1 Introduction

Impulsive dynamical systems provide a mathematical modeling framework for practical processes where a combination of continuous and discrete dynamics takes place. Such a hybrid dynamics appears in many applications, for example, in case of mechanical collisions or in control systems involving a combination of analog and digital controllers. As well in pandemic systems, a mass vaccination can be modeled as an impulsive action meaning a (nearly instantaneous) transition of a large amount of susceptible individuals to become immune.

A combination of discrete and continuous dynamics leads to a higher complexity in the behavior of solutions compared with a purely discrete or purely continuous system. Such unexpected effects as instability [27] or chaos [14] can arise. In particular, such properties as stability and robustness are more difficult to investigate, especially in case of nonlinear systems.

Stability in the sense of Lyapunov of nonlinear impulsive systems has a long history of investigations, see [17, 23]. Later, more general stability notions were developed for hybrid systems, which include impulsive ones as a particular case, see [12, 28]. These notions use the generalized (hybrid) time concept, which allows to develop rather general results for a wide class of hybrid systems, including impulsive, switched and sampled data systems.

In case of systems having input signals, the notion of input-to-state stability (ISS) was introduced in [26] and was found very fruitful in many applications [16]. This framework was also successfully used for studying robust stability of impulsive systems, see [4, 6, 13, 18, 22]. In particular, [6] derives dwell-time conditions to establish the ISS property for nonlinear impulsive systems on infinite dimensional state spaces. This result is based on certain stability assumptions imposed either on the continuous or on the discrete dynamics. The ISS is assured by the stability property of either of both dynamics using a suitable dwell-time condition. Stability of interconnected impulsive systems is then studied in the case, when the ISS-Lyapunov functions are known for the subsystems, which leads to a combination of the dwell-time and small-gain conditions. The ISS property of impulsive systems where impulsive actions depend on time was studied in [4], where new and rather general dwell-time conditions were developed.

A small-gain theorem for \(n\ge 2\) interconnected hybrid systems was established in [22] for the case where not all subsystems are assumed to be ISS, which extends the results of [6]. Similar results for nonlinear interconnected impulsive systems were developed in [8] for the case of absent external perturbations. This work derives sufficient stability conditions for interconnected systems by means of vector Lyapunov functions, which leads to conditions similar to the small-gain ones.

In most of works (e.g., in those mentioned above) studying stability of impulsive systems by means of Lyapunov methods, it is assumed that the discrete and continuous dynamics share a common Lyapunov function V which decays either on jumps or along the continuous flow. A dwell-time condition allows to compensate the destabilizing effect of one type of dynamics by the stabilizing property of the other one. Certainly, if V increases in both cases, then the system is unstable. However, in general, it can happen that the whole system is asymptotically stable even in the case when both discrete and continuous dynamics are unstable. Identification of such systems needs a more refined consideration of the interaction between both dynamics types. It is expected that stability conditions become more involved in this case. It should be noted that there are not many stability results in the literature that cover the case of simultaneous instability of discrete and continuous dynamics: [1, 9,10,11, 19, 24], see also [3] and [2] for the linear case. However, these results cannot be extended directly to the case of nonlinear infinite-dimensional systems with inputs, also we note that only local stability was studied in the first group of these papers.

Vector Lyapunov functions were used in [19] to establish stability results, where second-order derivatives of Lyapunov functions along solutions enter to the stability conditions. These results were generalized in [9, 10], where higher-order derivatives of Lyapunov functions are employed. This approach cannot be used in case of systems with inputs as one would need to require infinitely smooth disturbances, which is very restrictive in real applications.

Averaged dwell-time conditions were considered in [1, 24] to establish stability of a linear impulsive system on a Banach space. Based on the identities from the commutator calculus new comparison theorems, constructions of Lyapunov functions and conditions for the local asymptotic stability were developed there. In this work, we are interested in global stability properties for nonlinear systems with inputs.

With an exception of [22], in the most of works devoted to investigations of the ISS-like properties of hybrid systems, it is assumed that either the continuous or discrete dynamics satisfies the ISS property. Hence, it is interesting to further develop the direct Lyapunov method for the case, when both types of dynamics fail to be ISS. Our paper contributes into this direction, providing a new approach and new stability conditions.

In this work, we improve the results of [9, 10] and extend them to the case of ISS for nonlinear impulsive systems. We derive stability conditions by means of a series of Lyapunov-like functions. Instead of higher-order derivatives of Lyapunov functions employed in [9, 10], we use an infinite sequence of auxiliary functions to provide estimates of the dwell-time in order to guarantee the ISS property. The obtained results are then applied to the studying of the global asymptotic stability of linear impulsive systems with continuous dynamics governed by a parabolic PDE. The ISS property is also studied for this type of systems. Moreover, we derive conditions for the ISS property of nonlinear locally homogeneous finite-dimensional impulsive systems.

The paper consists of six sections. Section 2 introduces the notation and several auxiliary inequalities used in the paper. The problem statement is described in Sect. 3. Section 4 contains the main results with their proofs. Application of the results to the investigation of GAS and ISS properties of linear impulsive systems in infinite-dimensional spaces and of nonlinear finite-dimensional systems is provided in Sect. 5. A brief discussion and conclusions are collected in Sect. 6. Proofs of several technical results are placed in Appendix.

2 Notation and preliminaries

We use the following classes of comparison functions:

\({\mathcal {K}}=\{\gamma \,:\,\mathbb R_+\rightarrow \mathbb R_+\,:\,\text {continuous, strictly increasing and} \,\,\gamma (0)=0\},\)

\({\mathcal {K}}_{\infty }=\{\gamma \,:\,\mathbb R_+\rightarrow \mathbb R_+\,:\gamma \in {\mathcal {K}},\quad \gamma (s)\rightarrow \infty \quad \text {for}\quad s\rightarrow \infty \},\)

\({\mathcal {L}}=\{\gamma \,:\,\mathbb R_+\rightarrow \mathbb R_+\,:\,\text {continuous, strictly decreasing and}\,\, \lim \nolimits _{t\rightarrow \infty }\gamma (t)=0\},\)

\({\mathcal {K}}{\mathcal {L}}=\{\beta \,:\,\mathbb R_+\times \mathbb R_+\rightarrow \mathbb R_+\,:\,\text {continuous}\,\, \beta (\cdot ,t)\in {\mathcal {K}},\,\,\beta (s,\cdot )\in {\mathcal {L}}\}.\)

By C[0, l] we denote the space of continuous functions, defined on [0, l] with values in \(\mathbb R\) with norm \(\Vert f\Vert _{C[0,l]}=\max \nolimits _{x\in [0,l]}|f(x)|\), \(C^k[0,l]\) denotes of k-times continuously differentiable functions with the norm \(\Vert f\Vert _{C^k[0,l]}=\max \nolimits _{p=0,\dots ,k}\max \nolimits _{x\in [0,l]}|f^{(p)}(x)|\). \(H^0[0,l]=L^2[0,l]\) is the Hilbert space of measurable square integrable functions with scalar product \((f,g)_{L^2[0,l]}=\int \limits _0^lf(z)g(z)\,dz\). Let \({\mathfrak {L}}(L^2[0,l])\) be the Banach algebra of linear bounded operators defined on \(L^2[0,l]\).

For \(M\subset \mathbb R\) and a Banach space X by \(L^{\infty }(M,X)\), we denote the space of mappings \(f\,:\,M\rightarrow X\) with the norm \(\Vert f\Vert _{L^{\infty }}=ess\sup _{m\in M}\Vert f(m)\Vert _{X}\). For \(M=\mathbb Z_+{:=\mathbb N\cup \{0\}}\), we write \(L^{\infty }(\mathbb Z_+,X)=l^{\infty }(X)\). By \(B_r(x_0)\), we denote the open ball of radius \(r>0\) in X centered at \(x_0\).

\(C^1(U,\mathbb R^n)\), \(U\subset \mathbb R\) is the set of continuously differentiable mappings \(f:U\rightarrow \mathbb R^n\). \(C^1(\mathbb R^n)\) is the space of continuously differentiable functions \(f:\mathbb R^n\rightarrow \mathbb R^n\), and for \(f\in C^1(\mathbb R^n)\) by \(\partial _x f(x)\), we denote the corresponding Jacobi matrix.

For a linear bounded operator A acting on a Banach space, \(\sigma (A)\) denotes the spectrum of A and \(r_{\sigma }(A)\) denotes its spectral radius.

\(\mathbb R^{n\times m}\) is the space of \(n\times m\)-matrices, for \(m=n\) the set \(\mathbb R^{n\times n}\) is then a Banach algebra. We use the norm on \(\mathbb R^{n\times n}\) induced by the Euclidean norm in \(\mathbb R^n\): \(\Vert A\Vert =\sup _{\Vert x\Vert =1}\Vert Ax\Vert =\lambda _{\max }^{1/2}(A^{T}A)\).

We will use the following well-known inequalities. For any \(a,b\in \mathbb R_+\) and \(\varrho \in {\mathcal {K}}\) holds

$$\begin{aligned} \varrho (a+b)\le \varrho (2a)+\varrho (2b); \end{aligned}$$
(1)

for \(p_1,p_2\in (1,\infty )\) with \(\frac{1}{p_1}+\frac{1}{p_2}=1\), the Young’s inequality is

$$\begin{aligned} xy\le \frac{x^{p_1}}{p_1}+\frac{y^{p_2}}{p_2},\quad x\ge 0,\quad y\ge 0. \end{aligned}$$
(2)

3 Problem statement and related stability notions

We consider dynamical systems with inputs defined similarly to [5, 21, 26] as follows

Definition 1

Let X be the state space with the norm \(\Vert \cdot \Vert _X\) and \({\mathcal {U}}_1\subset \{f\,:\mathbb R\rightarrow U_1\}\) be the space of input signals normed by \(\Vert \cdot \Vert _{{\mathcal {U}}_1}\) 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 \({\mathcal {S}}_s\,:\,{\mathcal {U}}_1\rightarrow {\mathcal {U}}_1\), \(s\in \mathbb R\) is the linear operator defined by \({\mathcal {S}}_s u(t)=u(t+s)\).

The triple \(\varSigma _c=(X,{\mathcal {U}}_1,\phi _c)\) 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

(\(\varSigma _c1\)) 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 \(\lim _{t\rightarrow t_0+}\phi _c(t,t_0,x,d_1)=x\);

(\(\varSigma _c2\)) \(\phi _c(t,t,x,d_1)=x\) for any \((x,d_1)\in X\times {\mathcal {U}}_1\), \(t\in \mathbb R\);

(\(\varSigma _c3\)) 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 \(s\in [t_0,t]\), and it holds that \(\phi _c(t,t_0,x,d_1)=\phi _c(t,t_0,x,\widetilde{d}_1)\);

(\(\varSigma _c4\)) for any \((x,d_1)\in X\times {\mathcal {U}}_1\) and \(t\ge \tau \ge t_0\) with \(\tau \in [t_0,t_0+\epsilon _{t_0,x,d_1})\), \(t\in [\tau ,\tau +\epsilon _{\tau ,{\phi _c}(\tau ,t_0,x,d_1),d_1})\cap [t_0,t_0+\epsilon _{t_0,x,d_1})\) it holds that

$$\begin{aligned} \phi _c(t,t_0,x,d_1)=\phi _c(t,\tau ,{\phi _c}(\tau ,t_0,x,d_1),d_1), \end{aligned}$$

(\(\varSigma _c5\)) 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

$$\begin{aligned} \begin{aligned} \epsilon _{t_0+\tau ,x,d_1}&=\epsilon _{t_0,x,{\mathcal {S}}_{\tau }d_1},\\ \phi _c(t+\tau ,t_0+\tau ,x,d_1)&=\phi _c(t,\tau ,x,{\mathcal {S}}_{\tau }d_1). \end{aligned} \end{aligned}$$

Note that \((\varSigma _c5)\) implies that for all \(t\in [\tau ,\tau +\epsilon _{\tau ,x_0,d_1})\), \(\tau \le t\)

$$\begin{aligned} \phi _c(t,\tau ,x,d_1)=\phi _c(t-\tau ,0,x,{\mathcal {S}}_{\tau }d_1). \end{aligned}$$
(3)

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 \(\lim \limits _{k\rightarrow \infty }\tau _k=\infty \). Let \({\mathcal {U}}_2\subset \{f\,:\,\mathbb Z_+\rightarrow U_2\}\) be the space of input signals normed by \(\Vert \cdot \Vert _{{\mathcal {U}}_2}\) and taking values in a nonempty subset \(U_2\) of some linear normed space. Let \(g\,:\,X\times U_2\rightarrow X\) be a mapping defining impulsive actions and the mapping \(\phi \) be defined for all \((t,t_0,x,d_1,d_2)\in \mathbb R\times \mathbb R\times X\times {\mathcal {U}}_1\times {\mathcal {U}}_2\), \(t\ge t_0\).

