Input-to-state stability for impulsive switched nonlinear systems with unstable subsystems

Abstract

In this article, the input-to-state stability is investigated for impulsive switched systems. By means of the Lyapunov function and the average impulsive switched interval approach, the input-to-state stability properties are derived under the condition that all subsystems are stable, all subsystems are unstable and some subsystems are unstable. It is shown that if the continuous subsystems all have input-to-state stability and though the impulsive effects are destabilizing, the system has input-to-state stability with respect to a lower bound of the average impulsive switched interval. Moreover, if all the subsystems do not have input-to-state stability, the impulsive effects can still successfully stabilize the system but for an upper bound of the average impulsive switched interval. However, it is unveiled that if some continuous subsystems are not input-to-state stability, the impulsive effects can successfully stabilize the system for a lower bound of the average impulsive switched interval under specific conditions. It is worth noting that we introduce multiple jumps in this paper. Finally, three examples are illustrated with their simulations to manifest the validity of the main results.

Keywords

Impulsive switched system input-to-state stability Lyapunov function unstable subsystems

Introduction

Dynamical systems that are described by an interaction between continuous and discrete dynamics are usually called hybrid systems. As a subset of hybrid systems, impulsive systems which consist of three elements: A continuous dynamic, a discrete dynamic and a criterion that determines when the reset of the system state is to be implemented (Bainov and Simeonov, 1989; Dashkovskiy et al., 2012; Gao and Cai, 2016; Haddad et al., 2016; Perestyuk and Feketa, 2011; Perestyuk and Kapustyan, 2012; Xiang and Xiao, 2013; Yang et al., 2001; Zhu et al., 2014). In past decades, impulsive systems have attracted considerable attention in science and engineering, and have crucial applications in various fields, such as, control systems which have communication constraints (Chen and Zheng, 2011; Hespanha et al., 2008), sample-data systems (Naghshtabrizi et al., 2008; Van de Wouw et al., 2010), networked control systems with scheduling protocol (Ho et al., 2006; Nesic and Teel, 2004), mechanical systems (Chen et al., 2008; Zhang and Wu, 2016; Zhang et al., 2017), asynchronous switching (Gao and Wang, 2016), etc. Another important subset of hybrid systems is switched systems, which consist of a family of subsystems and a logical rule that orchestrates the switching between them (Gao et al., 2016; Liberzon et al., 1999; Wang et al., 2010; Zhao and Hill, 2008). We generally examine the nature of a switched system not under various classes of switching signals as in the stability analysis for switched systems (Hespanha, 2004). However, it is noteworthy that the switching signal and the impulsive effects are always simultaneously involved in the systems (Gao et al., 2015). In order to study these two types of hybrid systems in the same framework, we connected them to shape a more complex model, i.e. impulsive switched systems. The major motivation for investigating such systems comes partly from the fact that they have a wide range of applications in fields such as mechanical systems, control systems and many other fields.

As far as we know, how to describe the influences of external inputs on state quality is a significant subject when we investigate the stability. The notion of input-to-state stability (ISS), presented by Sontag (1989), is very valuable in this area and has been promoted to nonlinear systems by various works. The ISS implies that no matter what the size of the initial state is, the state will eventually be small if the external inputs are uniformly small. Recently, ISS was generalized in various works (Gao et al., 2015; Liu et al., 2010; Liu et al., 2016; Müller and Liberzon, 2012; Wang et al., 2015). In the work by Müller and Liberzon (2012), which investigated switched systems with unstable subsystems. Inspired by their work, we add impulsive effects to above work to improve its significance. Impulsive effects can be divided into destabilizing and stabilizing impulses, in the work by Wu et al.(2016), the works discuss the stability of systems under the condition of stabilizing and destabilizing impulses respectively. Based on the work by Liu et al. (2010), Wang et al. (2013) extended it to the case where the impulses are destabilizing, and some of the subsystems can be non-ISS. Compared with Wang et al. (2013), when some subsystems can be non-ISS, we discuss it in two conditions, i.e. stabilizing and destablizing impulses and get the desired results. Rate coefficients are actually on behalf of impulsive effects. In this work, we choose $d_{i} < 0$ or $d_{i} > 0$ , due to a different value of $d_{i}$ , i.e. $e^{- d_{i}} > 1$ or $0 < e^{- d_{i}} < 1$ , represents destabilizing impulses or stabilizing impulses. Furthermore, we extend to cases where all subsystems can be non-ISS and impulses are stabilizing. Applying results from Hespanha et al. and based on that, we put forward multiple jump maps as has been provided in the work by Dashkovskiy and Feketa (2016).

For impulsive switched systems, we assume that impulse and the switching happen simultaneously in this work (Zhu and Cao, 2010, 2012). At first, some definitions are introduced and ISS is in the forms of $K L$ functions. Based on these, some sufficient conditions are provided to testify the properties of ISS for impulsive switched nonlinear systems using the method of a Lyapunov function. Moreover, towards such systems, there have been no studies on the ISS properties where all subsystems are unstable for impulsive switched systems, and the conditions for ISS properties are derived under destabilizing and stabilizing impulses respectively. Indeed, impulses make contributions to stabilize unstable subsystems. In addition, we introduce a new definition: An average impulsive switched interval, which is a combination by average dwell-time (Hespanha et al., 1999) and average impulsive interval (Hespanha et al., 2008). Under such a technique, a sufficient ISS condition is obtained. Respect to a lower bound and an upper bound of an average impulsive switched interval, impulsive effects can successfully stabilize the unstable subsystems to make the systems stable. In this paper, the main objective is to establish some results in this direction.

In this work, we direct our attention to the ISS problems concerning the impulsive switched nonlinear systems with unstable subsystems, which may have been investigated by various works, but the results of this work still make contributions to the existing references. In summary, this work has the following contributions.

This work was exploited on the condition that all of the subsystems are ISS, all of the subsystems are not ISS and some of the subsystems are not ISS. To the best of our knowledge, the unstable subsystems and related ISS properties have not been considered yet in past literature, which increases the difficulty in checking the ISS properties.

Though some results from Hespanha et al. (2008) exist, we extend its conclusions to three more explicit cases, which cover all cases in that work. Especially the third case, which is more complex than the above two cases. In this paper, the average impulsive switched interval is a new and compound concept, which is presented for the first time. Based on a Lyapunov function, we show that if the dwell time satisfies a lower bound or an upper bound in different conditions, then the ISS of the impulsive switched system can be established.

This paper investigates the impulsive effects at the switching instants, making the study of the ISS more challenging. The impulse plays a very crucial role, which can successfully stabilize subsystems to make the whole system stable, even if it contains unstable subsystems.

The rest of this paper is organized as follows. In the ‘Problem formulation and preliminaries’ section, we provide some notions and introduce the definitions of ISS, K and $K_{\infty}$ functions. In the ‘Main results’ section, we prove the ISS property of above three cases. Three examples are illustrated in the ‘Numerical examples’ section. Finally, the conclusions are given in the final section.

Problem formulation and preliminaries

Throughout this article, let $Z^{+}$ denote the set of nonnegative integers, $R^{n}$ denote the n-dimensional Euclidean space; $R^{+}$ denotes the set of nonnegative real integers; |x| denotes the Euclidean norm $| x | = (\sum_{i = 1}^{n} x_{i}^{2})^{1 / 2}$ ; $P = {1, 2, \dots, p}$ is a index set, where p is a finite positive number.

For every $i = 1, 2, \dots, p, p \in N$ , let $T^{i} = {t_{1}^{i}, t_{2}^{i}, \dots}$ be a strictly increasing sequence of impulsive switched instants in $(t_{0}, \infty)$ , and

T = \underset{i = 1, \dots, p}{\cup} T^{i}, T^{i} \cap T^{j} = \emptyset, (i \neq j, i, j = 1, 2, \dots p)

Consider the following impulsive switched system with external inputs

{\begin{matrix} \overset{\cdot}{x} (t) = f_{σ (t)} (x (t), u (t)), t \notin T, \\ x (t) = h_{i} (x^{-} (t), u (t)), t \in T^{i}, i = 1, 2, \dots, p \end{matrix}

(1)

where the state $x \in R^{n}$ is absolutely continuous; $u \in R^{m}$ is locally bounded, Lebesgue-measurable input; the index function $σ (t)$ is the switching signal; $f : R^{n} \times R^{m} \to R^{n}$ and f is locally Lipschitz; $h : R^{n} \times R^{m} \to R^{n}$ . An initial condition $x (t_{0}) = x_{0}$ where the initial time is $t_{0} > 0$ . Here we let ${t_{1}^{i} < t_{2}^{i} < \dots < t_{k}^{i} < \dots}$ be a switching sequence and at time interval $(t_{k}^{i}, t_{k + 1}^{i})$ the $i_{k}$ th subsystem is active, where $t_{k}$ is the impulsive switching instant.

