RAZUMIKHIN-TYPE THEOREMS ON POLYNOMIAL STABILITY OF HYBRID STOCHASTIC SYSTEMS WITH PANTOGRAPH DELAY

. The main aim of this paper is to investigate the polynomial stability of hybrid stochastic systems with pantograph delay (HSSwPD). By us- ing the Razumikhin technique and Lyapunov functions, we establish several Razumikhin-type theorems on the p th moment polynomial stability and al- most sure polynomial stability for HSSwPD. For linear HSSwPD, suﬃcient conditions for polynomial stability are presented.

1. Introduction. Systems in many branches of science and industry do not only depend on the present state but also the past ones. Stochastic differential delay equations (SDDEs) have been widely used to model such systems. On the other hand, these systems may often experience abrupt changes in their structure and parameters and continuous time Markov chains have been used to model these abrupt changes. Hence, SDDEs with Markovian switching, known also as hybrid SDDEs, have appeared frequently in practice. The analysis and control of these systems involve investigating their stability, which is often regarded as one of the important issues of dynamical systems studied. There have been much work on the stability of hybrid SDDEs, we here mention Fei et al. [3], Hu et al. [8], Ji and Chizeck [11], Mao et al. [14,16,21,32], Shaikhet [25], Wu et al. [30] and Zhang et al. [35] among others.
It is well known that the classical and powerful techniques applied in the study of stability are based on the Lyapunov direct method. However, the greatest disadvantage of this approach is that it is difficult to construct the Lyapunov functionals effectively. In this case, the Razumikhin technique has been proposed and was used to study the stability of time delay systems. Razumikhin [24] was the first to develop this technique to study the stability of deterministic systems with delay. Mao [17,18] extended this technique to stochastic systems with delay and employed the Razumikhin approach to study both pth moment and almost sure exponential stability for stochastic functional differential equations (SFDEs) and neutral SFDEs. After that, the Razumikhin technique has become very popular and there have appeared many results based on it to discuss the stability of stochastic systems with time delay, see for example, [9,10,15,22,23,26,28,29,33] and the many references therein.
On the other hand, it is noted that the literature cited above mainly focuses on pth moment and almost sure exponential stability. However, in many cases we may find that the equation is not exponentially stable, but the solution does tend to zero asymptotically. Consequently, it appears to be necessary to study other stability, for instance, polynomial stability. Mao [19,20] considered the polynomial stability of stochastic system, which shows that solution tends to zero polynomially. Liu and Chen [12] studied the polynomial stability for stochastic differential equations (SDEs) with bounded delay. In recent years, Appleby and Buckwar [1] investigated the polynomial stability of SDEs with unbounded delay or pantograph delay. Milosevic [13] has established the sufficient conditions of almost sure polynomial stability of the solution for nonlinear SDEs with pantograph delay. Guo and Li [7] discuss the αth moment polynomial stability of the nonlinear SDEs with pantograph delay by using the Razumikhin-type technique.
Motivated by [19,18,12,1,13,7], in this paper, we are concerned with the polynomial stability of hybrid stochastic system with pantograph delay where 0 < q < 1, the coefficients f : R n ×R n ×R + ×S → R n and g : R n ×R n ×R + × S → R n×m are Borel-measurable, S = {1, 2, · · · , N }, and r(t) is a continuous-time Markov chain taking value in S, w(t) is an m-dimensional Brownian motion. In fact, as a class of special stochastic delay systems, stochastic system with pantograph delay has been investigated by many scholars, we can refer to Baker and Buckwar [2], Fan et al. [5,4], Xiao et al. [31], Guo and Li [6], Zhou and Xue [34], Shen et al. [27].
To the best of our knowledge, there are no results based on the Razumikhin technique referring to the polynomial stability of hybrid stochastic system with pantograph delay (1). The main aim of the present paper is to close this gap by extending the Razumikhin approach to the study of both the pth moment polynomial stability and almost sure polynomial stability for (1). Moreover, in [7], the authors only studied the polynomial stability in mean square, but in this paper, we shall also study the almost sure polynomial stability. On the other hand, since hybrid stochastic system (1) we are studied have continuous components as well as discrete components, their asymptotic behaviour is completely different from a single system of pantograph equation. Actually, (1) can be regarded as the result of the following N equations switching from one to the others according to the movement of the Markov chain. Due to the presence of the Markov chains, we will show that even if some subsystems (2) are not polymonially stable, the overall hybrid system (1) may still be polymonially stable.