The following data \(\varSigma =(X,\varSigma _c,{\mathcal {U}}_2, g,\phi ,{\mathcal {E}})\) defines an impulsive system if

\((\varSigma _1)\) for all \((k,x,d_1)\in \mathbb Z_+\times X\times {\mathcal {U}}_1\) the system \(\varSigma _c\) satisfies

$$\begin{aligned} \tau _{p(t_0)}-t_0<\epsilon _{t_0,x,d_1},\quad T_k:=\tau _{k+1}-\tau _k<\epsilon _{\tau _k,x,d_1} \end{aligned}$$

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

\((\varSigma _2)\) the mapping \(\phi \) satisfies

$$\begin{aligned} \begin{aligned} \phi (t,t_0,x,d_1,d_2)&=\phi _c(t,t_0,x,d_1),\quad \text { for all }\quad t\in [t_0,\tau _{p(t_0)}],\\ \phi (t,t_0,x,d_1,d_2)&=\phi _c(t,\tau _k,g(\phi (\tau _k,t_0,x,d_1,d_2),d_2(k)),d_1)\\&\quad \text { for all }\quad t\in (\tau _k,\tau _{k+1}],\quad k\in \mathbb Z_+, k\ge p(t_0). \end{aligned} \end{aligned}$$

We will denote for short

$$\begin{aligned} \phi (\tau _k^+,t_0,x,d_1,d_2)=g(\phi (\tau _k,t_0,x,d_1,d_2),d_2(k)),\quad k\ge p(t_0),\; \tau _k\ge t_0 \end{aligned}$$

The conditions \((\varSigma _c1)\) and \((\varSigma _2)\) imply

$$\begin{aligned} \begin{aligned} \lim \limits _{t\rightarrow \tau _k+}\phi (t,t_0,x,d_1,d_2)&=\phi (\tau _k^+,t_0,x,d_1,d_2),\\ \lim \limits _{t\rightarrow \tau _k-}\phi (t,t_0,x,d_1,d_2)&=\phi (\tau _k,t_0,x,d_1,d_2); \end{aligned} \end{aligned}$$

and \((\varSigma _c4)\), \((\varSigma _c5)\), \((\varSigma _2)\) imply that for \(t\ge \tau \ge t_0\), \((x,d_1,d_2)\in X\times {\mathcal {U}}_1\times {\mathcal {U}}_2\), the following holds:

$$\begin{aligned} \begin{aligned} \phi (t,t_0,x,d_1,d_2)=\phi (t,\tau ,\phi (\tau ,t_0,d_1,d_2),d_1,d_2). \end{aligned} \end{aligned}$$
(4)

The system \(\varSigma _c\) describes the continuous dynamics of the impulsive system \(\varSigma \). One can also consider its discrete dynamics separately as a system \(\varSigma _d\) defined next

Definition 3

A discrete dynamical system with input \(\varSigma _d=(X,g,\phi _d,{\mathcal {U}}_2)\) is given by a normed state space \((X,\Vert \cdot \Vert _X)\); a space of input signals \({\mathcal {U}}_2\subset \{f\,:\,\mathbb Z_+\rightarrow U_2\}\) with norm \(\Vert \cdot \Vert _{{\mathcal {U}}_2}\) and values in a nonempty subset \(U_2\) of a linear normed space; a mapping \(g\,:\,X\times U_2\rightarrow X\); and a mapping \(\phi _d\,\,:(k,l,x,d_2)\mapsto \phi _d(k,l,x,d_2)\), for \((k,l,x,d_2)\in \mathbb Z_+\times \mathbb Z_+\times X\times {\mathcal {U}}_2\), \(k\ge l\) such that

\((\varSigma _d1)\) \({\phi _d}(k,k,x,d_2)=x\), \(\phi _d(k+1,l,x,d_2)=g(\phi _d(k,l,x,d_2),d_2(k))\) for all \(k\ge l\).

Assumption

For any \(\tau \ge 0\), there exist \(\xi ,\,\xi _{\tau }\in {\mathcal {K}}_{\infty }\) and \(\chi _{\tau },\chi \in {\mathcal {K}}_{\infty }\) such that for all \((x,d_1,d_2)\in X\times {\mathcal {U}}_1\times {\mathcal {U}}_2\), it holds that

$$\begin{aligned} \Vert \phi _c(t,0,x,d_1)\Vert \le \xi _{\tau }(\Vert x\Vert )+\chi _{\tau }(\Vert d_1\Vert _{{\mathcal {U}}_1})\quad \text {for all}\quad t\in [0,\tau ] \end{aligned}$$
(5)

and

$$\begin{aligned} \Vert g(x,d_2)\Vert \le \xi (\Vert x\Vert )+\chi (\Vert d_2\Vert _{ U_2}). \end{aligned}$$
(6)

This assumption is not restrictive; for example, if g is continuous, then (6) is satisfied.

We are interested in the stability properties of the system \(\varSigma \) and its robustness with respect to the input signals \(d_1\) and \(d_2\). To this end, we use the notion of input-to-state stability (ISS). It was originally introduced in [25] for time invariant finite-dimensional systems. In our case, we adapt it as follows for \(\varSigma \)

Definition 4

For a fixed time sequence \({\mathcal {E}}\) of impulsive actions, the system \(\varSigma \) is called ISS if there exist \(\beta _{t_0}\in {\mathcal {K}}{\mathcal {L}}\), \(\gamma _{t_0}\in {\mathcal {K}}_{\infty }\), such that for any initial state \(x\in X\), any \(t\ge t_0\) and any \((d_1,d_2)\in {\mathcal {U}}_1\times {\mathcal {U}}_2\), it holds that

$$\begin{aligned} \Vert \phi (t, t_0,x,d_1,d_2)\Vert \le \beta _{t_0}(\Vert x\Vert ,t)+\gamma _{t_0}(d),\qquad d:=\Vert d_1\Vert _{{\mathcal {U}}_1}+\Vert d_2\Vert _{{\mathcal {U}}_2}. \end{aligned}$$
(7)

Next, we introduce a class of functions that we will use as Lyapunov functions to study the ISS property.

Definition 5

A function \(V:\,[t_0,\infty )\times X\rightarrow \mathbb R\) is said to be of class \({\mathcal {V}}({\mathcal {T}}_0)\), where \({\mathcal {T}}_0:=[t_0,\infty )\setminus {\mathcal {E}}\), if it satisfies the following properties:

  1. (1)

    V is continuous at any point \((t,x)\in {\mathcal {T}}_0\times X\), left continuous in t for \(t\in {\mathcal {E}}\), that is \(\lim \nolimits _{h\rightarrow 0-}V(t+h,x)=V(t,x)\) for all \((t,x)\in {\mathcal {E}}\times X\) and such that there exists \(\lim \nolimits _{h\rightarrow 0+}V(t+h,x):=V(t+0,x)\) for all \((t,x)\in {\mathcal {E}}\times X\);

  2. (2)

    The Lie derivative \({\dot{V}}(t,x,\zeta )\) exists for all \((t,x,\zeta )\in {\mathcal {T}}_0\times X\times U_1\) which is defined by

    $$\begin{aligned} {\dot{V}}(t,x,\zeta ):=\lim \limits _{h\rightarrow 0+}\frac{1}{h}(V(t+h,\phi _c(t+h,t,x,\zeta ))-{V(t,x)}), \end{aligned}$$
    (8)
  3. (3)

    For \((t,x,\zeta )\in {\mathcal {E}}\times X\times U_1\) the limits \({\dot{V}}(t\pm 0,x,\zeta )=\lim \nolimits _{h\rightarrow 0+}{\dot{V}}(t\pm h,x,\zeta )\) exist and \({\dot{V}}(t-0,x,\zeta )={\dot{V}}(t,x,\zeta )\)

4 Main results

In this section, we provide sufficient conditions to guarantee the ISS property for \(\varSigma \). Complementary stability conditions are given in Theorems 1 and 2, respectively. Application of these results are illustrated in Sect. 5.

Theorem 1

Assume that for \(\varSigma \) there are functions \(V_i\in {\mathcal {V}}({\mathcal {T}}_0)\), \(i\in \mathbb Z_+\) with

  1. (1)

    for some \(\alpha _1\), \(\alpha _2\in {\mathcal {K}}_{\infty }\), it holds that

    $$\begin{aligned} \alpha _1(\Vert x\Vert )\le V_0(t,x)\le \alpha _2(\Vert x\Vert ),\quad \text {for all}\quad (t,x)\in [t_0,\infty )\times X; \end{aligned}$$
    (9)
  2. (2)

    there is a sequence \(\eta _p\in {\mathcal {K}}_{\infty }\), \(p\in \mathbb Z_+\) such that for all \((t,x,\zeta )\in {\mathcal {T}}_0\times X\times U_1\) holds

    $$\begin{aligned} {\dot{V}}_0(t,x,\zeta )\le & {} V_1(t,x)+\eta _0(\Vert \zeta \Vert _{U_1}),\nonumber \\ -{\dot{V}}_p(t,x,\zeta )\le & {} V_{p+1}(t,x)+\eta _p(\Vert \zeta \Vert _{U_1}),\quad p\in \mathbb N; \end{aligned}$$
    (10)
  3. (3)

    there are \(\eta \in {\mathcal {K}}_{\infty }\) and \(W_k\,:X\rightarrow \mathbb R\), \(k\in \mathbb Z_+\) so that \(\forall \) \((k,x,\zeta )\in \mathbb Z_+\times X\times U_2\)

    $$\begin{aligned} V_0(\tau _k+0,g(x,\zeta ))-V_0(\tau _k,x)\le W_k(x)+\eta (\Vert \zeta \Vert _{U_2}); \end{aligned}$$
    (11)

    holds;

  4. (4)

    there exists \(\delta \in {\mathcal {K}}_{\infty }\) such that for all \((k,x)\in \mathbb Z_+\times X\), it holds that

    $$\begin{aligned} G_{k+1}(x):=W_{k+1}(x)+\sum \limits _{p=1}^{\infty }V_p(\tau _{k+1},x)\frac{(\tau _{k+1}-\tau _k)^p}{p!}\le -\delta (\Vert x\Vert ); \end{aligned}$$
    (12)
  5. (5)

    for any \(\rho >0\) exists \(q_{\rho }\in [0,1)\) such that \(\lim \nolimits _{p\rightarrow \infty }\frac{\eta _p(s)}{(p+1)\eta _{p-1}(s)}=q_{\rho }\) uniformly for \(s\in [0,\rho ]\) and for any \(k\in \mathbb Z_+\), there is \(\omega _k\in {\mathcal {K}}_{\infty }\) such that \(|V_p(s,x)|\le \omega _k(\Vert x\Vert )\) for all \((p,s)\in \mathbb Z_+\times (\tau _k,\tau _{k+1}]\).

Then, \(\varSigma \) satisfies the ISS property.