As a matter of convenience, we give the following definitions:

Definition 1. Given positive numbers $τ_{i}$ and $N_{0}$ which satisfies

\frac{t - s}{τ_{i}} - N_{0} \leq N_{i} (t, s) \leq N_{0} + \frac{t - s}{τ_{i}}, \forall t \geq s \geq 0

(2)

Let $N_{i} (t, s)$ be the number of impulsive switches of $t_{k}^{i} \in T^{i}$ during the interval $[t, s)$ , then $τ_{i}$ is called average impulsive switched interval of over $[t, s)$ .

Definition 2 (Khalil, 2002). Given a function $γ$ : $[0, a) \to [0, \infty)$ , which is said to belong to the class $K$ if it satisfies the condition that $γ$ is strictly increasing and $γ (0) = 0$ , then it is said to belong to class $K_{\infty}$ if $a = \infty$ and $γ (r) \to \infty$ as $r \to \infty$ .

Definition 3 (Khalil, 2002). Given a function $β$ : $[0, a) \times [0, \infty)$ , if for each fixeds, the mapping $β (r, s)$ belongs to class $K$ with respect to r and, for each fixed r, the mapping $β (r, s)$ is decreasing with respect to s and $β (r, s) \to 0$ as $s \to \infty$ , then $β$ is said to belong to class $K L$ .

Definition 4 (Khalil, 2002). Suppose that a sequence ${t_{k}^{i}}$ is given, if there exists a class $K L$ function $β$ and a class $K$ function $γ$ such that for any initial state $x (t_{0})$ and any locally essentially bounded input $u (t)$ , the solution $x (t)$ exists for all $t \geq t_{0}$ and satisfies

| x (t) | \leq β (| x (t_{0}) |, t - t_{0}) + γ (‖ u_{| t_{0}, t |} ‖), t \geq t_{0}

(3)

then the system given by equation (1) is said to be input-to-state stable (ISS).

Main results

In this section, we study under what Lyapunov conditions we can construct the ISS of the impulsive switched system given by equation (1). We begin with the situation in which all of the continuous subsystems are ISS. Next, we will discuss the situation where all of the subsystems are not ISS, and some of the subsystems are not ISS.

All subsystems are ISS

Theorem 1. For the system given by equation (1), if there exists functions $α_{1 i}$ , $α_{2 i}$ , $ψ \in K_{\infty}$ , let $V_{i} : R^{n} \to R$ be a candidate for the exponential ISS-Lyapunov function with rate coefficients $c_{si}$ , $d_{1}, d_{2}, \dots, d_{p} \in R^{n}$ , and constants $c_{si} > 0$ , $d_{i} < 0, (i = 1, \dots, p)$ , $η_{0} \geq 1$ , $η > 1$ such that for all $x \in R^{n}$ and all $i \in P$ , we have

α_{1 i} (| x |) \leq V_{i} \leq α_{2 i} (| x |)

(4)

\nabla V_{i} (x (t)) f_{i} (x, u) \leq - c_{si} V_{i} (x) + ψ (| u |)

(5)

If $σ (i)$ is an impulsive switching signal with average impulsive switched interval

τ_{i} > \frac{\ln η}{c_{si}}, η > 1,

(6)

and at each impulsive switching instant $t_{k}^{i}, (k = 1, 2, 3, \dots)$

V_{i} (x (t_{k}^{i})) \leq \max {α_{2} (ψ (‖ u ‖)), η_{0} V_{i - 1} (x (t_{k}^{i}))}

(7)

V_{i} (g_{i} (x, u)) \leq e^{- d_{i}} V_{i} (x^{-} (t_{k}^{i})) + ψ (| u^{-} (t_{k}^{i}) |)

(8)

For arbitrary constants $μ, λ > 0$ , let $S [μ, λ]$ denote the class of impulse time sequences satisfying

- (c_{si} - λ) (t - s) - \sum_{i = 1}^{p} d_{i} N_{i} (t, s) \leq μ, \forall t \geq s \geq t_{0}

(9)

then the system given by equation (1) is ISS, where $α_{1} = mi n_{i \in P} {α_{1 i}}$ , $α_{2} = ma x_{i \in P} {α_{2 i}}$ .

Proof of Theorem 1. Pick constants $ε > - 1$ , of the same sign as $d_{i}$ , and $δ > 0$ , both sufficiently close to 0 so that $d_{i} > \bar{d_{i}} : = d_{i} / (1 + ε)$ , $c_{si} > {\bar{c}}_{si} : = (c_{si} - δ) / (1 + ε)$ and $\bar{λ} : = (λ - δ) / (1 + ε) > 0$ . Adding $δ (t - s)$ to both sides of equation (9) and then dividing both sides by $1 + ε$ , we conclude that

- {\bar{c}}_{si} (t - s) - \sum_{i = 1}^{p} {\bar{d}}_{i} N_{i} (t, s) \leq \bar{μ} - \bar{λ} (t - s), \forall t \geq s \geq t_{0}

(10)

where $\bar{μ} : = μ / (1 + ε)$ . we can then rewrite equation (5) as

\nabla V_{i} (x) \cdot f_{i} (x, u) \leq - {\bar{c}}_{si} V_{i} (x) - (c_{si} - {\bar{c}}_{si}) V_{i} (x) + ψ (| u |)

and conclude from Lemma 1 in the Appendix (Hespanha et al., 2008) that between any two consecutive impulsive switched instants $t_{k - 1}^{i}, t_{k}^{i}$ the function $t \to V_{i} (x (t))$ is absolutely continuous and

{\overset{\cdot}{V}}_{i} (x (t)) \leq - {\bar{c}}_{si} V_{i} (x) - (c_{si} - {\bar{c}}_{si}) V_{i} (x) + ψ (| u |), \forall t \in (t_{k - 1}^{i}, t_{k}^{i})

This means that

\begin{array}{l} (c_{s i} - {\bar{c}}_{s i}) V_{i} (x (t)) \geq ψ (| u (t) |) \\ \Rightarrow \dot{V}_{i} (x (t)) \leq - {\bar{c}}_{s i} V_{i} (x (t)), \forall t \in (t_{k - 1}^{i}, t_{k}^{i}) \end{array}

(11)

Similarly, from equation (8) we can get that that at every impulsive switched instant $t_{k}^{i}$

\begin{matrix} (e^{- \bar{d_{i}}} - e^{- d_{i}}) V_{i} (x^{-} (t_{k}^{i})) \geq ψ (| u^{-} (t_{k}^{i}) |) \\ \Rightarrow V_{i} (x (t_{k}^{i})) \leq e^{- \bar{d_{i}}} V_{i} (x^{-} (t_{k}^{i})) \end{matrix}

(12)

Letting $a : = (\min {c_{si} - {\bar{c}}_{si}, e^{- \bar{d_{i}}} - e^{- d_{i}}})^{- 1} > 0$ . Denote the set $B = {x \in R^{n} : V (x (t)) \geq a ψ (| | u | |_{[t_{0}, t]})}$ and $B^{c} = {x \in R^{n} : V (x (t)) < a ψ (| | u | |_{[t_{0}, t]})}$ , which is the complementary set of B. Suppose there exists a sequence of times $t_{0} = {\hat{t}}_{0}^{i} \leq {\overset{⌣}{t}}_{1}^{i} < {\hat{t}}_{1}^{i} < {\overset{⌣}{t}}_{2}^{i} < {\hat{t}}_{2}^{i} < \dots$ which breaks the interval $[t_{0}, \infty)$ into a disjoint union of subintervals of two types. By the right-continuity of x and u, for the points from the first type of intervals ${x \in R^{n} : V (x (t)) \geq a ψ (| | u | |_{[t_{0}, t]})}$ holds, and for the points from the second type of intervals $B^{c} = {x \in R^{n} : V (x (t)) < a ψ (| | u | |_{[t_{0}, t]})}$ holds. Either this sequence is infinite and all subintervals are finite, or the sequence is finite and the last subinterval is infinite. We now analyse these subintervals separately.

Suppose that $t_{1}^{i} > t_{0}$ so that the subinterval $[t_{0}, t_{1}^{i})$ is non-empty; otherwise, skip forward to the line below equation (16).

