# On Input-to-State Stability of Impulsive Stochastic Systems with Time Delays

Authors: Fengqi Yao; Pei Cheng; Hao Shen; Li Qiu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper is concerned withpth moment input-to-state stability (p-ISS) and stochastic input-to-state stability (SISS) of impulsive stochastic systems with time delays. Razumikhin-type theorems ensuring p-ISS/SISS are established for the mentioned systems with external input affecting both the continuous and the discrete dynamics. It is shown that when the impulse-free delayed stochastic dynamics are p-ISS/SISS but the impulses are destabilizing, the p-ISS/SISS property of the impulsive stochastic systems can be preserved if the length of the impulsive interval is large enough. In particular, if the impulse-free delayed stochastic dynamics are p-ISS/SISS
and the discrete dynamics are marginally stable for the zero input, the impulsive stochastic system is p-ISS/SISS regardless of how often or how seldom the impulses occur. To overcome the difficulties caused by the coexistence of time delays, impulses, and stochastic effects, Razumikhin techniques and piecewise continuous Lyapunov functions as well as stochastic analysis techniques are involved together. An example is provided to illustrate the effectiveness and advantages of our results.


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## Body

## 1. Introduction

In practice, the performance of a real control system is affected more or less by uncertainties such as unmodeled dynamics, parameter perturbations, exogenous disturbances, and measurement errors [1]. To describe how solutions behave robustly to external inputs or disturbances, the concept of input-to-state stability (ISS) has been proven useful and effective in this regard. ISS was originally proposed by Sontag [2] for continuous-time systems. In view of its importance in the analysis and synthesis of nonlinear control systems [3–5], ISS and its variants such as local ISS, integral ISS, and exponential-weighted ISS have been investigated quite intensively and extended to different types of dynamical systems, for instance, discrete-time systems [6, 7], switched systems [1, 8–11], network control systems [12], neural networks [13–15], and so forth.As it is well known, impulsive effect is likely to exist in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time in the fields such as medicine and biology, economics, electronics, and telecommunications [16]. Recently, Hespanha initiated the study of ISS for impulsive systems [17]. It was proved therein that impulsive systems possessing an exponential ISS-Lyapunov function are uniformly ISS over a certain class of impulse time sequences. Since time delay phenomena are often encountered in real world systems and the existence of time delay is a significant cause of instability and deteriorative performance, [18] investigated the ISS property for nonlinear impulsive systems with time delays by using Razumikhin techniques. And [19] was also concerned with ISS of impulsive systems with time delays, where ISS theorems different from those in [18] were established by adopting both Razumikhin techniques and Lyapunov-Krosovskii functional method.In addition to the time delays and impulse effects, stochastic perturbations are always unavoidable in real systems (see [20–23] and references therein). Impulsive stochastic delayed systems incorporate impulses effects, stochastic perturbations, and time delays in one system simultaneously. During the last decade, there has been extensive interest in the study of force-free delayed impulsive stochastic systems; we refer to [24–28] and references therein. However, the corresponding theory for impulsive stochastic systems with external inputs has been relatively less developed.The present paper aims to generalize the ISS results for deterministic delayed impulsive systems to stochastic settings. Thepth moment input-to-state stability (p-ISS) and stochastic input-to-state stability (SISS) properties for impulsive stochastic delayed systems with external input affecting both the continuous dynamics and the impulses are investigated and Razumikhin-type theorems guaranteeing thep-ISS/SISS are established. The results indicate that when the delayed continuous stochastic dynamics arep-ISS/SISS and the discrete dynamics are destabilizing, thep-ISS/SISS properties of the original impulsive stochastic systems can be maintained if the length of impulsive interval is large enough. In particular, if the impulse-free delayed stochastic dynamics arep-ISS/SISS and the discrete dynamics are marginally stable for the zero input, the impulsive stochastic system isp-ISS/SISS regardless of how often or how seldom the impulses occur. As a byproduct, the criteria onpth moment global asymptotic stability (p-GAS) and global asymptotical stability in probability (GASiP) are also derived. The initial idea of this paper came from the works for deterministic impulsive delayed systems [18] and impulse-free stochastic systems [1, 29], but its extension to impulsive stochastic delayed systems will be much more challenging due to the simultaneous existence of time delays, impulses, and stochastic effects.The rest of this paper is organized as follows. In Section2, some basic notations and definitions used throughout the paper are introduced. In Section 3, criteria ensuring uniformp-ISS/SISS/p-GAS/GASiP are established and applied to linear impulsive stochastic delayed systems. Section 4 provides a numerical example to illustrate the effectiveness and advantages of our results. Finally, Section 5 includes a summary and a discussion of some extensions of the paper.


