Input-to-state stability of time-varying nonlinear systems with mode-dependent random delayed impulses

Abstract

This paper investigates the input-to-state stability (ISS) problem for a type of time-varying nonlinear systems with random delayed impulses. On the basis of the Lyapunov method, it provides sufficient conditions to obtain the ISS properties by indefinite Lyapunov functions with time-varying conditions. Besides, the mixed effect of random and delayed impulse is also fully considered. On this basis, we examine the ISS issue for impulsive systems in two different scenarios. One scenario is that the impulsive intensity is random, and the impulsive intensity is limited by the average impulsive interval (AII) and the mode-dependent average impulsive interval (MDAII). The other scenario is that the impulsive intensity is random, and the impulsive density satisfies certain stochastic conditions. Finally, two examples are provided to test the validity of the stability criterion proposed.

Keywords

the input-to-state stability the time-varying nonlinear systems random delayed impulses

Introduction

Hybrid systems, consisting of combinations of continuous and discrete dynamic systems, have examined their importance in control systems. Impulsive systems, as a type of hybrid system, consist of instantaneous state jumps and continuous time dynamic systems. There are a very wide range of applications for impulsive systems, such as multi-agent systems (Gu et al., 2026; Xiao et al., 2025), synchronization of complex dynamic networks (Liu and Li, 2026; Wan et al., 2025; Zhang et al., 2024), and Markov jump systems (Hua et al., 2025, 2022; Wu et al., 2024). During the course of development, systems naturally generate random factors (Hu and Mu, 2022) while also inevitably being subject to random disturbances from the external environment. Indeed, networked multi-intelligence systems can be susceptible to random disturbances at certain moments (Tang et al., 2016). In fact, qualitative description of impulses can limit to some extent the application of impulsive systems. In some dynamic networks, sampling delays occur at certain impulsive instants due to disturbances in the transmission process, which leads to the fact that the current state of the system will depend on the historical state of the sampling point (Cui et al., 2022; He et al., 2025). Currently, most of the existing research on impulsive systems still focuses on deterministic impulsive signals (Hespanha et al., 2008) or impulsive signals conforming to specific distributions (Tang et al., 2020; Teel et al., 2014). However, in the practical applications, not all impulses can satisfy specific regulations. In addition, during the transmission of impulsive information, time delay is unavoidable. For example, in communication security systems based on impulsive synchronization, both transmission and sampling involve time delay (Khadra et al., 2005). Therefore, it makes sense to consider random delayed impulses. Recently, the paper by Gao et al. (2025) analyzed the stability of impulsive systems with stochastic delayed impulses.

Stability analysis has been among the main concerns in the area of impulsive systems. The notion of input-to-state stability (ISS) was first proposed by Sontag (1989), and many excellent works have emerged under this concept since it portrays well how external inputs and initial values affect the stability of systems (Hua et al., 2023, 2021; Liu et al., 2022; Zheng and Zhu, 2025). In particular, for impulsive systems with stable and unstable impulsive effects, the article by Lu et al. (2010) proposes the use of the average impulsive interval (AII) approach to deal with the ISS problem. The concept of AII implicitly limits the average impulsive interval at which impulsive behavior occurs to a fixed interval, which limits the effects of different impulses to some extent. However, when jumping from different impulsive modes to the same impulsive mode, the resulting average impulsive intervals may vary due to the complexity of the impulsive behavior. Considering the different impulsive modes and the conservatism in the AII method, the literature (Xie et al., 2019) used the method of the mode-dependent average impulsive interval (MDAII) to establish the stability criterion for impulsive system. Besides, the time-delay effect can also have an impact on the stability of the system. With the development of impulse control theory, we know that the time delay exists not only in the part of the system under study (De La Sen and Luo, 2003) but also in the part of the impulse. The impulse that contains time delay is sometimes referred to as delayed impulse. This impulse describes a phenomenon where the impulsive transient is dependent on the historical state of the impulsive system. In the paper by Liu et al. (2011), hybrid time-delayed systems under delayed impulses were investigated.

From previous studies, we know (Wu et al., 2016a) that it is of interest to study the coefficients of the time-varying Lyapunov differential operator. Up to now, most of the available results investigating the effect of impulses on ISS in a stochastic infinite-dimensional system are still based on Lyapunov functions with constant rate coefficients. For example, the paper by Zhang and Zhu (2019) studied the ISS problem with deterministic delayed impulses, where the Lyapunov differential operators have constant coefficients. Although Tang et al. (2020) assessed the effect of random impulses on ISS, the Lyapunov differential operator’s coefficients are constant coefficients. However, the hypothesis of the Lyapunov differential operator coefficients being constant has not always held in reality. Owing to fluctuations in the performance of the system, the Lyapunov function’s rate coefficients may be time-varying (Liu, 2012), meaning that the evolution over time may be positive or negative (Ning et al., 2012; Peng and Zhang, 2010; Wu et al., 2012; Zhou, 2017). Recently, ISS and iISS of nonlinear systems with time-varying rate coefficients of the Lyapunov function have been considered in Ning et al. (2012) and Zhou (2017). The article by Liu et al. (2022) presents ISS properties of nonlinear systems with delayed impulses in which the coefficients of the Lyapunov differential operator are time-varying.

Inspired by the above-mentioned studies, the primary aim of this paper is to investigate the ISS properties of continuous time-varying nonlinear systems with random delayed impulses. The major content of this paper is shown as follows:

In this paper, the Lyapunov function is uncertain, that is, the Lyapunov function has time-varying rate coefficients, which is more common than constant coefficients. In addition, we also consider the influence of delayed impulses on the stability of the system, which was ignored in Tang et al. (2020).

We examine the ISS problem of nonlinear impulsive systems in two different cases. The first case considers random impulsive intensities and deterministic intervals by using AII and MDAII. The second case is that both impulsive intensities and impulsive intervals are random.

Some easily verifiable conditions have been established to facilitate ISS testing of nonlinear time-varying systems with random delayed impulses.

The rest of this paper has the following structure: Section “Problem formulation and preliminaries” presents some essential definitions and concepts. In section “Main results,” we assess the ISS problem for dynamical systems with random impulsive intensities and deterministic instantaneous impulses, and the ISS problem is treated with the consideration that the impulsive intensities and instants are random at the same time. Section “Simulation examples” gives two examples of our results and verifies the validity of our work. Section “Conclusions” summarizes the research in this paper.

