𝑝-Moment Stability of Stochastic Differential Delay Systems with Impulsive Jump and Markovian Switching

Abstract

This paper investigates 𝑝-moment stability of the stochastic differential delay systems with impulsive jump and Markovian switching. Some stability criteria are obtained based on Lyapunov functional method and stochastic theory. It is shown that, even if all the subsystems governing the continuous dynamics without impulse are not stable, as impulsive and switching signal satisfies a dwell-time upper bound condition, impulses can stabilize the systems in the 𝑝-moment stability sense. The opposite situation is also developed for which all the subsystems governing the continuous dynamics are 𝑝-moment stable. The results can be easily applied to stochastic systems with arbitrarily large delays. The efficiency of the proposed results is illustrated by two numerical examples.

1. Introduction

Stochastic differential systems which include stochastic delay differential systems have attracted much attention, owing to stochastic modeling having played an important role in many ways such as science and engineering and forecast of the growth of population [1, 2]. Stability analysis of different stochastic systems has been a subject of intense activities in the literature [3–9]. Switched systems are an important class of hybrid dynamical systems which are composed of a family of continuous time or discrete-time dynamical systems and a rule that orchestrates the switching among them [10]. A particular class of switched systems is named as markovian switched systems, whose system mode is governed by a Markov process. During the past few decades, many issues on Markovian switched systems such as stability and stabilization [11–14], 𝐻_∞ control and filtering [15], and adaptive control problem [16, 17] have been well investigated.

Beside stochastic effects and markovian switching, impulsive effect likewise exists in many evolution processed in which systems states change abruptly at certain moments of time, involving such fields as biology, engineering, and information science [18]. Therefore, the stability investigation of stochastic differential systems with impulsive jump is interesting to many investigators. The 𝑝-moment stability of stochastic differential systems is studied in [19] for the systems with impulsive jump, nonswitched, and no time delay. Thus, the results in [19] cannot be easily applied to the class of impulsive systems with markovian switching and time-varying delay. Liu and Peng [20] discussed the 𝑝-moment stability of stochastic differential delay systems with impulsive jump and markovian switching. It is shown that when the delayed continuous dynamics are 𝑝-moment stable, the stability properties is not destroyed by impulse at discrete instants irrespective of the length of delay. It is also noticed that the conclusions received in [20] are restricted to the case that are all the subsystems that govern the continuous dynamics are stable in 𝑝-moment sense. There is no attention has been paid to the class of hybrid systems in which all the subsystems that govern the continuous dynamics is not stable. Thus, how to establish a sufficient condition of 𝑝-moment stability for the class of stochastic delays hybrid systems is the key problem to be solved in future research.

Based on the above analysis, in this work, the 𝑝-moment stability of stochastic differential delay systems with impulsive jump and markovian switching is investigated. Motivated by the work of [20], we first relax the global Lipschitz condition of impulsive control law Δ𝑥(𝑡_𝑘). Then, we will propose some conditions of the 𝑝-moment stability for two classes of stochastic hybrid delay systems, that is, all the subsystems that govern the continuous dynamics are stable and not. Compared with [20], the criteria for the stable case in this paper are more general. Further, the results are applied to linear stochastic delay systems with arbitrarily large delays.

2. Preliminaries

Throughout this paper, unless otherwise specified, we let (Ω,𝐹,{𝐹_𝑡}_𝑡≥0,𝑃) be a complete probability space with a filtration {𝐹_𝑡}_𝑡≥0 satisfying the usual conditions, that is, it is right continuous and 𝐹₀ contains all 𝑃-null sets. 𝜔(𝑡) is an 𝑚-dimensional Brownian motion defined on (Ω,𝐹,{𝐹_𝑡}_𝑡≥0,𝑃). Let 𝑍⁺ define the set of nonnegative real numbers and 𝑅^𝑛 the 𝑛-dimensional real Euclidean space. |⋅| denotes the Euclidean norm for vectors or the spectral norm for matrices. For 𝑟>0, let PC([−𝑟,0],𝑅^𝑛) denote the class of functions from [−𝑟,0] to 𝑅^𝑛 satisfying the following: (i) it has at most a finite number of jump discontinuities on (−𝑟,0]; (ii) it is continuous from the right at all points in [−𝑟,0). For simplicity, PC is used for PC([−𝑟,0],𝑅^𝑛) for the rest of this paper. For function 𝜙∶[−𝑟,0]→𝑅^𝑛, a norm is defined as ‖𝜙‖_𝑟=sup_{−𝑟≤𝜃≤0}|𝜙(𝜃)|. Given 𝑥∈PC([−𝑟,∞],𝑅^𝑛) and for each 𝑡∈𝑅⁺, define 𝑥_𝑡,𝑥_𝑡−∈PC by 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠) for −𝑟≤𝑠≤0 and 𝑥_𝑡−(𝑠)=𝑥(𝑡+𝑠) for −𝑟≤𝑠<0, respectively. Denote by PC^𝑝_𝐹𝑡 the family of all 𝐹_𝑡-measurable PC-valued random variables 𝜙={𝜙(𝜃)∶−𝑟≤𝜃≤0}, satisfying sup_{−𝑟≤𝜃≤0}𝐸‖𝜙(𝜃)‖^𝑝<∞, where 𝐸 stands for the mathematical expectation. Let PC^𝑏_𝐹𝑡(𝛿)={𝜙∶𝜙∈PC^𝑝_𝐹𝑡([−𝑟,0],𝑅^𝑛)andsup_{−𝑟≤𝜃≤0}𝐸‖𝜙(𝜃)‖^𝑝<𝛿}.

The Markov process {𝑟(𝑡),𝑡≥0} represents the switching between the different modes taking values in a finite state space 𝑆={1,2,…,𝑁} with generator 𝜋=(𝜋_𝑖𝑗)_𝑁×𝑁 given by Pr𝑟()(𝑡)=𝜋𝑡+Δ=𝑗∣𝑟=𝑖^𝑖𝑗Δ+𝑜(Δ),if𝑖≠𝑗,1+𝜋_𝑖𝑖Δ+𝑜(Δ),if𝑖=𝑗, (1) where 𝜋_𝑖𝑗 is the transition rate from mode 𝑖 to 𝑗 and satisfies the following relations: 𝜋_𝑖𝑗≥0,𝜋_𝑖𝑖=−_𝑗≠𝑖𝜋_𝑖𝑗, (2) and 𝑜(Δ) is such that lim_Δ→0𝑜(Δ)/Δ=0. We assume that the Markov chain 𝑟(𝑡) is independent of the Brownian motion 𝜔(⋅). It is known that almost every sample path of 𝑟(𝑡) is a right-continuous step function with a finite number of simple jumps in any finite subinterval of 𝑅⁺=[0,+∞).