## 2. Preliminaries

Throughout this paper, unless otherwise specified, we will employ the following notations. Let(Ω,ℱ,{ℱt}t⩾0,ℙ) be a complete probability space with a filtration {ℱt}t⩾0 satisfying the usual conditions (i.e., it is right continuous and ℱ0 contains all ℙ-null sets) and let 𝔼[·] be the expectation operator with respect to the given probability measure ℙ. Let w(t)=(w1(t),…,wd(t))T be a d-dimensional Brownian motion defined on the probability space. ℝ=(-∞,+∞), ℝ+=[0,+∞), ℕ={1,2,3,…}, ℝn denotes the n-dimensional real space equipped with the Euclidean norm |·|, and ℝn×m denotes the n×m-dimensional real matrix space.Letτ⩾0 and PC([-τ,0];ℝn)={φ:[-τ,0]→ℝn∣φ(t) is continuous for all but at most a finite number of points t-, at which φ(t-+), φ(t--) exist and φ(t-+)=φ(t-)}, where φ(t-+) and φ(t--) denote the right-hand and left-hand limits of φ(t) at t-, respectively. We equip the linear space PC([-τ,0];ℝn) with the norm ∥φ∥ defined by ∥φ∥=sup⁡{|φ(θ)|:-τ⩽θ⩽0}. Let PCℱtb([-τ,0];ℝn) be the family of all ℱt-measurable and bounded PC([-τ,0];ℝn)-valued random variables ξ={ξ(θ):-τ⩽θ⩽0}.A functionα:ℝ+→ℝ+ is said to be of class 𝒦 if it is continuous and strictly increasing and satisfies α(0)=0; it is of class 𝒦∞ if in addition α(s)→∞ as s→∞. Note that if α is of class 𝒦∞, then the inverse function α-1 is well defined and is also of class 𝒦∞. v𝒦∞ and c𝒦∞ are the subsets of 𝒦∞ functions that are convex and concave, respectively. A function β:ℝ+×ℝ+→ℝ+ is said to be of class 𝒦ℒ if β(·,t)∈𝒦 for each fixed t⩾0 and β(r,t) decreases to 0 as t→∞ for each fixed r⩾0. The composition of two functions ϕ:A→B and ψ:B→C is denoted by ψ∘ϕ:A→C.IfA is a vector or a matrix, its transpose is denoted by AT. If P is a square matrix, P>0(P⩽0) means that P is a symmetric positive definite (negative semidefinite) matrix. λ_(·) and λ¯(·) represent the minimum and maximum eigenvalues of the corresponding matrix, respectively, and I stands for the identity matrix. The symbol * is used in symmetric matrices to denote the entries which can be inferred by symmetry. Unless explicitly stated, all matrices are assumed to have real entries and compatible dimensions.We consider the following impulsive stochastic nonlinear system with external inputs:(1)dx=f(t,xt,uc(t))dt+g(t,xt,uc(t))dw(t),dx=f(t,xt,uc(t))dt+dw(t)t≠tk,t⩾t0,x(tk)=Ik(tk,x(tk-),ud(tk-)),k∈ℕ,
with initial data xt0={x(t0+θ):-τ⩽θ⩽0}=ξ∈ℙCℱt0b([-τ,0];ℝn), where x∈ℝn and xt={x(t+θ):-τ⩽θ⩽0} is regarded as a PC([-τ,0];ℝn)-valued random variable; uc∈ℒ∞m1 is locally essentially bounded external input and ud∈ℒ∞m2 is the impulsive disturbance input; ℒ∞m denotes the set of all locally essentially bounded function u:ℝ+→ℝm with norm ∥u∥∞=esssup⁡t⩾t0|u(t)|; ∥u∥[a,b]=esssup⁡t∈[a,b]|u(t)|; both f:[t0,∞)×ℝn×ℝm1→ℝn and g:[t0,∞)×ℝn×ℝm1→ℝn×d are uniformly locally Lipschitz with respect to x and u; Ik:[t0,∞)×ℝn×ℝm2→ℝn represents the impulsive perturbation of x at tk and satisfies |Ik(tk,x,u)|<∞; {tk}k∈ℕ is a strictly increasing sequence of impulse times. We use 𝒮min⁡(β) and 𝒮all to denote the class of impulsive time sequences that satisfy inf⁡k∈ℕ{tk-tk-1}⩾β and the set containing all impulse time sequences, respectively.Moreover, we assume thatf(t,0,0)=g(t,0,0)=Ik(t,0,0)≡0 for all t⩾t0, k∈ℕ; then system (1) admits a trivial solution x(t)≡0. The input pair (uc,ud) is said to be admissible, denoted by (uc,ud)∈𝒰, if uc∈ℒ∞m1, ud∈ℒ∞m2 guarantee the the existence of a unique solution to system (1).On the foundation of the ISS concepts for impulse-free stochastic systems [1, 29, 30] and those for deterministic impulsive systems [18], we proposed the following definitions for impulsive stochastic delayed systems (1).Definition 1.