Notations

Notation	Significance
$R$	$(- \infty, + \infty)$
$R^{+}$	$[0, + \infty)$
$N$	${0, 1, 2, \dots}$
$R^{p}$	The p-dimensional Euclidean space
$R^{p \times q}$	The $p \times q -$ dimensional Euclidean space
Γ	${1, 2, . . ., ξ}$ , $ξ > 0$ is a finite integer
\|⋅\|	The Euclidean vector norm
$E (\cdot)$	The stochastic expectation
$J$	An interval in $R$
$O$	${ϰ : R^{+} \to R^{+} \| ϰ$ is continuous and $ϰ (0) = 0}$
$K$	${ϰ \in O \| ϰ$ is strictly increasing }
$K_{\infty}$	${ϰ \in K \| ϰ$ is unbounded }
$υ K (υ K_{\infty})$	${ϰ \in K (K_{\infty}) \| ϰ$ is convex }
$C K (C K_{\infty})$	${ϰ \in K (K_{\infty}) \| ϰ$ is concave }
$L$	${ϰ : R^{+} \to R^{+} \| ϰ$ is continuous that decreases
	strictly to 0 as $ı \to \infty}$
$KL$	${ϰ : R^{+} \times R^{+} \to R^{+} \| ϰ$ is continuous, $ϰ (\cdot, ı) \in K$ ,
	$ϰ (r, \cdot) \in L, \forall s \geq 0, \forall ı \geq 0}$

$C_{1}$ represents the class of non-negative differentiable functions $V$ on $R^{q} \to R^{+}$ . Define $(Ω, F, P)$ to be a complete probability space with a satisfying increasing and right-continuous filtration ${F_{ı}}_{{ı \geq 0}}$ . $PC (J; R^{p})$ is the set of piecewise continuous functions mapped from $J$ to $R^{p}$ .

Problem formulation and preliminaries

Throughout this paper, the following nonlinear time-varying systems with random delayed impulsive systems are considered:

\begin{matrix} {\begin{matrix} ẇ (ı) = Ψ (ı, w (ı), v (ı)), ı \neq ı_{l}, ı \geq 0; \\ w (ı) = Φ (w (ı - ε), v (ı^{-})), ı = ı_{l}, l \in Z^{+}, \end{matrix} \end{matrix}

(1)

where $w (ı) \in R^{p}$ defines the system state and $v (ı) \in R^{q}$ denotes a measurable locally bounded input; assume that the state $w (ı)$ and the input $v (ı)$ are right-continuous and have a left limit at all times; $ε > 0$ is the time delay; the sequence of impulse instants ${ı_{l}, l \in N}$ fulfills $0 \leq ı_{0} < ı_{1} < ı_{2} < \cdot \cdot \cdot$ and $\lim_{l \to + \infty} ı_{l} = + \infty$ ; the function $Ψ : R^{+} \times R^{p} \times R^{q} \to R^{p}$ is locally Lipschitz function and $Ψ (ı, 0, 0) = 0$ for $ı \in R^{+}$ implying that $w_{0} = w (ı_{0})$ is the initial value; for each $r (ı) = i \in Γ$ , the function $Φ_{i} : R^{p} \times R^{q} \to R^{p \times q}$ and $Φ_{i} (0, 0) = 0$ for $ı \in R^{+}$ ; we hypothesize that there always exists a unique solution $w (ı)$ .

Definition 1. (Wu et al., 2016b). If there are functions $β \in KL$ and $a, Υ \in K_{\infty}$ such that

\begin{matrix} a (E ∣ w (ı) ∣) \leq β (E | w_{0} |, ı - ı_{0}) + Υ (| v (ı) |_{[ı_{0}, ı]}) . \end{matrix}

(2)

Then, the system (1) with random delayed impulses is called ISS.

Lemma 1. (Zhou, 2016). If there are constants $λ > 0$ and $M_{0} \geq 0$ such that

\begin{matrix} \int_{ı_{0}}^{ı} ψ (σ) dσ \leq - λ (ı - ı_{0}) + M_{0}, \end{matrix}

(3)

where $M_{0}$ is independent of $ı_{0}$ . Then, the function $ψ (ı) \in PC (R^{+}, R)$ is $UES [WGDR λ]$ .

Definition 2. (Zhou, 2016). The function $ψ (ı) \in PC (R^{+}, R)$ is said to be uniformly exponentially stable with guaranteed decay rate λ (denoted as $UES [WGDR λ]$ ) if the following scalar linear time-varying system is uniformly exponentially stable with guaranteed decay rate λ:

\begin{matrix} ẏ (ı) = ψ (ı) y (ı), ı \geq ı_{0} . \end{matrix}

(4)

that is, there exist constants $γ_{0}$ ; $λ > 0$ such that

\begin{matrix} | y (ı) | \leq γ_{0} | y_{0} | \exp (- λ (ı - ı_{0})) . \end{matrix}

(5)

where $γ_{0}$ is independent of $ı_{0}$ .

Assumption 1. There is a sequence of mutually independent positive random variables ${μ_{i}}_{i \in Γ}$ with $E μ_{i} = θ_{i}$ , we let $θ = \max_{i \in Γ} θ_{i}$ .

Definition 3. (Tang et al., 2020). For the instant of occurrence, the sequence of impulses ${ı_{k}, k \in N_{+}}$ :

1. if there is $Γ_{a} > 0$ , $N_{0} \geq 0$ that satisfy

\frac{ı - s}{Γ_{a}} - N_{0} \leq N (ı, s) \leq \frac{ı - s}{Γ_{a}} + N_{0},

where $N (s, ı)$ represents the number of instants in $(s, ı]$ , then $Γ_{a}$ is said to be the AII and $N_{0}$ is said to be the elasticity number, and

2. if for the ith kind of impulsive intensity of the impulsive effects, there exist $Γ_{ai} > 0$ , $N_{0 i} \geq 0$ that satisfy

\frac{ı - s}{Γ_{ai}} - N_{0 i} \leq N_{i} (ı, s) \leq \frac{ı - s}{Γ_{ai}} + N_{0 i}, i \in Γ,

where $N_{i} (ı, s)$ represents the number of impulses for which the ith mode of the impulsive signal $r (ı)$ is activated during $(ı, s]$ , and then $Γ_{ai}$ is said to be the MDAII and $N_{0 i}$ is called the elasticity number for the ith kind of impulsive intensity $(i \in Γ)$ .

Assumption 2. We suppose that the impulsive effects can be produced by δ different distributions. The incidence of different distributions of impulsive effects is determined by the AII, and the incidence of the same distributions of impulsive effects is determined by the MDAII. The sequence ${r (ı_{k})}_{k \in N_{+}}$ denotes which the impulsive intensity ${μ_{i}}_{(i \in Γ)}$ will occur at each impulsive instant $ı_{k}$ .

Remark 1. The impulsive effect of a distribution can be any distribution type we know or discrete data, as long as the expectation of existence is satisfied.

Main results

ISS with only random impulsive intensities

Theorem 1. It is assumed that there exist Lyapunov functions $V \in C_{1}, \partial_{1} \in V K_{\infty}, \partial_{2}, ρ \in K_{\infty}$ , continuous functions $ψ : R^{+} \to R$ , constants $M_{0} \geq, λ > 0$ such that for $ı \geq ı_{0}$ ,

\begin{matrix} (H_{1}) \partial_{1} (| w (ı) |) \leq V (ı, w (ı)) \leq \partial_{2} (| w (ı) |); \\ (H_{2}) ▽ V (ı, w (ı)) \cdot Ψ (ı, w (ı), v (ı)) \leq ψ (ı) V (ı, w (ı)) + ρ (v (ı)); \\ (H_{3}) V (ı, w (ı)) \leq μ_{i} V (ı, w (ı - ε)), ı \in [ı_{k}, ı_{k + 1}); \\ (H_{4}) \sum_{i = 1}^{δ} \frac{\ln θ_{i}}{Γ_{ai}} + \frac{λε + M_{0}}{Γ_{a}} - λ < 0, \end{matrix}

where $θ_{i} = E μ_{r (i)}, ψ (ı)$ is $UES [WGDR λ]$ , $Γ_{a}$ and $Γ_{ai}$ are AII and MDAII, respectively.

