}$ there exists a
unique solution to the propblem (30)–(31). This follows from the fact the the
corresponding problem without impulsive actions

$\begin{gathered}\frac{d}{dt}\begin{pmatrix}x(\cdot,t)\\\
y(t)\end{pmatrix}+\mathcal{A}\begin{pmatrix}x(\cdot,t)\\\
y(t)\end{pmatrix}=f(t,x,y),\end{gathered}$ (33)

for each $(x_{0},y_{0})\in H_{0}^{1}(0,l)\times\mathbb{R}$ possesses a unique
solution defined for $[t_{0},t_{0}+\epsilon_{t_{0},x_{0},y_{0},d_{1}}]$,
$\epsilon_{t_{0},x_{0},y_{0},d_{1}}>\theta_{2}$, and that the mapping
$g(x,y,\mu)=\mathcal{B}(x,y)+\mu$ keeps the space $X$ invariant for any
$\mu\in U_{2}$.

The well-posedness of (33) can be established by means of Theorems 3.3.3 and
3.3.4 from [6]. In order to check the conditions of Theorem 3.3.3 (setting
$\alpha=0.5$ there) we need to show that the mapping $f$ is locally Hölder wrt
$t$ and locally Lipschitz wrt $(x,y)\in H_{0}^{1}(0,l)\times\mathbb{R}$, that
is for any $(t_{0},x_{0},y_{0})\in\mathbb{R}\times
H_{0}^{1}(0,l)\times\mathbb{R}$ there exists

$\begin{gathered}O_{\varrho}(t_{0},x_{0},y_{0}):=\\{(t,x,y)\in\mathbb{R}_{+}\times
H_{0}^{1}(0,l)\times\mathbb{R}\,\,|\\\
\,\,|t-t_{0}|<\varrho,\,\|x-x_{0}\|_{H_{0}^{1}(0,l)}<\varrho,\,|y-y_{0}|<\varrho\\}.\end{gathered}$

such that for any point $(t,x,y)\in O_{\varrho}(t_{0},x_{0},y_{0})$ it holds
that

$\begin{gathered}\|f(t,x,y)-f(t_{0},x_{0},y_{0})\|_{\widehat{X}}\leq
L(|t-t_{0}|^{\nu_{1}}\\\
+\|x-x_{0}\|_{H_{0}^{1}(0,l)}+|y-y_{0}|)\end{gathered}$

for some constants $L>0$, $\nu_{1}\in(0,1]$.

Indeed,

$\begin{gathered}\|f(t,x,y)-f(t_{0},x_{0},y_{0})\|_{\widehat{X}}\\\
\leq\frac{l}{\pi}\|\Phi\circ x-\Phi\circ
x_{0}+B(y-y_{0})+d_{11}(\cdot,t)-d_{11}(\cdot,t_{0})\|_{H_{0}^{1}(0,l)}\\\
+\Big{|}\int\limits_{0}^{l}D(z)(x(z)-x_{0}(z))\,dz+d_{12}(t)-d_{12}(t_{0})\Big{|}\\\
\leq\frac{l}{\pi}(\|\Phi\circ x-\Phi\circ
x_{0}\|_{H_{0}^{1}(0,l)}+\|B\|_{H^{1}_{0}(0,l)}|y-y_{0}|)\\\
+\|D\|_{L^{2}[0,l]}\|x-x_{0}\|_{L^{2}[0,l]}+|d_{12}(t)-d_{12}(t_{0})|\\\
+\frac{l}{\pi}\|d_{11}(\cdot,t)-d_{11}(\cdot,t_{0})\|_{H^{1}_{0}(0,l)}.\end{gathered}$
(34)

Consider the first summand separately

$\begin{gathered}\|\Phi\circ x-\Phi\circ
x_{0}\|_{H^{1}_{0}(0,l)}=\|\partial_{z}(\Phi\circ x-\Phi\circ
x_{0})\|_{L^{2}[0,l]}\\\ =\|(\Phi^{\prime}\circ
x)\,\partial_{z}x-(\Phi^{\prime}\circ
x_{0})\,\partial_{z}x_{0}\|_{L^{2}[0,l]}\\\ \leq\|(\Phi^{\prime}\circ
x-\Phi^{\prime}\circ x_{0})\partial_{z}x\|_{L^{2}[0,l]}\\\
+\|(\Phi^{\prime}\circ
x_{0})(\partial_{z}x-\partial_{z}x_{0})\|_{L^{2}[0,l]}\end{gathered}$

By the Sobolev embedding theorem we have $x\in C[0,l]$ and satisfies
$\|x-x_{0}\|_{C[0,l]}\leq C_{1}\|x-x_{0}\|_{H^{1}_{0}(0,l)}\leq C_{1}\varrho$,
for some $C_{1}>0$. Hance, using the condition (i), we obtain

$\begin{gathered}|\Phi^{\prime}(x(z))-\Phi^{\prime}(x_{0}(z))|\leq
L_{1}|x(z)-x_{0}(z)|\\\ \leq L_{1}\|x-x_{0}\|_{C[0,l]}\leq
L_{1}C_{1}\|x-x_{0}\|_{H^{1}_{0}(0,l)},\end{gathered}$

where $L_{1}$ is a positive constant, which can depend on $x_{0}$ and $\rho$.
Hence

