Comparison principles and input-to-state stability for stochastic impulsive systems with delays

In this paper, the input-to-state stability (ISS), the pth moment exponential ISS and the stochastic input-to-state stability issues are investigated for the stochastic impulsive systems with time delays via the comparison principlemethod. Firstly, several general comparison principles in vectorversion are proposed guaranteeing the existence, uniqueness and magnitude for solutions of the addressed system. Then based on these established comparison principles, the ISS-related properties are investigated for the stochastic impulsive delayed model. Finally, one example is given to illustrate effectiveness of the obtained results. ARTICLE HISTORY Received 31 December 2020 Accepted 12 March 2021


Introduction
In recent years, the input-to-state stability (ISS) and extensions of the ISS on different systems have attracted widespread attentions in the literature (Damak, 2021;Khalil, 2002;Sontag, 2004;Sontag & Wang, 1996) due to their extensive usage in characterizing the effects of external inputs (such as sensor noise (Liu et al., 2012), actuator disturbances (Liao et al., 2014), parameter perturbations, or measurement errors) on the considered systems. The notion of ISS, firstly proposed in Sontag (1989) for continuous-time nonlinear systems, is formed to investigate how the external disturbance affects the system stability. Roughly speaking, the property of ISS means that no matter what the magnitude of the initial state is, the system state will ultimately approach to a neighbourhood of the origin whose magnitude is proportional to the size of the input. Recently, various extensions of the ISS, such as integral ISS (iISS), stochastic ISS (SISS), and finite-time ISS, have been proposed successfully to different kinds of dynamical systems, for instance, impulsive systems (Liu & Hill, 2010;Wu et al., 2016;Yang et al., 2013), switched systems (Ren & Xiong, 2016;Wakaiki & Yamamoto, 2017), discretetime dynamical networks (Jiang et al., 2004;Teel, 1998;Zhao et al., 2015), and nonlinear systems with delays (Li et al., 2016;Liu, 2017;Wang et al., 2018).
Impulsive systems, as an important portion of the hybrid systems, combine the continuous dynamical systems with the discrete-time instantaneous state jumps or resets (also be referred to as impulses). Impulsive CONTACT Jinling Liang jinlliang@seu.edu.cn control models have attracted considerable concern in both the theoretical and the practical fields during the past few years, since they occur naturally from a wide range of areas such as biomechanical systems (Karafyllis et al., 2008), economic systems , and the like. Meanwhile, they can be extended as the impulsive time delay plants (Liu et al., 2011;Ren & Xiong, 2019) if there exists time delay in the continuous-time or discretetime dynamical impulsive systems. On the other hand, great efforts have been devoted to the stability analysis for stochastic systems (Hu et al., 2020;Liu, 2008;Mao, 2007;Song et al., 2020) because lots of natural processes and real-world systems are disturbed unavoidably by stochastic perturbation. For example, the networked systems are ubiquitously subject to random noises such as data dropouts or congestions (Antunes et al., 2013). By resorting to multiple-Lyapunov functions, the stability has been analysed for the deterministic and stochastic models (Chatterjee & Liberzon, 2006). With the method of matrix measure, the problem of reliable dissipative control is discussed in Zhang et al. (2008) for a class of stochastic systems. Based on the average dwell-time laws, global asymptotic stability in probability (GASiP) and SISS are investigated in Zhao, Feng et al. (2012) for the switched stochastic nonlinear systems. It should be noted that it is important and necessary to further develop the ISS-related results for the stochastic impulsive time delay systems (Niu et al., 2012;. However, up to now, only scattered efforts have been carried out for a comprehensive analysis on the ISS-related properties (including ISS, IS KB S, pth moment exponential ISS, and SISS) for the addressed stochastic impulsive time delay plants (Wu et al., 2016), which motivates the present study in this paper. Lyapunov method, one of the main techniques contributed to analysing the stability of dynamical systems (Hu et al., 2021;Kundu & Anghel, 2017;Li et al., 2020;Ren & Xiong, 2017;Xu et al., 2016), has also been employed extensively to investigate the ISS property for various