For a prescribed sequence{tk}k∈ℕ, system (1) is said to be pth (p>0) moment input-to-state stable (ISS) if there exist functions β∈𝒦ℒ,α,γc,γd∈𝒦∞ such that, for every initial condition ξ∈PCℱt0b and every input pair (uc,ud)∈𝒰,
(2)α(𝔼|x(t)|p)⩽β(𝔼∥ξ∥p,t-t0)+γc(∥uc∥[t0,t])+γd(max⁡tk∈(t0,t]{|ud(tk-)|}),t⩾t0.Definition 2.
For a given sequence{tk}k∈ℕ, system (1) is said to be stochastic input-to-state stable (SISS), if, for any ε>0, there exist functions β∈𝒦ℒ and α,γc,γd∈𝒦∞, such that, for every initial condition ξ∈PCℱt0b and every input pair (uc,ud)∈𝒰,
(3)P{(max⁡tk∈(t0,t]{|ud(tk-)|})α(|x(t)|)<β(∥ξ∥,t-t0)+γc(∥uc∥[t0,t])+γd(max⁡tk∈(t0,t]{|ud(tk-)|})}>1-ε,t⩾t0.Remark 3.
Redefiningβ and γc, γd, one can assume that α in (2) or (3) is the identity: if α(r)⩽β(s,t)+γc(υ)+γd(ν) holds, then also r⩽α-1(β(s,t)+γc(υ)+γd(ν))⩽α-1(3β(s,t))+α-1(3γc(υ))+α-1(3γd(ν)). We know by Lemma 4.2 in [31] that α-1(3β(·,·))∈𝒦ℒ and α-1(3γc(·)), α-1(3γd(·))∈𝒦∞. So estimates of the same type as (2) and (3) but with no “α” are obtained.In the following, we will definep-GAS and GASiP in the form of 𝒦ℒ function, which present very close analogy top-ISS and SISS, respectively.Definition 4.
For a prescribed sequence{tk}k∈ℕ, system (1) with input uc≡0, ud≡0 is said to be pth (p>0) moment globally asymptotically stable (GAS) if there exists a function β∈𝒦ℒ such that, for every initial condition ξ∈PCℱt0b,
(4)𝔼|x(t)|p⩽β(𝔼∥ξ∥p,t-t0),t⩾t0.Definition 5.
For a given sequence{tk}k∈ℕ, system (1) with input uc≡0, ud≡0 is said to be globally asymptotically stable in probability (GASiP), if, for any ε>0, there exists a function β∈𝒦ℒ, such that, for every initial condition ξ∈PCℱt0b,
(5)P{|x(t)|<β(∥ξ∥,t-t0)}>1-ε,t⩾t0.Remark 6.
By the vanishing ofγc(s) and γd(s) at s=0, (2) and (3) will degenerate to (4) and (5), respectively, when u≡0, which means thatp-ISS/SISS of system (1) impliesp-GAS/GASiP of the corresponding unforced system.System (1) is said to be uniformlyp-ISS or uniformly SISS over a given class of admissible impulsive time sequences 𝒮 if (2) or (3) holds for every sequence in 𝒮 with functions α, β, γc, and γd independent of the choice of the sequence. Uniformp-GAS and uniform GASiP can be defined similarly.Definition 7 (see [24]).
A functionV:ℝ+×ℝn→ℝ+ is said to be of class v0 if the following hold true. (i)
Vis continuous on each of the sets [tk-1,tk)×ℝn and for each x,y∈ℝn, t∈[tk-1,tk), k∈ℕ, and lim⁡(t,y)→(tk-,x)V(t,y)=V(tk-,x) exists.(ii)
V(t,x) is once continuously differentiable in t and twice in x in each of the sets (tk-1,tk)×ℝn, k∈ℕ.IfV∈v0, define an operator ℒV [24] with respect to system (1) by
(6)ℒV(t,φ,u)=Vt(t,φ(0))+Vx(t,x)f(t,φ,u)+12trace[gT(t,φ,u)Vxx(t,x)g(t,φ,u)],
where
(7)Vt(t,x)=∂V(t,x)∂t,Vx(t,x)=(∂V(t,x)∂x1,…,∂V(t,x)∂xn),Vxx(t,x)=(∂2V(t,x)∂xi∂xj)n×n.


## 3. Main Results

In this section, we will develop Lyapunov-Razumikhin methods and establish some criteria which provide sufficient conditions for thep-ISS and SISS properties of impulsive stochastic delayed systems (1).Theorem 8.