$\begin{gathered}\|\Phi^{\prime}\circ x-\Phi^{\prime}\circ
x_{0}\|_{C[0,l]}\leq C_{2}\|x-x_{0}\|_{H^{1}_{0}(0,l)}\leq
C_{2}\varrho,\end{gathered}$

where $C_{2}=L_{1}C_{1}$, and

$\begin{gathered}\|\Phi\circ x-\Phi\circ x_{0}\|_{H^{1}_{0}(0,l)}\\\
\leq\|\Phi^{\prime}\circ x-\Phi^{\prime}\circ
x_{0}\|_{C[0,l]}\|\partial_{z}x\|_{L^{2}[0,l]}\\\ +\|\Phi^{\prime}\circ
x_{0}\|_{C[0,l]}\|\partial_{z}x-\partial_{z}x_{0}\|_{L^{2}[0,l]}\\\ \leq
C_{2}\|x-x_{0}\|_{H^{1}_{0}(0,l)}\|x\|_{H^{1}_{0}(0,l)}+\|\Phi^{\prime}\circ
x_{0}\|_{C[0,l]}\|x-x_{0}\|_{H^{1}_{0}(0,l)}\\\
\leq(C_{2}(\|x_{0}\|_{H^{1}_{0}(0,l)}+\varrho)+\|\Phi^{\prime}\circ
x_{0}\|_{C[0,l]})\|x-x_{0}\|_{H^{1}_{0}(0,l)},\end{gathered}$

which together with (34) proves that $f$ is locally Lipschitz.

By means of Theorem 3.3.4 Exercise 1 from the Section 3.3 in [6]) we can show
that the solution to (33) exists globally. To this end it is sufficient to
check that

$\begin{gathered}\frac{\|f(t,x(\cdot,t),y(t),d_{1})\|_{\widehat{X}}}{1+\|x(t)\|_{H^{1}_{0}(0,l)}+|y(t)|}\end{gathered}$
(35)

is bounded on the domain of existence of the solution $(x(\cdot,t),y(t))\in
X$.

To verify this fact we introduce the following auxiliary function

$\begin{gathered}U(x,y)=\|x\|_{L^{2}[0,l]}^{2}+y^{2},\quad x\in
H_{0}^{1}[0,l],\quad y\in\mathbb{R}\end{gathered}$

and use the following

######  Proposition 2.

For any $(x,y,\xi)\in H_{0}^{1}(0,l)\times\mathbb{R}\times U_{1}$ it holds
that

$\begin{gathered}\dot{U}(x,y,\xi)\leq\zeta^{\operatorname{T\,}}(\widetilde{A}_{0}+\varepsilon\operatorname{id\,})\zeta+\varepsilon^{-1}\|\xi\|^{2}_{U_{1}},\end{gathered}$
(36)

where we denote

$\begin{gathered}\zeta=(\|x\|_{L^{2}[0,l]},|y|)^{\operatorname{T\,}},\\\
\widetilde{A}_{0}=\begin{pmatrix}-\frac{2\pi^{2}a^{2}}{l^{2}}&\|B+D\|_{L^{2}[0,l]}\\\
\|B+D\|_{L^{2}[0,l]}&2c^{2}\end{pmatrix},\end{gathered}$

The proof can be found in the Appendix.

######  Corollary 1.

Let $(x(\cdot,t),y(t))$ be a solution to (33) with initial conditions
$x(\cdot,t_{0})=x_{0}$, $y(t_{0})=y_{0}$ and defined on
$t\in[t_{0},t_{0}+\epsilon_{t_{0},x_{0},y_{0},d_{1}})$, then

$\begin{gathered}\sup\limits_{t\in[t_{0},t_{0}+\epsilon_{t_{0},x_{0},y_{0},d_{1}})}\|(x,y)\|_{\widehat{X}}\leq
C(x_{0},y_{0},d_{1}).\end{gathered}$

The proof can be found in the Appendix.

From this corollary and the locally Lipschitzness follows that (35) is
bounded, hence the solution $(x(\cdot,t),y(t))$ of (33) exists for all $t\geq
t_{0}$. The properties of $\alpha$, $\beta$ guarantee that
$g(H_{0}^{1}(0,l)\times\mathbb{R}\times U_{2})\subset
H_{0}^{1}(0,l)\times\mathbb{R}$, which demonstrates the well-posedness of the
problem (30)–(31).