In this paper, some preliminaries and notations on hybrid stochastic system with pantograph delay will be introduced in section 2. In sections 3, we will show the main results of our paper, where several useful criteria will be established on the polymonial stability in mean square as well as the almost sure polymonial stability for (1.1). In section 4, the general results established in sections 3 will be applied to deal with the polymonial stability of linear hybrid stochastic system with pantograph delay .

2.
Preliminaries. Let (Ω, F, P ) be a complete probability space with a filtration {F t } t≥t0 satisfying the usual conditions. Let w(t) be an m-dimensional Brownian motion defined on the probability space (Ω, F, P ). Let t ≥ t 0 > 0 and C([qt, t]; R n ) denote the family of the continuous functions ϕ from [qt, t] → R n with the norm ||ϕ|| = sup qt≤θ≤t |ϕ(θ)|, where |.| is the Euclidean norm in R n . If A is a vector or matrix, its transpose is denoted by A . If A is a matrix, its norm ||A|| is defined by ||A|| = sup{|Ax| : Let r(t), t ≥ 0 be a right-continuous Markov chain on the probability space (Ω, F, P ) taking values in a finite state space S = {1, 2 . . . N } with generator Γ = (γ ij ) N ×N given by: where ∆ > 0. Here γ ij ≥ 0 is the transition rate from i to j, i = j, while γ ii = − j =i γ ij . We assume that the Markov chain r(.) is independent of the Brownian motion w(.). Let us consider the nonlinear hybrid stochastic systems with pantograph delay Here In this paper, the following hypothesis are imposed on the coefficients f and g .
It is known that Assumption 2.1 only guarantees that (3) has a unique maximal solution, which may explode to infinity at a finite time. To avoid such a possible explosion, we need to impose an additional condition in terms of Lyapunov functions.
Let C(R n × [qt 0 , ∞) × S; R + ) denote the family of continuous functions from R n × [qt 0 , ∞) × S to R + . Also denote by C 2,1 = C 2,1 (R n × [qt 0 , ∞) × S; R + ) the family of all continuous non-negative functions V (x, t, i) defined on R n ×[qt 0 , ∞)×S such that they are continuously differentiable twice in x and once in t. Given V ∈ C 2,1 , we define the function LV : There exist a Lyapunov function V ∈ C 1,2 and some positive constants c 1 , c 2 , α 1 and α 2 such that for any x, y ∈ R n , t ≥ qt 0 , and i ∈ S, and Now, we present the definitions of the pth moment polynomial stability and almost surely polynomial stability of hybrid stochastic systems with pantograph delay (3).
Definition 2.4. The solution of (3) is said to be almost surely polynomially stable if there exists a constantγ such that 3. Main results. In this section, we shall investigate the pth moment polynomial stability and almost surely polynomial stability of (3) by using the Razumikhintype technique. Before analyzing the stability, we firstly show that (3) has a global solution.
Theorem 3.1. Let Assumptions 2.1 and 2.2 hold. Then for any given initial data ξ, there is a unique global solution x(t) to (3) on t ∈ [qt 0 , ∞). Moreover, the solution has the property that for any t ≥ qt 0 .
Proof. Since the coefficients of (2.1) are locally Lipschitz continuous, for any given initial data ξ, there is a maximal local solution x(t) on t ∈ [t 0 , σ ∞ ), where σ ∞ is the explosion time. Let k 0 > 0 be sufficiently large for ξ < k 0 . For each integer k ≥ k 0 , define the stopping time s. So we just need to show that τ ∞ = ∞ a.s. We shall first show that τ ∞ > t0 q a.s. By the Itô formula (see e.g. [14]) and condition (6), we can show that, for any k ≥ k 0 and t 1 ≥ t 0 , Let us now restrict t 1 ∈ [t 0 , t0 q ]. By condition (2.3), we then get where It then follows that for any k ≥ k 0 . In particular, Letting k → ∞ in (12) yields Let us now proceed to prove τ ∞ > t0 q 2 a.s. given that we have shown (13)- (14). For any k ≥ k 0 and t 1 ∈ [t 0 , t0 q 2 ], it follows from (10) that where In particular, E|x(τ k ∧ t0 (16) yields Repeating this procedure, we can show that, for any integer i ≥ 1, τ ∞ > t0 q i a.s., We must therefore have τ ∞ = ∞ a.s. and the required assertion (9) holds as well.