Then, the nonlinear time-varying system (1) with the random delayed impulses is ISS.

Proof. Since $\partial_{1} \in V K_{\infty}, \partial_{2}, ρ \in K_{\infty}$ , from condition $(H_{1})$ and the Jensen’s inequality in Rogers and Williams (2000), for $ı \geq ı_{0},$

\begin{matrix} \partial_{1} (E | w (ı) |) \leq E \partial_{1} (| w (ı) |) \leq E V (ı, w (ı)) \end{matrix}

(6)

and

\begin{matrix} E \partial_{2} (| w_{0}) |) \leq \partial_{2} (E | w_{0} |), \end{matrix}

(7)

where $w_{0} = w (ı_{0})$ .

Part I: In this part, the following inequality will be proved by mathematical induction for all $ı \geq ı_{0}$

\begin{matrix} E V (ı, w (ı)) \leq E V (ı_{0}, w_{0}) \prod_{m = 1}^{N (ı, ı_{0})} E μ_{m} \exp {\int_{ı_{0}}^{ı} ψ (ς) dς} \\ \times \prod_{m = 1}^{N (ı, ı_{0})} \exp {\int_{ı_{m}}^{ı_{m} - ε} ψ (ς) dς} + \sum_{p = 0}^{N (ı, ı_{0}) - 1} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) \\ \prod_{q = p + 1}^{N (ı, ı_{0})} E μ_{q} \exp {\int_{ς}^{ı} ψ (σ) dσ} \prod_{q = p + 1}^{N (ı, ı_{0})} \exp {\int_{ı_{q}}^{ı_{q} - ε} ψ (σ) dσ} dς} \\ + & \int_{ı_{N (ı, ı_{0})}}^{ı} ρ (| v (ς) |) \exp {\int_{ς}^{ı} ψ (σ) dσ} dς . \end{matrix}

(8)

Step $A_{1}$ : From condition $(H_{2})$ , it can be obtained for $ı \in [ı_{0}, ı_{1})$ ,

\begin{matrix} E V (ı, w (ı)) & \leq E V (ı_{0}, w_{0}) \exp {\int_{ı_{0}}^{ı} ψ (ς) dς} \\ + \int_{ı_{0}}^{ı} ρ (| v (ς) |) \exp {\int_{ς}^{ı} ψ (σ) dσ} dς . \end{matrix}

(9)

Thus, we obtain the inequality (8) for $ı \in [ı_{0}, ı_{1})$ .

Step $A_{2}$ : By mathematical induction, assume that (8) holds for $ı \in [ı_{k - 1}, ı_{k})$ ,

\begin{matrix} E V (ı, w (ı)) \leq E V (ı_{0}, w_{0}) \prod_{m = 1}^{k - 1} E μ_{m} \exp {\int_{ı_{0}}^{ı} ψ (ς) dς} \\ \times \prod_{m = 1}^{k - 1} \exp {\int_{ı_{m}}^{ı_{m} - ε} ψ (ς) dς} + \sum_{p = 0}^{k - 2} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) \\ \prod_{q = p + 1}^{k - 1} E μ_{q} \exp {\int_{ς}^{ı} ψ (σ) dσ} \\ \times \prod_{q = p + 1}^{k - 1} \exp {\int_{ı_{q}}^{ı_{q} - ε} ψ (σ) dσ} dς} \\ + & \int_{ı_{k - 1}}^{ı} ρ (| v (ς) |) \exp {\int_{ς}^{ı} ψ (σ) dσ} dς . \end{matrix}

(10)

Since $ı_{l} - ı_{l - 1} > ε$ , for any $l \in Γ$ , we know $ı_{k} - ε \in [ı_{k - 1}, ı_{k})$ . From inequality (10), we can get

\begin{matrix} E V (ı_{k} - ε, w (ı_{k} - ε)) \\ \leq E V (ı_{0}, w_{0}) \prod_{m = 1}^{k - 1} E μ_{m} \exp {\int_{ı_{0}}^{ı_{k} - ε} ψ (ς) dς} \\ \times \prod_{m = 1}^{k - 1} \exp {\int_{ı_{m}}^{ı_{m} - ε} ψ (ς) dς} + \sum_{p = 0}^{k - 2} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) \\ \prod_{q = p + 1}^{k - 1} E μ_{q} \exp {\int_{ς}^{ı_{k} - ε} ψ (σ) dσ} \\ \times \prod_{q = p + 1}^{k - 1} \exp {\int_{ı_{q}}^{ı_{q} - ε} ψ (σ) dσ} dς} \\ + \int_{ı_{k}}^{ı} ρ (| v (ς) |) \exp {\int_{ς}^{ı} ψ (σ) dσ} dς . \end{matrix}

(11)

Step $A_{3}$ : we will demonstrate that inequality (8) is true for $ı \in [ı_{k}, ı_{k + 1})$ . From conditions $(H_{3})$ and inequality (11), we obtain

\begin{matrix} E V (ı_{k}, w (ı_{k})) \leq E μ_{r (ı_{k})} E V (ı_{k} - ε, w (ı_{k} - ε)) \\ \leq E V (ı_{0}, w (τ_{0})) \prod_{m = 1}^{k} E μ_{m} \exp {\int_{ı_{0}}^{ı_{k} - ε} ψ (ς) dς} \\ \prod_{m = 1}^{k - 1} \exp {\int_{ı_{m}}^{ı_{m} - ε} ψ (ς) dς} \\ + \sum_{p = 0}^{k - 2} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) \prod_{q = p + 1}^{k} E μ_{q} \exp {\int_{ς}^{ı_{k} - ε} ψ (σ) dσ} \\ \times \prod_{q = p + 1}^{k - 1} \exp {\int_{ı_{q}}^{ı_{q} - ε} ψ (σ) dσ} dς} \\ + E μ_{r (ı_{k})} \int_{ı_{k}}^{ı_{k} - ε} ρ (| v (ς) |) \exp {\int_{ς}^{ı} ψ (σ) dσ} dς . \end{matrix}

(12)

From condition $(H_{2})$ and inequality (12), it can be obtained for $ı \in [ı_{k}, ı_{k + 1})$

\begin{matrix} E V (ı, w (ı)) \leq E V (ı_{k}, w (ı_{k})) \exp {\int_{ı_{k}}^{ı} ψ (ς) dς} \\ + \int_{ı_{k}}^{ı} ρ (| v (ς) |) \exp {\int_{ς}^{ı} ψ (σ) dσ} dς \\ \leq E V (ı_{0}, w (ı_{0})) \prod_{m = 1}^{k} E μ_{m} \exp {\int_{ı_{0}}^{ı} ψ (ς) dς} \prod_{m = 1}^{k} \exp {\int_{ı_{m}}^{ı_{m} - ε} ψ (ς) dς} \\ + \sum_{p = 0}^{k - 1} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) \prod_{q = p + 1}^{k} E μ_{q} \exp {\int_{ς}^{ı} ψ (σ) dσ} \\ \times \prod_{q = p + 1}^{k} \exp {\int_{ı_{q}}^{ı_{q} - ε} ψ (σ) dσ} dς} \\ + \int_{ı_{k}}^{ı} ρ (| v (ς) |) \exp {\int_{ς}^{ı} ψ (σ) dσ} dς . \end{matrix}

(13)

Thus, when $ı \geq ı_{0}$ , the inequality (8) is proved by mathematical induction.