Case 1. $x \in B$ . From equation (11), we conclude that between any two consecutive impulsive switched instants $t_{k - 1}^{i}, t_{k}^{i} \in (t_{0}, t_{1}^{i}]$ , we have that ${\overset{\cdot}{V}}_{i} (x (t)) \leq - {\bar{c}}_{si} V_{i} (x (t)), \forall t \in (t_{k - 1}^{i}, t_{k}^{i})$ a.e. Therefore

V_{i} (x^{-} (t_{k}^{i})) \leq e^{- {\bar{c}}_{si} (t_{k}^{i} - t_{k - 1}^{i})} V_{i} (x (t_{k - 1}^{i}))

(13)

Moreover, in view of equation (12) and $x \in B$ , we have

V_{i} (x (t_{k}^{i})) \leq e^{- \bar{d_{i}}} V_{i} (x^{-} (t_{k}^{i}))

(14)

Combining equations (7), (13) and (14), we conclude that

\begin{matrix} V_{i} (x (t_{k}^{i})) \leq e^{- {\bar{d}}_{i}} e^{- {\bar{c}}_{si} (t_{k}^{i} - t_{k - 1}^{i}}) V_{i} (x (t_{k - 1}^{i})) \\ \leq e^{- {\bar{d}}_{i}} e^{- {\bar{c}}_{si} (t_{k}^{i} - t_{k - 1}^{i}}) η_{0} V_{i - 1} (x (t_{k - 1}^{i})) \\ = e^{| {\bar{d}}_{i} |} e^{- {\bar{c}}_{si} (t_{k}^{i} - t_{k - 1}^{i}}) η_{0} V_{i - 1} (x (t_{k - 1}^{i})) \\ = η^{+} η_{0} e^{- {\bar{c}}_{si} (t_{k}^{i} - t_{k - 1}^{i}}) V_{i - 1} (x (t_{k - 1}^{i})) \\ = η e^{- {\bar{c}}_{si} (t_{k}^{i} - t_{k - 1}^{i}}) V_{i - 1} (x (t_{k - 1}^{i})), t \in (t_{k - 1}^{i}, t_{k}^{i}) \end{matrix}

(15)

where $η^{+} = e^{| \bar{d_{i}} |}$ , $η = η^{+} η_{0}$ .

So from the above inequality, for $t \in (t_{k - 1}, t_{k})$ , we have

\begin{matrix} V_{i} (x (t_{k})) \leq e^{\sum_{i}^{p} | {\bar{d}}_{i} |} e^{- {\bar{c}}_{si} (t_{k} - t_{k - 1}}) η_{0}^{p} V_{i - 1} (x (t_{k - 1})) \\ \leq (η^{+} η_{0})^{p} e^{- {\bar{c}}_{si} (t_{k} - t_{k - 1}}) V_{i - 1} (x (t_{k - 1})) \\ = η^{p} e^{- {\bar{c}}_{si} (t_{k} - t_{k - 1})} V_{i - 1} (x (t_{k - 1})), \forall t \in (t_{k - 1}, t_{k}) \end{matrix}

Noting that equation (15) is also for $k = 1$ , we can iterate equation (15) over $N_{i} (t_{0}, t)$ impulsive switching on $(t_{0}, t]$ to obtain the bound and because of the condition given by equation (6), we have

\begin{matrix} V_{i} (x (t)) \leq η^{N_{i} (t, t_{0})} e^{- {\bar{c}}_{si} (t_{k}^{i} - t_{k - 1}^{i})} V_{i} (x (t_{0})) \\ \leq η^{N_{0}} e^{(\frac{\ln η}{τ_{i}} - c_{si}) (t - t_{0})} V_{i} (x (t_{0})), \forall t \in (t_{0}, {\overset{⌣}{t}}_{1}^{i}] \end{matrix}

(16)

So for any $t \in (t_{0}, t_{1}]$ , we also get the following

\begin{matrix} V (x (t)) & \leq η^{\sum_{i = 1}^{p} N_{i} (t, t_{0})} e^{- c_{si} (t - t_{0})} V (x (t_{0})) \\ \leq η^{p N_{0}} e^{(\frac{\ln η}{τ_{i}} - c_{si}) (t - t_{0})} V (x (t_{0})) \\ \leq η^{p N_{0}} e^{- c (t - t_{0})} V (x (t_{0})) \end{matrix}

(17)

where $c : = c_{si} - \frac{\ln η}{τ_{i}} \in (0, c_{si} - \frac{\ln η}{τ_{i}})$ .

Case 2. $x \in B^{c}$ .

Here we assume that $t_{1}^{i} < \infty$ , otherwise the bound holds on $(t_{0}, \infty)$ .

Next we show that for $t \geq t_{1}^{i}$ , it is possible to construct an upper bound for $V_{i} (x (t))$ that only depends on $| | u | |_{| t_{0}, t |}$ . On every subinterval of the form $[t_{k}^{i}, {\hat{t}}_{k}^{i})$ , and for $x \in B^{c}$ . If ${\hat{t}}_{k}^{i}$ is not an impulsive switched instant, then the same bound holds for $t = {\hat{t}}_{k}^{i}$ . If ${\hat{t}}_{k}^{i}$ is an impulsive switched instant, then equation (8) gives

V_{i} (x ({\hat{t}}_{k}^{i})) \leq a e^{- d_{i}} ψ (| | u | |_{[t_{0}, {\hat{t}}_{k}^{i}]}) + ψ (| u^{-} ({\hat{t}}_{k}^{i}) |)

In other case, we have

V_{i} (x (t) \leq (a e^{| d_{i} |} + 1) ψ (| | u | |_{[t_{0}, t]}) \forall t \in [{\overset{⌣}{t}}_{k}^{i}, {\hat{t}}_{k}^{i}], i \geq 1

(18)

where again the bound holds $\forall t \geq t_{k}^{i}$ if ${\hat{t}}_{k}^{i} = \infty$ . Now, consider any subinterval of the form $[{\hat{t}}_{k}^{i}, t_{k + 1}^{i}), i \geq 1$ . Repeating the argument used to establish equation (16), with ${\hat{t}}_{k}^{i}$ in place of $t_{0}$ , and using equation (18) with $t = {\hat{t}}_{k}^{i}$ , we obtain

\begin{matrix} V_{i} (x (t)) \leq η^{N_{i} (t, {\hat{t}}_{k}^{i})} e^{- {\bar{c}}_{si} (t - {\hat{t}}_{k}^{i}))} V_{i} (x ({\hat{t}}_{k}^{i})) \\ \leq η^{N_{0}} (a e^{| d_{i} |} + 1) ψ (| | u | |_{[t_{0}, t]}), \forall t \in [t_{k}^{i}, {\hat{t}}_{k}^{i}], i \geq 1 \end{matrix}

So for $t \in [t_{k}, {\hat{t}}_{k}], i \geq 1$ , we have

V (x (t)) \leq \max_{i \in P} {η^{p N_{0}} (a e^{| d_{i} |} + 1) ψ (| | u | |_{[t_{0}, t]})}

(19)

Combining equations (17) and (19), $\forall t \geq t_{0}$ we have

V (x (t)) \leq η^{p N_{0}} e^{- c (t - t_{0})} V_{i} (x (t_{0})) + \max_{i \in P} {η^{p N_{0}} (a e^{| d_{i} |} + 1) ψ (| | u | |_{[t_{0}, t]})}

From the condition (4), we can obtain the following

\begin{matrix} β (| x (t_{0}) |, t - t_{0}) = α_{1}^{- 1} (η^{p N_{0}} e^{- c (t - t_{0})} α_{2} (| x (t_{0}) |)) \\ γ (| | u | |_{| t_{0}, t |}) = max_{i \in P} {α_{1}^{- 1} (η^{p N_{0}} (a e^{| d_{i} |} + 1) ψ (| | u | |_{[t_{0}, t]}))} \end{matrix}

i.e.

| x (t) | \leq β (| x (t_{0}) |, t - t_{0}) + γ (| | u | |_{| t_{0}, t |}) \forall t \geq t_{0}

So the system given by equation (1) is ISS. The proof is completed.

Remark 1. In the condition given by equation (8), the inequality represents the effects of impulse. We define $d_{i} < 0$ , i.e. $η^{+} > 1$ , which means destabilizing impulses. However, $η_{0} > 1$ , which means at each switching instant $t_{k}$ from mode k to $k + 1$ , the switching always causes a bounded increment of $V_{p} (x (t), t)$ . In this paper, though there exists destabilizing impulses, we can still make the system stable under certain conditions.

Remark 2. It is worth noting that, $η > 1$ , so we can only use the right half of the inequality given by equation (2).

All subsystems are not ISS

In this subsection, we discuss all subsystems that are not ISS. In order to prove its stability, we introduce the definition given by equation (2), an upper bound of the average impulsive switched interval to stabilize the system given by equation (1) even all of its subsystems are not ISS.

