Razumikhin-Type Theorems on General Decay Stability of Impulsive Stochastic Functional Differential Systems with Markovian Switching

In this paper, the Razumikhin approach is applied to the study of both p-th moment and almost sure stability on a general decay for a class of impulsive stochastic functional differential systems with Markovian switching. Based on the Lyapunov-Razumikhin methods, some sufficient conditions are derived to check the stability of impulsive stochastic functional differential systems with Markovian switching. One numerical example is provided to demonstrate the effectiveness of the results.


Introduction
Impulsive stochastic systems with Markovian switching is a class of hybrid dynamical systems, which is composed of both the logical switching rule of continuous-time finite-state Markovian process and the state represented by a stochastic differential system [1].Because of the presence of both continuous dynamics and discrete events, these types of models are capable of describing many practical systems in many areas, including social science, physical science, finance, control engineering, mechanical and industry.So this kind of systems have received much attention, recently (for instance, see [2]- [5]).
It is well-known that stability is the major issue in the study of control theory, one of the most important techniques applied in the investigation of stability for various classes of stochastic differential systems is based on a stochastic version of the Lyapunov direct method.However, the so-called Razumikhin technique combined with Lyapunov functions has also been a powerful and effective method in the study of stability.Recalled that Razumikhin developed this technique to study the stability of deterministic systems with delay in [6] [7], then, Mao extended this technique to stochastic functional differential systems [8].This technique has become very popular in recent years since it is extensively applied to investigate many phenomena in physics, biology, finance, etc. Mao incorporated the Razumikhin approach in stochastic functional differential equations [9] and in neutral stochastic functional differential equations [10] to investigate both p-th moment and almost sure exponential stability of these systems (see also [11]- [13], for instance).Later, this technique was appropriately developed and extended to some other stochastic functional differential systems, especially important in applications, such as stochastic functional differential systems with infinite delay [14]- [16], hybrid stochastic delay interval systems [17] and impulsive stochastic delay differential systems [18]- [20].Recently, some researchers have introduced ψ-type function and extended the stability results to the general decay stability, including the exponential stability as a special case in [21]- [23], which has a wide applicability.
In the above cited papers, both the p-th moment and almost sure stability on a general decay are investigated, but mostly used in stochastic differential equations.And As far as I know, a little work has been done on the impulsive stochastic differential equations or systems.In this paper, we will close this gap by extending the general decay stability to the impulsive stochastic differential systems.To the best of our knowledge, there are no results based on the general decay stability of impulsive stochastic delay differential systems with Markovian switching.And the main aim of the present paper is attempt to investigate the p-th moment and almost sure stability on a general decay of impulsive stochastic delay differential systems with Markovian switching.Since the delay phenomenon and the Markovian switching exists among impulsive stochastic systems, the whole systems become more complex and may oscillate or be not stable, we introduce Razumikhin-type theorems and Lyapunov methods to give the conditions that make the systems stable.By the aid of Lyapunov-Razumikhin approach, we obtain the p-th moment general decay stability of impulsive stochastic delay differential systems with Markovian.In order to establish the criterion on almost surely general decay stability of impulsive stochastic delay differential systems with Markovian, the Holder inequality, Burkholder-Davis-Gundy inequality and Borel-Cantelli's lemma are utilized in this paper.
The paper is organized as follows.Firstly, the problem formulations, definitions of general dacay stability and some lemmas are given in Section 2. In Section 3, the main results on p-th moment and almost sure stability on a general decay of impulsive stochastic delay differential systems with Markovian switching are obtained with Lyapunov-Razumikhin methods.An example is presented to illustrate the main results in Section 4. In the last section the conclusions are given.