systems (Chen & Zheng, 2009;Ren & Xiong, 2018). As an example, SISS has been tackled in Ren and Xiong (2019) for the stochastic impulsive systems by resorting of the Lyapunov function. The Lyapunov conditions are derived in Ren and Xiong (2018) for ascertaining the stability of stochastic impulsive switched systems. The ISS and iISS have been investigated in Chen and Zheng (2009) for nonlinear impulsive systems with delays by employing the Lyapunov method. However, it is still a challenging issue to construct suitable Lyapunov functions for the real-world complex delayed systems. Moreover, although existence of the SISS-Lyapunov function is equivalent to the SISS property for general nonlinear stochastic systems, such equivalence does not hold for the stochastic impulsive models. Based on this consideration, the comparison principle is another efficient choice, where stability of a complex delayed system can be guaranteed by comparing it with a scalar or lower-dimensional model with known certain stability property. In recent years, ISS-based comparison principles have been established for continuous/discrete-time systems with time delays (Luo, 2006;Luo & Shen, 2006). For example, the ISStype comparison principles have been proposed in Liu et al. (2013) to analyse the ISS for discrete-time delayed systems. The stability in probability and asymptotic stability in probability have been discussed in Luo (2006) by using the comparison principle method, and the iISS has been investigated in Rüffer et al. (2009) for interconnected systems by means of a lower-dimensional comparison system. To the best of the authors' knowledge, there are only few (if not none) relevant literatures concerning about the ISS-related results on impulsive stochastic time delay systems by resorting to the comparison principle method, which comprised the other motivation of the present study.
In this paper, we aim to study the ISS-related issues for stochastic impulsive systems with time delays and external inputs via the comparison principle method. Firstly the comparison principles in vector-version are established for the considered system. Then sufficient conditions are deduced to be input-to-state stable in mean, pth moment exponentially input-to-state stable in mean, stochastic input-to-state stable, and input-to-state KBstable in mean assuring the stochastic impulsive delayed systems. The ISS-related properties can be deduced by comparing the systems with a lower-dimension impulsive delayed system. The main novelties and significance of the present work are summarized from the following three aspects.
• A new comparison principle in vector-version is developed for solutions of the stochastic impulsive delayed system, which generalizes the scalar version of the classic comparison principle in Khalil (2002) and facilitates the examination of the ISS-related properties. • The existence and uniqueness of the solution are deduced for the stochastic impulsive delayed system with external inputs by comparing it with a lower-dimensional deterministic impulsive one which is assumed to have a global solution, where the linear growth constraints in Wu et al. (2016) are no longer necessary. • As for the stochastic impulsive delay-free (or time delay) systems with external inputs, it is the first time that the ISS-related properties are derived in terms of the vector-version comparison principles instead of the Lyapunov functions.
The rest of this paper is organized as follows. In Section 2, the problem to be addressed is formulated and some preliminaries are presented. In Section 3, we establish the general comparison principles in vector-version for the stochastic impulsive time delay system with external inputs, which guarantee the existence, uniqueness, and magnitude of the corresponding solutions. In Section 4, the ISS-related criteria are established for the considered stochastic impulsive model. In Section 5, one numerical example is provided to demonstrate the effectiveness of the proposed criteria. The conclusion is given in Section 6.
Notations: Standard notation is used in this paper. Let R ≥t 0 := [t 0 , +∞), R >t 0 := (t 0 , +∞), and N + be the set of positive integers. R n and R n×m denote, respectively, the n-dimensional real space and the n × m-dimensional real matrix space. For vector x ∈ R n , x represents its norm with x := n i=1 |x i |. PC([a, b]; R n ) stands for the class of piecewise continuous functions from [a, b] to R n and having finite right-hand continuous jumps on [a, b].