Consider the following stochastic nonlinear delay system with impulsive jump and markovian switching 𝑑𝑥(𝑡)=𝑓𝑥(𝑡),𝑥_𝑡(𝑡)𝑥(𝑡),𝑡,𝑟𝑑𝑡+𝑔,𝑥_𝑡,𝑡,𝑟(𝑡)𝑑𝜔(𝑡),𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=𝐼_𝑘(𝑥(𝑡⁻),𝑡),𝑡=𝑡_𝑘,𝑥𝑡₀+𝜃=𝜙(𝜃),𝜃∈[−𝑟,0], (3) where 𝜙(𝜃)∈PC is the initial data, 𝑥(𝑡)∈𝑅^𝑛 is the systems state, 𝑥(𝑡⁺) and 𝑥(𝑡⁻) denote the limit from the right and the left at point 𝑡, respectively, and {𝑡_𝑘∶𝑘∈𝑍⁺}⊂𝑅⁺ a strictly increasing sequence. We assume that for each 𝑖∈𝑆, given functional 𝑓∶𝑅^𝑛×PC×𝑆→𝑅^𝑛, 𝑔∶𝑅^𝑛×PC×𝑆→𝑅^𝑛×𝑚 satisfying 𝑓(0,0,𝑡,𝑖)≡𝑔(0,0,𝑡,𝑖)≡0. 𝐼_𝑘∶𝑅^𝑛×𝑅→𝑅^𝑛 with 𝐼_𝑘(0,𝑡)≡0 is the change of state variable at instant 𝑡_𝑘. In this paper, we always assume that there exists a unique stochastic process satisfying systems (3), and all solutions of systems (3) are continuous on the right-hand side and on the left-hand side. Moreover, by 𝑓(0,0,𝑡,𝑖)≡𝑔(0,0,𝑡,𝑖)≡0 and 𝐼_𝑘(0,𝑡)≡0, it is easily obtained that system (3) admits a trivial solution.

Definition 1 (see [20]). The trivial solution of system (3) is (1)𝑝-moment stable if for any initial data 𝑥_𝑡0=𝜙 and any 𝜀>0, there exists a 𝛿>0 such that 𝐸‖𝑥(𝑡)‖^𝑝<𝜀,𝑡≥𝑡₀, (4) whenever ‖𝜙‖^𝑝_𝑟<𝛿;(2) uniformly 𝑝-moment stable if the 𝛿 in (1) is independent of 𝑡₀.

Definition 2. Let 𝐶^2,1(𝑅^𝑛×[𝑡₀−𝑟,∞]×𝑆;𝑅⁺) denote the family of all nonnegative functions 𝑉(𝑥,𝑡,𝑖) that are continuously twice differentiable in 𝑥 and once differentiable with respect to 𝑡. For each 𝑉(𝑥,𝑡,𝑖)∈𝐶^2,1(𝑅^𝑛×[𝑡₀−𝑟,∞]×𝑆;𝑅⁺), define an operator 𝐿𝑉 from 𝑅^𝑛×[𝑡₀−𝑟,∞]×𝑆 to 𝑅⁺ as follows: 𝐿𝑉(𝑥,𝑡,𝑖)=𝑉_𝑡(𝑥,𝑡,𝑖)+𝑉_𝑥()𝑓+1𝑥,𝑡,𝑖𝑥,𝑦,𝑡,𝑖2trace𝑔^𝑇𝑥,𝑦,𝑡,𝑖𝑉_𝑥𝑥()𝑔+𝑥,𝑡,𝑖𝑥,𝑦,𝑡,𝑖_𝑁𝑗=1𝑞_𝑖𝑗𝑉𝑥,𝑡,𝑗, (5) where 𝑉_𝑡()=()𝑥,𝑡,𝑖𝜕𝑉𝑥,𝑡,𝑖,𝑉𝜕𝑡_𝑥()=()𝑥,𝑡,𝑖𝜕𝑉𝑥,𝑡,𝑖𝜕𝑥₁(),…,𝜕𝑉𝑥,𝑡,𝑖𝜕𝑥_𝑛,𝑉_𝑥𝑥()=𝜕𝑥,𝑡,𝑖²𝑉(𝑥,𝑡,𝑖)𝜕𝑥_𝑖𝜕𝑥_{𝑗𝑛×𝑛}. (6) Before giving the results, we need a lemma.

Lemma 3 (see [20]). If   𝑉∈𝐶^2,1(𝑅^𝑛×[𝑡₀−𝑟,∞]×𝑆;𝑅⁺), then for any stopping times 0≤𝑡₁≤𝑡₂<+∞, 𝐸𝑉𝑥𝑡₂,𝑡₂,𝑟𝑡₂=𝐸𝑉𝑥𝑡₁,𝑡₁,𝑟𝑡₁+𝐸^𝑡2_𝑡1𝐿𝑉(𝑥(𝑠),𝑠,𝑟(𝑠))𝑑𝑠, (7) as long as the integration involved exist and finite.

3. Main Results

In this section, Lyapunov-based sufficient conditions for 𝑝-moment stability of system (3) are developed. The first result is concerned with 𝑝-moment stability of system (3), in the case when all the subsystems governing the continuous dynamics of (3) are stable. Intuitively, the conditions in the following theorem consist of three aspects: (i) the Lyapunov-Krasovskii functionals satisfy certain positive definite and decrescent conditions; (ii) there exist some negative estimates of the upper right-hand derivatives of the functionals with respect to each stable mode of system (3); (iii) the jumps induced by the impulses and the estimates on the decay rate of continuous dynamics satisfy certain conditions.