To state the asymptotic stability conditions of the system (30)–(31) we define
the following symmetric matrices

$\begin{gathered}\scriptsize
A_{0}=\begin{pmatrix}-\frac{2\pi^{2}a^{2}}{l^{2}}&\|B\|_{H^{1}_{0}(0,l)}+\frac{l}{\pi}\|D\|_{L^{2}[0,l]}\\\
\|B\|_{H^{1}_{0}(0,l)}+\frac{l}{\pi}\|D\|_{L^{2}[0,l]}&2c^{2}\end{pmatrix},\end{gathered}$
$\begin{gathered}B_{0}=\\\
\scriptsize\begin{pmatrix}\|\alpha^{2}\|_{C[0,l]}+\frac{l^{2}}{\pi^{2}}\|\alpha\alpha_{zz}\|_{C[0,l]}+\frac{l^{2}}{\pi^{2}}\|\gamma\|_{L^{2}[0,l]}^{2}&*\\\
\|\alpha_{z}\beta_{z}+\gamma\delta\|_{L^{2}[0,l]}\frac{l}{\pi}+\|\alpha\beta_{z}\|_{L^{2}[0,l]}&\delta^{2}+\|\beta\|_{H_{0}^{1}(0,l)}^{2}\end{pmatrix},\\\
\end{gathered}$
$\begin{gathered}\sigma=\|\alpha^{2}\|_{C[0,l]}+\frac{l^{2}}{\pi^{2}}\|\alpha\alpha_{zz}\|_{C[0,l]}+\frac{l^{2}}{\pi^{2}}\|\gamma\|_{L^{2}[0,l]}^{2}\\\
+2(\|\alpha_{z}\beta_{z}+\gamma\delta\|_{L^{2}[0,l]}\frac{l}{\pi}+\|\alpha\beta_{z}\|_{L^{2}[0,l]})+\delta^{2}+\|\beta\|_{H_{0}^{1}(0,l)}^{2},\\\
\vartheta:=\frac{\pi^{2}a^{2}}{l^{2}}-\|B\|_{H^{1}_{0}(0,l)}-\frac{l}{\pi}\|D\|_{L^{2}[0,l]}-c^{2},\end{gathered}$

let $\varrho_{\max}^{A_{0}}\in\mathbb{R}^{2}$ and
$\varrho_{\max}^{B_{0}}\in\mathbb{R}^{2}$ be eigenvectors of $A_{0}$ and
$B_{0}$, respectively, corresponding to the maximal eigenvalues
$\lambda_{\max}(A_{0})$ and $\lambda_{\max}(B_{0})$, respectively.

######  Proposition 3.

Let the impulsive system (30) satisfy

$\begin{gathered}c^{2}+\frac{\pi^{2}a^{2}}{l^{2}}>\frac{l}{\pi}\|D\|_{L^{2}[0,l]}+\|B\|_{H^{1}_{0}(0,l)},\vartheta>0,\\\
(\varrho_{\max}^{A_{0}})^{\operatorname{T\,}}\operatorname{diag\,}\\{-1,1\\}\varrho_{\max}^{A_{0}}>0\end{gathered}$

and additionally let one of the two following conditions hold

(a)
$(\varrho_{\max}^{B_{0}})^{\operatorname{T\,}}\operatorname{diag\,}\\{-1,1\\}\varrho_{\max}^{B_{0}}\geq
0$, and $\theta_{1}$, $\theta_{2}$ satisfy

$\begin{gathered}\frac{1}{\vartheta}\ln\frac{\sigma}{2}<\theta_{1}\leq\theta_{2}<-\frac{1}{\lambda_{\max}(A_{0})}\ln\lambda_{\max}(B_{0}).\end{gathered}$

(b)
$(\varrho_{\max}^{B_{0}})^{\operatorname{T\,}}\operatorname{diag\,}\\{-1,1\\}\varrho_{\max}^{B_{0}}<0$,
and $\theta_{1}$, $\theta_{2}$ satisfy

$\begin{gathered}\frac{1}{\vartheta}\ln\lambda_{\max}(B_{0})<\theta_{1}\leq\theta_{2}<\frac{1}{\lambda_{\max}(A_{0})}\ln\frac{2}{\sigma}.\end{gathered}$

then (30) is ISS for all $\mathcal{E}$ satisfying the dwell-time condition
$\theta_{1}\leq T_{k}\leq\theta_{2}$.

To show how the last proposition can be applied we consider the following

###  5.1 Specific example

Consider (30) with $a=1$, $l=\pi$, $D(z)=0.05z$, $B(z)=0.05z(\pi-z)$, $c=0.5$,
$\alpha(z)=1$, $\beta(z)=0$, $\gamma(z)=0.05$, $\delta=0.25$,
$\Phi(s)=-s^{3}$. Then we have

$\begin{gathered}A_{0}=\begin{pmatrix}-2&0.3214875\\\
0.3214875&0.5\end{pmatrix},\\\
B_{0}=\begin{pmatrix}1.007853982&0.02215567314\\\
0.02215567314&0.0625\end{pmatrix},\end{gathered}$
$\begin{gathered}\lambda_{\max}(A_{0})=0.54067976,\quad\lambda_{\max}(B_{0})=1.0083729,\\\
\sigma=1.1146653,\quad\vartheta=0.42851243,\\\
\varrho_{\max}^{A_{0}}=(0.12553504,-0.9920891862)^{\operatorname{T\,}},\\\
\varrho_{\max}^{B_{0}}=(0.99972578,-0.023417096)^{\operatorname{T\,}}\end{gathered}$

Then all conditions of the Proposition 3 are satisfied and the dwell-time
condition reads as

$\begin{gathered}0.01945822<\theta_{1}\leq
T_{k}\leq\theta_{2}<1.0812185.\end{gathered}$

Let us note that with this parameters choice we have that both discrete and
continuous dynamics of (30) considered separately are not asymptotically
stable already for the unperturbed case $d_{1}=0$, $d_{2}=0$. Indeed, $W(x,y)$
can be used as a Chetaev function for the continuous dynamics of (30), and to
see that the discrete dynamics is unstable just check that the spectral radius
of the jump operator is larger than $1$.