Assume that there exist functionsV∈v0, χ1,χ2∈𝒦∞, α1∈c𝒦∞, α2∈v𝒦∞ and scalars q>1, c>0, μ∈[1,q) such that (i)
α1(|x|p)⩽V(t,x)⩽α2(|x|p);(ii)
𝔼ℒV(t,φ)⩽-c𝔼V(t,φ(0))+χ1(|uc(t)|), for all t⩾t0, t≠tk and φ∈PCℱt([-τ,0];ℝn) whenever 𝔼V(t+θ,φ(θ))⩽q𝔼V(t,φ(0));(iii)
𝔼V(tk,Ik(tk-,x,ud))⩽μ𝔼V(tk,x)+χ2(|ud|). Then for any given ρ>0 satisfying μe-cρ<1, system (1) is uniformlyp-ISS over 𝒮min⁡(ρ). In particular, when μ=1, system (1) is uniformlyp-ISS over 𝒮all.Proof.
Sinceμe-cρ<1, then 0<1+e-cρ-1/μ<1 and there exists c′>0 such that cmax⁡{μe-cρ,1+e-cρ-1/μ}<c′<c. We choose λ∈(0,c′) such that (q/μ)e-λτ>1, μe-(c-λ)ρ⩽1, λ⩽c-μ(c-c′). Let {tk}k∈ℕ be any impulsive time sequence belonging to 𝒮min⁡(ρ). For simplicity, we write V(t,x)=V(t). Define
(8)J(t)=eλ(t-t0)[𝔼V(t)-J0(t)],t⩾t0-τ,
where J0(t)=(1/(c-c′))χ1(∥uc∥[t0,t])+∑tk∈(t0,t]‍e-λ(t-tk)χ2(|ud(tk-)|) for t⩾t0 and J0(t)=0 for t∈[t0-τ,t0]. We claim that
(9)J(t)⩽α2(𝔼∥ξ∥p),t⩾t0.
We first prove that (9) holds for t∈[t0,t1). By condition (i) and Jensen's inequality, it is easy to see that
(10)J(t)=𝔼V(t)eλ(t-t0)⩽α2(𝔼|x(t)|p)⩽α2(𝔼∥ξ∥p),t∈[t0-τ,t0].
If (9) is not true for t∈[t0,t1), there must exist some t∈[t0,t1) such that J(t)>α2(𝔼∥ξ∥p). Let t*=inf⁡{t∈[t0,t1):J(t)>α2(𝔼∥ξ∥p)}. Then by the right continuity of J(t) in t∈[t0,t1) and noticing (10), we have t*⩾t0 and
(11)J(t*)=α2(𝔼∥ξ∥p),J(t)⩽α2(𝔼∥ξ∥p),α2(𝔼∥ξ∥p)t∈[t0-τ,t*),D+J(t*)>0.
Because J(t*)⩾J(t*+s),s∈[-τ,0] implies
(12)𝔼V(t*)⩾eλs𝔼V(t*+s)-eλsJ0(t*+s)+J0(t*)⩾e-λτ𝔼V(t*+s)>μq𝔼V(t*+s)⩾1q𝔼V(t*+s),s∈[-τ,0],
it follows from condition (ii) that
(13)𝔼ℒV(t*)⩽-c𝔼V(t*)+χ1(|uc(t*)|).
For ρ>0 sufficiently small satisfying t*+ρ<t1, by the Itô formula, we have
(14)𝔼V(t*+ρ)-𝔼V(t*)=∫t*t*+ρ‍𝔼ℒV(s,xs)ds
which yields
(15)limsup⁡ρ→0+𝔼V(t*+ρ)-𝔼V(t*)ρ=limsup⁡ρ→0+1ρ∫t*t*+ρ‍𝔼ℒV(s)ds;
that is,
(16)D+𝔼V(t*)=𝔼ℒV(t*)⩽-c𝔼V(t*)+χ1(|uc(t*)|),
where D+𝔼V(t)≜limsup⁡ρ→0+[𝔼V(t+ρ)-𝔼V(t)]/ρ. On the other hand, J(t*)=α2(𝔼∥ξ∥p)⩾0 implies
(17)𝔼V(t*)⩾J0(t*).
Therefore, from (16) and (17), and noticing J0(t)=(1/(c-c′))χ1(∥uc∥[t0,t]) and D+J0(t)⩾0 for t∈[t0,t1), we have
(18)D+J(t*)=eλ(t*-t0)[D+𝔼V(t*)+λ𝔼V(t*)=eλ(t*-t0)-λJ0(t*)-D+J0(t*)]⩽eλ(t*-t0)[-(c-λ)𝔼V(t*)+χ1(|uc(t*)|)⩽eλ(t*-t0)-λJ0(t*)]⩽eλ(t*-t0)[-cJ0(t*)+χ1(|uc(t*)|)]⩽-c′c-c′eλ(t*-t0)χ1(∥uc∥[t0,t*])<0,
which contradicts D+J(t*)>0. Therefore, (9) holds for t∈[t0-τ,t1).