Theorem 4. Assume that there exist function 𝑉(𝑥(𝑡),𝑡,𝑖)∈𝐶^2,1(𝑅^𝑛×[𝑡₀−𝑟,∞]×𝑆;𝑅⁺) and some positive constants 𝑝, 𝑎, 𝑏, 𝑒_𝑖, the nonnegative and continuous functions 𝑐_𝑖(𝑡) and 𝑑_𝑖(𝑡), such that(i)𝑎|𝑥|^𝑝≤𝑉(𝑥(𝑡),𝑡,𝑖)≤𝑏|𝑥_𝑡|^𝑝_𝑟; (ii)𝐸𝐿𝑉(𝑥(𝑡),𝑡,𝑖)≤−𝑐_𝑖(𝑡)𝐸𝑉(𝑥(𝑡),𝑡,𝑖)+𝑑_𝑖(𝑡)𝐸𝑉(𝑥_𝑡,𝑡,𝑖), 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘=1,2,…;(iii)𝐸𝑉(𝜙(0)+𝐼_𝑘(𝜙,𝑡_𝑘,𝑖)̃≤𝑒_𝑖𝑉(𝜙(0),𝑡⁻_𝑘,𝑖); (iv)0<𝑐^∗<1, where 𝑐^∗=sup{𝑐^∗_𝑖∣𝑐^∗_𝑖=𝑏𝑒_𝑖/𝑎,𝑖=1,2,…};(v)𝑐_𝑖(𝑡)>𝑑_𝑖(𝑡)/𝑐^∗,𝑡∈[𝑡_𝑘,𝑡_𝑘+1); then the trivial solution of system (3) is uniformly 𝑝-moment stable.

Proof. For any 𝜀>0, there exists a 𝛿=𝛿(𝜀)>0 such that 𝛿<(𝑎𝑐^∗/𝑏)𝜀 independent of 𝑡₀. For any 𝑡₀≥0 and 𝑥_𝑡0=𝜙∈PC^𝑏_𝐹𝑡(𝛿), let 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝜙) be the solution of (3).
In the following, we will show that (𝑥(𝑡))≤𝑏𝐸𝑉,𝑡,𝑖𝑐^∗𝛿,𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑘∈𝑍⁺. (8) It will be firstly proved that (𝑥(𝑡))≤𝑏𝐸𝑉,𝑡,𝑖𝑐^∗𝛿,𝑡∈𝑡₀,𝑡₁. (9) For any initial data 𝑥_𝑡0∈PC^𝑏_𝐹𝑡(𝛿), it follows from assumption (i) that 𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))≤𝑏𝐸𝑥_𝑡0^𝑝_𝑟𝑏≤𝑏𝛿<𝑐^∗𝛿,𝑡∈𝑡₀−𝑟,𝑡₀. (10) Suppose that (9) is not true, then there exists some 𝑡∈(𝑡₀,𝑡₁) such that (𝑥(𝑡)(𝑡>𝑏𝐸𝑉,𝑡,𝑟))𝑐^∗𝛿>𝑏𝛿≥𝐸𝑉𝑥𝑡₀,𝑡₀,𝑟𝑡₀. (11) Set 𝑡^∗=inf{𝑡∈[𝑡₀,𝑡₁)∶𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))>(𝑏/𝑐^∗)𝛿}. Observe that 𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)) is continuous on 𝑡∈[𝑡₀,𝑡₁), then 𝑡^∗∈(𝑡₀,𝑡₁) and 𝐸𝑉𝑥𝑡^∗,𝑡^∗,𝑟𝑡^∗=𝑏𝑐^∗𝛿,(𝑥(𝑡)(𝑡<𝑏𝐸𝑉,𝑡,𝑟))𝑐^∗𝛿,𝑡∈𝑡₀−𝑟,𝑡^∗. (12)(13) In view of (10), define 𝑡^∗∗=sup{𝑡∈[𝑡₀−𝑟,𝑡^∗]∶𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))≤𝑏𝛿}. Then 𝑡^∗∗∈(𝑡₀,𝑡^∗) and 𝐸𝑉𝑥𝑡^∗∗,𝑡^∗∗,𝑟𝑡^∗∗=𝑏𝛿,𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))>𝑏𝛿,𝑡∈𝑡^∗∗,𝑡^∗. (14)(15) Consequently, in view of (10)–(15), for all 𝑡∈[𝑡^∗,𝑡^∗∗] and 𝜃∈[−𝑟,0], one has (𝑥()(≤𝑏𝐸𝑉𝑡+𝜃,𝑡+𝜃,𝑟𝑡+𝜃))𝑐^∗1𝛿≤𝑐^∗𝐸𝑉𝑥𝑡^∗,𝑡^∗,𝑟𝑡^∗≤1𝑐^∗𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)). (16) Now, combining (16) and condition (ii), we have 𝐸𝐿𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))≤−𝑐_𝑖(𝑡)𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))+𝑑_𝑖(𝑡)𝐸𝑉𝑥_𝑡(𝑡)≤,𝑡,𝑟−𝑐_𝑖(𝑡)+𝑑_𝑖(𝑡)𝑐^∗𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)). (17) Applying Lemma 3, integrating (17) on [𝑡^∗∗,𝑡^∗], one obtains that 𝐸𝑉𝑥𝑡^∗,𝑡^∗,𝑟𝑡^∗≤𝐸𝑉𝑥𝑡^∗∗,𝑡^∗∗,𝑟𝑡^∗∗+^𝑡∗_𝑡∗∗−𝑐_𝑖(𝑡)+𝑑_𝑖(𝑡)𝑐^∗𝐸𝑉(𝑥(𝑠),𝑠,𝑟(𝑠))𝑑𝑠. (18) By condition (v) and (14) and the Gronwall inequality, one has 𝐸𝑉𝑥𝑡^∗,𝑡^∗,𝑟𝑡^∗≤𝐸𝑉𝑥𝑡^∗∗,𝑡^∗∗,𝑟𝑡^∗∗𝑒^{(−𝑐𝑖(𝑡)+(𝑑𝑖(𝑡)/𝑐∗))(𝑡∗−𝑡∗∗)}≤𝐸𝑉𝑥𝑡^∗∗,𝑡^∗∗,𝑟𝑡^∗∗𝑏=𝑏𝛿<𝑐^∗𝛿. (19) Since (19) contradicts with (12), inequality (9) holds and (8) is true for 𝑘=1.
Now, assume that (𝑥(𝑡))≤𝑏𝐸𝑉,𝑡,𝑖𝑐^∗𝛿,𝑡∈𝑡_𝑘−1,𝑡_𝑘, (20) for all 𝑘≤𝑚, where 𝑘,𝑚∈𝑍⁺. We proceed to show that (𝑥(𝑡))≤𝑏𝐸𝑉,𝑡,𝑖𝑐^∗𝛿,𝑡∈𝑡_𝑚,𝑡_𝑚+1. (21) Suppose that (21) is not true, set 𝑡=inf{𝑡∈[𝑡_𝑚,𝑡_𝑚+1)∶𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))>(𝑏/𝑐^∗)𝛿}. From condition (i), (iv), and (21), we know that 𝐸𝑉𝑥𝑡_𝑚,𝑡_𝑚,𝑟𝑡_𝑚≤𝑒_𝑖𝑚𝐸𝑉𝑥𝑡⁻_𝑚,𝑡⁻_𝑚,𝑟𝑡⁻_𝑚≤𝑏𝑒_𝑖𝑚𝑥_𝑡−𝑚^𝑝_𝑟≤𝑏𝑒_𝑖𝑚𝑎sup_{−𝑟≤𝜃≤0}𝐸𝑉𝑥𝑡⁻_𝑚+𝜃,𝑡⁻_𝑚+𝜃,𝑟𝑡⁻_𝑚≤+𝜃𝑏𝑒_𝑖𝑚𝑎𝑏𝑐^∗𝑏𝛿≤𝑏𝛿<𝑐^∗𝛿. (22) Owing to 𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)) is continuous on 𝑡∈[𝑡_𝑚,𝑡_𝑚+1), then 𝑡∈(𝑡_𝑚,𝑡_𝑚+1) and 𝐸𝑉𝑥𝑡,𝑡,𝑟𝑡=𝑏𝑐^∗𝛿,(𝑥(𝑡)(𝑡<𝑏𝐸𝑉,𝑡,𝑟))𝑐^∗𝛿,𝑡∈𝑡_𝑚,𝑡. (23)(24) Define 𝑡=sup{𝑡∈[𝑡₀−𝑟,𝑡]∶𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))≤𝑏𝛿}, then 𝑡∈(𝑡_𝑚,𝑡) and 𝐸𝑉𝑥𝑡,𝑡,𝑟𝑡=𝑏𝛿,𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))>𝑏𝛿,𝑡∈𝑡,𝑡. (25)(26) Fix any 𝑡∈[𝑡,𝑡], when 𝑡+𝜃≥𝑡_𝑚 for all 𝜃∈[−𝑟,0], then from (22)–(26), one has (𝑥()(≤𝑏𝐸𝑉𝑡+𝜃,𝑡+𝜃,𝑟𝑡+𝜃))𝑐^∗1𝛿≤𝑐^∗𝐸𝑉𝑥𝑡,𝑡,𝑟𝑡≤1𝑐^∗𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)). (27) When 𝑡+𝜃<𝑡_𝑚 for some 𝜃∈[−𝑟,0], without loss of generality, we assume that 𝑡+𝜃∈[𝑡_𝑙−1,𝑡_𝑙) for some 𝑙∈𝑍⁺, 𝑙<𝑚, then from (20) and (25), we obtain that(𝑥()(≤𝑏𝐸𝑉𝑡+𝜃,𝑡+𝜃,𝑟𝑡+𝜃))𝑐^∗1𝛿≤𝑐^∗𝐸𝑉𝑥𝑡,𝑡,𝑟𝑡≤1𝑐^∗𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)). (28) Therefore, from condition (ii), (27), and (28), one gets (𝑥()(≤𝐸𝐿𝑉𝑡+𝜃,𝑡+𝜃,𝑟𝑡+𝜃))−𝑐_𝑖𝑚(𝑡)+𝑑_𝑖𝑚(𝑡)𝑐^∗𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)). (29) Similar to the argument on [𝑡^∗∗,𝑡^∗], an application of Ito’s formula on [𝑡,𝑡] will lead to 𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))<(𝑏/𝑐^∗)𝛿, which would contradict with (23). So the inequality (21) is true. Therefore, by mathematical induction, one can obtain that (8) holds for all 𝑘∈𝑍⁺. Then from condition (i) and the definition of 𝛿, we have 𝐸‖𝑥(𝑡)‖^𝑝≤𝑏𝑎𝑐^∗𝛿<𝜀,𝑡≥𝑡₀. (30)
According to Definition 1, it is concluded that the trivial solution of system (3) is uniformly 𝑝-moment stable. The proof is complete.