##  6 Conclusions

Our results provide a dwell-time condition that guarantees the ISS property of
a nonlinear impulsive system. In contrary to the existing dwell-time
conditions in the literature our result can be in particular applied even to
the cases where both discrete and continuous dynamics are unstable
simultaneously. In contrary to the results of [4] the ISS property is assured
by the analysis of specific interaction of the discrete and continuous
dynamics instead of a compensation of the unstable discrete (continuous)
dynamics by means of the stable continuous (discrete) one. Our future research
will be devoted to the development of constructive approach in order to derive
the auxiliary Lyapunov $V$ and Chetaev $W$ functions. Another open problem
that needs to be investigated is the derivation of conditions under which a
combination of the simultaneously stable discrete and continuous dynamics
leads to an unstable dynamics of the overall impulsive system.

##  7 Appendix

###  7.1 Proof of Proposition 2

Consider the function

$\begin{gathered}U(x,y)=\|x\|_{L^{2}[0,l]}^{2}+y^{2},\quad\end{gathered}$

Its time derivative $\dot{U}$ with respect to the system (30) is

$\begin{gathered}\dot{U}(x,y,\xi)=2\int\limits_{0}^{l}x(z)(a^{2}\partial_{zz}x(z)+\Phi(x(z))+B(z)y\\\
+\xi_{1}(z))\,dz+2y(c^{2}y+\int\limits_{0}^{l}D(z)x(z)\,dz+\xi_{2})\end{gathered}$

Applying integration by parts and the Friedrich’s inequality (5) we get

$\begin{gathered}\int\limits_{0}^{l}x(z)\partial_{zz}x(z)\,dz=-\int\limits_{0}^{l}|\partial_{z}x(z)|^{2}\,dz\leq-\frac{\pi^{2}}{l^{2}}\|x\|_{L^{2}[0,l]}^{2}.\end{gathered}$

By means of the Cauchy inequality we can write

$\begin{gathered}\dot{U}(x,y,\xi)\leq-\frac{2\pi^{2}a^{2}}{l^{2}}\|x\|_{L^{2}[0,l]}^{2}+2c^{2}y^{2}\\\
+2\|B+D\|_{L^{2}[0,l]}\|x\|_{L^{2}[0,l]}|y|\\\
+2|y||\xi_{2}|+2\|\xi_{1}\|_{L^{2}[0,l]}\|x\|_{L^{2}[0,l]}\\\
\leq\Big{(}-\frac{2\pi^{2}a^{2}}{l^{2}}+\varepsilon\Big{)}\|x\|_{L^{2}[0,l]}^{2}+(2c^{2}+\varepsilon)y^{2}\\\
+2\|B+D\|_{L^{2}[0,l]}\|x\|_{L^{2}[0,l]}|y|+\varepsilon^{-1}(\|\xi_{1}\|^{2}_{L^{2}[0,l]}+\xi_{2}^{2}).\end{gathered}$

for all $(x,y,\xi)\in L^{2}[0,l]\times\mathbb{R}\times U_{1}$ and
$\varepsilon>0$.

Recall that $\zeta=(\|x\|_{L^{2}[0,l]},|y|)^{\operatorname{T\,}}$ hence the
last estimate can be written as

$\begin{gathered}\dot{U}(x,y,\xi)\leq\zeta^{\operatorname{T\,}}(\widetilde{A}_{0}+\varepsilon\operatorname{id\,})\zeta+\varepsilon^{-1}\|\xi\|^{2}_{U_{1}},\end{gathered}$
(37)

which finishes the proof.

###  7.2 Proof of the Corollary 1

Denote $u(t):=U(x(\cdot,t),y(t))$,
$t\in[t_{0},t_{0}+\epsilon_{t_{0},x_{0},y_{0},d_{1}})$,
$d_{1}\in\mathcal{U}_{1}$, then Proposition 2 implies that

$\begin{gathered}\dot{u}(t)\leq(\lambda_{\max}(\widetilde{A}_{0})+\varepsilon)u(t)+\varepsilon^{-1}\|d_{1}(t)\|_{U_{1}}^{2}\\\
\leq(\lambda_{\max}(\widetilde{A}_{0})+\varepsilon)u(t)+\varepsilon^{-1}\|d_{1}\|_{\mathcal{U}_{1}}^{2}.\end{gathered}$

