Razumikhin-Type Theorems on Exponential Stability of SDDEs Containing Singularly Perturbed Random Processes

This paper concerns Razumikhin-type theorems on exponential stability of stochastic differential delay equations with Markovian switching, where the modulating Markov chain involves small parameters. The smaller the parameter is, the rapider switching the system will experience. In order to reduce the complexity, we will “replace” the original systems by limit systems with a simple structure. Under Razumikhin-type conditions, we establish theorems that if the limit systems are th-moment exponentially stable; then, the original systems are th-moment exponentially stable in an appropriate sense.


Introduction
The stability of time delay systems is a field of intense research [1,2]. In [2], the global uniform exponential stability independent of time delay linear and time invariant systems subjected to point and distributed delays was studied. Moreover, noise and time delay are often the sources of instability, and they may destabilize the systems if they exceed their limits [3].
Hybrid delay systems driven by continuous-time Markov chains have been used to model many practical systems in which abrupt changes may be experienced in the structure and parameters caused by phenomena such as component failures or repairs. An area of particular interest has been the automatic control of the underlying systems, with consequent emphasis on the analysis of stability of the stochastic models. For systems with time delay, there are two approaches to proving stability that correspond to the conventional Lyapunov stability theory. The first is based on Lyapunov-Krasovski functionals, the second on Lyapunov-Razumikhin functions. The latter one originated with Razumikhin [4] for the ordinary differential delay equation which is called Razumikhin-type theorem and was developed by several people [5]. In his paper, Mao [6] was the first who established a Razumikhin-type theorem for stochastic functional differential equations (SFDEs). Roughly speaking, a Razumikhin-type theorem states that if the derivative of a Lyapunov function along trajectories is negative whenever the current value of the function dominates other values over the interval of time delay; then, the Lyapunov function along trajectories will converge to zero. The Razumikhin methods have been widely used in the study of stability for functional and differential-delay systems. In this work, we shall investigate stochastic differential delay equations with Markovian switching (SDDEwMSs). The switching we shall use will be a finite-state Markov chain, which incorporates various considerations into the models and often results in the underlying Markov chain having a large state space. To overcome the difficulties and to reduce the computational complexity, much effort has been devoted to the modeling and analysis of such systems, in which one of the main ideas is to split a large-scale system into several classes and lumping the states in each class into one state; see [7][8][9]. Starting from the work [10], by introducing a small parameter > 0, a number of asymptotic properties of the Markov chain (⋅) have been established. One of the main results in [9] is that a complicated system can be replaced by the corresponding limit system having a much simpler structure. In [11,12], long-term behavior of SDEwMSs and SDDEwMSs was investigated, respectively, while in [13,14] the stability of random delay system with two-time-scale Markovian switching was studied. Using the stability of the limit system as a bridge, the desired asymptotic properties of the original system is obtained using perturbed Lyapunov function methods. In this work, we shall establish a Razumikhin-type theorem for SDDEwMSs.
The remainder of this work is organised as follows: in the next section, we shall begin with the formulation of the problem. Section 3 investigates the Razumikhin-type theorem for SDDEs driven by Brownian motion. The exponential stability for SDDEs driven by pure jumps is discussed in Section 4.