Remark 1

The inequality (12) in condition (4) of the theorem restricts the time intervals between jumps, which is a dwell-time condition. If both continuous and discrete dynamics are stable, the existence of \(W_k\) and \(V_p\) so that \(W_{k+1}(x)\) and \(\sum \nolimits _{p=1}^{\infty }V_p(\tau _{k+1},x)\frac{(\tau _{k+1}-\tau _k)^p}{p!}\) are negative is guaranteed, so that (12) implies no restrictions on \({\mathcal {E}}\). However if one of the dynamics is unstable, the dwell-time is restricted. For example, if the discrete dynamics is stable but the continuous one is not, we have \(W_{k+1}(x)<0\), but the second summand in (12) can be positive, and hence it needs to be small enough in order to satisfy (12). This implies that the time distances \(\tau _{k+1}-\tau _k\) need to be small enough. Moreover, (12) allows the situation, where both dynamics types are not stable. This will be illustrated in our examples.

Let us fix any \((t_0,x_0)\in \mathbb R\times X\) as the initial data and assume without loss of generality that \(t_0\le \tau _0\). For any disturbances \((d_1,d_2)\in {\mathcal {U}}_1\times {\mathcal {U}}_2\), the corresponding solution to \(\varSigma \) will be denoted by \(x(t)=\phi (t,t_0,x_0,d_1,d_2)\).

Lemma 1

Under the conditions of Theorem 1 for any \(k\in \mathbb Z_+\), \(n\ge 2\) the following inequality holds true:

$$\begin{aligned} \begin{aligned} V_0(\tau _k+0,x(\tau _k+0))&\ge V_0(\tau _{k+1},x(\tau _{k+1}))-\sum \limits _{p=1}^{n-1}V_p(\tau _{k+1},x(\tau _{k+1}))\frac{(\tau _{k+1}-\tau _k)^p}{p!}\\&\quad -\int \limits _{\tau _k}^{\tau _{k+1}}\int \limits _{s_1}^{\tau _{k+1}}\dots \int \limits _{s_{n-2}}^{\tau _{k+1}}V_n(s_{n-1},x(s_{n-1}))\,ds_0\dots ds_{n-1}\\&\quad - \sum \limits _{p=1}^n\frac{\eta _{p-1}(\Vert d_1\Vert _{{\mathcal {U}}_1})\theta ^p}{p!}. \end{aligned} \end{aligned}$$
(13)

The proof can be found in Appendix.

Corollary 1

Under the conditions of Theorem 1, the following estimate is true:

$$\begin{aligned} \begin{aligned}&V_0(\tau _{k+1},x(\tau _{k+1}))-V_0(\tau _k+0,x(\tau _k+0))\\&\quad \le \sum \limits _{p=1}^{\infty }V_p(\tau _{k+1},x(\tau _{k+1}))\frac{(\tau _{k+1}-\tau _k)^p}{p!} +\sum \limits _{p=1}^{\infty }\frac{\eta _{p-1}(\Vert d_1\Vert _{{\mathcal {U}}_1})\theta ^p}{p!}. \end{aligned} \end{aligned}$$
(14)

Proof

The estimate follows after taking the limit for \(n\rightarrow \infty \) in the inequality (13), which is possible under the conditions of Theorem 1. \(\square \)

Proof

(of Theorem 1) The combination of (11) and (14) implies that

$$\begin{aligned} \begin{aligned}&V_0(\tau _{k+1}+0,x(\tau _{k+1}+0))-V_0(\tau _k+0,x(\tau _k+0))\\&\quad \le W_{k+1}(x(\tau _{k+1}))+\sum \limits _{p=1}^{\infty }V_p(\tau _{k+1},x(\tau _{k+1}))\frac{(\tau _{k+1}-\tau _k)^p}{p!} +\sum \limits _{p=1}^{\infty }\frac{\eta _{p-1}(\Vert d_1\Vert _{{\mathcal {U}}_1})\theta ^p}{p!}\\&\qquad +\eta (\Vert d_2\Vert _{{\mathcal {U}}_2}). \end{aligned} \end{aligned}$$

Let \({\widehat{\eta }}(s):=\sum \nolimits _{p=1}^{\infty }\frac{\eta _{p-1}(s)\theta ^p}{p!}+\eta (s)\), then \({\widehat{\eta }}\in {\mathcal {K}}_{\infty }\). Recall that \(d=\Vert d_1\Vert _{{\mathcal {U}}_1}+\Vert d_2\Vert _{{\mathcal {U}}_2}\), then for all \(k\in \mathbb Z_+\), we have

$$\begin{aligned} V_0(\tau _{k+1}+0,x(\tau _{k+1}+0))-V_0(\tau _k+0,x(\tau _k+0)) \le G_{k+1}(x(\tau _{k+1}))+{\widehat{\eta }}(d). \end{aligned}$$

From this inequality together with condition (4) of Theorem 1, it follows that for all \(k\in \mathbb Z_+\)

$$\begin{aligned} V_0(\tau _{k+1}+0,x(\tau _{k+1}+0))-V_0(\tau _k+0,x(\tau _k+0)) \le -\delta (\Vert x(\tau _{k+1})\Vert )+{\widehat{\eta }}(d).\qquad \end{aligned}$$
(15)

Let \(\epsilon \in (0,1)\), \(r=\delta ^{-1}(\frac{{\widehat{\eta }}(d)}{1-\epsilon })\). Let us show by contradiction that for some \(k^*\in \mathbb Z_+\), it holds that \(x(\tau _{k^*})\in B_r(0)\). Assume that this is not true, that is for all \(k\in \mathbb Z_+\), we have \(\Vert x(\tau _k)\Vert \ge r\), then from (15) follows:

$$\begin{aligned} \begin{aligned}&V_0(\tau _{k+1}+0,x(\tau _{k+1}+0))-V_0(\tau _k+0,x(\tau _k+0))\\&\quad \le -\delta (r)+{\widehat{\eta }}(d)=-\frac{{\widehat{\eta }}(d)}{1-\epsilon }+{\widehat{\eta }}(d)<0. \end{aligned} \end{aligned}$$

This means that the bounded from below sequence \(\{V_0(\tau _k+0,x(\tau _k+0))\}_{k\in \mathbb Z_+}\) is decreasing, hence there exists the limit \( m:=\lim \nolimits _{k\rightarrow \infty }V_0(\tau _k+0,x(\tau _k+0))\). From (15) follows then \(\lim \sup _{k\rightarrow \infty }\delta (\Vert x(\tau _{k+1})\Vert )\le {\widehat{\eta }}(d)\). This implies that

$$\begin{aligned} \begin{aligned} \frac{{\widehat{\eta }}(d)}{1-\epsilon }=\delta (r)\le \lim \sup _{k\rightarrow \infty }\delta (\Vert x(\tau _{k+1})\Vert )\le {\widehat{\eta }}(d), \end{aligned} \end{aligned}$$

which leads to a contradiction. Hence, for some \(k^*\), it holds that \(\Vert x(\tau _{k^*})\Vert <r\). Let \(R:=\max \{(\alpha _1^{-1}\circ \alpha _2)(\xi (r)+\chi (d)),r\}\). We show that for \(k\ge k^*\), the inequality \(\Vert x(\tau _k+0)\Vert \le R\) is true. Indeed, if for some \(m\ge k^*\)

$$\begin{aligned} \Vert x(\tau _m)\Vert <r,\quad \Vert x(\tau _{m+i})\Vert \ge r\quad \text {for all}\quad i=1,\dots ,j(m), \end{aligned}$$

where \(1\le j(m)\le \infty \), then from (15) follows that

$$\begin{aligned} \begin{aligned}&V_0(\tau _{m+i}+0,x(\tau _{m+i}+0))-V_0(\tau _{m+i-1}+0,x(\tau _{m+i-1}+0))\\&\quad \le -\delta (\Vert x(\tau _{m+i})\Vert )+{\widehat{\eta }}(d)<0; \end{aligned} \end{aligned}$$

hence, by condition (1) of Theorem 1, we obtain

$$\begin{aligned} \Vert x(\tau _{m+i}+0)\Vert \le (\alpha _1^{-1}\circ \alpha _2)(\Vert x(\tau _m+0)\Vert ). \end{aligned}$$

Taking (6) into account, we obtain \(\Vert x(\tau _m+0)\Vert \le \xi (\Vert x(\tau _m)\Vert )+\chi (d)\). Hence,

$$\begin{aligned} \begin{aligned} \Vert x(\tau _{m+i}+0)\Vert \le (\alpha _1^{-1}\circ \alpha _2)(\xi (r)+\chi (d))\le R. \end{aligned} \end{aligned}$$

Let \(S_r:=\{k\in \mathbb Z\,\,:\,\,\forall l,\,\,0\le l\le k\,\,\Vert x(\tau _l)\Vert \ge r\}\) and

$$\begin{aligned} \begin{aligned} N:={\left\{ \begin{array}{ll} \max S_r,\quad \text {for}\quad S_r\ne \emptyset ,\\ 0\quad \text {for}\quad S_r=\emptyset \end{array}\right. } \end{aligned} \end{aligned}$$

We need to consider the case \(N\ge 1\). Let \(k\in \mathbb N \) be such that \(1\le k\le N\), then \({\widehat{\eta }}(d)\le (1-\epsilon )\delta (\Vert x(\tau _k)\Vert )\) and from (15), it follows that

$$\begin{aligned} \begin{aligned}&V_0(\tau _{k}+0,x(\tau _{k}+0))-V_0(\tau _{k-1}+0,x(\tau _{k-1}+0))\\&\quad \le -\delta (\Vert x(\tau _{k})\Vert )+{\widehat{\eta }}(d)\le -\epsilon \delta (\Vert x(\tau _k)\Vert ). \end{aligned} \end{aligned}$$
(16)

The inequality (6) implies that

$$\begin{aligned} \begin{aligned} \Vert x(\tau _{k}+0)\Vert&\le \xi (\Vert x(\tau _k)\Vert )+\chi (d)\\&\le \xi (\Vert x(\tau _k)\Vert )+\chi ({\widehat{\eta }}^{-1}((1-\epsilon )\delta (\Vert x(\tau _k)\Vert ))):=\varphi (\Vert x(\tau _k)\Vert ). \end{aligned} \end{aligned}$$

It is easily seen that \(\varphi \in {\mathcal {K}}_{\infty }\), hence \(\Vert x(\tau _k)\Vert \ge \varphi ^{-1}(\Vert x(\tau _{k}+0)\Vert )\) and from (16) follows

$$\begin{aligned} \begin{aligned}&V_0(\tau _{k}+0,x(\tau _{k}+0))-V_0(\tau _{k-1}+0,x(\tau _{k-1}+0))\\&\quad \le -\epsilon (\delta \circ \varphi ^{-1})(\Vert x(\tau _{k}+0)\Vert )\\&\quad \le -\epsilon (\delta \circ \varphi ^{-1}\circ \alpha _2^{-1})(V_0(\tau _k+0,x(\tau _{k}+0))). \end{aligned} \end{aligned}$$
(17)

We denote \(\delta _1:=\delta \circ \varphi ^{-1}\circ \alpha _2^{-1}\in {\mathcal {K}}_{\infty }\), \(v_k:=V_0(\tau _{k}+0,x(\tau _{k}+0))\), and conclude that for all k, \(1\le k\le N\), the inequality (17) can be written as

$$\begin{aligned} v_k-v_{k-1}\le -\epsilon \delta _1(v_k). \end{aligned}$$
(18)

Let us define the sequence \({\widehat{v}}_k\) for \(k\in \mathbb N\) by \({\widehat{v}}_k:=v_k\) for \(1\le k\le N\) and so that for \(k\ge N+1\) the \({\widehat{v}}_k\) satisfies the difference equation \( {\widehat{v}}_k-{\widehat{v}}_{k-1}= -\epsilon \delta _1({\widehat{v}}_k). \)

Hence for all \(k\in \mathbb N\), the sequence \(\{{\widehat{v}}_k\}_{k\in \mathbb N}\) satisfies the inequality

$$\begin{aligned} {\widehat{v}}_k-{\widehat{v}}_{k-1}\le -\epsilon \delta _1({\widehat{v}}_k). \end{aligned}$$
(19)

Together with the inequality (19), we consider the comparison equation

$$\begin{aligned} w_k-w_{k-1}=-\epsilon \delta _1(w_k),\quad w_0={\widehat{v}}_0. \end{aligned}$$
(20)