Problem formulation
In this paper, we consider the following stochastic impulsive system with time delays: is a continuous function representing the time delay which satisfies 0 τ (t) τ with τ being a constant, and B(t) = (B 1 (t), B 2 (t), . . . , B n w (t)) T stands for an n w -dimensional F t -adapted Brownian motion defined on the complete probability space ( , F, {F t } t t 0 , P) in which F is a σ -field and {F t } t≥t 0 is a filtration. The given impulsive time sequence T := {t k : k ∈ N + } is strictly increasing such that t k → +∞ as k → +∞.
Remark 2.1: If there exists a positive integer k * such that t k * ∈ T , t k * < +∞ and t k * +1 = +∞, then system (1) reduces to the normal stochastic delayed system, of which the ISS and relevant issues have been extensively studied in the previous literature (Mao, 2007). Based on this fact, in this paper, the following assumption is proposed.
Assumption 2.1: Assume that the impulsive instances {t k , t ∈ N + } satisfy: Definition 2.1 (Khalil, 2002): A function γ (·) : R ≥0 → R ≥0 is said to belong to class-K (i.e. γ (·) ∈ K) if it is a continuous and strictly increasing function with γ (0) = 0. It is said to belong to class-K ∞ if it is of class-K and is unbounded. A vector function l(r) = (l 1 (r), l 2 (r), . . . , l m (r)) T : Definition 2.2: Given an impulsive time sequence In particular, when p = 1, system (1) is said to be inputto-state stable in mean.

Definition 2.4:
Given an impulsive time sequence T , system (1) is said to be stochastic input-to-state stable, if for any given ε ∈ (0, 1), there exist functions β(·, ·) ∈ KL and Remark 2.3: The above definition of ISS in mean is parallel to the one given in Wu et al. (2016) with minor modifications. When u(·) ≡ 0, inequality (2) with p = 1 reduces to E x(t) β(E ξ τ , t − t 0 ), which indicates the global asymptotic stability (GAS) in mean. As a result, the conclusion that system (1) is input-to-state stable in mean implies that the corresponding unforced system is globally asymptotically stable in mean. In addition, it is noticed that when u(·) ≡ 0, inequality (3) means that the corresponding unforced system is stable in mean. Furthermore, when u(·) ≡ 0, inequality (4) which infers that the corresponding unforced system of (1) is globally asymptotically stable in probability. Similar definition has been given in Zhao et al. (2015) for the discrete-time stochastic nonlinear systems.
The main aim of this paper is to propose several new comparison principles, based on which the existence and uniqueness of solutions are discussed for the stochastic impulsive system (1) with time delays. In addition, the ISS-related issues concerning model (1) are also analysed dexterously. To proceed, the following definitions and lemma are further introduced.

Definition 2.5: For a function
Definition 2.6 (Mao, 2007): C 1,2 represents the class of continuous functions Proof: It is a deduction of the Holder inequality so that the details are omitted for space consideration.

General comparison principle
In this section, two new comparison principles are established for the stochastic impulsive delayed system (1) with external inputs. For system (1), the following simple impulsive delayed model is considered as a comparison system: is a continuous Borel measurable function, which is concave and In the practical engineering, the target system (1) might be large-scaled. To effectively reduce the dimension of the comparison system (5), the following onedimensional system is further considered: is a continuous Borel measurable function and satisfies the Lipchitz condition. Function φ(y(t), w(t)) : R ≥0 × R ≥0 → R ≥0 satisfying φ(0, 0) = 0 is continuous, nondecreasing and concave in y(t). The initial function ζ 2 (·) ∈ PC([t 0 − τ , t 0 ]; R ≥0 ) is bounded. It follows from Luo and Shen (2006) that for any ζ 2 (t), system (6) has an unique solution. We further suppose that h 1 (t, 0, 0) ≡ 0, h 2 (t, 0, 0) ≡ 0 for all t ∈ R ≥t 0 , which mean that systems (5) and (6) admit the trivial solution. Let the solutions of systems (5) and (6) be denoted by z(t) := z(t; t 0 , ζ 1 , v) and y(t) := y(t; t 0 , ζ 2 , w), respectively.
If not, there should exist some t ∈ (t 0 , t 1 ) and certain subindex j ∈ N + and j n z such that component EV j (t, x(t)) satisfies EV j (t, x(t)) > z j (t). Set t * := inf{t ∈ (t 0 , t 1 ) : EV j (t, x(t)) > z j (t)}, and let j t * be the subindex corresponding to t * . Since EV(t, x(t)) and z(t) are continuous on [t 0 , t 1 ), it implies that EV j t * (t * , x(t * )) = z j t * (t * ) and EV j t * (t, x(t)) > z j t * (t) for t ∈ (t * , t * + ε), where ε > 0 is small enough and satisfies t * + ε < t 1 . Therefore, for all t ∈ (t * , t * + ε), we have On the other hand, noticing that h 1 (t, z t , v(t)) is concave and nondecreasing with respect to z t , it follows from the first inequality of (8) that which contradicts inequality (12). Therefore, it concludes that EV(t, x(t)) ≤ z(t) holds for all t ∈ [t 0 , t 1 ). Similarly, by using (9), we can conclude that z(t) l(y(t)) holds for all t ∈ [t 0 , t 1 ). Thus, inequality (11) is proved to be valid. Next, we prove (7) by using the method of mathematical induction. Suppose that, for r = 0, 1, . . . , k − 1 and k ∈ N + , then for a scalar ρ > 0 which is small enough, we have Further recalling that ψ(z(t), v(t)) is nondecreasing and concave in z(t), it follows from the second inequality of (8) that Similarly, by using the second inequality of (9) as well as condition (10), we could derive z(t k ) l(y(t k )). Thus one gets
Remark 3.1: Theorem 3.1 provides a new comparison principle in vector-version for solutions of the stochastic impulsive delayed system (1) with external inputs, which generalizes the scalar version of the classical comparison principle for nonlinear systems in Khalil (2002) that has been widely used in the literature (Chatterjee & Liberzon, 2006;Mao, 2007;Niu et al., 2012;Zhang et al., 2008;. Moreover, Theorem 3.1 contains the scalar results of Lemma 1 in Wu et al. (2016), where the conditions are proposed under linear assumption, which is relaxed in Theorem 3.1.