Part II: The ISS of the system (1) is presented in this part. Therefore, for any $ı \geq ı_{0}$ , we obtain

\begin{matrix} E V (ı, w (ı)) \\ \leq & E V (ı_{0}, w_{0}) \prod_{m = 1}^{N (ı, ı_{0})} E μ_{m} \exp {\int_{ı_{0}}^{ı} ψ (ς) dς} \prod_{m = 1}^{N (ı, ı_{0})} \exp {\int_{ı_{m}}^{ı_{m} - ε} ψ (ς) dς} \\ + \sum_{p = 0}^{N (ı, ı_{0}) - 1} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) \prod_{q = p + 1}^{N (ı, ı_{0})} E μ_{q} \exp {\int_{ς}^{ı} ψ (σ) dσ} \\ \times \prod_{q = p + 1}^{N (ı, ı_{0})} \exp {\int_{ı_{q}}^{ı_{q} - ε} ψ (σ) dσ} dς} \\ + \int_{ı_{N (ı, ı_{0})}}^{ı} ρ (| v (ς) |) \exp {\int_{ς}^{ı} ψ (σ) dσ} dς = Ψ_{1} + Ψ_{2}, \end{matrix}

(14)

where

\begin{matrix} Ψ_{1} = E V (ı_{0}, w_{0}) \prod_{m = 1}^{N (ı, ı_{0})} E μ_{r (m)} \exp {\int_{ı_{0}}^{ı} ψ (ς) dς} \\ \prod_{m = 1}^{N (ı, ı_{0})} \exp {\int_{ı_{m}}^{ı_{m} - ε} ψ (ς) dς}, \\ Ψ_{2} = & \sum_{p = 0}^{N (ı, ı_{0}) - 1} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) \prod_{q = p + 1}^{N (ı, ı_{0})} E μ_{q} \exp {\int_{ς}^{ı} ψ (σ) dσ} \\ \times \prod_{q = p + 1}^{N (ı, ı_{0})} \exp {\int_{ı_{q}}^{ı_{q} - ε} ψ (σ) dσ} dς} \\ + \int_{ı_{N (ı, ı_{0})}}^{ı} ρ (| v (ς) |) \exp {\int_{ς}^{ı} ψ (σ) dσ} dς . \end{matrix}

Based on Definition 3 of $N (ı, s)$ and $N_{i} (ı, s)$ , Assumption 1, Assumption 2, Lemma 1, and condition $(H_{4})$ , we get

\begin{matrix} Ψ_{1} = & E V (ı_{0}, w_{0}) \prod_{m = 1}^{N (ı, ı_{0})} E μ_{r (m)} \exp {\int_{ı_{0}}^{ı} ψ (ς) dς} \\ \prod_{m = 1}^{N (ı, ı_{0})} \exp {\int_{ı_{m}}^{ı_{m} - ε} ψ (ς) dς} \\ \leq & E V (ı_{0}, w_{0}) \prod_{i = 1}^{δ} θ_{i}^{N_{i} (ı, ı_{0})} e^{- λ (ı - ı_{0}) + M_{0}} \prod_{i = 1}^{N (ı, ı_{0})} e^{λε + M_{0}} \\ \leq & E V (ı_{0}, w_{0}) e^{M_{0} + (λε + M_{0}) N_{0}} e^{(\frac{λε + M_{0}}{Γ_{a}} - λ) (ı - ı_{0})} \\ \times e^{{\sum_{i = 1}^{δ} | \ln θ_{i} | N_{0 i} + \sum_{i = 1}^{δ} \frac{\ln θ_{i}}{Γ_{ai}} (ı - ı_{0})}} \\ \leq & ϖ_{1} E V (ı_{0}, w_{0}) e^{κ (ı - ı_{0})}, \end{matrix}

(15)

where

\begin{matrix} ϖ_{1} = \exp {\sum_{i = 1}^{δ} | \ln θ_{i} | N_{0 i} + (λε + M_{0}) N_{0} + M_{0}} \end{matrix}

and

\begin{matrix} κ = \sum_{i = 1}^{δ} \frac{\ln θ_{i}}{Γ_{ai}} + \frac{λε + M_{0}}{Γ_{a}} - λ . \end{matrix}

In addition, from Assumption 1, Assumption 2, and Lemma 1, it is easy to obtain

\begin{matrix} Ψ_{2} = \sum_{p = 0}^{N (ı, ı_{0}) - 1} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) \prod_{q = p + 1}^{N (ı, ı_{0})} E μ_{q} \exp {\int_{ς}^{ı} ψ (σ) dσ} \\ \times \prod_{q = p + 1}^{N (ı, ı_{0})} \exp {\int_{ı_{q}}^{ı_{q} - ε} ψ (σ) dσ} dς} \\ + \int_{ı_{N (ı, ı_{0})}}^{ı} ρ (| v (ς) |) \exp {\int_{s}^{ı} ψ (σ) dσ} dς \\ \leq \sum_{p = 0}^{N (ı, ı_{0}) - 1} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) e^{- λ (ı - s) + M_{0}} \prod_{i = 1}^{δ} θ_{i}^{N_{i} (ı_{p + 1}, ı)} \\ \times \prod_{i = 1}^{N (ı, ı_{p + 1})} e^{λε + M_{0}} dς} + \int_{ı_{N (ı, ı_{0})}}^{ı} ρ (| v (ς) |) \exp {\int_{ς}^{ı} ψ (σ) dσ} dς \\ \leq \sum_{p = 0}^{N (ı, ı_{0}) - 1} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) e^{{\sum_{i = 1}^{δ} | \ln θ_{i} | N_{0 i} + (λε + M_{0}) N_{0} + M_{0}}} \\ \times e^{(\sum_{i = 1}^{δ} \frac{\ln θ_{i}}{Γ_{ai}} + \frac{λε + M_{0}}{Γ_{a}} - λ) (ı - ı_{p + 1})} dς} + \int_{ı_{N (ı, ı_{0})}}^{ı} ρ (| v (ς) |) e^{- λ (ı - ς) + M_{0}} dς \\ \leq \sum_{p = 0}^{N (ı, ı_{0}) - 1} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) e^{{\sum_{i = 1}^{δ} | \ln θ_{i} | N_{0 i} + (λε + M_{0}) N_{0} + M_{0}}} dς} \\ + \int_{ı_{N (ı, ı_{0})}}^{ı} e^{M_{0}} ρ (| v (ς) |) dς \\ \leq \int_{ı_{0}}^{ı_{N (ı, ı_{0})}} ρ (| v (ς) |) e^{{\sum_{i = 1}^{δ} | \ln θ_{i} | N_{0 i} + (λε + M_{0}) N_{0} + M_{0}}} dς \\ + \int_{ı_{N (ı, ı_{0})}}^{ı} ρ (| v (ς) |) e^{M_{0}} dς \end{matrix}