First, let us show that for all \(k\in \mathbb Z_+\), the inequality \({\widehat{v}}_k\le w_k\) is true. Indeed, if for some \(k_1\in \mathbb N\) \({\widehat{v}}_{k_1-1}\le w_{k_1-1}\) and \({\widehat{v}}_{k_1}> w_{k_1}\), then from \({\text {id}}\,+\epsilon {\delta }_1\in {\mathcal {K}}_{\infty }\), it follows that

$$\begin{aligned} 0<{\widehat{v}}_{k_1}-w_{k_1}\le ({\text {id}}\,+\epsilon \delta _1)^{-1}({\widehat{v}}_{k_1-1})-({\text {id}}\,+\epsilon \delta _1)^{-1}(w_{k_1-1})\le 0, \end{aligned}$$

which is a contradiction. From (20) follows \(w_{k}=({\text {id}}\,+\epsilon \delta _1)^{-1}(w_{k-1})\), \(k\in \mathbb N\). By the properties of comparison functions, there exists \(\delta _2\in {\mathcal {K}}_{\infty }\) such that \(({\text {id}}\,+\epsilon \delta _1)^{-1}={\text {id}}\,-\delta _2\), hence (20) can be written as

$$\begin{aligned} w_k-w_{k-1}=-\delta _2(w_{k-1}),\quad w_0={\widehat{v}}_0. \end{aligned}$$
(21)

Let \({\widehat{\delta }}_2(s):=\min \{s,\delta _2(s)\}\), then

$$\begin{aligned} w_k-w_{k-1}\le -{\widehat{\delta }}_2(w_{k-1}),\quad w_0={\widehat{v}}_0. \end{aligned}$$
(22)

We define \(\varDelta _{v_0}(s):=\int \limits _{s}^{v_0}\frac{d\tau }{\widehat{\delta _2}(\tau )}\), then \(\varDelta _{v_0}(s)\rightarrow +\infty \) for \(s\rightarrow 0+\). By the mean value theorem for some \(s^*\in (w_k,w_{k-1})\) from (22), we obtain

$$\begin{aligned} \varDelta _{v_0}(w_k)-\varDelta _{v_0}(w_{k-1})=\int \limits _{w_k}^{w_{k-1}}\frac{ds}{\widehat{\delta _2}(s)} =\frac{w_{k-1}-w_k}{{\widehat{\delta }}_2(s^*)}\ge \frac{{\widehat{\delta }}_2(w_{k-1})}{{\widehat{\delta }}_2(s^*)}\ge 1. \end{aligned}$$

Hence, \(\varDelta _{v_0}(w_k)\ge \varDelta _{v_0}(v_0)+k=k\), which implies the estimate

$$\begin{aligned} \begin{aligned} V(\tau _k+0,x(\tau _k+0))\le \varDelta ^{-1}_{v_0}(k)\quad \text {for all}\quad k\in \mathbb Z_+. \end{aligned} \end{aligned}$$

This means that for all k, \(0\le k\le N\), we can estimate \( \Vert x(\tau _k+0)\Vert \le (\alpha _1^{-1}\circ \varDelta ^{-1}_{v_0})(k). \) From (5), we have that for all \(k\in \mathbb Z_+\) follows

$$\begin{aligned} \Vert x(t)\Vert \le \xi _{\theta }(\Vert x(\tau _k+0)\Vert )+\chi _{\theta }(d)\quad \text {for all}\quad t\in (\tau _k,\tau _{k+1}] \end{aligned}$$

That is for all \(t\in (\tau _k,\tau _{k+1}]\), \(0\le k\le N-1\), the following inequality holds:

$$\begin{aligned} \Vert x(t)\Vert \le (\xi _{\theta }\circ \alpha _1^{-1}\circ \varDelta ^{-1}_{v_0})(k)+\chi _{\theta }(d). \end{aligned}$$
(23)

Let \(\vartheta _k:=(\xi _2\circ \alpha _1^{-1}\circ \varDelta ^{-1}_{v_0})(k)\), and we define the function

$$\begin{aligned} \widetilde{\beta }(v_0,t)=\vartheta _{k-1}+\frac{t-\tau _{k-1}}{\tau _k-\tau _{k-1}}(\vartheta _{k}-\vartheta _{k-1})\quad \text {for}\quad t\in (\tau _{k-1},\tau _k],\quad k\in \mathbb N. \end{aligned}$$

It is easily seen that \(\widetilde{\beta }\in {\mathcal {K}}{\mathcal {L}}\). From (23), we get for all \(t\in (\tau _0,\tau _N]\), the inequality

$$\begin{aligned} \Vert x(t)\Vert \le \widetilde{\beta }(V(\tau _0+0,x(\tau _0+0)),t)+\chi _{\theta }(d)\le \widetilde{\beta }(\alpha _2(\Vert x(\tau _0+0)\Vert ),t)+\chi _{\theta }(d). \end{aligned}$$
(24)

Applying the estimates

$$\begin{aligned} \begin{aligned} \Vert x(\tau _0+0)\Vert \le \xi (\Vert x(\tau _0)\Vert )+\chi (d)\quad \text {and}\quad \Vert x(\tau _0)\Vert \le \xi _{\tau _0-t_0}(\Vert x_0\Vert )+\chi _{\tau _0-t_0}(d) \end{aligned} \end{aligned}$$

and the inequality (1), we obtain that for some functions \(\widetilde{\xi }_{\tau _0-t_0}\), \(\widetilde{\chi }_{\tau _0-t_0}\in {\mathcal {K}}_{\infty }\)

$$\begin{aligned} \Vert x(\tau _0+0)\Vert \le \widetilde{\xi }_{\tau _0-t_0}(\Vert x_0\Vert )+\widetilde{\chi }_{\tau _0-t_0}(d) \end{aligned}$$

Applying again (1), we derive

$$\begin{aligned} \alpha _2(\Vert x(\tau _0+0)\Vert )\le (\alpha _2\circ 2\widetilde{\xi }_{\tau _0-t_0})(\Vert x_0\Vert )+(\alpha _2\circ 2\widetilde{\chi }_{\tau _0-t_0})(d). \end{aligned}$$

From (24) using again (1), we get that for all \(t\in (\tau _0,\tau _N]\), the inequality

$$\begin{aligned} \begin{aligned} \Vert x(t)\Vert&\le \widetilde{\beta }(2(\alpha _2\circ 2\widetilde{\xi }_{\tau _0-t_0})(\Vert x_0\Vert ),t)+\widetilde{\beta }(2(\alpha _2\circ 2\widetilde{\chi }_{\tau _0-t_0})(d),t)+\chi _{\theta }(d)\\&\le {\widehat{\beta }}_{t_0}(\Vert x_0\Vert ,t)+{\widehat{\chi }}_{t_0}(d). \end{aligned} \end{aligned}$$
(25)

holds, where we have denoted

$$\begin{aligned} \begin{aligned}&{\widehat{\beta }}_{t_0}(s,t):=\widetilde{\beta }(2(\alpha _2\circ 2\widetilde{\xi }_{\tau _0-t_0})(s),t),\\&{\widehat{\chi }}_{t_0}(d)=\widetilde{\beta }(2(\alpha _2\circ 2\widetilde{\chi }_{\tau _0-t_0})(d),\tau _0+0)+\chi _{\theta }(d) \end{aligned} \end{aligned}$$

It easy to check by definition that \({\widehat{\beta }}_{t_0}\in {\mathcal {K}}{\mathcal {L}}\), \({\widehat{\chi }}_{t_0}\in {\mathcal {K}}_{\infty }\). There exists a function \(\beta _{t_0}\in {\mathcal {K}}{\mathcal {L}}\) such that \(\beta _{t_0}(s,t)\ge {\widehat{\beta }}_{t_0}(s,t)\) for \(t\in (\tau _0,\tau _N]\) and \(\beta _{t_0}(s,t)\ge \widetilde{\xi }_{\tau _0-t_0}(s)\) for \(t\in [t_0,\tau _0]\). Hence, from (25), it follows that for all \(t\in [t_0,\tau _N]\)

$$\begin{aligned} \Vert x(t)\Vert \le \beta _{t_0}(\Vert x_0\Vert ,t)+{{\widehat{\chi }}}_{t_0}(d)+\chi _{\tau _0-t_0}(d). \end{aligned}$$
(26)

Recall that for \(k\ge N\), we have \(\Vert x(\tau _k+0)\Vert \le R\). Hence from the estimate \(\Vert x(t)\Vert \le \xi _{\theta }(\Vert x(\tau _k+0)\Vert )+\chi _{\theta }(d)\), \(t\in (\tau _k,\tau _{k+1}]\) it follows that

$$\begin{aligned} \begin{aligned} \Vert x(t)\Vert \le \xi _{\theta }(R)+\chi _{\theta }(d) :={\widehat{\chi }}(d),\quad t>\tau _N, \end{aligned} \end{aligned}$$
(27)

where \({\hat{\chi }}\in {\mathcal {K}}_{\infty }\) by definition. Combining the inequalities (26) and (27), we see that for some \(\gamma _{t_0}\in {\mathcal {K}}_{\infty }\), the following estimate holds

$$\begin{aligned} \begin{aligned} \Vert x(t)\Vert \le \beta _{t_0}(\Vert x_0\Vert ,t)+\gamma _{t_0}(d),\quad t\ge t_0, \end{aligned} \end{aligned}$$

which proves the theorem. \(\square \)

Remark 2

Condition (5) of Theorem 1 assures the convergence of three last terms in (14) for \(n\rightarrow \infty \). However, if \(V_{n}(t,x)\le 0\) for some \(n\ge 1\), then we can set \(V_p(t,x)\equiv 0\) for all \(p>n\) and this condition (5) can be dropped. Stability investigation in this case is essentially easier because we deal with a finite number of auxiliary functions instead of an infinite sequence. The class of systems, where this simplification is possible becomes wider due to the next result (see the difference in the sign before \({\dot{V}}_p\) in conditions (2) of the previous and the next theorem). Such simplification will be used in some of our examples later.

Theorem 2

Assume that for the system \(\varSigma \), there are \(V_i\in {\mathcal {V}}({\mathcal {T}}_0)\) such that

  1. (1)

    for some \(\alpha _1\), \(\alpha _2\in {\mathcal {K}}_{\infty }\) it holds that

    $$\begin{aligned} \alpha _1(\Vert x\Vert )\le V_0(t,x)\le \alpha _2(\Vert x\Vert ),\quad \text {for all}\quad (t,x)\in [t_0,+\infty )\times X, \end{aligned}$$
    (28)
  2. (2)

    there is a sequence \(\eta _p\in {\mathcal {K}}_{\infty }\), \(p\in \mathbb Z_+\) such that \(\forall \,(t,x,\zeta )\in {\mathcal {T}}_0\times X\times U_1\)

    $$\begin{aligned} {\dot{V}}_p(t,x,\zeta )\le V_{p+1}(t,x)+\eta _p(\Vert \zeta \Vert _{U_1}),\quad p\in \mathbb Z_+, \end{aligned}$$
    (29)
  3. (3)

    there are \(\eta \in {\mathcal {K}}_{\infty }\) and \(W_k\,:X\rightarrow \mathbb R\), \(k\in \mathbb Z_+\) so that \(\forall \,(k,x,\zeta )\in \mathbb Z_+\times X\times U_2\)

    $$\begin{aligned} V_0(\tau _k+0,g(x,\zeta ))-V_0(\tau _k,x)\le W_k(x)+\eta (\Vert \zeta \Vert _{U_2}), \end{aligned}$$
    (30)
  4. (4)

    there are \(Q_{kp}\,:\,X\rightarrow \mathbb R\) and \(\pi _p\in {\mathcal {K}}_{\infty }\), \((k,p)\in \mathbb Z_+\times \mathbb N\), \((x,\zeta )\in X\times U_2\) with

    $$\begin{aligned} V_p(\tau _k+0,g(x,\zeta ))-V_p(\tau _k,x)\le Q_{kp}(x)+\pi _p(\Vert \zeta \Vert _{U_2}), \end{aligned}$$
    (31)
  5. (5)

    there is \(\delta \in {\mathcal {K}}_{\infty }\) such that for all \((k,x)\in \mathbb Z_+\times X\), the next inequality holds

    $$\begin{aligned} G_{k}(x):=W_{k}(x)+\sum \limits _{p=1}^{\infty }(V_p(\tau _{k},x)+Q_{kp}(x))\frac{(\tau _{k+1}-\tau _k)^p}{p!}\le -\delta (\Vert x\Vert ), \end{aligned}$$
    (32)
  6. (6)

    For any \(\rho >0\), there exists \(q_{\rho }\in [0,1)\) such that \( \lim \nolimits _{p\rightarrow \infty }\frac{\eta _p(s)}{(p+1)\eta _{p-1}(s)}\le q_{\rho } \) and \( \lim \nolimits _{p\rightarrow \infty }\frac{\pi _p(s)}{(p+1)\pi _{p-1}(s)}\le q_{\rho } \) exist uniformly for \(s\in [0,\rho ]\), and for each \(k\in \mathbb Z_+\) exists \(\omega _k\in {\mathcal {K}}_{\infty }\) such that

    $$\begin{aligned} |V_p(s,x)|\le \omega _k(\Vert x\Vert ),\quad |Q_{kp}(x)|\le \omega _k(\Vert x\Vert ) \end{aligned}$$

    for all \((p,s,x)\in \mathbb Z_+\times (\tau _k,\tau _{k+1}]\times X\). Then, system \(\varSigma \) is ISS.

