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On Input-to-State Stability of Impulsive Stochastic Systems with Time Delays
This paper is concerned with pth 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.
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 . 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  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 , 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 . Recently, Hespanha initiated the study of ISS for impulsive systems . 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,  investigated the ISS property for nonlinear impulsive systems with time delays by using Razumikhin techniques. And  was also concerned with ISS of impulsive systems with time delays, where ISS theorems different from those in  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. The pth 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 the p-ISS/SISS are established. The results indicate that when the delayed continuous stochastic dynamics are p-ISS/SISS and the discrete dynamics are destabilizing, the p-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 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. As a byproduct, the criteria on pth 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  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 Section 2, some basic notations and definitions used throughout the paper are introduced. In Section 3, criteria ensuring uniform p-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.
Throughout this paper, unless otherwise specified, we will employ the following notations. Let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets) and let be the expectation operator with respect to the given probability measure . Let be a -dimensional Brownian motion defined on the probability space. , , , denotes the -dimensional real space equipped with the Euclidean norm , and denotes the -dimensional real matrix space.
Let and is continuous for all but at most a finite number of points , at which , exist and , where and denote the right-hand and left-hand limits of at , respectively. We equip the linear space with the norm defined by . Let be the family of all -measurable and bounded -valued random variables .
A function is said to be of class if it is continuous and strictly increasing and satisfies ; it is of class if in addition as . Note that if is of class , then the inverse function is well defined and is also of class . and are the subsets of functions that are convex and concave, respectively. A function is said to be of class if for each fixed and decreases to as for each fixed . The composition of two functions and is denoted by .
If is a vector or a matrix, its transpose is denoted by . If is a square matrix, means that is a symmetric positive definite (negative semidefinite) matrix. and represent the minimum and maximum eigenvalues of the corresponding matrix, respectively, and 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: with initial data , where and is regarded as a -valued random variable; is locally essentially bounded external input and is the impulsive disturbance input; denotes the set of all locally essentially bounded function with norm ; ; both and are uniformly locally Lipschitz with respect to and ; represents the impulsive perturbation of at and satisfies ; is a strictly increasing sequence of impulse times. We use and to denote the class of impulsive time sequences that satisfy and the set containing all impulse time sequences, respectively.
Moreover, we assume that for all , ; then system (1) admits a trivial solution . The input pair is said to be admissible, denoted by , if , 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 , we proposed the following definitions for impulsive stochastic delayed systems (1).
Definition 1. For a prescribed sequence , system (1) is said to be th () moment input-to-state stable (ISS) if there exist functions such that, for every initial condition and every input pair ,
Definition 2. For a given sequence , system (1) is said to be stochastic input-to-state stable (SISS), if, for any , there exist functions and , such that, for every initial condition and every input pair ,
Remark 3. Redefining and , , one can assume that in (2) or (3) is the identity: if holds, then also . We know by Lemma 4.2 in  that and , . So estimates of the same type as (2) and (3) but with no “” are obtained.
In the following, we will define p-GAS and GASiP in the form of function, which present very close analogy to p-ISS and SISS, respectively.
Definition 4. For a prescribed sequence , system (1) with input , is said to be th () moment globally asymptotically stable (GAS) if there exists a function such that, for every initial condition ,
Definition 5. For a given sequence , system (1) with input , is said to be globally asymptotically stable in probability (GASiP), if, for any , there exists a function , such that, for every initial condition ,
Remark 6. By the vanishing of and at , (2) and (3) will degenerate to (4) and (5), respectively, when , which means that p-ISS/SISS of system (1) implies p-GAS/GASiP of the corresponding unforced system.
System (1) is said to be uniformly p-ISS or uniformly SISS over a given class of admissible impulsive time sequences if (2) or (3) holds for every sequence in with functions , , , and independent of the choice of the sequence. Uniform p-GAS and uniform GASiP can be defined similarly.
Definition 7 (see ). A function is said to be of class if the following hold true. (i)is continuous on each of the sets and for each , , , and exists.(ii) is once continuously differentiable in and twice in in each of the sets , .
3. Main Results
In this section, we will develop Lyapunov-Razumikhin methods and establish some criteria which provide sufficient conditions for the p-ISS and SISS properties of impulsive stochastic delayed systems (1).
Theorem 8. Assume that there exist functions , , , and scalars , , such that (i);(ii), for all , and whenever ;(iii). Then for any given satisfying , system (1) is uniformly p-ISS over . In particular, when , system (1) is uniformly p-ISS over .
Proof. Since , then and there exists such that . We choose such that , , . Let be any impulsive time sequence belonging to . For simplicity, we write . Define
where for and for . We claim that
We first prove that (9) holds for . By condition (i) and Jensen's inequality, it is easy to see that
If (9) is not true for , there must exist some such that . Let . Then by the right continuity of in and noticing (10), we have and
it follows from condition (ii) that
For sufficiently small satisfying , by the Itô formula, we have
where . On the other hand, implies
Therefore, from (16) and (17), and noticing and for , we have
which contradicts . Therefore, (9) holds for .
Suppose that (9) holds for , where , . We will prove that (9) holds for . To this end, we claim that where . If not, then . We consider the following two cases.