Preliminaries
Let (Ω, F, {F } ≥0 , P) be a complete probability space with a filtration {F } ≥0 satisfying the usual conditions (i.e. it is increasing and right continuous, and F 0 contains all P-null sets). Throughout the paper, we let ( ) = ( 1 ( ), . . . , ( )) be an -dimensional Brownian motion defined on the probability space (Ω, F, {F } ≥0 , P). If is a vector or matrix, its transpose is denoted by . Let | ⋅ | denote the Euclidean norm in R as well as the trace norm of a matrix.
where > 0 and is the transition rate from to satisfying > 0 if ̸ = and = − ∑ ̸ = . We assume the Markov (⋅) is independent of the Brownian motion (⋅). It is well known that almost every sample path (⋅) is a rightcontinuous step function with finite number of simple jumps in any finite subinterval of R + := [0, ∞). As a standing hypothesis, we assume that the Markov chain is irreducible. This is equivalent to the condition that for any , ∈ S, we can find 1 , 2 , . . . , ∈ S such that Thus, Γ always has an eigenvalue 0. The algebraic interpretation of irreducibility is rank(Γ) = − 1. Under this condition, the Markov chain has a unique stationary (probability) distribution Γ = 0, subject to ∑ =1 = 1 and > 0 for all ∈ S. For a real valued function (⋅) defined on S, we define for each ∈ S.
To highlight the effect of the fast switching, we rewrite the system (4) as To assure the existence and uniqueness of the solution, we give the following standard assumptions.
The averaged system of (8) is defined as follows:

Moment Exponential Stability
In this section, we shall establish the Razumikhin-type theorem on the exponential stability for (8). Let (R ×S; R + ) be the class of nonnegative real-valued functions defined on R × S that are -times continuously differentiable with respect to . We give the following assumption about ( , ) ∈ (R × S; R + ) for some ≥ 4.
denotes the ℓth derivative of ( , ) with respect to and ( ) denotes the function of satisfying sup | ( )|/ < ∞. Theorem 1. Let (H1)-(H3) hold; there is a function ( , ) ∈ (R ×S; R + ) satisfying (H4), and there are positive constants , 1 , 2 , and > 1 such that where L ( , , ) = ( , ) ( , , ) where Remark 2. Note that the conditions of Theorem 1 are sufficient conditions for the average system (16) ( ) (or the limit process ( )). However the conclusion of Theorem 1 is about the process ( ). Since the structure of the the average system (16) is much simpler than that of ( ), this theorem has reduced the computational complexity for the system (8).
Since ( ( ), ( )) converges to ( ( ), ( )) with probability one (see Lemma 2.3 in [12]), by condition (i), we can derive Consequently, there exists a sufficiently small 0 > 0, such that, for any ∈ (0, 0 ), By condition (ii), then, Noting that ] < ] ≤ , we have We now consider By the argument of Lemma 7.14 in [9], the right side of above inequality is equivalent to to 0; that is, 2 = 0. Similarly, we can show By assumption (H4) and the argument of Lemma 7.14 in [9], we have the right side of above inequality is equivalent to 0, that is, 4 = 0. Therefore by the condition (ii) It is easy to see that we can find a > 1 such that (1/4) − ( /16) > 0. Therefore, for any Hence, by Theorem 1, the solution ( ) is mean square stable when is sufficient small.

Stochastic Delay System with Pure Jumps
In this section we discuss the stability of the following stochastic delay system with pure jumps: where { , = 1, . . . , } are independent one-dimensional Poisson random measures with characteristic measure { , = 1, . . . , } coming from independent one-dimensional Poisson point processes.
To assure the existence and uniqueness of the solution of (52), we also give the following standard assumptions.
(H2 ) For any integer , there is a constant ℎ > 0, such that for all ∈ S and those 1 , There is an ℎ > 0, such that for any , ∈ R , ∈ S, Given ∈ (R × S; R + ), we define the operator L by We need the following lemma, for details see [16]. We now state our main result in this section. where Proof. As the proof of Theorem 1, define Consequently, there exists a sufficiently small 0 > 0, such that for any ∈ (0, 0 ), By the argument of Lemma 7.14 in [9], the right side of the inequality above is equivalent to 0, that is, 4 By the argument of Lemma 7.14 in [9], we have 3 = 0. Similar to the proof of Theorem 1, we can derive 2 = 0, 5 = 0. Therefore we arrive at This contradicts the definition of . The proof is therefore completed.
We shall give an example to illustrate our theory: We can find a > 1 such that 3−2 > 0. Therefore, for any ∈ Hence, by Theorem 6, the solution ( ) is mean square stable.