Theorem 2. For the system given by equation (1), if there exist functions $α_{1 i}$ , $α_{2 i}$ and $ψ \in K_{\infty}$ , let $V_{i} : R^{n} \to R$ be a candidate exponential ISS-Lyapunov function with rate coefficients $c_{ui}$ , $d_{1}, d_{2}, \dots, d_{p} \in R^{n}$ , and constants $c_{ui} > 0$ , $d_{i} > 0, (i = 1, \dots, p)$ , $0 < η_{0} < 1$ , $0 < η < 1$ such that equation (4) holds for all $x \in R^{n}$ and all $i \in P$ , we have

\nabla V_{i} (x (t)) f_{i} (x, u) \leq c_{ui} V_{i} (x) + ψ (| u |)

(20)

If $σ (i)$ is an impulsive switching signal with average impulsive switched interval

τ_{i} < \frac{\ln η}{- c_{ui}}, 0 < η < 1

(21)

and at each impulsive switching instant $t_{k}^{i}, (k = 1, 2, 3, \dots)$

V_{i} (x (t_{k}^{i})) \leq \max {α_{2} (ψ (| | u | |)), η_{0} V_{i - 1} (x (t_{k}^{i}))}

(22)

V_{i} (g_{i} (x, u)) \leq e^{- d_{i}} V_{i} (x^{-} (t_{k}^{i})) + ψ (| u^{-} (t_{k}^{i}) |)

(23)

For arbitrary constants $μ, λ > 0$ , let $S [μ, λ]$ denote the class of impulse time sequences satisfying

(c_{ui} + λ) (t - s) - \sum_{i = 1}^{p} d_{i} N_{i} (t, s) \leq μ \forall t \geq s \geq t_{0}

(24)

then the system given by equation (1) is ISS, where $α_{1} = mi n_{i \in P} {α_{1 i}}$ , $α_{2} = ma x_{i \in P} {α_{2 i}}$ .

Proof of Theorem 2. Pick constants $ε > - 1$ , of the same sign as $d_{i}$ , and $δ > 0$ , both sufficiently close to 0 so that $d_{i} > \bar{d_{i}} : = d_{i} / (1 + ε)$ , $c_{ui} > {\bar{c}}_{ui} : = (c_{ui} - δ) / (1 + ε)$ , $\bar{λ} : = (λ - δ) / (1 + ε) > 0$ . Adding $δ (t - s)$ to both sides of equation (24) and then dividing both sides by $1 + ε$ , we conclude that

- {\bar{c}}_{ui} (t - s) + \sum_{i = 1}^{p} {\bar{d}}_{i} N_{i} (t, s) \leq \bar{μ} + \bar{λ} (t - s), \forall t \geq s \geq t_{0}

(25)

where $\bar{μ} : = μ / (1 + ε)$ . we can then rewrite equation (20) as

\nabla V_{i} (x) \cdot f_{i} (x, u) \leq {\bar{c}}_{ui} V_{i} (x) + (c_{ui} - {\bar{c}}_{ui}) V_{i} (x) + ψ (| u |)

{\overset{\cdot}{V}}_{i} (x (t)) \leq {\bar{c}}_{ui} V_{i} (x) + (c_{ui} - {\bar{c}}_{ui}) V_{i} (x) + ψ (| u |), \forall t \in (t_{k - 1}^{i}, t_{k}^{i})

This means that

\begin{array}{l} ({\bar{c}}_{u i} - c_{u i}) V_{i} (x (t)) \geq ψ (| u (t) |) \\ \Rightarrow \dot{V}_{i} (x (t)) \leq {\bar{c}}_{u i} V_{i} (x (t)), \forall t \in (t_{k - 1}^{i}, t_{k}^{i}) \end{array}

(26)

Similarly, from equation (23) we can get that at every impulsive switched instant $t_{k}^{i}$

\begin{matrix} (e^{- d_{i}} - e^{- \bar{d_{i}}}) V_{i} (x^{-} (t_{k}^{i})) \geq ψ (| u^{-} (t_{k}^{i}) |) \\ \Rightarrow V_{i} (x (t_{k}^{i})) \leq e^{- \bar{d_{i}}} V_{i} (x^{-} (t_{k}^{i})) \end{matrix}

(27)

Letting $a : = (\min {{\bar{c}}_{ui} - c_{ui}, e^{- d_{i}} - e^{- \bar{d_{i}}}})^{- 1} > 0$ . Denote the set $B = {x \in R^{n} : V (x (t)) \geq a ψ (| | u | |_{[t_{0}, t]})}$ and $B^{c} = {x \in R^{n} : V (x (t)) < a ψ (| | u | |_{[t_{0}, t]})}$ , which is the complementary set of B. Suppose there exists a sequence of times $t_{0} = {\hat{t}}_{0}^{i} \leq t_{1}^{i} < {\hat{t}}_{1}^{i} < t_{2}^{i} < {\hat{t}}_{2}^{i} < \dots$ which breaks the interval $[t_{0}, \infty)$ into a disjoint union of subintervals of two types. By the right-continuity of x and u, for the points from the first type of intervals ${x \in R^{n} : V (x (t)) \geq a ψ (| | u | |_{[t_{0}, t]})}$ holds, and for the points from the second type of intervals ${x \in R^{n} : V (x (t)) < a ψ (| | u | |_{[t_{0}, t]})}$ holds. Either this sequence is infinite and all subintervals are finite, or the sequence is finite and the last subinterval is infinite. We now analyse these subintervals separately.

Suppose that $t_{1}^{i} > t_{0}$ so that the subinterval $[t_{0}, t_{1}^{i})$ is non-empty; otherwise, skip forward to the line below equation (31).

Case 1. $x \in B$ . From equation (26), we conclude that between any two consecutive impulsive switched instants $t_{k - 1}^{i}, t_{k}^{i} \in (t_{0}, t_{1}^{i}]$ , we have that ${\overset{\cdot}{V}}_{i} (x (t)) \leq {\bar{c}}_{ui} V_{i} (x (t)), \forall t \in (t_{k - 1}^{i}, t_{k}^{i})$ a.e. Therefore

V_{i} (x^{-} (t_{k}^{i})) \leq e^{{\bar{c}}_{ui} (t_{k}^{i} - t_{k - 1}^{i})} V_{i} (x (t_{k - 1}^{i}))

(28)

Moreover, in view of equation (27) and $x \in B$ , we have

V_{i} (x (t_{k}^{i})) \leq e^{- \bar{d_{i}}} V_{i} (x^{-} (t_{k}^{i}))

(29)

Combining equations (22), (28) and (29), we conclude that

\begin{matrix} V_{i} (x (t_{k}^{i})) \leq e^{- {\bar{d}}_{i}} e^{{\bar{c}}_{ui} (t_{k}^{i} - t_{k - 1}^{i}}) V_{i} (x (t_{k - 1}^{i})) \\ \leq e^{- {\bar{d}}_{i}} e^{{\bar{c}}_{ui} (t_{k}^{i} - t_{k - 1}^{i}}) η_{0} V_{i - 1} (x (t_{k - 1}^{i})) \\ = η^{+} η_{0} e^{{\bar{c}}_{ui} (t_{k}^{i} - t_{k - 1}^{i}}) V_{i - 1} (x (t_{k - 1}^{i})) \\ = η e^{{\bar{c}}_{ui} (t_{k}^{i} - t_{k - 1}^{i}}) V_{i - 1} (x (t_{k - 1}^{i})), t \in (t_{k - 1}^{i}, t_{k}^{i}) \end{matrix}

(30)

where $η^{+} = e^{- \bar{d_{i}}}$ , $η = η^{+} η_{0}$ .

So from the above inequality, for $t \in (t_{k - 1}, t_{k})$ , we have

\begin{matrix} V_{i} (x (t_{k})) & \leq e^{- \sum_{i}^{p} \bar{d_{i}}} e^{{\bar{c}}_{ui} (t_{k} - t_{k - 1}}) η_{0}^{p} V_{i - 1} (x (t_{k - 1})) \\ \leq (η^{+} η_{0})^{p} e^{{\bar{c}}_{ui} (t_{k} - t_{k - 1}}) V_{i - 1} (x (t_{k - 1})) \\ = η^{p} e^{{\bar{c}}_{ui} (t_{k} - t_{k - 1})} V_{i - 1} (x (t_{k - 1})), \forall t \in (t_{k - 1}, t_{k}) \end{matrix}

Noting that equation (30) is also for $k = 1$ , we can iterate equation (30) over $N_{i} (t_{0}, t)$ impulsive switching on $(t_{0}, t]$ to obtain the bound, we have

\begin{matrix} V_{i} (x (t)) \leq η^{N_{i} (t, t_{0})} e^{{\bar{c}}_{ui} (t_{k}^{i} - t_{k - 1}^{i})} V_{i} (x (t_{0})) \\ \leq η^{- N_{0}} e^{(\frac{\ln η}{τ_{i}} + c_{ui}) (t - t_{0})} V_{i} (x (t_{0})) \forall t \in (t_{0}, {\overset{⌣}{t}}_{1}^{i}] \end{matrix}

(31)

So for any $t \in (t_{0}, t_{1}]$ , and by the condition given by equation (21), we also get the following

\begin{matrix} V (x (t)) \leq η^{\sum_{i = 1}^{p} N_{i} (t, t_{0})} e^{c_{ui} (t - t_{0})} V (x (t_{0})) \\ \leq η^{- p N_{0}} e^{(\frac{\ln η}{τ_{i}} + c_{ui}) (t - t_{0})} V (x (t_{0})) \\ \leq η^{- p N_{0}} e^{- c (t - t_{0})} V (x (t_{0})) \end{matrix}

(32)

where $c \in (0, \frac{\ln η}{τ_{i}} + c_{ui})$ .