Suppose that (9) holds for t∈[t0-τ,tm), where m⩾1, m∈ℕ. We will prove that (9) holds for t∈[tm,tm+1). To this end, we claim that
(19)J1(tm-)⩽α2(𝔼∥ξ∥p),
where J1(t)=eλ(t-t0)[μ𝔼V(t)-J0(t)]. If not, then J1(tm-)>α2(𝔼∥ξ∥p). We consider the following two cases.
Case
1. J1(t)>α2(𝔼∥ξ∥p) for all t∈[tm-1,tm). It is easy to see that J1(t)>α2(𝔼∥ξ∥p)⩾J(t+θ) for t∈[tm-1,tm) and θ∈[-τ,0]. It follows that
(20)𝔼V(t+θ)<e-λθ[μ𝔼V(t)-J0(t)]+J0(t+θ)⩽eλτμ𝔼V(t)-e-λθJ0(t)+J0(t+θ)<q𝔼V(t),t∈[tm-1,tm),θ∈[-τ,0].
The last inequality comes from the fact that (q/μ)e-λτ>1, and
(21)J0(t+θ)=1c-c′χ1(∥uc∥[t0,t+θ])+∑tk∈(t0,t+θ]‍e-λ(t+θ-tk)χ2(|ud(tk-)|)⩽e-λθc-c′χ1(∥uc∥[t0,t])+∑tk∈(t0,t]‍e-λ(t+θ-tk)χ2(|ud(tk-)|)=e-λθJ0(t).
By condition (ii), (20) indicates that
(22)𝔼ℒV(t)⩽-c𝔼V(t)+χ1(|uc(t)|),t∈[tm-1,tm).
By Itô’s formula, we have
(23)ectm𝔼V(tm-)=ectm-1𝔼V(tm-1)+∫tm-1tm‍ecs[c𝔼V(s)+𝔼ℒV(s)]ds⩽ectm-1𝔼V(tm-1)+∫tm-1tm‍ecsχ1(|uc(s)|)ds⩽ectm-1𝔼V(tm-1)+1cectmχ1(∥uc∥[t0,tm));
thus,
(24)𝔼V(tm-)⩽e-c(tm-tm-1)𝔼V(tm-1)+1cχ1(∥uc∥[t0,tm)).
On the other hand, J(tm-1)⩽α2(𝔼∥ξ∥p) implies
(25)𝔼V(tm-1)⩽e-λ(tm-1-t0)α2(𝔼∥ξ∥p)+J0(tm-1)⩽e-λ(tm-1-t0)α2(𝔼∥ξ∥p)+1c-c′χ1(∥uc∥[t0,tm-1])+∑k=1m-1‍e-λ(tm-1-tk)χ2(|ud(tk-)|).
Substituting (25) into (24) and noticing the fact that inf⁡{tk-tk-1}⩾ρ, we have
(26)𝔼V(tm-)⩽e-c(tm-tm-1)-λ(tm-1-t0)α2(∥ξ∥p)+(1c+e-cρc-c′)χ1(∥uc∥[t0,tm))+e-c(tm-tm-1)∑k=1m-1‍e-λ(tm-1-tk)χ2(|ud(tk-)|).
Substituting (26) into J1(tm-) yields
(27)J1(tm-)=eλ(tm-t0)[μ𝔼V(tm-)-J0(tm-)]⩽μeλ(tm-t0)[(e-cρc-c′+1c)e-c(tm-tm-1)-λ(tm-1-t0)α2(𝔼∥ξ∥p)⩽μeλ(tm-t0)+(e-cρc-c′+1c)χ1(∥uc∥[t0,tm))⩽μeλ(tm-t0)+e-c(tm-tm-1)∑k=1m-1‍e-λ(tm-1-tk)⩽μeλ(tm-t0)×χ2(|ud(tk-)|)(e-cρc-c′+1c)]-eλ(tm-t0)[∑k=1m-1‍1c-c′χ1(∥uc∥[t0,tm))-eλ(tm-t0)+∑k=1m-1‍e-λ(tm-tk)χ2(|ud(tk-)|)]⩽μe-(c-λ)ρα2(𝔼∥ξ∥p)+eλ(tm-t0)[μ(e-cρc-c′+1c)-1c-c′]χ1(∥uc∥[t0,tm))+(μe-(c-λ)ρ-1)∑k=1m-1‍eλ(tk-t0)χ2(|ud(tk-)|)⩽α2(𝔼∥ξ∥p).
The last inequality holds because μe-(c-λ)ρ⩽1 and μ(((e-cρ)/(c-c′))+(1/c))-(1/(c-c′))⩽0. This is a contradiction.