Lemma 2

Under the conditions of Theorem 2, we have for all \(k\in \mathbb Z_+\), \(n\ge 2\)

$$\begin{aligned} V_0(\tau _{k+1},x(\tau _{k+1}))\le & {} V_0(\tau _{k}+0,x(\tau _{k}+0))\nonumber \\&\quad +\sum \limits _{p=1}^{n-1}V_p(\tau _{k}+0,x(\tau _{k}+0))\frac{(\tau _{k+1}-\tau _k)^p}{p!}\nonumber \\&\quad +\int \limits _{\tau _k}^{\tau _{k+1}}\int \limits _{\tau _{k}}^{s_1}\dots \int \limits _{\tau _{k}}^{s_{n-2}}V_n(s_{n-1},x(s_{n-1}))\,ds_0\dots ds_{n-1}\nonumber \\&\quad + \sum \limits _{p=1}^n\frac{\eta _{p-1}(\Vert d\Vert _{{\mathcal {U}}_1})\theta ^p}{p!}. \end{aligned}$$
(33)

Corollary 2

Under the conditions of Theorem 2, the following holds:

$$\begin{aligned} \begin{aligned} V_0(\tau _{k+1},x(\tau _{k+1}))&\le V_0(\tau _{k}+0,x(\tau _{k}+0))\\&\quad +\sum \limits _{p=1}^{\infty }V_p(\tau _{k}+0,x(\tau _{k}+0))\frac{(\tau _{k+1}-\tau _k)^p}{p!}\\&\quad +\sum \limits _{p=1}^{\infty }\frac{\eta _{p-1}(\Vert d_1\Vert _{{\mathcal {U}}_1})\theta ^p}{p!}. \end{aligned} \end{aligned}$$
(34)

Proof

Condition (6) of Theorem 2 assures the possibility to take the limit for \(n\rightarrow \infty \) in (33) which implies the assertion. \(\square \)

Proof

(of Theorem 2) From (30) and (34) follows

$$\begin{aligned} \begin{aligned} V_0(\tau _{k+1},x(\tau _{k+1}))&\le V_0(\tau _{k},x(\tau _{k})) +W_k(x(\tau _{k}))\\&\quad +\sum \limits _{p=1}^{\infty }V_p(\tau _{k}+0,x(\tau _{k}+0))\frac{(\tau _{k+1}-\tau _k)^p}{p!}\\&\quad +\eta (\Vert d_2\Vert _{{\mathcal {U}}_2})+\sum \limits _{p=1}^{\infty }\frac{\eta _{p-1}(\Vert d_1\Vert _{{\mathcal {U}}_1})\theta ^p}{p!}. \end{aligned} \end{aligned}$$
(35)

Taking (31) into account, we obtain

$$\begin{aligned} V_0(\tau _{k+1},x(\tau _{k+1}))\le & {} V_0(\tau _{k},x(\tau _{k})) +W_k(x(\tau _{k}))\nonumber \\&+\sum \limits _{p=1}^{\infty }(V_p(\tau _k,x(\tau _k))+Q_{kp}(x(\tau _k)))\frac{(\tau _{k+1}-\tau _k)^p}{p!}\nonumber \\&+\eta (\Vert d_2\Vert _{{\mathcal {U}}_2})+\sum \limits _{p=1}^{\infty }\frac{(\pi _p(\Vert d_2\Vert _{{\mathcal {U}}_2})+\eta _{p-1}(\Vert d_1\Vert _{{\mathcal {U}}_1}))\theta ^p}{p!}.\qquad \end{aligned}$$
(36)

Let \( {\widehat{\eta }}(s):=\eta (s)+\sum \nolimits _{p=1}^{\infty }\frac{(\pi _p(s)+\eta _{p-1}(s))\theta ^p}{p!}. \) Obviously \({\widehat{\eta }}\in {\mathcal {K}}_{\infty }\) and from (32) and (36) follows (recall that \(d=\Vert d_1\Vert _{{\mathcal {U}}_1}+\Vert d_{2}\Vert _{{\mathcal {U}}_2}\))

$$\begin{aligned} V_0(\tau _{k+1},x(\tau _{k+1})) \le V_0(\tau _{k},x(\tau _{k}))-\delta (\Vert x(\tau _k)\Vert )+{\widehat{\eta }}(d). \end{aligned}$$
(37)

Let \(\epsilon \in (0,1)\), \(r=\delta ^{-1}(\frac{{\widehat{\eta }}(d)}{1-\epsilon })\). First we show by contradiction that there exists \(k^*\in \mathbb Z_+\) such that \(\Vert x(\tau _{k^*})\Vert <r\). Indeed, otherwise, for all \(k\in \mathbb Z_+\) \(\Vert x(\tau _k)\Vert \ge r\) and hence from (37), it follows that

$$\begin{aligned} \begin{aligned} V_0(\tau _{k+1},x(\tau _{k+1}))-V_0(\tau _{k},x(\tau _{k})) \le -\delta (r)+{\widehat{\eta }}(d)= -\frac{{\widehat{\eta }}(d)}{1-\epsilon }+{\widehat{\eta }}(d) <0. \end{aligned} \end{aligned}$$

This means that the sequence \(\{V_0(\tau _{k},x(\tau _{k}))\}_{k\in \mathbb Z_+}\) is strictly decreasing and is bounded from below, hence it possesses a nonnegative limit \(m^*=\lim _{k\rightarrow \infty }V_0(\tau _{k},x(\tau _{k}))\). From (37) follows

$$\begin{aligned} \begin{aligned} \frac{{\widehat{\eta }}(d)}{1-\epsilon }\le \delta (r)\le \lim \sup \limits _{k\rightarrow \infty }\delta (\Vert x(\tau _k)\Vert )\le {\widehat{\eta }}(d), \end{aligned} \end{aligned}$$

which leads to a contradiction.

We denote \(R:=\max \{(\alpha _1^{-1}\circ \alpha _2)(\xi _{\theta }(\xi (r)+\chi (d))+\chi _{\theta }(d)),r\}\). Let us show that for \(k\ge k^*\) the inequality \(\Vert x(\tau _k)\Vert \le R\) is true. Indeed, if for some \(m\ge k^*\)

$$\begin{aligned} \Vert x(\tau _m)\Vert <r,\quad \Vert x(\tau _{m+i})\Vert \ge r\quad \text {for all}\quad i=1,\dots ,j(m), \end{aligned}$$

where \(1\le j(m)\le \infty \), then from (37), we have that for \(i\ge 2\)

$$\begin{aligned} \begin{aligned} V_0(\tau _{m+i},x(\tau _{m+i}))-V_0(\tau _{m+i-1},x(\tau _{m+i-1})) \le -\delta (\Vert x(\tau _{m+i-1})\Vert )+{\widehat{\eta }}(d)<0, \end{aligned} \end{aligned}$$

and hence from (5) and by the condition (1) of Theorem 2, it follows that

$$\begin{aligned} \Vert x(\tau _{m+i})\Vert \le (\alpha _1^{-1}\circ \alpha _2)(\Vert x(\tau _{m+1})\Vert )<(\alpha _1^{-1}\circ \alpha _2)(\xi _{\theta }(\Vert x(\tau _m+0)\Vert )+\chi _{\theta }(d)). \end{aligned}$$

Taking (6) into account, we obtain

$$\begin{aligned} \Vert x(\tau _{m+i})\Vert \le (\alpha _1^{-1}\circ \alpha _2)(\xi _{\theta }(\xi (r)+\chi (d))+\chi _{\theta }(d))\le R. \end{aligned}$$

Similarly, for \(i=1\) we get

$$\begin{aligned} \begin{aligned} \Vert x(\tau _{m+1})\Vert&\le \xi _{\theta }(\Vert x(\tau _{m}+0)\Vert )+\chi _{\theta }(d)\le \xi _{\theta }(\xi (\Vert x(\tau _{m})\Vert )+\chi (d))+\chi _{\theta }(d)\\&\le \xi _{\theta }(\xi (r)+\chi (d))+\chi _{\theta }(d)\le R. \end{aligned} \end{aligned}$$

Let \(S_r:=\{k\in \mathbb Z_+\,\,:\,\,\forall l,\,\,0\le l\le k\,\,\Vert x(\tau _l)\Vert \ge r\}\) and

$$\begin{aligned} \begin{aligned} N={\left\{ \begin{array}{ll} \max S_r,\quad \text {for}\quad S_r\ne \emptyset ,\\ 0\quad \text {for}\quad S_r=\emptyset \end{array}\right. } \end{aligned} \end{aligned}$$

It is enough to consider the case \(N\ge 1\). Let k be such that \(0\le k\le N\), then \({\widehat{\eta }}(d)\le (1-\epsilon )\delta (\Vert x(\tau _k)\Vert )\), and from (37), the next inequality follows:

$$\begin{aligned} V_0(\tau _{k+1},x(\tau _{k+1})) \le V_0(\tau _{k},x(\tau _{k}))-\epsilon \delta (\Vert x(\tau _k)\Vert ). \end{aligned}$$
(38)

By condition (1) of Theorem 2, we obtain \(\Vert x(\tau _k)\Vert \ge \alpha _2^{-1}(V_0(\tau _{k},x(\tau _{k})))\) and hence (38) can be written as

$$\begin{aligned} V_0(\tau _{k+1},x(\tau _{k+1})) \le V_0(\tau _{k},x(\tau _{k}))-\epsilon \delta _1(V_0(\tau _{k},x(\tau _{k}))), \end{aligned}$$

where \(\delta _1=\delta \circ \alpha _2^{-1}\in {\mathcal {K}}_{\infty }\). Let us denote \(v_k:=V_0(\tau _{k},x(\tau _{k}))\), then for all \(k\in \mathbb Z_+\), we have the inequality \( v_{k+1}-v_k\le -\epsilon \delta _1(v_k). \) Let \({\widehat{\delta }}_1(s):=\min \{s,\delta _1(s)\}\in {\mathcal {K}}_{\infty }\), then

$$\begin{aligned} v_{k+1}-v_k\le -\epsilon {\widehat{\delta }}_1(v_k). \end{aligned}$$

We define the \(\{{\widehat{v}}_k\}_{k\in \mathbb Z_+}\) sequence recurrently setting \({\widehat{v}}_k:=v_k\) for \(0\le k\le N\) and

$$\begin{aligned} {\widehat{v}}_{k+1}-{\widehat{v}}_k=-\epsilon {\widehat{\delta }}_1({\widehat{v}}_k) \end{aligned}$$

for \(k\ge N\). From this definition follows that for all \(k\in \mathbb Z_+\)

$$\begin{aligned} {\widehat{v}}_{k+1}-{\widehat{v}}_k\le -\epsilon {\widehat{\delta }}_1({\widehat{v}}_k) \end{aligned}$$
(39)

and \(0\le {\widehat{v}}_k\le v_0\).