Preliminaries
Throughout this paper, let ( ) , ,P Ω  be a complete probability space with some filtration { } 0 t t≥  satisfying the usual condition (i.e., the filtration is increasing and right continuous while 0  contain all P-null sets).Let  [ ] E ⋅ means the expectation of a stochastic process; We assume that the Markov chain ( ) r ⋅ is independent of the Brownian motion ( ) In this paper, we consider the following impulsive stochastic delay differential systems with Markovian switching where the initial value represents the impulsive perturbation of x at time k t .The fixed moments of impulse times k t satisfy  ( ) is said to be ψ-type function, if it satisfies the following conditions: (1) It is continuous, monotone increasing and differentiable; (2) ( ) ( ) when ( ) e t t ψ = , we say that it is almost surely exponential stable, when ( ) denote the family of all nonnegative functions ( ) that are continuously once differentiable in t and twice in x.For each ( ) That is, there exist a set o Ω ∈ with ( ) ( ( ) That is, there exists a set θ Ω ∈ with ( ) 1

Main Results
In this section, we shall establish some criteria on the p-th moment exponential stability and almost exponential stability for system (1) by using the Razumikhin technique and Lyapunov functions.
Theorem 1 For systems (1), let (H) hold, and ψ is a ψ-type function, Assume that there exist a function , , where ( ) where ( ) Then, for any initial , there exists a solution ( ) ( ) (1).Moreover, the system (1) is p-th moment exponentially stable with decay ( ) arbitrarily and write ( ) ( ) simply.When µ is replaced by γ , if we can prove that the system (1) is p-th moment exponentially stable with decay ( ) , and thus we can have the following fact: ( ) Then it follows from condition (H In the following, we will use the mathematical induction method to show that ( In order to do so, we first prove that This can be verified by a contradiction.Hence, suppose that inequality (9) is not true, than there exist some Consequently, for all , , min 0 , , , , .
Applying the Îto formula to By condition ( 14), we obtain On the other hand, a direct computation yields Then, we will prove that (8) holds for , , .
Then, it follows from the condition (H 3 ) and ( 17) that which implies that the m t dose not satisfy the inequality (19).And from this, set , then ( 20)-( 22) imply that , we assume that, without loss of generality, , then from ( 17) and ( 20)-( 22), we obtain by condition (H 2 ) we have Similar to (15), applying the Îto formula to   20) and ( 22), we have   )  ,  x t x t ξ = simply.We claim that ( ) ( ) ) Choose δ sufficiently small and , for the fixed δ , let , where [ ] x is the maximum integer not more than x.Then for any For each i when 1 By Theorem 1, we have By Holder inequality, condition (26) and Theorem 1, we derives that ( ) ( ) Similarly, by the Lemma 1 and (32), we obtain ( ) where p C is a positive constant dependent on p only.Substituting (31), (32) and (33) into (30) yields

Conclusion
In this paper, p-th moment and almost surely stability on a general decay have been investigated for a class of impulsive stochastic delay systems with Markovian switching.Some sufficient conditions have been derived to check the stability criteria by using the Lyapunov-Razumikhin methods.A numerical example is provided to verify the effectiveness of the main results.

oP
Ω = and an integer valued random variable o k such that for every o ω ∈ Ω we have such that for every θ ω ∈ Ω , there exists a sub-seq- uence { } i k A such that the ω belongs to every i k A .

Figure 1 .
Figure 1.State of the example.

Figure 2 .
Figure 2. Markovian switching of the example.

of order 2 . 1 ,
The simulation result of system (35) is shown in Figure and the Markovian switching of system (35) is described in Figure 2.
the existence and uniqueness of the solution we impose a hypothesis: impulsive stochastic delay differential systems with Markovian switching (1) is said to be p-th moment stable with decay ( ) t ψ of order γ , if there exist positive constants γ and function ( ) t ψ of order γ , if there exist positive constant γ and function ( ) ψ ⋅ , such that →∞ ≤ − ) (1)or system(1), suppose the conditions of Theorem 1 are satisfied.Let t ψ of order γ .The proof is complete. Theorem 2 Theorem 1 are satisfied.So the impulsive stochastic delay system with Markovian switching is p-th moment stable with decay