Theorem 3.2 (Comparison principle for the existence and uniqueness of the solution):
Consider the stochastic impulsive delayed system (1) and the comparison system (5). Assume that there exist functions V(·, ·) ∈ C 1,2 and χ i (·) ∈ K ∞ (1 ≤ i ≤ n z ) such that condition (i) of Theorem 3.1 and the following inequalities hold:

then the existence of the global solution z(t) of system (5) implies that system (1) has an unique global solution x(t) for any given initial function
Proof: Due to the fact that functions f (t, x t , u(t)) and g(t, x t , u(t)) both satisfy the Lipchitz condition, it follows from Mao (2007) that the continuous dynamic system corresponding to model (1) has an unique solution is the maximal existence interval of the solution. Then system (1) itself also has an unique solution x(t) on (t 0 − τ , σ ∞ ) for the given initial condition ξ ∈ PC([t 0 − τ , t 0 ]; R n x ) by the approach of solution's continuation. Next, we only need to prove that σ ∞ = +∞. For any k ∈ N + , let s k = {x(t) : x(t) ≥ k} and define the stopping time ρ(k) as Obviously, the sequence {ρ(k) : k ∈ N + } is increasing with k. Assume σ ∞ < +∞, then lim k→∞ ρ(k) = ρ(∞) exists, and ρ(k) ≤ ρ(∞) ≤ σ ∞ for any k ∈ N + . It is known that there exists some integer k 0 ∈ N + such that σ ∞ ∈ (t k 0 −1 , t k 0 ), and for any k ∈ N + , some i k exists satisfying i k ≤ k 0 and ρ(k) ∈ [t i k −1 , t i k ). It follows from the first inequality of (8) and Itô's formula that Let t := min{t, ρ(k)} with t ∈ (t i k −1 , t i k ), based on (18) we have (t, x(t)) and m t = EV t . Combine (8) with (18) and (19) follows that Comparing system (5) with equation (20), we immediately get Based on which we have that, for any t ∈ (t i k −1 , t i k ),

EV( t, x( t)) = m(t) ≤ z(t).
(21) It then yields from inequalities (16) and (21) that, for any i = 1, 2, . . . , n z , Let k → ∞ and t → min{σ ∞ , t i ∞ } in the above expression, it achieves This is a contradiction with the existence condition of the global solution z(t) for system (5). Therefore we achieve σ ∞ = ∞, which means the system (1) has an unique global solution x(t) for any given initial function ξ Remark 3.2: Theorem 3.2 presents a new comparison principle in vector-version for the existence and uniqueness of solution for the stochastic impulsive delayed system (1) with external inputs. As is known to all, the linear growth assumption is needed in many literatures (Wu et al., 2016) to guarantee the existence and uniqueness of solution for the stochastic impulsive systems, or sometimes the existence/uniqueness of solutions are directly assumed without any proof (Ren & Xiong, 2019). By employing Theorem 3.2, the existence and uniqueness of solutions for model (1) can be deduced by comparing it with a lower-dimensional deterministic impulsive system (5) which is assumed to have a global solution, and the linear growth constraints on model (1) are no longer necessary.