(16)

\begin{matrix} \leq ϖ_{1} \int_{ı_{0}}^{ı} ρ (| v (ς) |) dς \\ \leq ϖ_{1} ρ (| v (ς) |_{[ı_{0}, ı]}) . \end{matrix}

From inequalities (16) and (15), we can conclude that

\begin{matrix} E V (ı, w (ı)) & = Ψ_{1} + Ψ_{2} \leq ϖ_{1} E V (ı_{0}, w_{0}) e^{κ (ı - ı_{0})} \\ + ϖ_{1} ρ (| v (ς) |_{[ı_{0}, ı]}), \end{matrix}

where

\begin{matrix} ϖ_{1} = \exp {\sum_{i = 1}^{δ} | \ln θ_{i} | N_{0 i} + (λε + M_{0}) N_{0} + M_{0}} \end{matrix}

and

\begin{matrix} κ = \sum_{i = 1}^{δ} \frac{\ln θ_{i}}{Γ_{ai}} + \frac{λε + M_{0}}{Γ_{a}} - λ < 0 . \end{matrix}

Based on condition $(H_{1})$ and inequality (7), we get

\begin{matrix} \partial_{1} (E | w (ı) |) \leq & E V (ı, w (ı)) \\ \leq & ϖ_{1} E V (ı_{0}, w_{0}) e^{κ (ı - ı_{0})} + ϖ_{1} ρ (| v (ς) |_{[ı_{0}, ı]}), \\ \leq & ϖ_{1} \partial_{2} (| w_{0} |) e^{κ (ı - ı_{0})} + ϖ_{1} ρ (| v (ς) |_{[ı_{0}, ı]}) . \end{matrix}

(17)

which means that

\begin{matrix} E | w (ı) | & \leq \partial_{1}^{- 1} [ϖ_{1} \partial_{2} (| w_{0} |) e^{κ (ı - ı_{0})} + ϖ_{1} ρ (| v (ς) |_{[ı_{0}, ı]})] \\ \leq β (| w_{0} |, ı - ı_{0}) + Υ (| v (ς) |_{[ı_{0}, ı]}), \end{matrix}

where $β (| w_{0} |, ı - ı_{0}) = \partial_{1}^{- 1} (ϖ_{1} \partial_{2} (| w_{0} |) e^{κ (ı - ı_{0})})$ is a $KL$ -function and $Υ (| v (ı) |_{[ı_{0}, ı]}) = \partial_{1}^{- 1} (ϖ_{1} ρ (| v (ς) |_{[ı_{0}, ı]}))$ is a $K_{\infty}$ -function. This completes the proof that the random delayed impulsive system (1) is ISS.

Remark 2. As we know, when $0 < μ_{i} < 1$ , the impulsive effect is a stable effect, and when $μ_{i} > 1$ , the impulsive effect shows an unstable effect. It is assumed that when $0 < θ_{i} < 1$ , the impulsive effect may be stable or unstable at some instants. As can be seen from the examples we have given, it does not matter whether the impulsive intensity $μ_{i}$ is $μ_{i} > 1$ or $0 < μ_{i} < 1$ , but it does not affect the judgment of our sufficient condition for stability.

ISS with random impulsive intensities and impulsive densities

Definition 4. (Tang et al., 2020). Assume that $Y_{n}, n \geq 1$ are non-negative, independent identically distributed random variable with a common distribution function Φ, where $Φ (0) = P {Y (n) = 0} < 1$ . Let $T_{0} = ı_{0}, and T_{n} = \sum_{i = 1}^{n} Y_{i}$ . Then, the counting process $ℕ (ı_{0}, ı) = \sup {n, T_{n} \leq ı}, ı \geq ı_{0}$ is known as a renewal process.

Assumption 3. We assume that the impulses with stochastic strength and stochastic instantly meet the conditions:

The impulsive instant ${ı_{k}}_{{k \in N}}$ is the kth update time of the renewal process $ℕ (ı, ı_{0})$ , which satisfies that $\exists S \in N \cup 0$ such that $\forall j \geq S$ , we have

\begin{matrix} P (ℕ (ı, ı_{0}) = j) \leq \frac{{(\bar{λ} (ı - ı_{0}))}^{j} e^{- \tilde{λ} (ı - ı_{0})}}{j!} . \end{matrix}

(18)

And the impulsive density $T_{k} = ı_{k} - ı_{k - 1}$ is an independent identically distributed random sequence with $E T_{k} = τ < \infty$ .

The impulsive intensity is a random variable, which is defined in Assumption 1.

The impulsive intensity $μ_{i}$ and impulsive interval $T_{k}$ are independent of each other.

Lemma 2. (Chatterjee and Liberzon, 2007). Suppose that $\exists S \in N \cup 0$ such that $\forall j \geq S$ , we have

\begin{matrix} P (ℕ (ı, ı_{0}) = j) \leq \frac{{(\bar{λ} (ı - ı_{0}))}^{j} e^{- \tilde{λ} (ı - ı_{0})}}{j!} \end{matrix}

(19)

holds. So, $W \geq 0$ such that the moment generating function $E [e^{sℕ (ı, ı_{0})}]$ of $ℕ (ı, ı_{0})$ fulfills

\begin{matrix} E [e^{sℕ (ı, ı_{0})}] \leq W + e^{(\bar{λ} e^{s} - \tilde{λ}) (ı - ı_{0})}, \forall s \geq 0 . \end{matrix}

(20)

Proof. Using condition (19), for $s \geq 0$ , a small amount of calculation will give us

\begin{matrix} E [e^{sℕ (ı, ı_{0})}] = & \sum_{j = 0}^{\infty} e^{sj} P (ℕ (ı, ı_{0}) = j) \\ \leq & \sum_{j = 0}^{S - 1} e^{sj} P (ℕ (ı, ı_{0}) = j) + \sum_{j = S}^{\infty} e^{sj} \frac{{(\bar{λı})}^{j} e^{- \tilde{λ} (ı - ı_{0})}}{j!} \\ \leq & W + e^{(\bar{λ} e^{s} - \tilde{λ}) (ı - ı_{0})}, \end{matrix}

where $W = \sum_{j = 0}^{S - 1} e^{sj} \geq 0 .$

Remark 3. Our renewal process $ℕ (ı, ı_{0})$ is defined in the same way as in Tang et al. (2020); it is a counting process. The left side of inequality (19) in Chatterjee and Liberzon (2007) refers to the probability measure of the number of event switches. Based on the definition in Tang et al. (2020), we find that we can apply Lemma 6 in Chatterjee and Liberzon (2007) to our renewal process $ℕ (ı, ı_{0})$ . Therefore, we generalize Lemma 6 to impulsive systems and give a simple proof.