By the comparison principle we obtain

$\begin{gathered}u(t)\leq
e^{(\lambda_{\max}(\widetilde{A}_{0})+\varepsilon)(t-t_{0})}u(t_{0})\\\
+\varepsilon^{-1}\int\limits_{t_{0}}^{t}e^{(\lambda_{\max}(\widetilde{A}_{0})+\varepsilon)(t-s)}\,ds\|d\|_{\mathcal{U}_{1}}^{2}.\end{gathered}$

By the basic inequality $\sqrt{a+b}\leq\sqrt{a}+\sqrt{b}$, $a$, $b\geq 0$ we
derive

$\begin{gathered}\|(x(\cdot,t),y(t))\|_{\widehat{X}}\leq
e^{(\lambda_{\max}(\widetilde{A}_{0})+\varepsilon)\epsilon_{t_{0},x_{0},y_{0},d_{1}}/2}\|(x_{0},y_{0})\|_{\widehat{X}}\\\
+\sqrt{\frac{e^{(\lambda_{\max}(\widetilde{A}_{0})+\varepsilon)\epsilon_{t_{0},x_{0},y_{0},d_{1}}}-1}{\varepsilon(\lambda_{\max}(\widetilde{A}_{0})+\varepsilon)}}\|d_{1}\|_{\mathcal{U}_{1}},\end{gathered}$

for all $t\in[t_{0},t_{0}+\epsilon_{t_{0},x_{0},y_{0},d_{1}})$ which finishes
the proof.

###  7.3 Proof of Proposition 3

Let us define

$\begin{gathered}V(x,y)=\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2},\quad
W(x,y)=y^{2}-\|x\|_{H^{1}_{0}(0,l)}^{2}\end{gathered}$

and calculate

$\begin{gathered}\dot{V}(x,y,\xi)=-2\int\limits_{0}^{l}\partial_{zz}x(z)(a^{2}\partial_{zz}x(z)+\Phi(x(z))\\\
+B(z)y+\xi_{1}(z))\,dz+2y(c^{2}y+\int\limits_{0}^{l}D(z)x(z)\,dz+\xi_{2})\end{gathered}$

Using the integration by parts

$\begin{gathered}\int\limits_{0}^{l}\partial_{zz}x(z)\Phi(x(z))\,dz=-\int\limits_{0}^{l}\Phi^{\prime}(x(z))\,(\partial_{z}x(z))^{2}\,dz,\end{gathered}$
(38)
$\begin{gathered}\int\limits_{0}^{l}\partial_{zz}x(z)B(z)\,dz=-\int\limits_{0}^{l}\partial_{z}x(z)\partial_{z}B(z)\,dz,\end{gathered}$
$\begin{gathered}\int\limits_{0}^{l}\partial_{zz}x(z)\xi_{1}(z)\,dz=-\int\limits_{0}^{l}\partial_{z}x(z)\partial_{z}\xi_{1}(z)\,dz,\end{gathered}$

as well as the property $\Phi^{\prime}(s)\leq 0$ for all $s\in\mathbb{R}$ and
(6), we obtain

$\begin{gathered}\dot{V}(x,y,\xi)\leq-\frac{2\pi^{2}a^{2}}{l^{2}}\|x\|_{H^{1}_{0}(0,l)}^{2}\\\
+2\int\limits_{0}^{l}\partial_{z}x(z)\partial_{z}B(z)\,dzy+2\int\limits_{0}^{l}\partial_{z}x(z)\partial_{z}\xi_{1}(z)\,dz\\\
+2c^{2}y^{2}+2\int\limits_{0}^{l}D(z)x(z)\,dzy+2\xi_{2}y\end{gathered}$

By the Friedrich’s inequality (5) we get

$\begin{gathered}\dot{V}(x,y,\xi)\leq-\frac{2\pi^{2}a^{2}}{l^{2}}\|x\|_{H^{1}_{0}(0,l)}^{2}\\\
+2(\|B\|_{H^{1}_{0}(0,l)}+\frac{l}{\pi}\|D\|_{L^{2}[0,l]})\|x\|_{H^{1}_{0}(0,l)}|y|\\\
+2c^{2}y^{2}+2|\xi_{2}||y|+2\|x\|_{H^{1}_{0}(0,l)}\|\xi_{1}\|_{H^{1}_{0}(0,l)}\\\
\leq\Big{(}-\frac{2\pi^{2}a^{2}}{l^{2}}+\varepsilon\Big{)}\|x\|_{H^{1}_{0}(0,l)}^{2}+2(\|B\|_{H^{1}_{0}(0,l)}\\\
+\frac{l}{\pi}\|D\|_{L^{2}[0,l]})\|x\|_{H^{1}_{0}(0,l)}|y|+(2c^{2}+\varepsilon)y^{2}\\\
+\varepsilon^{-1}(\|\xi_{1}\|_{H^{1}_{0}(0,l)}^{2}+\xi_{2}^{2}).\end{gathered}$
(39)

and by the Cauchy inequality with (6) and (38) we obtain

$\begin{gathered}\dot{W}(x,y,\xi)=2y(c^{2}y+\int\limits_{0}^{l}D(z)x(z)\,dz+\xi_{2})\\\
+2\int\limits_{0}^{l}\partial_{zz}x(z)(a^{2}\partial_{zz}x(z)+\Phi(x(z))+B(z)y+\xi_{1}(z))\,dz\\\
\geq
2c^{2}y^{2}+2a^{2}\int\limits_{0}^{l}(\partial_{zz}x(z))^{2}\,dz+2y\int\limits_{0}^{l}D(z)x(z)\,dz\\\
-2y\int\limits_{0}^{l}\partial_{z}B(z)\partial_{z}x(z)\,dz+2y\xi_{2}-2\int\limits_{0}^{l}\partial_{z}x(z)\partial_{z}\xi_{1}(z)\,dz\\\
\end{gathered}$