Case
2. There exists some t∈[tm-1,tm) such that J1(t)⩽α2(𝔼∥ξ∥p). Set t′=sup⁡{t∈[tm-1,tm):J1(t)⩽α2(𝔼∥ξ∥p)}. Then J1(t′)=α2(𝔼∥ξ∥p) and J1(t)>α2(𝔼∥ξ∥p) for t∈(t′,tm). Thus, for t∈[t′,tm), J1(t)⩾α2(𝔼∥ξ∥p)⩾J(t+θ) for θ∈[-τ,0]. This implies that (20) holds for θ∈[-τ,0], t∈[t′,tm). Thus, by condition (ii),
(28)D+𝔼V(t)=𝔼ℒV(t)⩽-c𝔼V(t)+χ1(|uc(t)|),D+𝔼V(t)=𝔼ℒV(t)⩽-c𝔼V(t)+t∈[t′,tm).
Hence, noticing the fact that J0(t)⩾0, D+J0(t)⩾0 for t∈[t′,tm), we have
(29)D+J1(t)=eλ(t-t0)[λμ𝔼V(t)-λJ0(t)+μD+𝔼V(t)=eλ(t-t0)-D+J0(t)]⩽eλ(t-t0)[-μ(c-λ)𝔼V(t)-λJ0(t)⩽eλ(t-t0)+μχ1(|uc(t)|)-D+J0(t)]⩽μeλ(t-t0)[-(c-λ)𝔼V(t)+χ1(|uc(t)|)],22222222222222222222222222t∈[t′,tm).
Because J1(t)⩾α2(𝔼∥ξ∥p)>0 for t∈[t′,tm), there holds 𝔼V(t)>(1/μ)J0(t)>(1/μ(c-c′))χ1(∥uc∥[t0,t]) for t∈[t′,tm). Substituting this inequality with (29), and recalling the choice of λ, it follows that
(30)D+J1(t)⩽-eλ(t-t0)(c-λc-c′-μ)χ1(∥uc∥∞)⩽0,
which yields the following contradiction: α2(𝔼∥ξ∥p)<J1(tm-)⩽J1(t′)=α2(𝔼∥ξ∥p).
Therefore, we haveJ1(tm-)⩽α2(𝔼∥ξ∥p). Using condition (iii), we obtain that J(tm)⩽J1(tm-)⩽α2(𝔼∥ξ∥p). Repeating the argument used in the proof of J(t)⩽α2(𝔼∥ξ∥p) for t∈[t0,t1), we can get J(t)⩽α2(𝔼∥ξ∥p) for t∈[tm,tm+1). By the mathematical induction, we know that (9) holds for all t⩾t0. For any given t∈[tm,tm+1), one can get
(31)∑tk∈(t0,t]‍e-λ(t-tk)χ2(|ud(tk-)|)⩽11-e-λρχ2(max⁡tk∈(t0,t]{|ud(tk-)|}).
It follows from (9), (31), and the definition of J(t) that
(32)𝔼V(t)⩽e-λ(t-t0)α2(𝔼∥ξ∥p)+1c-c′χ1(∥uc∥∞)+11-e-λρχ2(max⁡tk∈(t0,t]{|ud(tk-)|}).
By casualty,
(33)𝔼V(t)⩽e-λ(t-t0)α2(𝔼∥ξ∥p)+1c-c′χ1(∥uc∥[t0,t])+11-e-λρχ2(max⁡tk∈(t0,t]{|ud(tk-)|}).
Then by condition (i) and Jensen's inequality, the required assertion (2) holds with β(r,s)=e-λsα2(r), γc(r)=(1/(c-c′))χ1(r) and γd(r)=(1/(1-e-λρ))χ2(r). By Lemma 4.2 in [31], it is easy to see that β∈𝒦ℒ,γc,γd∈𝒦∞. As β, γc, γd are independent of the particular choice of the impulse time sequence, system (1) is uniformlyp-ISS over 𝒮min⁡(ρ).
For the special caseμ=1, μe-cρ<1 holds for any ρ>0, so system (1) is uniformlyp-ISS over 𝒮min⁡(ρ) for any ρ>0. In other words, system (1) is uniformlyp-ISS over 𝒮all. The proof is complete.Remark 9.
Whenμ>1, condition (iii) implies that the impulses may be destabilizing. So, in order to maintain thep-ISS property of system (1), the impulse interval is required to be large enough to reduce the effect of the impulses. When μ=1, the discrete dynamics are marginally stable for the zero input. In this case, thep-ISS of system (1) is not affected by the impulses.With minor modification to the conditions of Theorem8, a criterion on SISS can be obtained as follows.Theorem 10.
Assume that conditions (ii) and (iii) of Theorem8 hold, while condition (i) is replaced by (i*)
α1(|x|)⩽V(t,x)⩽α2(|x|), where α1,α2∈𝒦∞. Then, for any given ρ>0 satisfying μe-cρ<1, system (1) is uniformly SISS over 𝒮min⁡(ρ). In particular, when μ=1, system (1) is uniformly SISS over 𝒮all.Proof.
