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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 854743, 12 pages
Razumikhin-Type Theorems on Exponential Stability of SDDEs Containing Singularly Perturbed Random Processes
1College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China
2Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK
3Department of Mathematics, Swansea University, Swansea SA2 8PP, UK
Received 13 December 2012; Revised 25 April 2013; Accepted 25 April 2013
Academic Editor: Chuangxia Huang
Copyright © 2013 Junhao Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
The stability of time delay systems is a field of intense research [1, 2]. In , 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 .
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  for the ordinary differential delay equation which is called Razumikhin-type theorem and was developed by several people . In his paper, Mao  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 [7–9]. Starting from the work , by introducing a small parameter , a number of asymptotic properties of the Markov chain have been established. One of the main results in  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.
Let be a complete probability space with a filtration satisfying the usual conditions (i.e. it is increasing and right continuous, and contains all -null sets). Throughout the paper, we let be an -dimensional Brownian motion defined on the probability space . If is a vector or matrix, its transpose is denoted by . Let denote the Euclidean norm in as well as the trace norm of a matrix. For , denotes the family of continuous functions from to with the norm . Denote by the family of all measurable and bounded -valued random variable. We will denote the indicator function of a set by .
Let be a right-continuous Markov chain on taking values in a finite state space with the generator given by where and is the transition rate from to satisfying if and . We assume the Markov is independent of the Brownian motion . It is well known that almost every sample path is a right-continuous step function with finite number of simple jumps in any finite subinterval of . As a standing hypothesis, we assume that the Markov chain is irreducible. This is equivalent to the condition that for any , we can find such that Thus, always has an eigenvalue . The algebraic interpretation of irreducibility is . Under this condition, the Markov chain has a unique stationary (probability) distribution , subject to and for all . For a real valued function defined on , we define for each .
Consider the following stochastic delay system with Markovian swtching: where and .
To highlight the fast and slow motions, we introduce a parameter and rewrite the Markov chain as and the generator as . is given by where represents the fast varying motions, and represents the slowly changing dynamics. We denote , , and . To the reduction of complexity, needs to have a certain structure. Suppose that with and , and that where for each and is a generator of a Markov chain taking values in . We impose the following hypothesis:
(H1) For each , is irreducible.
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.
(H2) For any integer , there is a constant , such that for all and those with .
(H3) There is an , such that for any ,
Under the assumptions (H2) and (H3), system (8) has a unique solution denoted by on , where the notation emphasizes the dependence on the initial data . Moreover, for every and any compact subset of , there exists a positive constant which is independent of such that
We will consider the stability of system (8), but the state space of the Markov chain is large, and it is difficult to handle (8) directly. So we will consider the average system of (8). To proceed, lump the states in each into a single state and define an aggregated process as Denote the state space of by , the stationary distribution by and . Define with and . It has been known that converges weakly to as , where is a continuous-time Markov chain with generator and state space (cf. ).
Define for each with and . It is easily seen that and are the averages with respect to the stationary distribution of the Markov chain. Note that for any are nonnegative definite matrices, so we find its “square root” of (15), which is denoted by . For degenerate diffusions, we can see the argument in .
The averaged system of (8) is defined as follows:
3. Moment Exponential Stability
In this section, we shall establish the Razumikhin-type theorem on the exponential stability for (8).
Let be the class of nonnegative real-valued functions defined on that are -times continuously differentiable with respect to . We give the following assumption about for some .
(H4) For each , as . Moreover, for , where denotes the th derivative of with respect to and denotes the function of satisfying .
Theorem 1. Let (H1)–(H3) hold; there is a function satisfying (H4), and there are positive constants , and such that(i), (ii) provided ,
where Then, for all , 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).
Remark 3. does exist by (11).
Proof of Theorem 1. Define
We extend to by setting ; then, is right continuous on .
Let be arbitrary, and define If we can show that , then the proof is completed.
If , by condition (i),
If , we will prove that . Otherwise, there exists the smallest such that all and as well as for all suffieciently small .
For , If , then , .
Since converges to with probability one (see Lemma 2.3 in ), by condition (i), we can derive Recalling the fact , and using the uniqueness of the equation, we then have , a.e. . Therefore we have Then , which is a contradiction. Hence we see that . For , there exists a such that Consequently, there exists a sufficiently small , such that, for any , By condition (ii), then, Noting that , we have We now consider By the definition of operator , we have
So By the definition of , This, together with assumption (H2), implies By the argument of Lemma 7.14 in , the right side of above inequality is equivalent to to . Similarly, we can show By the definition of and , we have hence By assumption (H4) and the argument of Lemma 7.14 in , we have the right side of above inequality is equivalent to .
Therefore by the condition (ii) this is This contradicts the definition of . The proof is now completed.
Example 4. Let be a Markov chain generated by given in (5) with The generator consists of two irreducible blocks. The stationary distributions are , , and Consider a one-dimensional equation with Then the limit equation is where is the Markov chain generated by and Let ; then, Consequently It is easy to see that we can find a such that . Therefore, for any satisfying on , (49) yields Hence, by Theorem 1, the solution is mean square stable when is sufficient small.
4. Stochastic Delay System with Pure Jumps
In this section we discuss the stability of the following stochastic delay system with pure jumps: where , . We assume that the each column of the matrix depends on only through the th coordinate ; that is, is a -dimensional Poisson process, and the compensated Poisson, process is defined by where are independent one-dimensional Poisson random measures with characteristic measure coming from independent one-dimensional Poisson point processes.
The averaged system of (18) is defined as follows: where , . Similar to the definition of , we define For each with and .
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 , such that for all and those with .
(H3′) There is an , such that for any , , Given , we define the operator by where We need the following lemma, for details see .
Lemma 5. Let (H1) and (H2′), (H3′) hold, as ; then, converges weakly to in , where is the space of functions defined on that are right continuous and have left limits taking values in and endowed with the Skorohod topology.
We now state our main result in this section.
Theorem 6. Let (H1) and (H2′), (H3′) hold; there is a function satisfying (H4), and there are positive constants , and such that(i), (ii) provided ,Then, for all , where
Proof. As the proof of Theorem 1, define
We extend to by setting . Then, is right continuous on .
Let be arbitrary, and define If we can show that , then the proof is completed.
If , by condition (i), is the same as the proof of Theorem 1, we have .
In the following we shall prove that if . Otherwise, there exists the smallest such that all , and as well as for all suffieciently small .
As the same in the proof of Theorem 1 we can have that . Hence for , there exists a such that Consequently, there exists a sufficiently small , such that for any ,
By condition (ii), we then have for ,
We now consider By the definition of the operator similar to that of the proof of Theorem 1, we have This implies