Corollary 5. Assume that there exist function 𝑉(𝑥(𝑡),𝑡,𝑖)∈𝐶^2,1(𝑅^𝑛×[𝑡₀−𝑟,∞]×𝑆;𝑅⁺) and some positive constants 𝑝, 𝑎, 𝑏, 𝑒_𝑖, 𝑐_𝑖, and 𝑑_𝑖 such that(i)𝑎|𝑥|^𝑝≤𝑉(𝑥(𝑡),𝑡,𝑖)≤𝑏|𝑥_𝑡|^𝑝_𝑟; (ii)𝐸𝐿𝑉(𝑥(𝑡),𝑡,𝑖)≤−𝑐_𝑖𝐸𝑉(𝑥(𝑡),𝑡,𝑖)+𝑑_𝑖𝐸𝑉(𝑥_𝑡,𝑡,𝑖), 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘=1,2,…;(iii)𝐸𝑉(𝜙(0)+𝐼_𝑘(𝜙,𝑡_𝑘,𝑖)̃≤𝑒_𝑖𝑉(𝜙(0),𝑡⁻_𝑘,𝑖); (iv)0<𝑐^∗<1, where 𝑐^∗=sup{𝑐^∗_𝑖∣𝑐^∗_𝑖=(𝑏𝑒_𝑖/𝑎),𝑖=1,2,…};(v)𝑐_𝑖>𝑑_𝑖/𝑐^∗, 𝑡∈[𝑡_𝑘,𝑡_𝑘+1); then the trivial solution of system (3) is uniformly 𝑝-moment stable.