Case 2. $x \in B^{c}$ .

Here, we assume that $t_{1}^{i} < \infty$ , otherwise the bound holds on $(t_{0}, \infty)$ .

V_{i} (x ({\hat{t}}_{k}^{i})) \leq a e^{- d_{i}} ψ (| | u | |_{[t_{0}, {\hat{t}}_{k}^{i}]}) + ψ (| u^{-} ({\hat{t}}_{k}^{i}) |)

In either case, we have

V_{i} (x (t) \leq (a e^{d_{i}} + 1) ψ (| | u | |_{[t_{0}, t]}), \forall t \in [t_{k}^{i}, {\hat{t}}_{k}^{i}], i \geq 1

(33)

where again the bound holds $\forall t \geq t_{k}^{i}$ if ${\hat{t}}_{k}^{i} = \infty$ . Now, consider any subinterval of the form $[{\hat{t}}_{k}^{i}, t_{k + 1}^{i}), i \geq 1$ . Repeating the argument used to establish equation (32), with ${\hat{t}}_{k}^{i}$ in place of $t_{0}$ , and using equation (33) with $t = {\hat{t}}_{k}^{i}$ , we obtain

\begin{matrix} V_{i} (x (t)) \leq η^{N_{i} (t, {\hat{t}}_{k}^{i})} e^{{\bar{c}}_{ui} (t - {\hat{t}}_{k}^{i}))} V_{i} (x ({\hat{t}}_{k}^{i})) \\ \leq η^{- N_{0}} (a e^{d_{i}} + 1) ψ (| | u | |_{[t_{0}, t]}) \forall t \in [t_{k}^{i}, {\hat{t}}_{k}^{i}], i \geq 1 \end{matrix}

So for $t \in [t_{k}, {\hat{t}}_{k}], i \geq 1$ , we have

V (x (t)) \leq \max_{i \in P} {η^{- p N_{0}} (a e^{d_{i}} + 1) ψ (| | u | |_{[t_{0}, t]})}

(34)

Combining equations (32) and (34), $\forall t \geq t_{0}$ we have

\begin{array}{l} V (x (t)) \leq η^{- p N_{0}} e^{- c (t - t_{0})} \\ V_{i} (x (t_{0})) + \max_{i \in P} {η^{- p N_{0}} (a e^{d_{i}} + 1) ψ (| | u | |_{[t_{0}, t]})} \end{array}

From the condition given by equation (4), therefore we have the following

\begin{matrix} β (| x (t_{0}) |, t - t_{0}) = α_{1}^{- 1} (η^{- p N_{0}} e^{- c (t - t_{0})} α_{2} (| x (t_{0}) |)) \\ γ (| | u | |_{| t_{0}, t |}) = \max_{i \in P} {α_{1}^{- 1} (η^{- p N_{0}} (a e^{d_{i}} + 1) ψ (| | u | |_{[t_{0}, t]}))} \end{matrix}

i.e.

| x (t) | \leq β (| x (t_{0}) |, t - t_{0}) + γ (| | u | |_{| t_{0}, t |}), \forall t \geq t_{0}

So the system given by equation (1) is ISS. The proof is completed.

Remark 3. According to the above proof, $d_{i} > 0$ , i.e. $0 < η^{+} = e^{- d_{i}} < 1$ , which represents a stabilizing impulse, we can get $0 < η < 1$ , so we can only use the left half of the inequality given by equation (2). Even the subsystems are all unstable, under the condition of the stabilizing impulse, we can still guarantee the ISS property of the system.

Some subsystems are not ISS

In this subsection, we extend the last two analyses to the case where not all subsystems are ISS. This case is more complex than above two cases. Let $P = P_{s} \cup P_{u}$ hold that $P_{s} \cap P_{u} = \emptyset$ . Define $T_{u} (t_{0}, t)$ as the whole activation time of the systems in $P_{u}$ and $T_{s} (t_{0}, t)$ as the whole activation time of the systems in $P_{s}$ at the time interval $[t_{0}, t)$ , in which $t_{0} \leq t$ . Obviously, $T_{s} (t_{0}, t) = t - t_{0} - T_{u} (t_{0}, t)$ .

Theorem 3. For the system given by equation (1), if there exist functions $α_{1 i}$ , $α_{2 i}$ , $ψ \in K_{\infty}$ , let $V_{i} : R^{n} \to R$ be a candidate exponential ISS-Lyapunov function with rate coefficients $c_{si}$ , $c_{ui}$ , $d_{1}, d_{2}, \dots, d_{p} \in R^{n}$ , and constants $c_{si} > 0$ , $c_{ui} > 0$ , $η_{0} > 0$ such that for equation (4) all $x \in R^{n}$ and all $i \in P$ , we have

{\begin{matrix} \nabla V_{i} (x (t)) f_{i} (x, u) \leq - c_{si} V_{i} (x) + ψ (| u |), \forall s \in P_{s} \\ \nabla V_{i} (x (t)) f_{i} (x, u) \leq c_{ui} V_{i} (x) + ψ (| u |), \forall u \in P_{u} \end{matrix}

(35)

If there exists constants $ρ, T_{0}$ , satisfy

ρ < \frac{c_{si}}{c_{si} + c_{ui}}

(36)

\forall t \geq t_{0} : T_{u}^{i} (t, t_{0}) \leq T_{0} + ρ (t - t_{0})

(37)

If $σ (i)$ is an impulsive switching signal with average impulsive switched interval

{\begin{matrix} τ_{i} > 0, 0 < η \leq 1 \\ τ_{i} > \frac{\ln η}{c_{si} (1 - ρ) - c_{ui} ρ}, η > 1 \end{matrix}

(38)

and at each impulsive switching instant $t_{k}^{i}, (k = 1, 2, 3, \dots)$

V_{i} (x (t_{k}^{i})) \leq \max {α_{2} (ψ (| | u | |)), η_{0} V_{i - 1} (x (t_{k}^{i}))}

(39)

V_{i} (g_{i} (x, u)) \leq e^{- d_{i}} V_{i} (x^{-} (t_{k}^{i})) + ψ (| u^{-} (t_{k}^{i}) |)

(40)

For arbitrary constants $μ, λ > 0$ , let $S [μ, λ]$ denote the class of impulse time sequences satisfying

(c_{ui} - c_{si} - 2 λ) (t - s) - 2 \sum_{i = 1}^{p} d_{i} N_{i} (t, s) \leq μ, \forall t \geq s \geq t_{0}

(41)

then the system given by equation (1) is ISS, where $η = η^{+} η_{0}$ , $η^{+} = e^{- d_{i}}$ , $α_{1} = mi n_{i \in P} {α_{1 i}}$ and $α_{2} = ma x_{i \in P} {α_{2 i}}$ .

Proof of Theorem 3. Pick constants $ε > - 1$ , of the same sign as $d_{i}$ , and $δ > 0$ , so they are both sufficiently close to 0 so that $d_{i} > \bar{d_{i}} : = d_{i} / (1 + ε)$ , $(c_{ui} - c_{si}) > ({\bar{c}}_{ui} - {\bar{c}}_{ui}) : = (δ + c_{ui} - c_{si}) (1 + ε)$ , $\bar{λ} : = (2 λ - δ) (1 + ε) > 0$ . Adding $δ (t - s)$ to both sides of equation (41) and then dividing both sides by $1 + ε$ , we conclude that

({\bar{c}}_{ui} - {\bar{c}}_{si}) (t - s) - 2 \sum_{i = 1}^{p} {\bar{d}}_{i} N_{i} (t, s) \leq \bar{μ} - \bar{λ} (t - s), \forall t \geq s \geq t_{0}

(42)

where $\bar{μ} : = μ / (1 + ε)$ . From equation (35), we can conclude the following

\nabla V_{i} (x (t)) f_{i} (x, u) \leq (c_{ui} - c_{si}) V_{i} (x) + ψ (| u |)

(43)

Then we can rewrite equation (43) as

\nabla V_{i} (x) \cdot f_{i} (x, u) \leq [(c_{ui} - c_{si}) - ({\bar{c}}_{ui} - {\bar{c}}_{si})] V_{i} (x) + ({\bar{c}}_{ui} - {\bar{c}}_{si}) V_{i} (x) + ψ (| u |)

and conclude from Lemma 1 in the Appendix (Hespanha et al., 2008) that between any two consecutive impulsive switched instants $t_{i_{k - 1}}^{i}, t_{i_{k}}^{i}$ the function $t \to V_{i} (x (t))$ is absolutely continuous and