Case 1. for all . It is easy to see that for and . It follows that The last inequality comes from the fact that , and By condition (ii), (20) indicates that By Itô’s formula, we have thus, On the other hand, implies Substituting (25) into (24) and noticing the fact that , we have Substituting (26) into yields The last inequality holds because and . This is a contradiction.
Case 2. There exists some such that . Set . Then and for . Thus, for , for . This implies that (20) holds for , . Thus, by condition (ii), Hence, noticing the fact that , for , we have Because for , there holds for . Substituting this inequality with (29), and recalling the choice of , it follows that which yields the following contradiction: .
Therefore, we have . Using condition (iii), we obtain that . Repeating the argument used in the proof of for , we can get for . By the mathematical induction, we know that (9) holds for all . For any given , one can get It follows from (9), (31), and the definition of that By casualty, Then by condition (i) and Jensen's inequality, the required assertion (2) holds with , and . By Lemma 4.2 in , it is easy to see that . As , , are independent of the particular choice of the impulse time sequence, system (1) is uniformly p-ISS over .
For the special case , holds for any , so system (1) is uniformly p-ISS over for any . In other words, system (1) is uniformly p-ISS over . The proof is complete.
Remark 9. When , condition (iii) implies that the impulses may be destabilizing. So, in order to maintain the p-ISS property of system (1), the impulse interval is required to be large enough to reduce the effect of the impulses. When , the discrete dynamics are marginally stable for the zero input. In this case, the p-ISS of system (1) is not affected by the impulses.
With minor modification to the conditions of Theorem 8, a criterion on SISS can be obtained as follows.
Theorem 10. Assume that conditions (ii) and (iii) of Theorem 8 hold, while condition (i) is replaced by (), where . Then, for any given satisfying , system (1) is uniformly SISS over . In particular, when , system (1) is uniformly SISS over .
Proof. By condition (i*), (10) can be replaced by Then, following the same lines of the proof of Theorem 8, it is easy to see that holds for all . Consequently, by Chebyshev's inequality, it follows that where can be made arbitrarily small by an appropriate choice of . That is, which yields where , , . By condition (i*), we know that (3) holds. Therefore, system (1) is uniformly SISS over and the proof is complete.
Corollary 11. Assume that there exist functions , , and scalars , , such that (i);(ii), for all , and whenever ;(iii). Then, for any given satisfying , system (1) is uniformly p-GAS over . In particular, when , system (1) is uniformly p-GAS over .
Corollary 12. Assume that conditions (ii) and (iii) of Corollary 11 hold, while condition (i) is replaced by (), where . Then, for any given satisfying < 1, system (1) is uniformly GASiP over . In particular, when , system (1) is uniformly GASiP over .
Now let us apply the obtained results to the linear impulsive stochastic delayed system with the following form: on with initial data , where and , are system state and inputs, respectively; is short for ; , , , , , , , are constant matrices with appropriate dimensions.
Corollary 13. Assume that there exist a matrix and constants , , , , satisfying such that the following matrix inequalities hold: Then system (39) is uniformly ISS in mean square and uniformly SISS over .
Proof. We choose the candidate ISS-Lyapunov function . By using (40), and in view of the fact that , we can obtain by simple calculation that So, whenever , we have On the other hand, So, It is obvious that all conditions of Theorem 8 are satisfied, with and . Therefore, we conclude by Theorems 8 and 10 that system (39) is uniformly p-ISS and uniformly SISS over .
Remark 14. It is noted that (40) are not linear with the combined variables , and, therefore, they are not linear matrix inequalities (LMIs). This makes the computation difficult but also flexible. We can first assign , , and 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 input , the state will not converge to zero but will remain bounded (in the sense of mean square or in probability), which is consistent with the definition of p-ISS/SISS.
Example 1. Consider system (39) with the following parameters:
Setting , , , and solving LMIs (40) by using the Matlab LMI Toolbox, then is a group of feasible solutions. Choosing , , 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 . The sample path and the mean square of the solution are displayed in Figures 1 and 2, respectively, where , initial data for , and impulse interval and external inputs .
As p-ISS/SISS implies p-GAS/GASiP of the corresponding unforced system, we conclude that the system with is uniform GAS in mean square and GASiP for arbitrary sequence of impulse times satisfying . The simulations of the unforced system are shown in Figures 3 and 4.
This paper has investigated the p-ISS/SISS of impulsive stochastic systems with external inputs. By combining stochastic analysis techniques, piecewise continuous Lyapunov functions, and Razumikhin techniques, sufficient conditions for uniform p-ISS/SISS over a given class of impulse times sequences have been established. As a byproduct, the criteria on p-GAS/GASiP are also derived. For future research, interesting topics may include establishing p-ISS/SISS theorems with stabilizing impulses, as well as p-ISS/SISS analysis by exploring new techniques such as Lyapunov-Krosovskii functional method.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Excellent Youthful Talent Foundation of Colleges and Universities of Anhui Province of China (2013SQRL024ZD), the Research Foundation for Young Scientists of Anhui University of Technology (QZ201314), the National Natural Science Foundation of China (11301004), and Anhui Provincial Nature Science Foundation (1308085QA15).
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