Proof. Replacing 𝑐_𝑖(𝑡) and 𝑑_𝑖(𝑡) in Theorem 4 with 𝑐_𝑖 and 𝑑_𝑖, respectively, we find that it is a direct conclusion of Theorem 4.

Remark 6. From condition (ii) in Theorem 4, it is seen that each of the continuous dynamics is stable. On the other hand, from condition (iv), we see that each of the discrete dynamics is also stabilizing (0<𝑐^∗<1 implies 𝑒_𝑖<1). It implies that the stability properties of a time-delay impulsive markovian switched system with stable continuous dynamics can be preserved under stabilizing impulsive perturbations irrespective of the times of impulses and switching. Hence, the bound of dwell-time is not necessary.

Remark 7. If condition (iii) is omitted in Corollary 5, the result is consistent with that of Theorem 1 in [20]. Thus, the results in this work are an extension of that in [20].
We proceed to consider in next the 𝑝-moment stability of systems (3). It is supposed that all the subsystems governing the continuous dynamics of (3) can be unstable while the impulses are stabilizing. Intuitively, the conditions in the following theorem consist of four aspects: (i) the Lyapunov functionals satisfy certain positive definite and decrescent conditions; (ii) there exist some positive estimates of the upper right-hand derivatives of the functionals with respect to each unstable mode of system (3); (iii) the dwell time of each mode of system (3) satisfies some supper bounds; (iv) the jumps induced by the stabilizing impulses satisfy certain diminishing conditions.

Theorem 8. Assume that there exist function 𝑉(𝑥(𝑡),𝑡,𝑖)∈𝐶^2,1(𝑅^𝑛×[𝑡₀−𝑟,∞]×𝑆;𝑅⁺) and some positive scalars 𝑝, 𝑎, 𝑏, 𝑒_𝑖, 𝜌<1, 𝛼, and 𝛽 such that(i)𝑎|𝑥|^𝑝≤𝑉(𝑥(𝑡),𝑡,𝑖)≤𝑏|𝑥_𝑡|^𝑝_𝑟; (ii)𝐸𝑉(𝜙(0)+𝐼_𝑘(𝜙,𝑡_𝑘,𝑖)̃≤𝜌𝑒_𝑖𝑉(𝜙(0),𝑡⁻_𝑘,𝑖); (iii)0<𝑐^∗<1, where 𝑐^∗=sup{𝑐^∗_𝑖∣𝑐^∗_𝑖=(𝑏/𝑎)𝜌𝑒_𝑖,𝑖=1,2,…};(iv) there exist nonnegative and piecewise continuous functions 𝑐_𝑖(𝑡)∶[𝑡₀,∞)→𝑅^𝑛 satisfying ∫_𝑡^𝑡+𝛼𝑐_𝑖(𝑠)𝑑𝑠≤𝛼𝛽 for all 𝑡≥𝑡₀, such that 𝐸𝐿𝑉(𝑥(𝑡),𝑡,𝑖)≤𝑐_𝑖(𝑡)(𝑥(𝑡)),𝑡𝐸𝑉,𝑡,𝑖𝑡∈_𝑘−1,𝑡_𝑘,𝑘=1,2,…, (31) whenever 𝑡≥𝑡₀ and 𝜙∈PC^𝑏_𝐹𝑡 are such that 𝑐^∗𝐸𝐿𝑉(𝜙(𝑠),𝑡+𝑠,𝑖)̃≤𝐸𝑉(𝜙(0),𝑡,𝑖);(v)sup_𝑘∈𝑍⁺{𝑡_𝑘−𝑡_𝑘−1}=𝛼<−(1/𝛽)ln((𝑏/𝑎)𝜌𝑒_𝑖), 𝑘∈𝑍⁺; then the trivial solution of system (3) is uniformly 𝑝-moment stable.