Remark 4. Due to the fact that the left side of inequality (19) is a probability measure of an event, the right side can be substituted for $\min {\frac{{(\bar{λ} (ı - ı_{0}))}^{j} e^{- \tilde{λ} (ı - ı_{0})}}{j!}, 1}$ . In the special case $\bar{λ} = \tilde{λ}$ , the counting process $ℕ (ı_{0}, ı)$ is a Poisson process, and the right side of inequality (19) represents a probabilistic estimate of the amount of impulses that occur in the interval $[ı_{0}, ı]$ .

Theorem 2. Suppose that there are functions $\partial_{1} \in V K_{\infty}, \partial_{2}, ρ \in K_{\infty}$ , continuous functions $ψ : R^{+} \to R$ and a locally Lipschitz continuous function $V \in C_{1}$ such that conditions $(H_{1}) - (H_{3})$ in Theorem 1 are satisfied such that for all $ı \geq ı_{0}$ ,

(H′₄) (a) there are constants $η > 0$ and $M \geq 0$ such that $\int_{s}^{ı} ψ (σ) dσ + \sum_{l = 1}^{ℕ (ı, s)} \int_{ı_{l}}^{ı_{l} - ε} ψ (σ) dσ \leq - η (ı - s) + M$ ;

(b) $\int_{ı_{0}}^{\infty} ψ^{+} (σ) dσ < \infty$ , and there is a positive constant $α > 0$ such that for sufficiently large ı, $\int_{ı_{0}}^{ı} ψ^{-} (σ) dσ \geq α (ı - ı_{0})$ .

(H₅) $\bar{λ} θ e^{- ητ} - \tilde{λ} < 0$ .

Then, the nonlinear time-varying system (1) with the random delayed impulses is ISS.

Proof. Due to the stochastic nature of the impulsive intensity and the impulsive instant, we use mathematical induction analogous to Theorem 1, we obtain the following inequality:

\begin{matrix} E V (ı, w (ı)) \\ \leq & E V (ı_{0}, w_{0}) \prod_{m = 1}^{ℕ (ı, ı_{0})} E μ_{m} E [\exp {\int_{ı_{0}}^{ı} ψ (ς) dς} \\ \prod_{m = 1}^{ℕ (ı, ı_{0})} \exp {\int_{ı_{m}}^{ı_{m} - ε} ψ (ς) dς}] + \sum_{p = 0}^{ℕ (ı, ı_{0}) - 1} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) \\ \prod_{q = p + 1}^{ℕ (ı, ı_{0})} E μ_{q} E [\exp {\int_{ς}^{ı} ψ (σ) dσ} \\ \times \prod_{q = p + 1}^{ℕ (ı, ı_{0})} \exp {\int_{ı_{q}}^{ı_{q} - ε} ψ (σ) dσ}] dς} \\ + \int_{ı_{ℕ (ı, ı_{0})}}^{ı} ρ (| v (ς) |) \exp {\int_{ς}^{ı} ψ (σ) dσ} dς = Θ_{1} + Θ_{2}, \forall ı > ı_{0} \end{matrix}

where

\begin{matrix} Θ_{1} = E V (ı_{0}, w_{0}) \prod_{m = 1}^{ℕ (ı, ı_{0})} E μ_{m} E [\exp {\int_{ı_{0}}^{ı} ψ (ς) dς} \\ \prod_{m = 1}^{ℕ (ı, ı_{0})} \exp {\int_{ı_{m}}^{ı_{m} - ε} ψ (ς) dς}], \end{matrix}

\begin{matrix} Θ_{2} = & \sum_{p = 0}^{ℕ (ı, ı_{0}) - 1} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) \\ \prod_{q = p + 1}^{ℕ (ı, ı_{0})} E μ_{q} E [\exp {\int_{ς}^{ı} ψ (σ) dσ} \\ \times \prod_{q = p + 1}^{ℕ (ı, ı_{0})} \exp {\int_{ı_{q}}^{ı_{q} - ε} ψ (σ) dσ}] dς} \\ + \int_{ı_{ℕ (ı, ı_{0})}}^{ı} ρ (| v (ς) |) \exp {\int_{s}^{ı} ψ (σ) dσ} dς . \end{matrix}

According to (a) of condition $(H_{4}^{'})$ , condition $(H_{5})$ , Assumption 1, and Lemma 2, we have

\begin{matrix} Θ_{1} = & E V (ı_{0}, w_{0}) \prod_{m = 1}^{ℕ (ı, ı_{0})} E μ_{m} E [\exp {\int_{ı_{0}}^{ı} ψ (ς) dς} \\ \prod_{m = 1}^{ℕ (ı, ı_{0})} \exp {\int_{ı_{m}}^{ı_{m} - ε} ψ (ς) dς}] \\ \leq & E V (ı_{0}, w_{0}) θ^{ℕ (ı, ı_{0})} E [e^{- η \sum_{i = 1}^{ℕ (ı, ı_{0})} T_{i} + M}] \\ = & e^{M} E V (ı_{0}, w_{0}) E [e^{(\ln θ - ητ) ℕ (ı, ı_{0})}] \\ \leq & e^{M} E V (ı_{0}, w_{0}) [W + e^{(\bar{λ} θ e^{- ητ} - \tilde{λ}) (ı - ı_{0})}] . \end{matrix}

(21)

In addition, it is easy to obtain, from condition $(H_{4}^{'})$ , Assumption 1, and Lemma 2, we get