Let \(\varDelta _{v_0}(s):=\int \limits _s^{v_0}\frac{d\tau }{{\widehat{\delta }}_1(\tau )}\). By the mean value theorem \(\exists \,s^*\in ({\widehat{v}}_{k+1},{\widehat{v}}_{k})\) with

$$\begin{aligned} \varDelta _{v_0}({\widehat{v}}_{k+1})-\varDelta _{v_0}({\widehat{v}}_k)=\int \limits _{{\widehat{v}}_{k+1}}^{{\widehat{v}}_k}\frac{ds}{{\widehat{\delta }}_1(s)} =\frac{{\widehat{v}}_k-{\widehat{v}}_{k+1}}{{\widehat{\delta }}_1(s^*)}\ge \epsilon \frac{{\widehat{\delta }}_1({\widehat{v}}_k)}{{\widehat{\delta }}_1(s^*)} \ge \epsilon . \end{aligned}$$

Hence, \(\varDelta _{v_0}({\widehat{v}}_k)\ge k\epsilon \) for all \(k\in \mathbb Z_+\). From its definition, it follows that \(\varDelta _{v_0}(s)\) is strictly growing with respect to \(v_0\) for a fixed s and strictly decreasing in s for a fixed \(v_0\) with \(\varDelta _{v_0}(s)\rightarrow \infty \) for \(s\rightarrow 0+\). This implies that

$$\begin{aligned} V(\tau _k,x(\tau _k))\le \varDelta _{v_0}^{-1}(k\epsilon )\quad \text {for all}\quad k\in \mathbb Z_+. \end{aligned}$$

The desired ISS estimate follows similarly as in the proof of Theorem 1, which proves Theorem 2. \(\square \)

Remark 3

If for some \(n\ge 1\) we have \(V_{n}(t,x)\le 0\), then we can take \(V_p(t,x)\equiv 0\) for all \(p>n\), and in this case, condition (5) of Theorem 2 can be dropped.

5 Examples

5.1 Heat equation with variable coefficients

Let X be the normed linear vector space defined by

$$\begin{aligned} \begin{aligned} X=\{f\,\,:\,\,[0,l]\rightarrow \mathbb R\,|\,f\in C^1(0,l), f|_{(0,2\delta )}\in C^2(0,2\delta ),\quad f|_{(2\delta ,l)}\in C^2(2\delta ,l)\}, \end{aligned} \end{aligned}$$

with \(l>0\), \(0<2\delta <l\) and the norm \(\Vert \cdot \Vert _X:=\Vert \cdot \Vert _{L^2[0,l]}\).

Consider the following linear system with impulsive actions

$$\begin{aligned} \begin{aligned} u_t(z,t)&=a^2u_{zz}(z,t)+b(z)u(z,t),\quad t\notin {\mathcal {E}}\\ u(z,t^+)&=u(z,t)+c(z)u(z,t),\quad t\in {\mathcal {E}} \end{aligned} \end{aligned}$$
(40)

with the following initial and boundary conditions

$$\begin{aligned} \begin{aligned} u(0,t)&=u(l,t)=0,\\ u(z,0+0)&=\varphi (z),\quad \varphi \in X,\quad \varphi (0)=\varphi (l)=0, \end{aligned} \end{aligned}$$
(41)

where \((z,t)\in [0,l]\times [0,\infty )\), \(u\,\,:\,\,[0,l]\times [0,\infty )\rightarrow \mathbb R\). We assume that \(a>0\), \(b\,:[0,l]\rightarrow \mathbb R\) is piecewise continuous bounded function which can be discontinuous only at \(z=2\delta \), and \(c\,:[0,l]\rightarrow \mathbb R\), \(\,c\in C^2[0,l]\).

We are interested in the properties of classical solutions to (40)–(41) defined next. Let us denote \({\mathcal {T}}_T=[0,T]\cap {\mathcal {T}}_0\), \({\mathcal {E}}_T=[0,T]\cap {\mathcal {E}}\) for \(T>0\).

Definition 6

A function \(u\,:[0,l]\times [0,T]\rightarrow \mathbb R\) is called classical solution to (40)-(41), if

$$\begin{aligned} \begin{aligned} u_t(z,t)=a^2u_{zz}(z,t)+b(z)u(z,t),\quad (z,t)\in ((0,2\delta )\cup (2\delta ,l))\times {\mathcal {T}}_T; \end{aligned} \end{aligned}$$

For \(z=2\delta \), it holds that

$$\begin{aligned} \begin{aligned} u(2\delta +0,t)&=u(2\delta -0,t),\quad u_z(2\delta +0,t)=u_z(2\delta -0,t),\quad t\in \mathbb R_+;\\ u(z,t^+)&=u(z,t)+c(z)u(z,t),\quad (z,t)\in [0,l]\times {\mathcal {E}}_T; \end{aligned} \end{aligned}$$

and u satisfies the initial and boundary conditions (41).

Since \(c\in C^2[0,l]\), it follows that the state space X is invariant under jumps, that is, \(f\in X\) implies \((1+c)f\in X\). It is also clear that the jump operator is consistent with the boundary conditions. Theorem 1 from [15] implies the existence and uniqueness of solutions to (40)—(41) so that \(u(\cdot ,t)\in X\) for \(t\ge 0\).

We consider the question of asymptotic stability of solutions to (40)–(41) with respect to the norm in \(L^2[0,l]\). The next proposition verifies the estimates (5)–(6).

Proposition 1

Solutions of (40)—(41) satisfy the estimates (5)–(6) with the following choice of functions

$$\begin{aligned} \xi _{\tau }(s)=e^{\tau b_{\max }}s,\quad \xi (s)=(1+c_{\max })s,\quad \chi _{\tau }(s)=0\quad \chi (s)=0, \end{aligned}$$

where \(b_{\max }=\sup \nolimits _{z\in [0,l]}|b(z)|\), \(c_{\max }=\sup \nolimits _{z\in [0,l]}|c(z)|\).

The proof can be found in Appendix.

To study stability properties, we introduce the following Lyapunov function:

$$\begin{aligned} V_0(u(\cdot ,t))=\int \limits _0^lu^2(z,t)\,dz, \end{aligned}$$

for which we have

$$\begin{aligned} \begin{aligned} {\dot{V}}_0(u(\cdot ,t))&=2\int \limits _0^lu(z,t)u_t(z,t)\,dz=2\int \limits _0^lu(z,t) (a^2u_{zz}(z,t)+b(z)u(z,t))\,dz\\&=-2a^2\int \limits _0^lu_z^2(z,t)\,dz+2\int \limits _0^lb(z)u^2(z,t)\,dz:=V_1(u(\cdot ,t)). \end{aligned} \end{aligned}$$
(42)

Let us calculate the full time derivative of \(V_1(u(\cdot ,t))\) along solutions to (40)–(41):

$$\begin{aligned} {\dot{V}}_1(u(\cdot ,t))= & {} -4a^2\int \limits _0^lu_z(z,t)u_{zt}(z,t)\,dz+4\int \limits _0^lb(z)u(z,t)u_t(z,t)\,dz \\= & {} 4a^2\int \limits _0^lu_{zz}(z,t)(a^2u_{zz}(z,t)+b(z)u(z,t))\,dz\\&+4\int \limits _0^lb(z)u(z,t)(a^2u_{zz}(z,t)+b(z)u(z,t))\,dz \\= & {} 4\int \limits _0^l(a^2u_{zz}(z,t)+b(z)u(z,t))^2\,dz\ge 0. \end{aligned}$$

Hence, we can apply Theorem 1 taking \(V_2=0\). From (40) for \(t\in {\mathcal {E}}\), we obtain

$$\begin{aligned} \begin{aligned}&V_0((1+c(z))u(\cdot ,t))-V_0(u(\cdot ,t))\\&\quad \le \int \limits _0^l(2c(z)+c^2(z))u^2(z,t)\,dz =W(u(\cdot ,t)). \end{aligned} \end{aligned}$$
(43)

With help of Theorem 1, we arrive to the following conditions for the global asymptotic stability

$$\begin{aligned} \begin{aligned} G(u(\cdot ,z))=V_1(u(\cdot ,z))T_k+W(u(\cdot ,t))<-\varepsilon _0\Vert u(\cdot ,t)\Vert ^2_{L^2[0,l]} \end{aligned} \end{aligned}$$
(44)

for some \(\varepsilon _0>0\), where \(T_k=\tau _{k+1}-\tau _k\) denotes the dwell-time between two impulsive actions. By means of the Friedrich’s inequality, we arrive to the following condition guaranteeing the GAS property of the system (40)–(41)

$$\begin{aligned} \begin{aligned} \sup \limits _{k\in \mathbb Z_+}\max \limits _{z\in [0,l]} \Big (T_k\big (b(z)-\frac{\pi ^2a^2}{l^2}\big )+c(z)+\frac{1}{2}c^2(z)\Big )<0. \end{aligned} \end{aligned}$$
(45)

For illustration, we apply this condition to the following particular case of (40)-(41): \(a=1\), \(l=\pi \) and

$$\begin{aligned} b(z)= & {} {\left\{ \begin{array}{ll} 1-2\varepsilon , z\in [0,2\delta ],\\ 1+\varepsilon , z\in (2\delta ,\pi ], \end{array}\right. }\\ c(z)= & {} {\left\{ \begin{array}{ll} \varepsilon ,\quad z\in [0,\delta ],\\ \varepsilon (-18(\frac{z}{\delta })^5+135(\frac{z}{\delta })^4-390(\frac{z}{\delta })^3+540(\frac{z}{\delta })^2-360\frac{z}{\delta }+94), z\in (\delta ,2\delta ],\\ -2\varepsilon ,\quad z\in (2\delta ,\pi ], \end{array}\right. } \end{aligned}$$

where \(\varepsilon >0\), \(\delta <\pi /2\), \(c\in C^2\), \(-2\varepsilon \le c(z)\le \varepsilon \) . We will see that for some choice of parameters \(\varepsilon ,\delta \) both continuous and discrete dynamics have unstable behavior.

In this case, the dwell-time condition (45) reduced to the following two inequalities:

$$\begin{aligned} \begin{aligned} -2\varepsilon T_k+\varepsilon +\frac{1}{2}\varepsilon ^2<0,\quad T_k\varepsilon -2\varepsilon +2\varepsilon ^2<0 \end{aligned} \end{aligned}$$

or equivalently to the following explicit condition applied on the dwell-times

$$\begin{aligned} \begin{aligned} \varepsilon \in (0,2/3),\quad 0.5+0.25\varepsilon<\inf \limits _{k\in \mathbb Z_+}T_k\le \sup \limits _{k\in \mathbb Z_+}T_k<2-2\varepsilon . \end{aligned} \end{aligned}$$
(46)

Let us consider continuous and discrete dynamics of (40)–(41) separately.

First, we consider the stability properties of the differential equation

$$\begin{aligned} \begin{aligned} u_t(z,t)=u_{zz}(z,t)+b(z)u(z,t), \end{aligned} \end{aligned}$$
(47)

with the following boundary and initial conditions

$$\begin{aligned} \begin{aligned} u(0,t)=u(\pi ,t)=0,\quad t\in \mathbb R_+,\quad u(z,0+0)=\varphi (z),\quad \varphi \in X. \end{aligned} \end{aligned}$$
(48)

The corresponding self-adjoint spectral problem is

$$\begin{aligned} \begin{aligned} \psi ^{\prime \prime }(z)+b(z)\psi (z)=\lambda \psi (z),\quad \psi (0)=\psi (\pi )=0. \end{aligned} \end{aligned}$$
(49)

We can show that at least for \(\varepsilon \) small enough there exists some critical \(\delta ^*\), (\(\delta ^*\approx 0.651331\)) such that for all \(\delta <\delta ^*\) the linear system (47)–(48) is not stable, and for \(\delta >\delta ^*\), this system is asymptotically stable.