Proof. For any 𝜀>0, there exists a 𝛿=𝛿(𝜀)>0 such that 𝛿<(𝑎𝑐^∗/𝑏)𝜀 independent of 𝑡₀. For any 𝑡₀≥0 and 𝑥_𝑡0=𝜙∈PC^𝑏_𝐹𝑡(𝛿), let 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝜙) be the solution of (3).
By generalized Ito’s formula (Lemma 3), when 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), we get (𝑥(𝑡)(𝑡𝑥𝑡𝐸𝑉,𝑡,𝑟))=𝐸𝑉_𝑘,𝑡_𝑘,𝑟𝑡_𝑘+𝐸^𝑡_𝑡𝑘𝐿𝑉(𝑥(𝑠),𝑠,𝑟(𝑠))𝑑𝑠+𝐸^𝑡_𝑡𝑘𝜕𝑉(𝑥(𝑠),𝑠,𝑟(𝑠))(𝑠)𝑔(𝑥(𝑠)(𝑠(𝑠)𝑥𝑡𝜕𝑥,𝑠,𝑟))𝑑𝜔=𝐸𝑉_𝑘,𝑡_𝑘,𝑟𝑡_𝑘+𝐸^𝑡_𝑡𝑘𝐿𝑉(𝑥(𝑠),𝑠,𝑟(𝑠))𝑑𝑠. (32) Given small enough Δ𝑡>0 such that 𝑡+Δ𝑡∈[𝑡_𝑘,𝑡_𝑘+1), one has (𝑥()(𝑥𝑡𝐸𝑉𝑡+Δ𝑡,𝑡+Δ𝑡,𝑟𝑡+Δ𝑡))=𝐸𝑉_𝑘,𝑡_𝑘,𝑟𝑡_𝑘+𝐸_𝑡^𝑡+Δ𝑡_𝑘(𝑥(𝑠)(𝑠,𝑡𝐿𝑉,𝑠,𝑟))𝑑𝑠𝑡∈_𝑘,𝑡_𝑘+1. (33) In view of condition (iv), (32), and (33), it follows that(𝑥()((𝑥(𝑡)(𝑡𝐸𝑉𝑡+Δ𝑡,𝑡+Δ𝑡,𝑟𝑡+Δ𝑡))−𝐸𝑉,𝑡,𝑟))=_𝑡^𝑡+Δ𝑡(𝑥(𝑠)(𝑠≤𝐸𝐿𝑉,𝑠,𝑟))𝑑𝑠_{𝑡𝑡+Δ𝑡}𝑐_𝑖(𝑡)(𝑥(𝑠)(𝑠𝑡𝐸𝑉,𝑠,𝑟))𝑑𝑠,𝑡∈_𝑘,𝑡_𝑘+1. (34) From (34), one gets 𝐷⁺𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))≤𝑐_𝑖(𝑡)(𝑥(𝑡)(𝑡,𝑡𝐸𝑉,𝑡,𝑟))𝑡∈_𝑘,𝑡_𝑘+1, (35) where 𝐷⁺ is the Dini-derivative defined as 𝐷⁺𝑉(𝑡)=lim_ℎ→0+𝑉()(𝑡)sup𝑡+ℎ−𝑉ℎ, (36) in which 𝑉(𝑡) is continuous function.
In the following, we will show that (𝑥(𝑡))≤𝑏𝐸𝑉,𝑡,𝑖𝑐^∗𝛿,𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑘∈𝑍⁺. (37) For this purpose, we first prove that (𝑥(𝑡))≤𝑏𝐸𝑉,𝑡,𝑖𝑐^∗𝛿,𝑡∈𝑡₀,𝑡₁. (38) From condition (i) and 𝑥_𝑡0∈PC^𝑏_𝐹𝑡(𝛿), it follows that 𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))≤𝑏𝐸𝑥_𝑡0^𝑝_𝑟𝑏≤𝑏𝛿<𝑐^∗𝛿,𝑡∈𝑡₀−𝑟,𝑡₀. (39) If (38) is not true, there must exist some 𝑡∈[𝑡₀,𝑡₁) such that (𝑥(𝑡)(𝑡>𝑏𝐸𝑉,𝑡,𝑟))𝑐^∗𝛿>𝑏𝛿≥𝐸𝑉𝑥𝑡₀,𝑡₀,𝑟𝑡₀. (40) Let 𝑡^∗=inf{𝑡∈[𝑡₀,𝑡₁)∶𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))>(𝑏/𝑐^∗)𝛿}. From (39), one has 𝑡^∗∈(𝑡₀,𝑡₁) and 𝐸𝑉𝑥𝑡^∗,𝑡^∗,𝑟𝑡^∗=𝑏𝑐^∗(𝑥(𝑡)(𝑡<𝑏𝛿,𝐸𝑉,𝑡,𝑟))𝑐^∗𝛿,𝑡∈𝑡₀−𝑟,𝑡^∗,𝐷⁺𝐸𝑉𝑥𝑡^∗,𝑡^∗,𝑟𝑡^∗≥0. (41) Owing to (𝑏/𝑐^∗)𝛿>𝑏𝛿 and 𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))≤𝑏𝛿 for 𝑡∈[𝑡₀−𝑟,𝑡₀], then there exists a 𝑡^∗∗∈[𝑡₀,𝑡^∗) such that 𝐸𝑉𝑥𝑡^∗∗,𝑡^∗∗,𝑟𝑡^∗∗(𝑥(𝑡)(𝑡𝑡=𝑏𝛿,𝐸𝑉,𝑡,𝑟))>𝑏𝛿,𝑡∈^∗∗,𝑡^∗,𝐷⁺𝐸𝑉𝑥𝑡^∗,𝑡^∗,𝑟𝑡^∗≥0. (42) Consequently, by (41) and (42), for 𝑡∈[𝑡^∗,𝑡^∗∗], we have 𝐸𝑉(𝑥(𝑡+𝑠),𝑡+𝑠,𝑟(𝑡+𝑠))≤𝑏𝛿𝑐^∗≤1𝑐^∗𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)). (43) So, for 𝑡∈[𝑡^∗,𝑡^∗∗], it has 𝐷⁺𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))≤𝑐_𝑖(𝑡)𝐸𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)). (44) Integrating (44) from 𝑡^∗∗ to 𝑡^∗𝑡∗_𝑡∗∗𝐷⁺𝐸𝑉(𝑥(𝑠),𝑠,𝑟(𝑠))𝐸𝑉(𝑥(𝑠),𝑠,𝑟(𝑠))𝑑𝑠≤^𝑡1_𝑡0𝑐_𝑖(𝑡)𝑑𝑠≤𝛼𝛽. (45) Meanwhile, by condition (v), one has ^𝑡∗_𝑡∗∗𝐷⁺𝐸𝑉(𝑥(𝑠),𝑠,𝑟(𝑠))(𝑥(𝑠)(𝑠=𝐸𝑉,𝑠,𝑟))𝑑𝑠_{𝐸𝑉(𝑥(𝑡∗),𝑡∗,𝑟(𝑡∗(𝑥(𝑡))𝐸𝑉∗∗),𝑡∗∗(𝑡,𝑟∗∗))}1𝑢=𝑑𝑢_{(𝑏/𝑐∗)𝛿𝑏𝛿}1𝑢𝑑𝑢=−ln𝑐^∗>𝑐_𝑖𝑡₁−𝑡₀, (46) which is contradictory with (45), so (38) is true.
By condition (i) and (38), we have 𝐸𝑉𝑥𝑡₁,𝑡₁,𝑟𝑡₁≤𝜌𝑒_𝑖1𝐸𝑉𝑥𝑡⁻₁,𝑡⁻₁,𝑟𝑡⁻₁≤𝑏𝑒_𝑖1𝑥_𝑡⁻₁^𝑝_𝑟≤𝑏𝑎𝜌𝑒_𝑖1sup_{−𝑟≤𝜃≤0}𝐸𝑉𝑥𝑡⁻₁+𝜃,𝑡⁻₁+𝜃,𝑟𝑡⁻₁≤𝑏+𝜃𝑎𝑏𝑐^∗𝜌𝑒_𝑖1𝑏𝛿≤𝑏𝛿<𝑐^∗𝛿. (47) Now, we assume that (37) holds for 𝑘=1,2,…,𝑚, that is, (𝑥(𝑡)(𝑡≤𝑏𝐸𝑉,𝑡,𝑟))𝑐^∗𝛿,𝑡∈𝑡_𝑘−1,𝑡_𝑘. (48) We will prove that, for 𝑘=𝑚+1, (𝑥(𝑡)(𝑡≤𝑏𝐸𝑉,𝑡,𝑟))𝑐^∗𝛿,𝑡∈𝑡_𝑚,𝑡_𝑚+1. (49) Suppose that (49) does not hold, by the same procedure as in [𝑡₀,𝑡₁), we can get a contradiction and (49) follows. Finally, we get 𝐸‖𝑥(𝑡)‖^𝑝≤𝑏𝑎𝑐^∗𝛿<𝜀,𝑡≥𝑡₀. (50) From Definition 1, it is concluded that the trivial solution of system (3) is uniformly 𝑝-moment stable. The proof is complete.