\begin{matrix} Θ_{2} = \sum_{p = 0}^{ℕ (ı, ı_{0}) - 1} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) \\ \prod_{q = p + 1}^{ℕ (ı, ı_{0})} E μ_{q} E [\exp {\int_{ς}^{ı} ψ (σ) dσ} \\ \times \prod_{q = p + 1}^{ℕ (ı, ı_{0})} \exp {\int_{ı_{q}}^{ı_{q} - ε} ψ (σ) dσ}] dς} \\ + \int_{ı_{ℕ (ı, ı_{0})}}^{ı} ρ (| v (ς) |) E [e^{\int_{s}^{ı} ψ (σ) dσ}] dς \\ \leq & \sum_{p = 0}^{ℕ (ı, ı_{0}) - 1} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) θ^{ℕ (ı, ς)} E [e^{- η \sum_{q = p + 1}^{ℕ (ı, ı_{0})} T_{q} + M}] dς} \\ + \int_{ı_{ℕ (ı, ı_{0})}}^{ı} ρ (| v (ς) |) E [e^{\int_{ς}^{ı} ψ^{+} (σ) dσ - \int_{ς}^{ı} ψ^{-} (σ) dσ}] dς \\ \leq & \sum_{p = 0}^{ℕ (ı, ı_{0}) - 1} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) e^{M} E [e^{(\ln θ - ητ) ℕ (ı, ς)}] dς} \\ + \int_{τ_{N (ı, ı_{0})}}^{ı} ρ (| v (ς) |) E [e^{\int_{ς}^{ı} ψ^{+} (σ) dσ - α (ı - ς)}] dς \\ \leq & \sum_{p = 0}^{ℕ (ı, ı_{0}) - 1} {\int_{ı_{p}}^{ı_{p + 1} - ε} ρ (| v (ς) |) e^{M} [W + e^{(\bar{λ} θ e^{- ητ} - \tilde{λ}) (ı - ς)}] dς} \\ + \int_{ı_{ℕ (ı, ı_{0})}}^{ı} ρ (| v (ς) |) E [e^{\int_{s}^{ı} ψ^{+} (σ) dσ - α (ı - ς)}] dς \\ \leq & \int_{ı_{0}}^{ı_{ℕ (ı, ı_{0})}} ρ (| v (ς) |) e^{M} [W + e^{(\bar{λ} θ e^{- ητ} - \tilde{λ}) (ı - ς)}] dς \\ + \int_{ı_{ℕ (ı, ı_{0})}}^{ı} ρ (| v (ς) |) E [e^{\int_{ς}^{ı} ψ^{+} (σ) dσ - α (ı - ς)}] dς \\ \leq & \bar{ω} ρ (| v (ς) |_{[ı_{0}, ı]}), \end{matrix}

(22)

where $\bar{ω} = e^{M} (W + 1) \land e^{\int_{ı_{ℕ (ı, ı_{0})}}^{ı} ψ^{+} (σ) dσ}$ .

From inequalities (21) and (22), we can conclude that

\begin{matrix} E V (ı, w (ı)) = Θ_{1} + Θ_{2} \\ \leq & e^{M} E V (ı_{0}, w_{0}) [W + e^{(\bar{λ} θ e^{- ητ} - \tilde{λ}) (ı - ı_{0})}] + \bar{ω} ρ (| v (ς) |_{[ı_{0}, ı]}) . \end{matrix}

Based on the condition $(H_{1})$ and inequality (7), we get

\begin{matrix} E | w (ı) | & \leq \partial_{1}^{- 1} {e^{M} \partial_{2} (| w_{0} |) [W + e^{(\bar{λ} θ e^{- ητ} - \tilde{λ}) (ı - ı_{0})}] \\ + \bar{ω} ρ (| v (ς) |_{[ı_{0}, ı]})} \\ \leq β (| w_{0} |, ı - ı_{0}) + Υ (| v (ς) |_{[ı_{0}, ı]}), \end{matrix}

where $β (| w_{0} |, ı - ı_{0}) = \partial_{1}^{- 1} (e^{M} \partial_{2} (| w_{0} |) [W + e^{(\bar{λ} θ e^{- ητ} - \tilde{λ}) (ı - ı_{0})}])$ is a $KL$ -function and $Υ (| v (ı) |_{[ı_{0}, ı]}) = \partial_{1}^{- 1} (\bar{ω} ρ (| v (ς) |_{[ı_{0}, ı]}))$ is a $K_{\infty}$ -function. This completes the proof that the random delayed impulsive system (1) is ISS.

Remark 5. In Table 1, we compare the differences between the results of this paper and those already available, mainly the differences in the systems, the different Lyapunov functions constructed, and the differences in the impulsive signals.

Table 1.

Comparison with some existing ISS schemes.^a

Schemes	Theorem 1	Tang et al. (2020)	Gao et al. (2025)
System	Ordinary differential equation	Ordinary differential equation	Discrete stochastic differential equation
Time delays	Fixed	No time delay	Fixed
Impulse effects	AII and MDAII	AII and MDAII	AED-AII
Lyapunov function	$▽ V (ı, w) \cdot Ψ (ı, w, v) \leq$ $ψ (ı) V (ı, w) + ρ (v)$	$V (x) \geq φ (∥ u ∥_{[0, t]}),$ $▽ V (x) \cdot f (x, u) \leq λV (x)$	$V (l + 1) - V (l) \leq - μ_{δ (l)} V (l) + χ (\| u (l + 1) \|)$
Main conditions	$\sum_{i = 1}^{δ} \frac{\ln θ_{i}}{Γ_{ai}}$ $+ \frac{λε + M_{0}}{Γ_{a}} - λ < 0$	$\sum_{i = 1}^{δ} \frac{\ln θ_{i}}{T_{ai}} + λ < 0$	$\sum_{j, j^{'}} \frac{\ln P_{j j^{'}}}{τ_{j j^{'}}} +$ $\sum_{j} \ln (1 - μ_{δ (j)}) < 0$

Although all variables in this table are rewritten in order to be consistent with those in this paper, some specific assumptions or exact restrictions for these variables may still be different.

Simulation examples

Example 1. We consider the time-varying nonlinear system with random delayed impulses:

{\begin{matrix} w (ı) = Ψ (ı, w (ı), v (ı)), ı \neq ı_{l}, ı \geq 0 \\ w (ı) = Φ (ı, w (ı - ε), v (ı^{-})), ı = ı_{l}, l \in Z^{+} \end{matrix}

where $w (τ) = {(w_{1} (ı), w_{2} (ı))}^{T}$ ,

\begin{matrix} Ψ (ı, w (ı), v (ı)) = \\ [\begin{matrix} (\frac{| \cos ı | w_{2}^{2} (ı)}{1 + w_{2}^{2} (ı)} - | \sin ı | - \frac{1}{2}) w_{1} (ı) - \frac{| \cos ı | w_{2} (ı)}{2} + \frac{v (ı)}{2} \\ (| \cos ı | - | \sin ı | - \frac{1}{2}) w_{2} (ı) - \frac{| \cos ı | w_{1} (ı)}{2} + \frac{v (ı)}{2} \end{matrix}] \end{matrix}

w (ı) = w (ı - ε) .

Constructing $V (ı, w (ı)) = | w_{1} (ı) | + | w_{2} (ı) |$ , we are able to derive that

\begin{matrix} \overset{\cdot}{V} (ı, w (ı)) \leq (\frac{| \cos ı | w_{2}^{2} (ı)}{1 + w_{2}^{2} (ı)} - | \sin ı | - \frac{1}{2}) | w_{1} (ı) | \\ - \frac{| \cos ı |}{2} | w_{2} (ı) | + \frac{1}{2} v (ı) \\ + (| \cos ı | - | \sin ı | - \frac{1}{2}) | w_{2} (ı) | - \frac{| \cos ı |}{2} | w_{1} (ı) | + \frac{1}{2} v (ı) \\ \leq (\frac{| \cos ı |}{2} - | \sin ı | - \frac{1}{2}) (| w_{1} (ı) | + | w_{2} (ı) |) + v (ı) \\ = (\frac{| \cos ı |}{2} - | \sin ı | - \frac{1}{2}) V (ı, w (ı)) + v (ı), \\ V (ı, w (ı)) \leq μ_{i} V (ı, w (ı - ε)) . \end{matrix}

It means that $ψ (ı) = \frac{| \cos ı |}{2} - | \sin ı | - \frac{1}{2}$ . Based on Lemma 1, we are able to calculate

\begin{matrix} \int_{s}^{ı} ψ (σ) dσ = \int_{s}^{ı} (\frac{| \cos σ |}{2} - | \sin σ | - \frac{1}{2}) dσ \\ \leq - \frac{2 + π}{2 π} (ı - s) + 2, \end{matrix}

we know that $ψ (ı)$ is $UES [WGDR \frac{2 + π}{2 π}]$ , where $λ = \frac{2 + π}{2 π}$ and $M_{0} = 2$ .