{\overset{\cdot}{V}}_{i} (x (t)) \leq [(c_{ui} - c_{si}) - ({\bar{c}}_{ui} - {\bar{c}}_{si})] V_{i} (x) + ({\bar{c}}_{ui} - {\bar{c}}_{si}) V_{i} (x) + ψ (| u |), \forall t \in (t_{k - 1}^{i}, t_{k}^{i})

This means that

\begin{matrix} [({\bar{c}}_{ui} - {\bar{c}}_{si}) - (c_{ui} - c_{si})] V_{i} (x (t)) \geq ψ (| u (t) |) \Rightarrow \\ {\overset{\cdot}{V}}_{i} (x (t)) \leq ({\bar{c}}_{ui} - {\bar{c}}_{si}) V_{i} (x (t)), \\ \forall t \in (t_{k - 1}^{i}, t_{k}^{i}) \end{matrix}

(44)

Similarly, from equation (40) we can get that that at every impulsive switched instant $t_{i_{k}}^{i}$

\begin{matrix} (e^{- \bar{d_{i}}} - e^{- d_{i}}) V_{i} (x^{-} (t_{k}^{i})) & \geq ψ (| u^{-} (t_{k}^{i}) |) \\ \Rightarrow V_{i} (x (t_{k}^{i})) & \leq e^{- \bar{d_{i}}} V_{i} (x^{-} (t_{k}^{i})) \end{matrix}

(45)

Letting $a : = (\min {[(c_{ui} - c_{si}) - ({\bar{c}}_{ui} - {\bar{c}}_{si})], e^{- \bar{d_{i}}} - e^{- d_{i}}})^{- 1} > 0$ . Denote the set $B = {x \in R^{n} : V (x (t)) \geq a ψ (| | u | |_{[t_{0}, t]})}$ and $B^{c} = {x \in R^{n} : V (x (t)) < a ψ (| | u | |_{[t_{0}, t]})}$ , which is the complementary set of B. Suppose there exists a sequence of times $t_{0} = {\hat{t}}_{0}^{i} \leq t_{1}^{i} < {\hat{t}}_{1}^{i} < t_{2}^{i} < {\hat{t}}_{2}^{i} < \dots$ which breaks the interval $[t_{0}, \infty)$ into a disjoint union of subintervals of two types. By the right-continuity of x and u, the points from the first type of intervals ${x \in R^{n} : V (x (t)) \geq a ψ (| | u | |_{[t_{0}, t]})}$ hold, and the points from the second type of intervals ${x \in R^{n} : V (x (t)) < a ψ (| | u | |_{[t_{0}, t]})}$ hold. Either this sequence is infinite and all subintervals are finite, or the sequence is finite and the last subinterval is infinite. We now analyse these subintervals separately.

Suppose that $t_{1}^{i} > t_{0}$ so that the subinterval $[t_{0}, t_{1}^{i})$ is non-empty; otherwise, skip forward to the line below Case 1 and Case 2.

Case 1. $x \in B$ . From equation (44), we conclude that between any two consecutive impulsive switched instants $t_{k - 1}^{i}, t_{k}^{i} \in (t_{0}, t_{1}^{i}]$ , we have ${\overset{\cdot}{V}}_{i} (x (t)) \leq ({\bar{c}}_{ui} - {\bar{c}}_{si}) V_{i} (x (t)), \forall t \in (t_{k - 1}^{i}, t_{k}^{i})$ a.e. Therefore,

V_{i} (x^{-} (t_{k}^{i})) \leq e^{({\bar{c}}_{ui} - {\bar{c}}_{si}) (t_{k}^{i} - t_{k - 1}^{i})} V_{i} (x (t_{k - 1}^{i}))

(46)

Moreover, in view of equation (45) and $x \in B$ , we have

V_{i} (x (t_{k}^{i})) \leq e^{- \bar{d_{i}}} V_{i} (x^{-} (t_{k}^{i}))

(47)

Combining equations (39), (46) and (47), we conclude that

\begin{matrix} V_{i} (x (t_{k}^{i})) \leq e^{- {\bar{d}}_{i}} e^{({\bar{c}}_{ui} - {\bar{c}}_{si}) (t_{k}^{i} - t_{k - 1}^{i}}) V_{i} (x (t_{k - 1}^{i})) \\ \leq e^{- {\bar{d}}_{i}} e^{({\bar{c}}_{ui} - {\bar{c}}_{si}) (t_{k}^{i} - t_{k - 1}^{i})} η_{0} V_{i - 1} (x (t_{k - 1}^{i})), t \in (t_{k - 1}^{i}, t_{k}^{i}) \end{matrix}

Next, analogous to above two proofs, we will discuss $d_{i}$ in two cases.

1. When $d_{i} < 0$ , i.e. $η^{+} = e^{| \bar{d_{i}} |} > 1$ , if $η = η_{0} η^{+} > 1$ , by above inequality, for $t \in (t_{k - 1}, t_{k})$ , we have

\begin{matrix} V_{i} (x (t_{k})) \leq e^{\sum_{i}^{p} | \bar{d_{i}} |} e^{({\bar{c}}_{ui} - {\bar{c}}_{si}) (t_{k} - t_{k - 1}}) (η_{0})^{p} V_{i - 1} (x (t_{k - 1})) \\ = (η_{0} η^{+})^{p} e^{({\bar{c}}_{ui} - {\bar{c}}_{si}) (t_{k} - t_{k - 1}}) V_{i - 1} (x (t_{k - 1})) \\ = η^{p} e^{({\bar{c}}_{ui} - {\bar{c}}_{si}) (t_{k} - t_{k - 1})} V_{i - 1} (x (t_{k - 1})), \forall t \in (t_{k - 1}, t_{k}) \end{matrix}

(48)

And by the conditions given by equations (36) and (37), for any $t \in (t_{0}, t_{1}]$ , we get the following

\begin{matrix} V_{i} (x (t)) \leq η^{\sum_{i = 1}^{p} N_{i} (t, t_{0})} e^{c_{ui} \sum_{i = 1}^{p} T_{u}^{i}} e^{- c_{si} \sum_{i = 1}^{p} T_{s}^{i}} V_{k - 1} (x (t_{k - 1}^{i}) \\ \leq η^{\sum_{i = 1}^{p} N_{i} (t, t_{0})} e^{c_{ui} T_{u}} e^{- c_{si} T_{s}} V_{k - 1} (x (t_{k - 1}^{i}) \\ \leq η^{p N_{0}} e^{(c_{si} + c_{ui}) T_{0}} e^{(\frac{\ln η}{τ_{i}} + ρ (c_{ui} + c_{si}) - c_{si}) (t - t_{0})} V_{i} (x (t_{0})) \\ \leq η^{p N_{0}} e^{(c_{si} + c_{ui}) T_{0}} e^{c (t - t_{0})} V_{i} (x (t_{0})), \forall t \in (t_{0}, {\overset{⌣}{t}}_{1}^{i}] \end{matrix}

(49)

where $c = (c_{s} - (c_{si} + c_{ui}) ρ - \frac{\ln η}{τ_{i}}) \in (0, c_{si} - (c_{si} + c_{ui}) ρ)$ , $T_{u} = \sum_{i = 1}^{p} T_{u}^{i}$ , $T_{s} = \sum_{i = 1}^{p} T_{s}^{i}$ .

2. When $d_{i} > 0$ , i.e. $0 < η^{+} = e^{- \bar{d_{i}}} < 1$ , if $0 < η = η_{0} η^{+} < 1$ , analogous to 1., for any $t \in (t_{0}, t_{1}]$ we have

\begin{matrix} V_{i} (x (t)) \leq η^{\sum_{i = 1}^{p} N_{i} (t, t_{0})} e^{c_{ui} \sum_{i = 1}^{p} T_{u}^{i}} e^{- c_{si} \sum_{i = 1}^{p} T_{s}^{i}} V_{i - 1} (x (t_{k - 1}^{i}) \\ \leq η^{\sum_{i = 1}^{p} N_{i} (t, t_{0})} e^{c_{ui} T_{u}} e^{- c_{si} T_{s}} V_{i - 1} (x (t_{k - 1}^{i}) \\ \leq η^{- p N_{0}} e^{(c_{si} + c_{ui}) T_{0}} e^{(\frac{\ln η}{τ_{i}} + ρ (c_{ui} + c_{si}) - c_{si}) (t - t_{0})} V_{i} (x (t_{0})) \end{matrix}

(50)

If $τ_{i} > 0$ , then $V_{i} (x (t))$ still decays exponentially.

It is worth noting that, in 1., we can get $0 < η < 1$ if $η_{0}$ is sufficiently small; in 2., if we can get $η > 1$ if $η_{0}$ is sufficiently large. The results still can be applied above.