Corollary 9. Assume that there exist positive scalars 𝑝, 𝑎, 𝑏, 𝜌<1, 𝑒_𝑖, 𝑐_𝑖(𝑖∈𝑆), 𝛼, and 𝛽 such that conditions (iv) in Theorem 8 are replaced by the following:(iv)^∗ there exists positive number 𝑐_𝑖 such that 𝐸𝐿𝑉(𝑥(𝑡),𝑡,𝑖)≤𝑐_𝑖(𝑥(𝑡)),𝑡𝐸𝑉,𝑡,𝑖𝑡∈_𝑘−1,𝑡_𝑘,𝑘=1,2,…, (51)whenever 𝑡≥𝑡₀ and 𝜑∈PC^𝑏_𝐹𝑡([−𝑟,0];𝑅^𝑛) are such that 𝑐^∗𝐸𝐿𝑉(𝜑(𝑠),𝑡+𝑠,𝑖)̃≤𝐸𝑉(𝜑(0),𝑡,𝑖); and all other conditions remain the same, then the trivial solution of system (3) is uniformly 𝑝-moment stable.

Proof. Similar to the procedure of Theorem 8, correspondingly, the item 𝑐_𝑖(𝑡) in the proof is replaced by 𝑐_𝑖.

Remark 10. It is clear that Theorem 8 implies that each of the continuous dynamics can be unstable, since the item 𝑐_𝑖(𝑡) in condition (iv), which characterizes the changing rate of Lyapunov function 𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)) at 𝑡, is assumed to be nonnegative. Theorem 8 shows that an unstable stochastic delay system can be successfully stabilized by impulse.

Remark 11. Constant 𝑒_𝑖 in condition (ii) characterizes certain perturbations in the overall impulsive stabilization process, that is, it is not strictly required by Theorem 8 that each impulse contributes to stabilize the system, as long as the overall contribution of the impulses is stabilizing. When 𝑒_𝑖≡1, it is easily obtained that each impulse must be stabilizing (𝜌<1), which is more restrictive.
Now consider the linear stochastic system with impulses and markovian switching (𝑡)=[𝐴(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥(+[𝐶(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑡),𝑑𝑥))+𝐵))𝑡−𝑟)]𝑑𝑡))+𝐷))𝑡−𝑟)]𝑑𝜔𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=𝐸_𝑘(𝑥(𝑡⁻),𝑟(𝑡)),𝑡=𝑡_𝑘,𝑥𝑡₀+𝜃=𝜙(𝜃),𝜃∈[−𝑟,0], (52) where 𝐴_𝑖, 𝐵_𝑖, 𝐶_𝑖, 𝐷_𝑖, and 𝐸_𝑘 are all 𝑛×𝑛 matrices and 𝜔 is a one-dimensional standard wiener process. 𝑟(𝑡) is a Markov chain taking values in 𝑆={1,2,…,𝑁} in which the transition rate is 𝜋_𝑖𝑗.

Theorem 12. When 𝑟(𝑡)=𝑖∈𝑆, define 𝐴(𝑟(𝑡))=𝐴_𝑖, 𝐵(𝑟(𝑡))=𝐵_𝑖, 𝐶(𝑟(𝑡))=𝐶_𝑖, 𝐷(𝑟(𝑡))=𝐷_𝑖. (i) If there exist constants 𝑝_𝑖, 𝑐_𝑖, 𝑑_𝑖, 𝜀₁, and 𝜀₂ such that 𝑝_𝑖𝜆_max𝐴^𝑇_𝑖+𝐴_𝑖+𝜀₁𝐼+𝑝_𝑖1+𝜀₂𝐷_𝑖²+^𝑁_𝑗=1𝜋_𝑖𝑗𝑝_𝑗≤−𝑐_𝑖,𝑝_𝑖𝜀₁⁻¹𝐵_𝑖²+1+𝜀₂⁻¹𝐺_𝑖²≤𝑑_𝑖;𝑝_𝑖𝑐_𝑖𝐼+𝐸_𝑘²>𝑑_𝑖, (53) then system (52) is uniformly 𝑝-moment stable.(ii) If there exist constants 𝑐_𝑖, 𝜀₁, and 𝜀₂ such that 𝑝_𝑖𝜆_max𝐴^𝑇_𝑖+𝐴_𝑖+𝜀₁𝐼+𝑝_𝑖1+𝜀₂𝐷_𝑖²+^𝑁_𝑗=1𝜋_𝑖𝑗𝑝_𝑗+2𝑝_𝑖ln𝐼+𝐸_𝑘×𝜀₁⁻¹𝐵_𝑖²+1+𝜀₂⁻¹𝐺_𝑖²≤𝑐_𝑖;ln𝑝_𝑖𝐼+𝐸_𝑘1≤−2𝑐_𝑖𝑡_𝑘+1−𝑡_𝑘, (54) then system (52) is uniformly 𝑝-moment stable.

Proof. It is a direct consequence of Corollaries 5 and 9 with 𝑉(𝑥(𝑡),𝑟(𝑡))=𝑝_𝑖‖𝑥(𝑡)‖².

4. Numerical Examples

The applicability of the results derived in the preceding section is illustrated by the following two examples.