ISS-related analysis
In this section, based on the comparison principles established in the previous section, criteria are to be derived for the stochastic impulsive system (1) concerning the properties of ISS in mean, SISS and the pth moment exponential ISS in mean.
Proof: Firstly, it follows from Theorem 3.2 and condition (22) that system (1) has an unique global solution x(t) if system (5) has a global solution. Secondly, we need to prove the ISS in mean of system (1) provided that model (5) is input-to-state stable. Another case is similar and will be omitted for space consideration. If system (5) is input-to-state stable, then there exist functions β z (·, ·) ∈ KL and γ z (·) ∈ K ∞ such that, for For any given initial function . Then by Theorem 3.1, we obtain that EV(t, x(t)) ≤ z(t) for all t t 0 , i.e.
Remark 4.1: Theorem 4.1 can be viewed as a comparison principle for the ISS-related properties of the stochastic impulsive delayed system (1). By using it, the ISS-related criteria concerning model (1) can be derived by the corresponding ISS-related properties of the deterministic impulsive systems. On another front, condition (23) in Theorem 4.1 should be relaxed to the case with u ∞ < ∞, v ∞ < ∞ and w ∞ < ∞ when the term u [t 0 ,t] in Definitions 2.2-2.4 is replaced by u ∞ , provided that the input u(·) is known to be globally bounded, that is, u ∞ < ∞.

Remark 4.2:
Based on the Lyapunov method, the ISS issue has been analysed for stochastic delay-free systems in  and then discussed in Yao et al. (2014) for stochastic impulsive models without delays. Soon afterwards, the ISS-type issues for stochastic impulsive systems with time delays are considered in Wu et al. (2016), where a scalar Lyapunov function is also resorted to under the assumption of the linear growth constraint which is no longer necessary here in Theorem 4.1. To be more specific, when investigating the ISS-related characteristics of the stochastic impulsive delayed model (1), influence of the time delay should be carefully/extensively taken into account, especially on the magnitude of the solutions of the target plant. This paper tries its first attempt to analyse these properties by introducing the lower-dimensional comparison systems which are also delay-dependent, and the comparison principles in vector version are established which are much more easier to be implemented in practice.
where c 1 , c 2 , p are given positive constants, then system (1) is pth moment exponentially input-to-state stable in mean.

Proof:
The conclusion can be derived by using Theorems 3.1 and 4.1.  (40) is proposed under the linear growth assumption and condition (41) is given to guarantee the ISS property of the one-dimensional comparison system defined in the following form:

Corollary 4.2:
Assume that all conditions of Theorem 3.1 and constraints (22)-(23) in Theorem 4.1 hold. Then the IS KB S property of system (5) implies the IS KB S in mean of model (1), and the IS KB S property of system (6) implies the IS KB S in mean of model (1) and model (5). (5) is input-to-state KB-stable, then employ the similar method as that of Theorem 4.1, it will be directly proved that system (1) is input-to-state KBstable in mean, where β z (·) ∈ KB in inequality (31).
Let N 0 = 1, T a < 0.1523. Then (43) is input-to-state stable in mean and stochastic input-to-state stable according to Theorem 4.1 as well as Corollary 4.1.

Conclusion
This article has addressed the ISS-related issues for stochastic impulsive systems with time-varying delay and external inputs. Firstly, a reduced-order deterministic impulsive delayed system and an one-dimensional impulsive system with time delays are selected as the comparison models, based on which the generalized vector-version comparison principles have been established assuring the existence and uniqueness of solutions for the addressed system as well as the magnitude of the solutions. Then the ISS in mean problem, the pth moment exponential ISS problem and SISS problem have been separately analysed for the considered stochastic impulsive system by employing the comparison principles. One example has been provided finally to demonstrate feasibility of the acquired theoretical results.

Disclosure statement
No potential conflict of interest was reported by the author(s).