Applying (6) we have

$\begin{gathered}\dot{W}(x,y,\xi)\geq
2c^{2}y^{2}+\frac{2\pi^{2}a^{2}}{l^{2}}\|x\|_{H^{1}_{0}(0,l)}^{2}\\\
-2(\frac{l}{\pi}\|D\|_{L^{2}[0,l]}+\|B\|_{H^{1}_{0}(0,l)})\|x\|_{H^{1}_{0}(0,l)}|y|\\\
-2|y||\xi_{2}|-2\|x\|_{H^{1}_{0}(0,l)}\|\xi_{1}\|_{H^{1}_{0}(0,l)}\\\
\geq(2c^{2}-\varepsilon)y^{2}+\Big{(}\frac{2\pi^{2}a^{2}}{l^{2}}-\varepsilon\Big{)}\|x\|_{H^{1}_{0}(0,l)}^{2}\\\
-2(\frac{l}{\pi}\|D\|_{L^{2}[0,l]}+\|B\|_{H^{1}_{0}(0,l)})\|x\|_{H^{1}_{0}(0,l)}|y|\\\
-\varepsilon^{-1}(\|\xi_{1}\|^{2}_{H^{1}_{0}(0,l)}+\xi_{2}^{2}),\end{gathered}$

for all $(x,y,\xi)\in X\times U_{1}$. Further we have

$\begin{gathered}V(g(x,y,\mu))\leq\int\limits_{0}^{l}(\alpha_{z}(z)x(z)+\alpha(z)x_{z}(z)+\beta_{z}(z)y\\\
+\mu_{1z}(z))^{2}\,dz+\Big{(}\int\limits_{0}^{l}\gamma(z)x(z)\,dz+\delta
y+\mu_{2}\Big{)}^{2}\\\
\leq\|\alpha^{2}\|_{C[0,l]}\|x_{z}\|_{L^{2}[0,l]}^{2}+\|\beta_{z}\|_{L^{2}[0,l]}^{2}y^{2}+\|\mu_{1z}\|_{L^{2}[0,l]}^{2}\\\
+\|\alpha\alpha_{zz}\|_{C[0,l]}\|x\|_{L^{2}[0,l]}^{2}+2\|\alpha_{z}\beta_{z}+\gamma\delta\|_{L^{2}[0,l]}\|x\|_{L^{2}[0,l]}|y|\\\
+2\|\alpha\beta_{z}\|_{L^{2}[0,l]}\|x\|_{H_{0}^{1}(0,l)}|y|+2\|\alpha_{z}\|_{C[0,l]}\|\mu_{1}\|_{H_{0}^{1}(0,l)}\|x\|_{L^{2}[0,l]}\\\
+2\|\beta\|_{H^{1}_{0}(0,l)}\|\mu_{1}\|_{H_{0}^{1}(0,l)}|y|+2\|\alpha\|_{C[0,l]}\|\mu_{1}\|_{H_{0}^{1}(0,l)}\|x\|_{H_{0}^{1}(0,l)}\\\
+\|\gamma\|_{L^{2}[0,l]}^{2}\|x\|_{L^{2}[0,l]}^{2}+\delta^{2}y^{2}+\mu_{2}^{2}+2|\delta||\mu_{2}||y|\\\
+2|\mu_{2}|\|\gamma\|_{L^{2}[0,l]}\|x\|_{L^{2}[0,l]}\end{gathered}$

By the Friedrich’s inequality (5) we have

$\begin{gathered}V(g(x,y,\mu))\leq\Big{(}\|\alpha^{2}\|_{C[0,l]}+\frac{l^{2}}{\pi^{2}}\|\alpha\alpha_{zz}\|_{C[0,l]}+\frac{l^{2}}{\pi^{2}}\|\gamma\|_{L^{2}[0,l]}^{2}\\\
+\varepsilon(\frac{l^{2}}{\pi^{2}}\|\alpha_{z}\|_{C[0,l]}+\|\alpha\|_{C[0,l]}+\|\gamma\|_{L^{2}[0,l]}\frac{l^{2}}{\pi^{2}})\Big{)}\|x\|_{H^{1}_{0}(0,l)}^{2}\\\
+2\Big{(}\|\alpha_{z}\beta_{z}+\gamma\delta\|_{L^{2}[0,l]}\frac{l}{\pi}+\|\alpha\beta_{z}\|_{L^{2}[0,l]}\Big{)}\|x\|_{H_{0}^{1}(0,l)}|y|\\\
+(\|\beta\|_{H^{1}_{0}(0,l)}^{2}+\delta^{2}+\varepsilon(\|\beta\|_{H_{0}^{1}(0,l)}+|\delta|))y^{2}\\\
+(1+\varepsilon^{-1}(\|\alpha_{z}\|_{C[0,l]}+\|\beta\|_{H_{0}^{1}(0,l)}+\|\alpha\|_{C[0,l]}))\|\mu_{1}\|_{H_{0}^{1}(0,l)}^{2}\\\
+(1+\varepsilon^{-1}(|\delta|+\|\gamma\|_{L^{2}[0,l]}))\mu_{2}^{2}\end{gathered}$