Example 1. Consider the following linear impulsive switching delay system: (𝑡)=[𝐴(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥(+[𝐶(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑡),𝑑𝑥))+𝐵))𝑡−1)]𝑑𝑡))+𝐷))𝑡−1)]𝑑𝜔𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=𝐸_𝑘(𝑥(𝑡⁻),𝑟(𝑡)),𝑡=𝑡_𝑘, (55) where 𝐴₁=0.10.2−0.10.20.150.300.240.1,𝐵₁=,𝐷−0.240.0400.24−0.40.100.24−0.2₁=0.50.40.2−0.240.320.30.60.26−0.5,𝐺₁=,𝐴0.140−0.210.20.10.160.10.150.17₂=0.20.3−0.10.30.10.300.20.1,𝐵₂=,𝐷−0.20.0400.2−0.40.100.2−0.2₂=0.50.40.2−0.20.30.30.60.2−0.5,𝐺₂=,𝐸0.240−0.210.20.10.160.10.10.17_𝑘=−−0.50010−0.4000−0.3,𝜋=31323−23. (56) Set 𝜀₁=𝜀₂=0.5, 𝑝_𝑖=1. It is easy to see that 𝜆_max(𝐴^𝑇₁+𝐴₁+0.5𝐼)=1.3819, 𝜆_max(𝐴^𝑇₂+𝐴₂+0.5𝐼)=1.5020, ‖𝐵₁‖²=0.3141, ‖𝐵₂‖²=0.2764, ‖𝐷₁‖²=0.9175, ‖𝐷₂‖²=0.8737, ‖𝐺₁‖²=0.1360, ‖𝐺₂‖²=0.1219, and ‖𝐼+𝐸_𝑘‖²=4.9. Taking 𝑐₁=3.1, 𝑐₂=3.2, 𝑡_𝑘−𝑡_𝑘−1=0.22, we can verify that all the conditions (ii) of Theorem 12 are satisfied: 𝜆_max𝐴^𝑇₁+𝐴₁+𝜀₁𝐼+1+𝜀₂𝐷₁²+^𝑁_𝑗=1𝜋_1𝑗+2ln𝐼+𝐸_𝑘×𝜀₁⁻¹𝐵₁²+1+𝜀₂⁻¹𝐺₁²=3.0190≤𝑐₁𝜆=3.1;_max𝐴^𝑇₂+𝐴₂+𝜀₁𝐼+1+𝜀₂𝐷₁²+^𝑁_𝑗=1𝜋_2𝑗+2ln𝐼+𝐸_𝑘×𝜀₁⁻¹𝐵₂²+1+𝜀₂⁻¹𝐺₂²=3.1574≤𝑐₂=3.2;ln𝐼+𝐸_𝑘²=−0.7133≤−𝑐_𝑖𝑡_𝑘+1−𝑡_𝑘=−3.2×0.22=−0.7040. (57) So, system (55) is 𝑝-moment stable. Numerical simulations for this example are shown in Figures 1 and 2. It is clearly demonstrated that impulses can successfully stabilize an otherwise unstable stochastic delay system.

Figure 1 

State response of system (55) under impulsive perturbations.

(a)

(b)

Figure 2 

State response of system (55) without impulsive perturbations.

Example 2. Consider the following linear impulsive switching delay system: (𝑡)=[𝐴(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥(+[𝐶(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑡),𝑑𝑥))+𝐵))𝑡−1)]𝑑𝑡))+𝐷))𝑡−1)]𝑑𝜔𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=𝐸_𝑘(𝑥(𝑡⁻),𝑟(𝑡)),𝑡=𝑡_𝑘, (58) where 𝐴₁=−400−5,𝐵₁=,𝐷0.20.30.50.1₁=0.30.40.50.2,𝐺₁=,𝐸0.80.10.10.3_𝑘=,𝐴−0.60.50.3−0.5₂=−300−4,𝐵₂=,𝐷0.30.50.40.3₂=0.50.10.40.1,𝐺₂=,0.20.20.10.4𝑃=.0.50.50.50.5 (59)(60) Choosing 𝜀₁=𝜀₂=2,𝑝_𝑖=1. It is easy to see that 𝜆_max(𝐴^𝑇₁+𝐴₁+2𝐼)=−6, 𝜆_max(𝐴^𝑇₂+𝐴₂+2𝐼)=−4, ‖𝐵₁‖²=0.3403, ‖𝐵₂‖²=0.5687, ‖𝐷₁‖²=0.5009, ‖𝐷₂‖²=0.4298, ‖𝐺₁‖²=0.6712, ‖𝐺₂‖²=0.2347, ‖𝐼+𝐸_𝑘‖²=0.7467. Taking 𝑐₁=3.3, 𝑐₂=1.6, 𝑑₁=1.2, 𝑑₂=0.65, we can verify that all the conditions (i) of Theorem 12 are satisfied:𝜆_max𝐴^𝑇_𝑖+𝐴_𝑖+𝜀₁𝐼+1+𝜀₂𝐷_𝑖²+^𝑁_𝑗=1𝜋_𝑖𝑗=−3.4973<−3.3=𝑐₁;−1.7166<−1.6=𝑐₂;𝜀₁⁻¹𝐵_𝑖²+1+𝜀₂⁻¹𝐺_𝑖²=1.1770<1.2=𝑑₁;0.6364<0.65=𝑑₂;𝐼+𝐸_𝑘²𝑐_𝑖=0.7467×3.3=2.4641>1.2=𝑑₁;1.6=1.1947>0.65=𝑑₂. (61)(62) Theorem 12 guarantees that the trivial solution of system (58) is 𝑝-moment stable. Numerical simulations are shown in Figure 3. It is clearly demonstrated that the 𝑝-moment stability properties can be preserved irrespective of the length of the delay.

Figure 3 

State response of system (58) under impulsive perturbations

5. Conclusions

In this paper, a method of multiple Lyapunov functionals has been applied to deal with the effects of impulses on 𝑝-moment stability of stochastic differential delay systems with impulsive jump and markovian switching. Some stability criteria are obtained based on Lyapunov functional method and stochastic theory. It is shown that, even if all the subsystems governing the continuous dynamics without impulse are not stable, as impulsive and switching signal satisfies a dwell-time upper bound condition, impulses can stabilize the systems in the 𝑝-moment stability sense. The opposite situation is also developed for which all the subsystems governing the continuous dynamics without impulse are 𝑝-moment stable. Applying the derived results to a class of stochastic systems with arbitrarily large delays, the deduced new stability criteria can relax some restrictions on impulses imposed by the existing results. Two illustrative examples have been provided to demonstrate the main theoretical analysis.

Acknowledgments

This work was supported by National Natural Science Foundation of China (61273091, 61104007), Natural Science Foundation of Shandong Province (ZR2011FM033), the Young and Middle-Aged Scientists Research Foundation of Shandong Province (BS2011DX013, BS2012SF008), and Taishan Scholar Project of Shandong Province of China.