The proof is therefore complete.
Applying the Itô formula to U (t) yields However, this contradicts U (t 0 +t + δ) > U (t 0 +t), so (21) must hold. The proof is therefore complete. Now, we use this theorem to establish a useful result on pth moment polynomial stability.
Theorem 3.4. Let p > 0, α 1 > α 2 ≥ 0 and let Assumptions 2.1 and 2.2 hold. Assume that, for all x, y ∈ R n , i ∈ S and t ≥ t 0 , Then (3) is pth moment polynomially stable, that is, for any initial data ξ ∈ L p Ft 0 where λ is the unique root to the following equation λ = α 1 − α 2 q −λ .
Proof. If the solution x(t) of (3) satisfies that for all t ≥ t 0 , then by (23) and (25), we have This shows that condition (17) is satisfied and (24) follows from Theorem 3.3.
The following theorem gives the sufficient conditions for the almost sure polynomial stability of (3).
Then (24) implies In other words, the pth moment polynomial stability implies almost sure polynomial stability.
Remark 3.6. In general, the pth moment stability and almost sure stability of the solution do not imply each other. In [17,18,26,10,23], the authors proved that the pth moment stability imply the almost sure stability when the diffusion term and the drift term of stochastic systems obey the linear growth condition. It should be mentioned that the drift term of stochastic systems (3) does not satisfy the linear growth condition in this paper. Therefore, the proposed method in [17,18,26,10,23] can not be used here. Of course, w(t) and r(t) are assumed to be independent. Consider the following scalar hybrid stochastic systems with pantograph delay dx(t) = f (x(t), t, r(t))dt + g(x(0.5t), t, r(t))dw(t), t ≥ 1, with q = 0.5, the initial data ξ(t) = x 0 (0.5 ≤ t ≤ 1) and r(1) = 1. Moreover, for (x, y, t, i) ∈ R × R × [0.5, ∞) × S, f (x, t, 1) = x, g(y, t, 1) = 0.5y, f (x, t, 2) = −5x − x 3 , g(y, t, 2) = √ 2 2 y. We note that (33) can be regarded as the result of the two equations dx(t) = x(t)dt + 0.5x(0.5t)dw(t) (34) and switching among each other according to the movement of the Markov chain r(t).
It is easy to see that (35) is polynomially stable but (34) is unstable. However, we shall see that due to the Markovian switching, the overall system (33) will be polynomially stable. In fact, the coefficients f and g satisfy the local Lipschitz condition but the coefficient f do not satisfy the linear growth condition. To find out whether hybrid stochastic systems with pantograph delay (33) is polynomial stability in mean square, we use the Lyapunov function It is easy to see that the operator LV from R × R × [0.5, ∞) × S to R has the form LV (x, y, t, 1) ≤ −2.8x 2 + 0.25y 2 , and LV (x, y, t, 2) ≤ −(4 − 0.6γ)x 2 + 0.2y 2 .
By Theorem 3.4, we can conclude that if the transition rate γ ∈ (2, 5.625), then for any initial data x 0 , (33) will be mean square polynomially stable. On the other hand, by Theorem 3.5, we can conclude that if the transition rate γ ∈ (2, 2.917), (33) will be almost surely polynomially stable. But, if the transition rate γ ∈ (2.917, 5.625), (33) will not be almost surely polynomially stable. For example, if we choosing γ = 3, we can conclude that (33) will be mean square polynomially stable but not be almost surely polynomially stable. In fact, the root λ of the following equation λ = 2.2 − 0.625 × 0.5 −λ is not great than 1 which contradicts the fact λ > 1.
Proof. Let V (x(t), t, i) = θ i |x(t)| 2 , where θ i > 0 for all i ∈ S. Clearly, The operator LV has the form Note that and It follows that For any t ≥ t 0 and V satisfying V (x(t), t, i)], we have λ 1 E|x(qt)| 2 ≤ q −λ λ 2 E|x(t)| 2 on t ≥ t 0 .
Conclusion. This paper is devoted to the pth moment polynomial stability and almost sure polynomial stability of hybrid stochastic systems with pantograph delay. The Razumikhin technique and Lyapunov functions is used to derive sufficient conditions for stabilities of nonlinear hybrid stochastic systems with pantograph delay, which further helps to derive easily verifiable conditions for linear hybrid stochastic systems with pantograph delay. Finally, two examples are provided to verify the effectiveness of the main results.