Let us denote

$\begin{gathered}\scriptsize
B_{1}=\begin{pmatrix}\frac{l^{2}}{\pi^{2}}\|\alpha_{z}\|_{C[0,l]}+\|\alpha\|_{C[0,l]}+\|\gamma\|_{L^{2}[0,l]}\frac{l^{2}}{\pi^{2}}&0\\\
0&\|\beta\|_{H_{0}^{1}(0,l)}+|\delta|\end{pmatrix},\\\
\kappa(\varepsilon)=\max\\{1+\varepsilon^{-1}(\|\alpha_{z}\|_{C[0,l]}+\|\beta\|_{H_{0}^{1}(0,l)}+\|\alpha\|_{C[0,l]}),\\\
1+\varepsilon^{-1}(|\delta|+\|\gamma\|_{L^{2}[0,l]})\\},\end{gathered}$

then the last inequality can be written as

$\begin{gathered}V(g(x,y,\mu))\leq\zeta^{T}(B_{0}+\varepsilon
B_{1})\zeta+\kappa(\varepsilon)\|\mu\|^{2}.\end{gathered}$ (40)

The inequalities (39) and (40) imply the estimates (9) and (10): Denote
$w(t)=\|x(\cdot,t)\|_{H^{1}_{0}(0,l)}^{2}+y^{2}(t)$, $t\geq 0$, then from (9)
follows:

$\begin{gathered}\dot{w}(t)\leq(\lambda_{\max}(A_{0})+\varepsilon)w(t)+\varepsilon^{-1}\|d_{1}(t)\|_{U_{1}}^{2}\\\
\leq(\lambda_{\max}(A_{0})+\varepsilon)w(t)+\varepsilon^{-1}\|d\|_{\mathcal{U}_{1}}^{2}.\end{gathered}$

By the comparison principle we have

$\begin{gathered}w(t)\leq e^{(\lambda_{\max}(A_{0})+\varepsilon)t}w(0)+\\\
\varepsilon^{-1}\int\limits_{0}^{t}e^{(\lambda_{\max}(A_{0})+\varepsilon)(t-s)}\,ds\|d_{1}\|_{\mathcal{U}_{1}}^{2}.\end{gathered}$

and with help of $\sqrt{a+b}\leq\sqrt{a}+\sqrt{b}$, $a$, $b\geq 0$ we get

$\begin{gathered}\|(x(\cdot,t),y(t))\|_{X}\leq
e^{(\lambda_{\max}(A_{0})+\varepsilon)\tau/2}\|(x_{0},y_{0})\|_{X}+\\\
\sqrt{\frac{e^{(\lambda_{\max}(A_{0})+\varepsilon)\tau}-1}{\varepsilon(\lambda_{\max}(A_{0})+\varepsilon)}}\|d_{1}\|_{\mathcal{U}_{1}},\end{gathered}$

for all $t\in[0,\tau]$, which proves (9). Similarly (10) follows from (40).

Let us estimate $\dot{V}(x,y,\xi)$, $V(g(x,y,\mu))$ on the sets

$\begin{gathered}G^{+}=\\{(x,y)\in
X\,\,:\,\,|y|\geq\|x\|_{H^{1}_{0}(0,l)}\\},\\\ G^{-}=\\{(x,y)\in
X\,\,:\,\,|y|\leq\|x\|_{H^{1}_{0}(0,l)}\\}\end{gathered}$

and $\dot{W}(x,y,\xi)$ on the set, where $W(x,y)=0$ holds.

By the conditions of Proposition 3 we have
$(\varrho_{\max}^{A_{0}})^{\operatorname{T\,}}\operatorname{diag\,}\\{-1,1\\}\varrho_{\max}^{A_{0}}>0$
and

$\begin{gathered}-\vartheta=-\frac{\pi^{2}a^{2}}{l^{2}}+\|B\|_{H^{1}_{0}(0,l)}+\frac{l}{\pi}\|D\|_{L^{2}[0,l]}+c^{2}<0.\end{gathered}$

Hence by (39) for all $(x,y)\in G^{+}$ we have

$\begin{gathered}\dot{V}(x,y,\xi)\leq(\lambda_{\max}(A_{0})+\varepsilon)(\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2})+\varepsilon^{-1}\|\xi\|^{2}_{U_{1}}\\\
\leq(\lambda_{\max}(A_{0})+2\varepsilon)(\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2}),\end{gathered}$

if $\sqrt{\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2}}>\varepsilon^{-1}\|\xi\|_{U_{1}}$.