In conclusion, from 1. and 2., let $Δ = \max {η^{p N_{0}}, η^{- p N_{0}}}$ . If $ρ$ and $τ_{i}$ satisfy the conditions given by equations (36) and (37), for any $t \in (t_{0}, t_{1})$ , we have

V (x (t)) \leq Δ V (x (t_{0})) e^{- c (t - t_{0})}

(51)

Case 2. $x \in B^{c}$ .

Here we assume that $t_{1}^{i} < \infty$ , otherwise the bound holds on $(t_{0}, \infty)$ .

V_{i} (x ({\hat{t}}_{k}^{i})) \leq a e^{- d_{i}} ψ (| | u | |_{[t_{0}, {\hat{t}}_{k}^{i}]}) + ψ (| u^{-} ({\hat{t}}_{k}^{i}) |)

In either case, we have

V_{i} (x (t) \leq (a e^{| d_{i} |} + 1) ψ (| | u | |_{[t_{0}, t]}), \forall t \in [t_{k}^{i}, {\hat{t}}_{k}^{i}], i \geq 1

(52)

\begin{matrix} V_{i} (x (t)) \leq η^{N_{i} (t, {\hat{t}}_{k}^{i})} e^{({\bar{c}}_{ui} - {\bar{c}}_{si}) (t - {\hat{t}}_{k}^{i}))} V_{i} (x ({\hat{t}}_{k}^{i})) \\ \leq Δ (a e^{| d_{i} |} + 1) ψ (| | u | |_{[t_{0}, t]}), \forall t \in [t_{k}^{i}, {\hat{t}}_{k}^{i}], i \geq 1 \end{matrix}

So for $t \in [t_{k}, {\hat{t}}_{k}], i \geq 1$ , we have

V (x (t)) \leq \max_{i \in P} {Δ (a e^{| d_{i} |} + 1) ψ (| | u | |_{[t_{0}, t]})}

(53)

Combining equations (51) and (53), $\forall t \geq t_{0}$ we have

V (x (t)) \leq Δ V (x (t_{0})) e^{- c (t - t_{0})} + \max_{i \in P} {Δ (a e^{| d_{i} |} + 1) ψ (| | u | |_{[t_{0}, t]})}

From the condition given by equation (4), we can therefore obtain the following

\begin{array}{l} β (| x (t_{0}) |, t - t_{0}) = α_{1}^{- 1} (Δ e^{- c (t - t_{0})} α_{2} (| x (t_{0}) |)) \\ γ (| | u | |_{| t_{0}, t |}) = \max_{i \in p} α_{1}^{- 1} (Δ (a e^{| d_{i} |} + 1) ψ (| | u | |_{[t_{0}, t]}))} \end{array}

i.e.

| x (t) | \leq β (| x (t_{0}) |, t - t_{0}) + γ (| | u | |_{| t_{0}, t |}), \forall t \geq t_{0}

So the system given by equation (1) is ISS. The proof is completed.

Remark 4. In the condition given by equation (39), $η_{0} > 0$ , which means the system is composed of two parts: Stable and unstable subsystems. When $d_{i} > 0$ and $0 < η^{+} = e^{\min {- d_{i}}} < 1$ , this means stabilizing impulses; when $d_{i} < 0$ and $η^{+} = e^{| d_{i} |} > 1$ , this means destabilizing impulse. In this paper, no matter what destabilizing or stabilizing impulses, we can make the system stable under certain conditions. Definition 2 indeed makes a contribution to ISS.

Remark 5. Compared with the previous work by Yang et al. (2014) and Theorem 3.2 in the work by Yang et al. (2014), the relationship between $m_{s}$ and $m_{u} s$ has constraints for stable subsystems and unstable subsystems. Due to the existence of impulses, it can make a system obtain the desired ISS property, which reduces the conservatism property.

Remark 6. When $d_{i} = 0$ , $η^{+} = e^{\min {- d_{i}}} = 1$ , which still means stabilizing impulses, as a special case, one can be regarded as a case which belongs to case 1. in this paper.

Numerical examples

In this section, three examples are presented to illustrate the validity of our main results.

Consider the impulsive switched nonlinear systems as follows

{\begin{matrix} \overset{\cdot}{x} (t) = f_{σ (t)} (x (t), u (t)), t \notin T \\ x (t) = h_{i} (x (t^{-}), u (t^{-})), t \in T_{i}, k \in Z^{+} \ {0} \end{matrix}

where $σ (t)$ is the switching signal. Let $T^{i} = {t_{0}^{i} < t_{1}^{i} < \dots < t_{k}^{i} < \dots}$ be a switching sequence and the $i_{k}$ th subsystem is active during time interval $[t_{k}^{i}, t_{k + 1}^{i})$ , where $t_{k}$ is the switching instant, $k = 0, 1, 2, 3, \dots$ .

Example 1. In this example, for simplicity, we choose a special case: $i = 1$ , i.e. $T^{1} = {t_{0}^{1} < t_{1}^{1} < \dots < t_{k}^{1} < \dots}$ . Choosing $f (x, u) = - x - x^{\frac{2}{3}} u$ , $V = | x |$ , then $\frac{1}{2} | x | \leq V \leq | x |$ , and we have

\nabla V (x) f (x, u) = sign (x) (- x - x^{\frac{2}{3}} u) = - | x | - u | x |_{3}^{2}

If $| u_{t} | \leq \frac{1}{2} | x |^{\frac{1}{3}}$ , above inequality becomes $\nabla Vf (x, u) \leq - 1.5 | x | = - 1.5 V$ .

Choosing $h_{1} (x^{-} (t), u (t)) = 0.5 x^{-} (t) + 0.5 x (t) u (t)$ .

From Theorem 1, we can get $c_{s 1} = 1.5$ . Let $η^{+} = 3$ , $η_{0} = 2$ , then $η = 6$ , we may choose $τ_{i} = 1.2 > \frac{\ln 6}{1.5}$ . When $u (t) = e^{- t} \sin (t)$ , the corresponding state trajectory is portrayed in Figures 1 and 2 shows the impulsive and switching signals.

Remark 7. In Example 1, by choosing a Lyapunov function which satisfies the condition given by equation (4), corresponding $c_{si}$ satisfies the condition given by equation (5). In simulation, the impulsive signal and the switching signal happen simultaneously, and the state trajectory converges to zero, which meets the ISS property.

Example 2. Choosing $T^{i} = {t_{0}^{i} < t_{1}^{i} < \dots < t_{k}^{i} < \dots}, i = 1, 2, 3, 4$ . So at each subinterval, we choose

{\begin{matrix} f (x, u) = 0.5 x + 0.2 xsin (x) + 0.5 x^{\frac{1}{3}} u \\ h_{1} (x^{-} (t), u (t)) = 0.2 x^{-} (t) + 0.2 x (t) u (t) \\ h_{2} (x^{-} (t), u (t)) = 0.4 x^{-} (t) + 0.2 x (t) u (t) \\ h_{3} (x^{-} (t), u (t)) = 0.3 x^{-} (t) + 0.5 x (t) u (t) \\ h_{4} (x^{-} (t), u (t)) = 0.1 x^{-} (t) + 0.4 x (t) u (t) \end{matrix}

Choosing $V = | x |$ , then $\frac{1}{2} | x | \leq V \leq | x |$ , and we have $\nabla V (x) f (x, u) = sign (x) (0.5 x + 0.2 x \sin (x) + x^{\frac{1}{3}} u) = 0.5 | x | + 0.2 | x | \sin x + 0.5 u | x |^{\frac{1}{3}}$ If $| u_{t} | \leq | x |^{\frac{2}{3}}$ , above inequality becomes $\nabla Vf (x, u) \leq 1.2 | x | = 1.2 V$ .

From Theorem 2, we can get $c_{ui} = 1.2$ . Let $η^{+} = 0.4$ , $η_{0} = 0.8$ , then $η = 0.32$ . We may choose $τ_{i} = 0.94 < \ln 0.32 / - 1.2$ . When $u (t) = e^{- t}$ , the corresponding state trajectory is portrayed in Figures 3 and 4 and shows the impulsive and switching signals.

Remark 8. In the simulation for Example 2, because of all the unstable subsystems, the state trajectory goes to divergency in the beginning, but after a time verging to zero. Because $i = 4$ , from Figure 3, we can see that four impulses emerge repeatedly.