The impulsive intensity is revealed in Figure 1. As we can see, the intensity of impulses is random. We let the random variable $μ_{1}$ satisfies the exponential distribution $Exp (2.5)$ , and the random variable $μ_{2}$ satisfies the uniform distribution $U (0.5, 1.5)$ . It can be calculated that $θ_{1} = 0.4713$ and $θ_{2} = 1.0144$ . In order to illustrate the effectiveness of Theorem 1, we set $Γ_{a} = 4, Γ_{ai} = 4, ε = 0.1$ . From condition $(H_{4})$ of Theorem 1, we can calculate that

\frac{\ln 1.0144}{4} + \frac{\ln 0.4713}{4} + \frac{0.1 \times \frac{2 + π}{2 π} + 2}{4} - \frac{2 + π}{2 π} < 0 .

That is, all conditions of Theorem 1 are fulfilled. Then, according to Theorem 1, it follows that the system (1) is ISS. After that, we show the trajectories of the states with the input $v (ı) = \frac{1}{4} \cos ı$ in Figure 2.

Example 2. We consider the time-varying nonlinear system with random delayed impulses:

{\begin{matrix} w (ı) = Ψ (ı, w (ı), v (ı)), ı \neq ı_{l}, ı \geq 0 \\ w (ı) = Φ (ı, w (ı - ε), v (ı^{-})), ı = ı_{l}, l \in Z^{+}, \end{matrix}

where $w (ı) = {(w_{1} (ı), w_{2} (ı))}^{T}$ ,

\begin{matrix} Ψ (ı, w (ı), v (ı)) = \\ [\begin{matrix} (\frac{3 w_{2}^{2} (ı)}{(1 + ı^{2}) (1 + w_{2}^{2} (ı))} - | \sin ı | - \frac{1}{π}) w_{1} (ı) - \frac{w_{2} (ı)}{1 + ı^{2}} + \frac{v (ı)}{2} \\ (\frac{3}{1 + ı^{2}} - | \sin ı | - \frac{1}{π}) w_{2} (ı) - \frac{w_{1} (ı)}{1 + ı^{2}} + \frac{v (ı)}{2} \end{matrix}] \end{matrix}

w (ı) = w (ı - ε) .

Constructing $V (ı, w (ı)) = | w_{1} (ı) | + | w_{2} (ı) |$ , we get

\begin{matrix} \overset{\cdot}{V} (ı, w (ı)) \leq & (\frac{3 w_{2}^{2} (ı)}{(1 + ı^{2}) (1 + w_{2}^{2} (ı))} - | \sin ı | - \frac{1}{π}) | w_{1} (ı) | \\ - \frac{| w_{2} (ı) |}{1 + ı^{2}} + \frac{v (ı)}{2} \\ + (\frac{3}{1 + ı^{2}} - | \sin ı | - \frac{1}{π}) | w_{2} (ı) | - \frac{| w_{1} (ı) |}{1 + ı^{2}} + \frac{v (ı)}{2} \end{matrix}

\begin{matrix} \leq & (\frac{2}{1 + ı^{2}} - | \sin ı | - \frac{1}{π}) (| w_{1} (ı) | + | w_{2} (ı) |) + v (ı) \\ = & (\frac{2}{1 + ı^{2}} - | \sin ı | - \frac{1}{π}) V (ı, w (ı)) + v (ı), \\ V (ı, w (ı)) \leq & μ_{i} V (ı, w (ı - ε)) . \end{matrix}

It means that $ψ (ı) = \frac{2}{1 + ı^{2}} - | \sin ı | - \frac{1}{π}$ . From condition $(H_{4}^{'})$ , we can calculate

\int_{ς}^{ı} ψ (σ) dσ + \sum_{k = 1}^{ℕ (ı, ı)} \int_{ı_{k}}^{ı_{k} - ε} ψ (σ) dσ \leq - η (ı - ς) + M,

where $η = \frac{3}{π}$ and $M = 2 π, ε = 0.0038$ .

\begin{matrix} \int_{ς}^{ı} ψ^{+} (σ) dσ & = \int_{ς}^{ı} \frac{2}{1 + σ^{2}} dσ = 2 π < \infty, \\ \int_{ς}^{ı} ψ^{-} (σ) dσ & = \int_{ς}^{ı} (| \sin σ | + \frac{1}{π}) dσ > \frac{3}{π} (ı - ς) . \end{matrix}

The impulsive density and the impulsive intensity are revealed in Figures 3 and 4, respectively. As we can see, the density and intensity of impulses are random.

It can be calculated that $θ = \max_{i \in Γ} θ_{i} = 1.0227$ . The sequence $T_{k}$ is an independent sequence of exponential stochastic variables of parameter $ℏ = 3$ . $E T_{k} = τ = 0.3867 < \infty$ . It is an simple calculation that shows that it fulfills condition $(H_{4}^{'})$ of Theorem 2 with $ℏ = \bar{λ} = \hat{λ}$ . An easy calculation shows that

\bar{λ} θ e^{- ητ} - \tilde{λ} = 3 \times 1.0227 \times e^{- \frac{3}{π} \times 0.3867} - 3 < 0 .

That is, all conditions of Theorem 2 are fulfilled. Hence, based on Theorem 2, it follows that the system (1) is ISS. Afterwards, we show the states $w (ı)$ trajectories for $v (ı) = 0$ and $v (ı) = \frac{1}{3} \cos ı$ in Figures 5 and 6, respectively.

Conclusions

This paper investigates the ISS properties of nonlinear time-varying systems with random delayed impulses based on the Lyapunov-type functions with indefinite derivatives. We examine the effect of delayed impulses with random impulsive intensities on ISS when they occur at the fixed impulsive instant and at the random impulsive instant, respectively. It is shown that ISS is guaranteed in both cases under the conditions we give. Two numerical examples verify the correctness of the results. Throughout this paper, the delayed impulse remains fixed, and it would be interesting to investigate the direction in which the delayed impulse is random in the future.

Footnotes

ORCID iD

Meng Zhang

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (62103231 and 62173139) and the Support Plan for Outstanding Youth Innovation Team in Shandong Higher Education Institutions under grant 2022KJ184.

Declaration of conflicting interests

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

Data availability statement

Data sharing is not applicable to this article, as no datasets were generated or analyzed during the current study.

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