This follows immediately from the next proposition proved in Appendix:

Proposition 2

The largest eigenvalue \(\lambda _{\max }(\varepsilon )\) of the spectral equation (49) can be represented asymptotically as

$$\begin{aligned} \begin{aligned} \lambda _{\max }(\varepsilon )=\frac{2}{\pi }\Big (\frac{3}{4}\sin (4\delta )+\frac{\pi }{2}-3\delta \Big )\varepsilon +O(\varepsilon ^2),\quad \varepsilon \rightarrow 0+. \end{aligned} \end{aligned}$$

Indeed, the function \(f(\delta )=\frac{3}{4}\sin (4\delta )+\frac{\pi }{2}-3\delta \) is decreasing on \([0,\pi /2]\), and we have \(f(0)f(\pi /2)<0\), hence the equation \(f(\delta )=0\) has a unique solution in \([0,\pi /2]\), which we denote by \(\delta ^*\). Then for \(\delta <\delta ^*\) for sufficiently small \(\varepsilon \), we have \(\lambda _{\max }(\varepsilon )>0\), and for \(\delta >\delta ^*\), we have \(\lambda _{\max }(\varepsilon )<0\).

This mens that for sufficiently small \(\varepsilon \), the parameter \(\delta \) can be chosen so that the partial differential equation (47)–(48), which describes the continuous dynamics of the original impulsive system (40)–(41), is unstable.

We consider the difference equation in the state space X, that describes the discrete dynamics of the original impulsive system (40)—(41), and demonstrate its unstable behavior. Let \(C\psi :=(1+c(z))\psi \) for \(\psi \in X\), then \(C\in {\mathfrak {L}}(X)\) and \( \Vert C^n\Vert =\sup \nolimits _{\Vert \psi \Vert _{L^2[0,\pi ]}=1}\Vert (1+c(z))^n\psi \Vert _{L^2[0,l]} \ge \sup \nolimits _{\Vert \psi \Vert _{L^2[0,\pi ]}=1,{\text {supp}}\,\psi \subset [0,\delta ]}\Vert (1+\varepsilon )^n\psi \Vert _{L^2[0,l]}=(1+\varepsilon )^n, \) which implies that \(r_{\sigma }(C)\ge 1+\varepsilon >1\), showing the instability property.

Remark 4

Let us note that in case of an unstable scalar ODE of the first order subjected to destabilizing impulsive actions, the overall dynamics of the whole impulsive system is always unstable. In contrary to this, our example shows that for the considered unstable PDE with destabilizing impulses the overall dynamics is stable under certain condition on dwell times.

5.2 Nonlinear impulsive ODE

Consider the following nonlinear ODE system with impulsive actions

$$\begin{aligned} \begin{aligned} \dot{x}(t)&=\frac{f(x(t))+d_1(t)}{\sqrt{1+\Vert x(t)\Vert ^{2m}}},\quad t\ne k\theta ,\\ x(t^+)&=x(t)+\frac{g(x(t))+d_2(k)}{\sqrt{1+\Vert x(t)\Vert ^{2m}}},\quad t=k\theta , \end{aligned} \end{aligned}$$
(50)

where \(x\in \mathbb R^n\), \(f,g\,:\mathbb R^n\rightarrow \mathbb R^n\) are homogeneous functions of the order \(m>1\) with odd m, that is, \(f(\lambda x)=\lambda ^m f(x)\), \(g(\lambda x)=\lambda ^m g(x)\) for all \((\lambda ,x)\in \mathbb R^{n+1}\), \(f\in C^1(\mathbb R^n)\). Let \(U_1=U_2=\mathbb R^n\), \({\mathcal {U}}_1=L^{\infty }(\mathbb R_+,\mathbb R^n)\), \({\mathcal {U}}_2=L^{\infty }(\mathbb Z_+,\mathbb R^n)=l^{\infty }(\mathbb R^n)\).

Let there exist a positive definite matrix P such that for all \(x\in \mathbb R^n\) such that the function f, g satisfies the inequalities

$$\begin{aligned} \Vert g(x)\Vert&\le b_0\Vert x\Vert ^m,\quad x^{{\text {T}}\,}P(\theta f(x)+g(x))\le -c_0\Vert x\Vert ^{m+1},\nonumber \\&\quad \times (Pf(x)+\partial _xf^{{\text {T}}\,}(x)Px)^{{\text {T}}\,}f(x)\ge \nu _0\Vert x\Vert ^{2m},\nonumber \\&\quad \times \Vert x\Vert ^2(Pf(x)+\partial _xf^{{\text {T}}\,}(x)Px)^{{\text {T}}\,}f(x)-mx^{{\text {T}}\,}Pf(x)x^{{\text {T}}\,}f(x)\ge \nu _1\Vert x\Vert ^{2m+2} \end{aligned}$$
(51)

for some positive constants \(b_0\), \(c_0\), \(\nu _0\) and \(\nu _1\). For short, we denote

$$\begin{aligned} \begin{aligned} a_0:=\sup \limits _{\Vert x\Vert =1}\Vert f(x)\Vert ,\quad a_1:=\sup \limits _{\Vert x\Vert =1}\Vert \partial _xf(x)\Vert . \end{aligned} \end{aligned}$$

Proposition 3

Solutions of the system (50) satisfy the estimates (5)—(6) with functions

$$\begin{aligned} \begin{aligned} \xi _{\tau }(s)=\omega _1(\tau )s,\quad \chi _{\tau }(s)=\omega _2(\tau )s^{(m+1)/2m},\quad \xi (s)=(1+b_0)s,\quad \chi (s)=s, \end{aligned} \end{aligned}$$

where \(\omega _i\), \(i=1,2\) are certain positive functions (provided explicitly in the proof).

The proof can be found in Appendix.

Stability properties will be studied with help of the following Lyapunov function \(V_0(x)=x^{{\text {T}}\,}Px\), \(P\succ 0\). Its time derivative along solutions to (50) for \(t\ne k\theta \) is

$$\begin{aligned} \begin{aligned} {\dot{V}}_0(x,d_1)=\frac{2x^{{\text {T}}\,}Pf(x)+2x^{{\text {T}}\,}Pd_1}{\sqrt{1+\Vert x\Vert ^{2m}}}. \end{aligned} \end{aligned}$$
(52)

By means of the Young’s inequality (2), we have

$$\begin{aligned} \begin{aligned} 2|x^{{\text {T}}\,}Pd_1|\le 2\Vert P\Vert \Vert x\Vert \Vert d_1\Vert \le 2\Vert P\Vert \left( \frac{\tau ^{m+1}}{m+1}\Vert x\Vert ^{m+1}+\frac{m\tau ^{-(m+1)/m}}{m+1}\Vert d_1\Vert ^{1+1/m}\right) \end{aligned} \end{aligned}$$

for any \(\tau >0\). Hence from (52), we obtain

$$\begin{aligned} {\dot{V}}_0(x,d_1)\le \frac{2x^{{\text {T}}\,}Pf(x)+2\Vert P\Vert \frac{\tau ^{m+1}}{m+1}\Vert x\Vert ^{m+1}}{\sqrt{1+\Vert x\Vert ^{2m}}} +2\Vert P\Vert \frac{m\tau ^{-(m+1)/m}}{m+1}\Vert d_1\Vert ^{1+1/m}. \end{aligned}$$
(53)

We chose

$$\begin{aligned} \begin{aligned} V_1(x):=\frac{2x^{{\text {T}}\,}Pf(x)+2\Vert P\Vert \frac{\tau ^{m+1}}{m+1}\Vert x\Vert ^{m+1}}{\sqrt{1+\Vert x\Vert ^{2m}}},\quad \eta _0(s):=2\Vert P\Vert \frac{m\tau ^{-(m+1)/m}}{m+1}s^{1+1/m}.\nonumber \end{aligned}\\ \end{aligned}$$
(54)

On the jumps due to impulsive actions from the definition of \(V_0\), we calculate

$$\begin{aligned} \begin{aligned}&V_0\Big (x+\frac{g(x)+d_2}{\sqrt{1+\Vert x\Vert ^{2m}}}\Big )-V_0(x)= \frac{2x^{{\text {T}}\,}Pg(x)}{\sqrt{1+\Vert x\Vert ^{2m}}}+\frac{g^{{\text {T}}\,}(x)Pg(x)}{1+\Vert x\Vert ^{2m}}\\&\quad +\frac{2x^{{\text {T}}\,}Pd_2}{\sqrt{1+\Vert x\Vert ^{2m}}}+\frac{2g^{{\text {T}}\,}(x)Pd_2+d_2^{{\text {T}}\,}Pd_2}{1+\Vert x\Vert ^{2m}}. \end{aligned} \end{aligned}$$
(55)

Again by the Young’s inequality (2), we obtain

$$\begin{aligned} \begin{aligned} 2|x^{{\text {T}}\,}Pd_2|&\le 2\Vert P\Vert \left( \frac{\tau ^{m+1}}{m+1}\Vert x\Vert ^{m+1}+\frac{m\tau ^{-(m+1)/m}}{m+1}\Vert d_2\Vert ^{1+1/m}\right) ,\\ 2|g^{{\text {T}}\,}(x)Pd_2|&\le 2\Vert P\Vert b_0\Vert x\Vert ^m\Vert d_2\Vert \le \Vert P\Vert b_0\Big (\tau \Vert x\Vert ^{2m}+\tau ^{-1}\Vert d_2\Vert ^2\Big ). \end{aligned} \end{aligned}$$

From (51), it follows that \(|g^{{\text {T}}\,}(x)Pg(x)|\le \Vert P\Vert b_0^2\Vert x\Vert ^{2m}\). Hence from (55), the following estimate follows:

$$\begin{aligned} \begin{aligned}&V_0\Big (x+\frac{g(x)+d_2}{\sqrt{1+\Vert x\Vert ^{2m}}}\Big )-V_0(x) \le \frac{2x^{{\text {T}}\,}Pg(x)+2\Vert P\Vert \frac{\tau ^{m+1}}{m+1}\Vert x\Vert ^{m+1}}{\sqrt{1+\Vert x\Vert ^{2m}}}\\&\quad +(b_0+\tau )\Vert P\Vert b_0\frac{\Vert x\Vert ^{2m}}{1+\Vert x\Vert ^{2m}} +\frac{2\Vert P\Vert m\tau ^{-(m+1)/m}\Vert d_2\Vert ^{1+1/m}}{(m+1)\sqrt{1+\Vert x\Vert ^{2m}}}\\&\quad +\frac{b_0\tau ^{-1}+1}{1+\Vert x\Vert ^{2m}}\Vert P\Vert \Vert d_2\Vert ^2\\&\quad \le W(x)+\eta (\Vert d_2\Vert ), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} W(x)= & {} \frac{2x^{{\text {T}}\,}Pg(x)+2\Vert P\Vert \frac{\tau ^{m+1}}{m+1}\Vert x\Vert ^{m+1}}{\sqrt{1+\Vert x\Vert ^{2m}}} +(b_0+\tau )\Vert P\Vert b_0\frac{\Vert x\Vert ^{2m}}{1+\Vert x\Vert ^{2m}},\nonumber \\ \eta (s)= & {} 2\Vert P\Vert \frac{m}{m+1}\tau ^{-(m+1)/m}s^{1+1/m}+(b_0\tau ^{-1}+1)\Vert P\Vert s^2. \end{aligned}$$
(56)