For all $(x,y)\in G^{-}$ we have

$\begin{gathered}\dot{V}(x,y,\xi)\leq(-\vartheta+\frac{\varepsilon}{2})(\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2})+\varepsilon^{-1}\|\xi\|^{2}_{U_{1}}\\\
\leq(-\vartheta+\varepsilon)(\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2}),\end{gathered}$

if
$\sqrt{\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2}}>\sqrt{2}\varepsilon^{-1}\|\xi\|_{U_{1}}$.

Now choosing $\varepsilon>0$ small enough ($\varepsilon<\vartheta$), we can
take $\varphi_{1}(s)=(\vartheta-\varepsilon)s$,
$\varphi_{2}(s)=(\lambda_{\max}(A_{0})+2\varepsilon)s$.

If the condition (a) is satisfied, that is
$(\varrho_{\max}^{B_{0}})^{\operatorname{T\,}}\operatorname{diag\,}\\{-1,1\\}\varrho_{\max}^{B_{0}}\geq
0$, then for all $(x,y)\in G^{+}$ we have

$\begin{gathered}V(g(x,y,\mu))\leq(\lambda_{\max}(B_{0})+\varepsilon\|B_{1}\|)(\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2})\\\
+\kappa(\varepsilon)\|\mu\|^{2}_{U_{2}}\leq(\lambda_{\max}(B_{0})+\varepsilon(1+\|B_{1}\|))(\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2}).\end{gathered}$

if
$\sqrt{\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2}}>\sqrt{\varepsilon^{-1}\kappa(\varepsilon)}\|\mu\|_{U_{2}}$.

For all $(x,y)\in G^{-}$ we have

$\begin{gathered}V(g(x,y,\mu))\leq\Big{(}\frac{\sigma}{2}+\varepsilon(1+\|B_{1}\|)\Big{)}V(x,y),\end{gathered}$

if
$\sqrt{\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2}}>\sqrt{\varepsilon^{-1}\kappa(\varepsilon)}\|\mu\|_{U_{2}}$.

In this case
$\psi_{1}(s)=\Big{(}\frac{\sigma}{2}+\varepsilon(1+\|B_{1}\|)\Big{)}s$,
$\psi_{2}(s)=(\lambda_{\max}(B_{0})+\varepsilon(1+\|B_{1}\|))s$.

If the condition (b) is satisfied, that is
$(\varrho_{\max}^{B_{0}})^{\operatorname{T\,}}\operatorname{diag\,}\\{-1,1\\}\varrho_{\max}^{B_{0}}<0$,
then for all $(x,y)\in G^{+}$ we have

$\begin{gathered}V(g(x,y,\mu))\leq\Big{(}\frac{\sigma}{2}+\varepsilon(1+\|B_{1}\|)\Big{)}V(x,y),\end{gathered}$

if
$\sqrt{\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2}}>\sqrt{\varepsilon^{-1}\kappa(\varepsilon)}\|\mu\|_{U_{2}}$.

For all $(x,y)\in G^{-}$ we have

$\begin{gathered}V(g(x,y,\mu))\leq(\lambda_{\max}(B_{0})+\varepsilon\|B_{1}\|)(\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2})\\\
+\kappa(\varepsilon)\|\mu\|^{2}_{U_{2}}\leq(\lambda_{\max}(B_{0})+\varepsilon(1+\|B_{1}\|))(\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2}).\end{gathered}$

if
$\sqrt{\|x\|_{H^{1}_{0}(0,l)}^{2}+y^{2}}>\sqrt{\varepsilon^{-1}\kappa(\varepsilon)}\|\mu\|_{U_{2}}$.
In this case $\psi_{1}(s)=(\lambda_{\max}(B_{0})+\varepsilon(1+\|B_{1}\|))s$,
$\psi_{2}(s)=\Big{(}\frac{\sigma}{2}+\varepsilon(1+\|B_{1}\|)\Big{)}s$.

If

$\begin{gathered}c^{2}+\frac{\pi^{2}a^{2}}{l^{2}}>\frac{l}{\pi}\|D\|_{L^{2}[0,l]}+\|B\|_{H^{1}_{0}(0,l)},\end{gathered}$

then $W(x,y)=0$ implies

$\begin{gathered}\dot{W}(x,y,\mu)\geq\Big{(}c^{2}+\frac{\pi^{2}a^{2}}{l^{2}}-\big{(}\frac{l}{\pi}\|D\|_{L^{2}[0,l]}\\\
+\|B\|_{H^{1}_{0}(0,l)}\big{)}-2\varepsilon\Big{)}(\|x\|_{H_{0}^{1}(0,l)}^{2}+y^{2})>0,\end{gathered}$

if $\sqrt{\|x\|_{H_{0}^{1}(0,l)}^{2}+y^{2}}>\varepsilon^{-1}\|\xi\|_{U_{1}}$,
with small enough $\varepsilon$.

Now applying Theorem 1 and choosing positive constant $\varepsilon$ and
$\delta$ small enough, we see that the conditions of Proposition 3 guarantee
the ISS property for the system (30), which finishes the proof.

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