By condition (i*), (10) can be replaced by
(34)J(t)=𝔼V(t)eλ(t-t0)⩽𝔼α2(|x(t)|)⩽𝔼α2(∥ξ∥),J(t)=𝔼V(t)eλ(t-t0)⩽𝔼α2(|x(t)|)t∈[t0-τ,t0].
Then, following the same lines of the proof of Theorem 8, it is easy to see that
(35)𝔼V(t)⩽e-λ(t-t0)𝔼α2(∥ξ∥)+1c-c′χ1(∥uc∥[t0,t])+11-e-λρχ2(max⁡tk∈(t0,t]{|ud(tk-)|})
holds for all t⩾t0. Consequently, by Chebyshev's inequality, it follows that
(36)P{11-e-λρχ2(max⁡tk∈(t0,t]{|ud(tk-)|}))V(t)-e-λ(t-t0)α2(∥ξ∥)111⩾δ((max⁡tk∈(t0,t]{|ud(tk-)|})1c-c′χ1(∥uc∥[t0,t])111⩾δ2+11-e-λρχ2(max⁡tk∈(t0,t]{|ud(tk-)|}))}⩽𝔼V(t)-𝔼α2(∥ξ∥)e-λ(t-t0)×(+11-e-λρχ2(max⁡tk∈(t0,t]⁡{|ud(tk-)|}))δ((max⁡tk∈(t0,t]⁡{|ud(tk-)|}22)1c-c′χ1(∥uc∥[t0,t])+11-e-λρχ2(max⁡tk∈(t0,t]⁡{|ud(tk-)|})))-1⩽ε,
where ε can be made arbitrarily small by an appropriate choice of δ∈𝒦∞. That is,
(37)P{111⩾δ2+11-e-λρχ2(max⁡tk∈(t0,t]{|ud(tk-)|}))V(t)<e-λ(t-t0)α2(∥ξ∥)111+δ((max⁡tk∈(t0,t]{|ud(tk-)|})1c-c′χ1(∥uc∥[t0,t])111⩾δ2+11-e-λρχ2(max⁡tk∈(t0,t]{|ud(tk-)|}))}>1-ε,
which yields
(38)P{(max⁡tk∈(t0,t]{|ud(tk-)|})V(t)<β(∥ξ∥,t-t0)+γc(∥uc∥[t0,t])22+γd(max⁡tk∈(t0,t]{|ud(tk-)|})}>1-ε,
where β(r,s)=e-λsα2(r), γc(r)=δ((2/(c-c′))χ1(r)), γd(r)=δ((2/(1-e-λρ))χ2(r)). By condition (i*), we know that (3) holds. Therefore, system (1) is uniformly SISS over 𝒮min⁡(ρ) and the proof is complete.In view of Definitions1–5, it is easy to obtain the following criteria onp-GAS and GASiP according to Theorems 8 and 10.Corollary 11.
Assume that there exist functionsV∈v0, α1∈c𝒦∞, α2∈v𝒦∞ and scalars q>1, c>0, μ∈[1,q) such that (i)
α1(|x|p)⩽V(t,x)⩽α2(|x|p);(ii)
𝔼ℒV(t,φ)⩽-c𝔼V(t,φ(0)), for all t⩾t0, t≠tk and φ∈PCℱt([-τ,0];ℝn) whenever 𝔼V(t+θ,φ(θ))⩽q𝔼V(t,φ(0));(iii)
𝔼V(tk,Ik(tk-,x,ud))⩽μ𝔼V(tk,x). Then, for any given ρ>0 satisfying μe-cρ, system (1) is uniformlyp-GAS over 𝒮min⁡(ρ). In particular, when μ=1, system (1) is uniformlyp-GAS over 𝒮all.Corollary 12.
Assume that conditions (ii) and (iii) of Corollary11 hold, while condition (i) is replaced by (i*)
α1(|x|)⩽V(t,x)⩽α2(|x|), where α1,α2∈𝒦∞. Then, for any given ρ>0 satisfying μe-cρ < 1, system (1) is uniformly GASiP over 𝒮min⁡(ρ). In particular, when μ=1, system (1) is uniformly GASiP over 𝒮all.Now let us apply the obtained results to the linear impulsive stochastic delayed system with the following form:(39)dx=(Ax(t)+A1xτ+Buc(t))dt+(Cx(t)+C1xτ+Duc(t))dw,t≠tk,x(tk)=Ex(tk-)+Fud(tk-),k∈ℕ,
on t⩾t0 with initial data xt0=ξ, where x∈ℝn and uc∈ℒ∞m1, ud∈ℒ∞m2 are system state and inputs, respectively; xτ is short for x(t-τ); A, A1, B, C, C1, D, E, F are constant matrices with appropriate dimensions.Corollary 13.