Now, consider \({\dot{V}}_1(x,d_1)\) for \(t\ne k\theta \)

$$\begin{aligned} \begin{aligned} {\dot{V}}_1(x,d_1)&=\frac{(2f^{{\text {T}}\,}(x)P+2x^{{\text {T}}\,}P\partial _xf(x)+2\tau ^{m+1}\Vert P\Vert \Vert x\Vert ^{m-1}x^{{\text {T}}\,})(f(x)+d_1)}{1+\Vert x\Vert ^{2m}}\\&\quad -2m\left( x^{{\text {T}}\,}Pf(x)+\Vert P\Vert \frac{\tau ^{m+1}}{m+1}\Vert x\Vert ^{m+1}\right) \frac{\Vert x\Vert ^{2m-2}x^{{\text {T}}\,}(f(x)+d_1)}{(1+\Vert x\Vert ^{2m})^2}\\&=\frac{2(Pf(x)+\partial _xf^{{\text {T}}\,}(x)Px)^{{\text {T}}\,}f(x)}{1+\Vert x\Vert ^{2m}}- \frac{2mx^{{\text {T}}\,}Pf(x)\Vert x\Vert ^{2m-2}x^{{\text {T}}\,}f(x)}{(1+\Vert x\Vert ^{2m})^2}\\&\quad +\frac{2\tau ^{m+1}\Vert P\Vert \Vert x\Vert ^{m-1}x^{{\text {T}}\,}f(x)}{1+\Vert x\Vert ^{2m}} -2m\Vert P\Vert \frac{\tau ^{m+1}}{m+1}\Vert x\Vert ^{3m-1}\frac{x^{{\text {T}}\,}f(x)}{(1+\Vert x\Vert ^{2m})^2}\\&\quad +\frac{(2f^{{\text {T}}\,}(x)P+2x^{{\text {T}}\,}P\partial _xf(x)+2\tau ^{m+1}\Vert P\Vert \Vert x\Vert ^{m-1}x^{{\text {T}}\,})d_1}{1+\Vert x\Vert ^{2m}}\\&\quad -2m\left( x^{{\text {T}}\,}Pf(x)+\Vert P\Vert \frac{\tau ^{m+1}}{m+1}\Vert x\Vert ^{m+1}\right) \frac{\Vert x\Vert ^{2m-2}x^{{\text {T}}\,}d_1}{(1+\Vert x\Vert ^{2m})^2}. \end{aligned} \end{aligned}$$

By means of the Cauchy inequality and from (51), it follows that

$$\begin{aligned} \begin{aligned} |x^{{\text {T}}\,}f(x)|&\le a_0\Vert x\Vert ^{m+1},\Vert f^{{\text {T}}\,}(x)P+x^{{\text {T}}\,}P\partial _xf(x)\\&\quad +\tau ^{m+1}\Vert P\Vert \Vert x\Vert ^{m-1}x^{{\text {T}}\,}\Vert \le \Vert P\Vert (a_0+a_1+\tau ^{m+1})\Vert x\Vert ^m. \end{aligned} \end{aligned}$$

Hence,

$$\begin{aligned} \begin{aligned} {\dot{V}}_1(x,d_1)&\ge \frac{2\nu _0\Vert x\Vert ^{2m}+2\nu _1\Vert x\Vert ^{4m}}{(1+\Vert x\Vert ^{2m})^2}\\&\quad -\frac{2\Vert P\Vert a_0\tau ^{m+1}\Vert x\Vert ^{2m}}{1+\Vert x\Vert ^{2m}}-\frac{2ma_0\tau ^{m+1}\Vert P\Vert \Vert x\Vert ^{4m}}{(m+1)(1+\Vert x\Vert ^{2m})^2}\\&\quad -\frac{2\Vert P\Vert (a_0+a_1+\tau ^{m+1})\Vert x\Vert ^m\Vert d_1\Vert }{1+\Vert x\Vert ^{2m}}-\frac{2m\Vert P\Vert (a_0 +\frac{\tau ^{m+1}}{m+1})\Vert x\Vert ^{3m}\Vert d_1\Vert }{(1+\Vert x\Vert ^{2m})^2}. \end{aligned} \end{aligned}$$

Now, we apply the Young’s inequality with the parameter \(\tau >0\)

$$\begin{aligned} \begin{aligned} \Vert x\Vert ^m\Vert d_1\Vert&\le \frac{\tau }{2}\Vert x\Vert ^{2m}+\frac{\tau ^{-1}}{2}\Vert d_1\Vert ^2,\\ \Vert x\Vert ^{3m}\Vert d_1\Vert&\le \frac{3\tau ^{4/3}}{4}\Vert x\Vert ^{4m}+\frac{\tau ^{-4}}{4}\Vert d_1\Vert ^4 \end{aligned} \end{aligned}$$

and obtain

$$\begin{aligned} \begin{aligned} {\dot{V}}_1(x,d_1)&\ge \frac{(2\nu _0-\tau \Vert P\Vert (a_0+a_1+\tau ^{m+1})-2a_0\Vert P\Vert \tau ^{m+1})\Vert x\Vert ^{2m}}{(1+\Vert x\Vert ^{2m})^2}\\&\quad + \frac{(2\nu _1-\frac{3}{2}m\tau ^{4/3}\Vert P\Vert (a_0+\frac{\tau ^{m+1}}{m+1})-\frac{4m+2}{m+1}a_0\tau ^{m+1}\Vert P\Vert -\tau \Vert P\Vert (a_0+a_1+\tau ^{m+1}))\Vert x\Vert ^{4m}}{(1+\Vert x\Vert ^{2m})^2}\\&\quad -\tau ^{-1}\Vert P\Vert (a_0+a_1+\tau ^{m+1})\Vert d_1\Vert ^2-\frac{m\tau ^{-4}\Vert P\Vert (a_0+\frac{\tau ^{m+1}}{m+1})}{2}\Vert d_1\Vert ^4. \end{aligned} \end{aligned}$$

Denoting

$$\begin{aligned} \begin{aligned} \vartheta _1(\tau )&=2\nu _0-\tau \Vert P\Vert (a_0+a_1+\tau ^{m+1})-2a_0\Vert P\Vert \tau ^{m+1},\\ \vartheta _2(\tau )&=2\nu _1-1.5m\tau ^{4/3}\Vert P\Vert \left( a_0+\frac{\tau ^{m+1}}{m+1}\right) \\&\quad -\frac{2(2m+1)}{m+1}a_0\tau ^{m+1}\Vert P\Vert -\tau \Vert P\Vert (a_0+a_1+\tau ^{m+1}),\\ \eta _2(s)&=\tau ^{-1}\Vert P\Vert (a_0+a_1+\tau ^{m+1})s^2+\frac{m\tau ^{-4}\Vert P\Vert (a_0+\frac{\tau ^{m+1}}{m+1})}{2}s^4, \end{aligned} \end{aligned}$$

we can write

$$\begin{aligned} \begin{aligned} {\dot{V}}_1(x,d_1)\ge \frac{\vartheta _1(\tau )\Vert x\Vert ^{2m}+\vartheta _2(\tau )\Vert x\Vert ^{4m}}{(1+\Vert x\Vert ^{2m})^2} -\eta _2(\Vert d_1\Vert ):=-V_2(x)-\eta _2(\Vert d_1\Vert ). \end{aligned} \end{aligned}$$

For sufficiently small \(\tau >0\), we have \(\vartheta _1(\tau )>0\), \(\vartheta _2(\tau )>0\). Since \(V_2(x)\le 0\), we have

$$\begin{aligned} \begin{aligned} -{\dot{V}}_1(x,d_1)\le \eta _2(\Vert d_1\Vert ) \end{aligned} \end{aligned}$$
(57)

and we can set \(V_3(x)=0\).

In order to apply Theorem 1 (see Remark 2), we need to estimate the function

$$\begin{aligned} \begin{aligned} G(x)=\theta V_1(x)+W(x). \end{aligned} \end{aligned}$$

From the estimates (54) and (56), we obtain

$$\begin{aligned} \begin{aligned} G(x)&\le \left( -2c_0+2\Vert P\Vert (\theta +1)\frac{\tau ^{m+1}}{m+1}\right) \frac{\Vert x\Vert ^{m+1}}{\sqrt{1+\Vert x\Vert ^{2m}}}\\&\quad + \Vert P\Vert (b_0^2+b_0\tau )\frac{\Vert x\Vert ^{2m}}{1+\Vert x\Vert ^{2m}}\\&=-\frac{\Vert x\Vert ^{m+1}}{\sqrt{1+\Vert x\Vert ^{2m}}}\left( \sigma _3(\tau )-\sigma _4(\tau )\frac{\Vert x\Vert ^{m-1}}{\sqrt{1+\Vert x\Vert ^{2m}}}\right) , \end{aligned} \end{aligned}$$
(58)

where \(\sigma _3(\tau )=2c_0-2\Vert P\Vert (\theta +1)\frac{\tau ^{m+1}}{m+1}\), \(\sigma _4(\tau )=\Vert P\Vert (b_0^2+b_0\tau )\).

Proposition 4

If

$$\begin{aligned} \sigma _4(0)\frac{(m-1)^{(m-1)/2m}}{\sqrt{m}}<\sigma _3(0), \end{aligned}$$

then for \(\tau >0\) small enough and for all \(s\in \mathbb R_+\), the next inequality is true

$$\begin{aligned} \begin{aligned} \sigma _3(\tau )-\sigma _4(\tau )\frac{s^{m-1}}{\sqrt{1+s^{2m}}}>0. \end{aligned} \end{aligned}$$

The proof can be found in Appendix.

Proposition 5

Let system (50) satisfy the conditions of (51) as well as the inequality

$$\begin{aligned} \begin{aligned} \Vert P\Vert b_0^2\frac{(m-1)^{(m-1)/2m}}{\sqrt{m}}<2c_0. \end{aligned} \end{aligned}$$

Then, system (50) is ISS.

Proof

Number \(\tau >0\) can be chosen small enough, so that from the inequality

$$\begin{aligned} \begin{aligned} \Vert P\Vert b_0^2\frac{(m-1)^{(m-1)/2m}}{\sqrt{m}}<2c_0 \end{aligned} \end{aligned}$$

the following estimation \(G(x)\le -\delta (\Vert x\Vert )\) c \(\delta (s)=\varepsilon _1s^{m+1}/\sqrt{1+s^{2m}}\), follows where \(\varepsilon _1>0\) is small enough. Hence, system (50) satisfies all conditions of Theorem 1 (see Remark 2) from which desired assertion follows. \(\square \)

To illustrate this result, we take

$$\begin{aligned} \begin{aligned} f(x)&=\begin{pmatrix} -2.5x_1^3+0.1x_1^2x_2\\ 0.1x_2^2x_1+0.3x_2^3 \end{pmatrix},\quad g(x)=\begin{pmatrix} 0.5x_1^3-0.1x_1^2x_2\\ -0.1x_2^2x_1-x_2^3 \end{pmatrix}, \end{aligned} \end{aligned}$$

\(P=I\), \(\theta =1\), then \(b_0^2=1.003329653\), \(c_0=0.5185185185\), \(\nu _0=0.2511999583\), \(\nu _1=0.08780118723\).

It is easy to see that conditions of the last proposition are satisfied. Hence, the nonlinear impulsive system (50) is ISS. Note that in this case, both continuous and discrete dynamics are unstable.

6 Discussion and conclusions

The main results of this paper are given by Theorems 1 and 2. They allow us to study the ISS property of nonlinear impulsive systems with different assumptions imposed on the discrete and continuous dynamics. These theorems can establish the ISS property even for the case when neither discrete nor continuous dynamics are ISS. Our approach enables usage of a wider class of ISS-Lyapunov functions to study the ISS property of nonlinear impulsive systems. One advantage of our approach, demonstrated in the examples, is that a rather simple (energetic) Lyapunov function equipped with a sequence of auxiliary functions allows to derive desired stability condition to assure the ISS or GAS property.

An interesting direction for future research would be to develop an approach of stability investigation of nonlinear impulsive systems by means of a combination of a Lyapunov and Chetaev functions, which was used for studying local stability of finite-dimensional systems without inputs [11]. This can provide stability conditions alternative to the ones developed in the current paper. First steps in this direction can be found in [7].

Another interesting direction of research is to explore an extension possibility of our results to the strong ISS notion in the context of time varying impulsive systems (see [20]).