Example 3 Choosing $T^{i} = {t_{0}^{i} < t_{1}^{i} < \dots < t_{k}^{i} < \dots}, i = 1, 2, 3, 4$ . And at each subinterval, we choose

{\begin{matrix} f (x, u) = {\begin{matrix} - 0.1 x - 0.6 x^{\frac{2}{3}} u \\ 0.25 x_{2} + 0.2 x_{1}^{\frac{2}{3}} u \end{matrix} \\ h_{1} (x^{-} (t), u (t)) = 0.2 x^{-} (t) 0.2 x (t) u (t) \\ h_{2} (x^{-} (t), u (t)) = 0.4 x^{-} (t) - 0.15 x (t) u (t) \\ h_{3} (x^{-} (t), u (t)) = 0.3 x^{-} (t) + 0.1 x (t) u (t) \\ h_{4} (x^{-} (t), u (t)) = 0.15 x^{-} (t) + 0.4 x (t) u (t) \end{matrix}

Choosing $V = | x |$ , if $| u_{t} | \leq \frac{1}{2} | x |^{\frac{1}{3}}$ ,we have $\nabla V (x) f (x, u) \leq - 0.7 V$ and $\nabla V (x) f (x, u) \leq 0.45 V$ .

From Theorem 3, we can get $c_{si} = 0.7$ and $c_{ui} = 0.45$ , then $ρ = 0.6 \leq 0.7 / (0.7 + 0.45)$ . Let $η^{+} = 0.5$ , $η_{0} = 2.2$ , then $η = 1.1$ , we may choose $τ_{i} = 0.98 > \ln 1.1 / (0.7 * 0.4 - 0.45 * 0.6)$ . When $u (t) = e^{- t}$ , the corresponding state trajectory is portrayed in Figures 5 and 6, showing the impulsive and switching signals.

Remark 9. Similarly in Example 3, because there exists unstable subsystems, the time of state converging to zero becomes longer, but figures tend towards equilibrium eventually, and four impulses emerge repeatedly.

Conclusions

In this paper, three theorems are presented concerning three cases where all subsystems are stable, some subsystems are not stable, and all subsystems are not stable. Impulsive effects at switching instants using a Lyapunov function method for impulsive switched nonlinear systems. The ISS property can be retained if the impulsive switching law satisfies a lower bound of time intervals or an upper bound time intervals. On the other hand, if the subsystems are unstable, impulses can successfully stabilize the system in the ISS sense. Examples have been given to show that the results are effective and have proved that the conclusions have a better performance than the existing results.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the NNSF of China (grant numbers 61403228 and 61304066), China Postdoctoral Science Foundation (grant number 2015M580579), Program for New Century Excellent Talents in University (grant number NCET-13- 0878) and the Taishan Scholar Project of Shandong Province of China.

References

Bainov

Simeonov

(1989) Systems with Impulse Effects: Stability, Theory and Applications. Chichester: Ellis Horwood.

Chen

Wang

Tang

et al . (2008) Robust H_∞ control of uncertain linear impulsive stochastic systems. International Journal of Robust and Nonlinear Control 18: 1348–1371.

Chen

Zheng

(2011) Exponential stability of nonlinear time-delay systems with delayed impulse effects. Automatica 45: 1075–1083.

Dashkovskiy

Kosmykov

Mironchenko

et al . (2012) Stability of interconnected impulsive systems with and without time delays using Lyapunov methods. Nonlinear Analysis: Hybrid Systems 6: 899–915.

Dashkovskiy

Feketa

(2016) Input-to-state stability of impulsive systems with different jump maps. IFAC-Papers Online 49(18): 1073–1078.

Gao

Wang

(2015) Further results on exponential stability for impulsive switched nonlinear time-delay systems with delayed impulse effects. Applied Mathematics and Computation 268: 186–200.

Gao

Cai

(2016) Finite-time stability of time-delay switched systemswith delayed impulse effects. Circuits, Systems, and Signal Processing 35(9): 3135–3151.

Gao

Wang

(2016) Input-to-state stability and integral input-to-state stability for impulsive switched systems with time-delay under asynchronous switching. Nonlinear Analysis: Hybrid Systems 20: 55–71.

Gao

Jiang

Wang

(2016) Observer-based robust finite time H sliding mode control for Markovian switching systems with mode-dependent time-varying delay and incomplete transition rate. ISA Transactions 61: 29–48.

10.

Gao

Yang

Shen

et al . (2015) Input-to-state stability analysis for a class of nonlinear switched descriptor systems. International Journal of Robust and Nonlinear Control 46: 2973–2981.

11.

Hespanha

Morse

(1999) Stability of switched systems with average dwell time. In: Proceedings of the 38th conference on decision and control, pp. 2655–2660.

12.

Hespanha

(2004) Uniform stability of switched linear systems: Extension of LaSalle’s invariance principle. IEEE Transactions on Automatic Control 49: 470–482.

13.

Hespanha

Liberzon

Teel

(2008) Lyapunov conditions for input-to-state stability of impulsive systems. Automatica 44: 2735–2744.

14.

Haddad

Chellaboina

Nersesov

(2006) Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. Princeton, NJ: Princeton University.

15.

DWC

Liang

Lam

(2006) Global exponential stability of impulsive high order BAM neural networks with time-varying delays. Neural Networks 19: 1581–1590.

16.

Khalil

(2002) Nonlinear Systems. 3rd ed.Englewood Cliffs, NJ: Prentice-Hall.

17.

Liberzon

Hespanha

Morse

(1999) Stability of switched systems: A lie-algebraic condition. Systems and Control Letters 37: 117–122.

18.

Liu

Xie

(2011) Input-to-state stability of impulsive and switching hybrid systems with time-delay. Automatica 47: 899–908.

19.

Liu

Kao

Karimi

et al . (2016) Input-to-state stability for discrete-time nonlinear switched singular systems. Information Sciences 358–359: 18–28.

20.

Müller

Liberzon

(2012) Input/output-to-state stability and state-norm estimators for switched nonlinear systems. Automatica 48: 2029–2039.

21.

Naghshtabrizi

Hespanha

Teel

(2008) Exponential stability of impulsive systems with application to uncertain sampled-data systems. Systems and Control Letters 57: 378–385.

22.

Nesic

Teel

(2004) Input-to-state stability of networked control systems. Automatica 40: 2121–2128.

23.

Perestyuk

Feketa

(2011) Invariant sets of impulsive differential equations with particularities in -limit set. Abstract and Applied Analysis 2011: 249–265.

24.

Perestyuk

Kapustyan

(2012) Long-time behavior of evolution inclusion with non-damped impulsive effects. Memoirs on Differential equations and Mathematical Physics 52: 89–113.

25.

Sontag

(1989) Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control 34: 93–100.

26.

van de Wouw

Naghshtabrizi

Cloosterman

MBG

et al .(2010) Tracking control for sampled-data systems with uncertain time-varying sampling intervals and delays. International Journal of Robust and Nonlinear Control 20: 387–411.

27.

Wang

Gao

Zong

(2015) Input-to-state stability of impulsive switched hybrid time-delay systems with delayed impulse effects. Transactions of the Institute of Measurement and Control 37: 970–982.

28.

Wang

Zhao

(2013) Input-to-state stability of non-linear impulsive and switched delay systems. Control Theory and Applications, IET 7: 1179–1185.

29.

Wang

Liu

(2010) Global synchronization for discrete-time stochastic networks with randomly occured nonlinearities and mixed time delays. IEEE transactions on Neural Network 21: 11–25.

30.

Tang

Zhang

(2016) Input-to-state stability of impulsive stochastic delayed systems under linear assumptions. Automatica 66: 195–204.

31.

Xiang

Xiao

(2013) H_∞ filtering for switched non-linear systems with polytopic uncertainties via filter state impulsive jump approach. Transactions of the Institute of Measurement and Control 35: 540–550.

32.

Yang

Jiang

Cocquempot

(2014) A survey of results and perspectives on stabilization of switched nonlinear systems with unstable modes. Nolinear Analysis: Hybrid Systems 13: 45–60.

33.

Yang

(2001) Impulsive Control Theory. Heidelberg: Springer-Verlag Berlin.

34.

Zhao

Hill

(2008) Passivity theory for switched systems. IEEE Transactions on Automatic Control 53: 941–953.

35.

Zhu

(2014) pth Moment exponential stability of impulsive stochastic functional dierential equations with Markovian switching. Journal of The Franklin Institute 351(7): 3965–3986.

36.

Zhu

Cao

(2010) Robust exponential stability of Markovian jump impulsive stochastic Cohen-Grossberg neural networks with mixed time delays. IEEE Transactions on Neural Networks 21(8): 1314–1325.

37.

Zhu

Cao

(2012) Stability analysis of Markovian jump stochastic BAM neural networks with impulse control and mixed time delays. IEEE Transactions on Neural Networks and Learning Systems 23(3): 467–479.

38.

Zhang

(2016) Adaptive motion and force tracking control for nonholonomic dynamic systems subject to affine constraints. Transactions of the Institute of Measurement and Control 38(4): 482–491.

39.

Zhang

Huang

(2017) Differential-flatness-based finite-time anti-swing control of underactuated crane systems. Nonlinear Dynamics 87(3): 1749–1761.