Assume that there exist a matrixP>0 and constants λ1<0, λ2>0, λ3>0, λ4>1, λ5>0 satisfying λ4e(λ1+λ2p)ρ<1 such that the following matrix inequalities hold:
(40)[ATP+PA+CTPC-λ1PPA1+CTPC1PB+CTPD*C1TPC1-λ2PC1TPD**DTPD-λ3I]⩽0,[ETPE-λ4PETPF*FTPF-λ5I]⩽0.
Then system (39) is uniformly ISS in mean square and uniformly SISS over 𝒮min⁡(ρ).Proof.
We choose the candidate ISS-Lyapunov functionV(t,x)=xTPx. By using (40), and in view of the fact that λmin⁡(P)|x|2⩽xTPx⩽λmax⁡(P)|x|2, we can obtain by simple calculation that
(41)ℒV(t,x,u)=(xxτuc)T×(ATP+PA+CTPCPA1+CTPC1PB+CTPD*C1TPC1C1TPD**DTPD)×(xxτuc)⩽λ1xTPx+λ2xτTPxτ+λ3ucTuc.
So, whenever 𝔼V(t+θ)⩽q𝔼V(t), we have
(42)𝔼ℒV(t,x,u)⩽(λ1+λ2q)𝔼V(t,x)+λ3|uc|2.
On the other hand,
(43)V(tk,x(tk))=(xud)T(ETPEETPF*FTPF)(xud)⩽λ4xTPx+λ5udTud.
So,
(44)𝔼V(tk,x(tk))⩽λ4𝔼V(tk-)+λ5|ud|2.
It is obvious that all conditions of Theorem 8 are satisfied, with c=-(λ1+λ2p) and μ=λ4. Therefore, we conclude by Theorems 8 and 10 that system (39) is uniformlyp-ISS and uniformly SISS over 𝒮min⁡(ρ).Remark 14.
It is noted that (40) are not linear with the combined variables (P,λ1,λ2,λ4), and, therefore, they are not linear matrix inequalities (LMIs). This makes the computation difficult but also flexible. We can first assign λ1, λ2, and λ4 and then solve LMIs (40) by using the Matlab LMI Toolbox.


## 4. Illustrative Example

In this section, to illustrate the validity of our results, we give the following linear numerical example. We point out that, due to the effect of the inputu, the state x(t) will not converge to zero but will remain bounded (in the sense of mean square or in probability), which is consistent with the definition ofp-ISS/SISS.Example 1.
Consider system (39) with the following parameters:
(45)A=diag⁡(-3,-2.5),A1=[0.30.10.10.2],B=[0.10.1],C=[0.20.40.30.1],C1=[-0.20.100.3],D=[0.10.1],E=[1.10.2-0.31.2],F=[0.20.5].
Settingλ1=-3.8, λ2=0.2, λ4=1.4, and solving LMIs (40) by using the Matlab LMI Toolbox, then
(46)P=[395.5597-72.7854395.5597269.6470],λ3=0.9546,λ5=3.5515
is a group of feasible solutions. Choosing p=1.41>μ=λ4=1.4, ρ=0.1, it is easy to check that all the conditions of Corollary 13 are satisfied, which means that the system is uniform ISS in mean square and uniform SISS for arbitrary sequence of impulse times satisfying inf⁡{tk-tk-1}⩾0.1. The sample path and the mean square of the solution are displayed in Figures 1 and 2, respectively, where τ=0.5, initial data ξ(θ)=[1-1]T for θ∈[-0.5,0], and impulse interval tk-tk-1=0.1 and external inputs uc(t)=ud(t)=sint.
Asp-ISS/SISS impliesp-GAS/GASiP of the corresponding unforced system, we conclude that the system with uc=ud≡0 is uniform GAS in mean square and GASiP for arbitrary sequence of impulse times satisfying inf⁡{tk-tk-1}⩾0.1. The simulations of the unforced system are shown in Figures 3 and 4.Figure 1
The mean square of the solution with external input (2000 samples).Figure 2
The state trajectory of the system with external input (single sample).Figure 3
The mean square of the solution without external input (2000 samples).Figure 4
The state trajectory of the system without external input (single sample).


## 5. Conclusions

This paper has investigated thep-ISS/SISS of impulsive stochastic systems with external inputs. By combining stochastic analysis techniques, piecewise continuous Lyapunov functions, and Razumikhin techniques, sufficient conditions for uniformp-ISS/SISS over a given class of impulse times sequences have been established. As a byproduct, the criteria onp-GAS/GASiP are also derived. For future research, interesting topics may include establishingp-ISS/SISS theorems with stabilizing impulses, as well asp-ISS/SISS analysis by exploring new techniques such as Lyapunov-Krosovskii functional method.



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