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Research Article | Open Access
Exponential Stability of Neutral Stochastic Functional Differential Equations with Two-Time-Scale Markovian Switching
We develop exponential stability of neutral stochastic functional differential equations with two-time-scale Markovian switching modeled by a continuous-time Markov chain which has a large state space. To overcome the computational effort and the complexity, we split the large-scale system into several classes and lump the states in each class into one class by the different states of changes of the subsystems; then, we give a limit system to effectively “replace” the large-scale system. Under suitable conditions, using the stability of the limit system as a bridge, the desired asymptotic properties of the large-scale system with Brownian motion and Poisson jump are obtained by utilizing perturbed Lyapunov function methods and Razumikhin-type criteria. Two examples are provided to demonstrate our results.
In many practical dynamical systems such as neural networks, computer aided design, population ecology, chemical process simulation, and automatic control, stochastic differential equations represent the class of important dynamics (see [1–4]). During the recent several years, the asymptotic properties of neutral stochastic functional differential equations have been investigated by many authors (see [5–14]). Mao [10, 11] gave the exponential stability of neutral stochastic functional differential equations by using the Razumikhin-type theorems. Zhou and Hu  used the same argument to discuss the exponential stability in th moment of neutral stochastic functional differential equations and neutral stochastic functional differential equations with Markovian switching. Wu et al.  examined the almost sure robust stability of nonlinear neutral stochastic functional differential equations with infinite delay, including the exponential stability and the polynomial stability. Song and Shen  investigated the asymptotic behavior of neutral stochastic functional differential equations under the more general conditions than the classical linear growth condition. Chen et al.  considered the exponential stability in mean square moment of mild solution for impulsive neutral stochastic partial functional differential equations by employing the inequality technique. The attraction and quasi-invariant sets of neutral stochastic partial functional differential equations were also studied in the recent paper .
In this paper, we will consider neutral stochastic functional differential equations with two-time-scale Markovian switching modeled by a continuous-time Markov chain which has a large state space. The computational effort and the complexity become a real concern. To overcome the difficulties, we have devoted much effort 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 lump the states in each class into one state (see [3, 15–21]). Khasminskii et al. for the first time established the asymptotic properties of the Markov chain by introducing a small parameter (see ). Yin and Zhang developed the method in their book  that a complicated system can be replaced by the corresponding limit system that has a much simpler structure. Motivated by the papers [16, 21], under suitable conditions, using the stability of the limit system as a bridge, we will study the exponential stability of neutral stochastic functional differential equations with Brownian motion and Poisson jump by utilizing perturbed Lyapunov function methods and Razumikhin-type criteria.
The remainder of this paper is organized as follows. In Section 2, we introduce some notations and notions needed in our investigation. In Section 3, we state our main results, that is, exponential stability of neutral stochastic functional differential equations with two-time-scale Markovian switching. The exponential stability for neutral stochastic functional differential equations driven by pure jumps is also discussed in Section 4. Finally, two examples are presented to justify and illustrate applications of the theory in Section 5.
Throughout this paper, unless otherwise specified, 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). Let be an -dimensional Brownian motion defined on the probability space. For , let denote the family of continuous functions from to with norm , where is the Euclidean norm in . If is a vector or matrix, its transpose is denoted by , while its trace norm is denoted by . Denote by the family of all measurable and bounded -valued random variables. For and , denote by the family of all -measurable -valued random variables such that . We will denote the indicator function of a set by .
Consider an -dimensional neutral stochastic functional differential equation with Markovian switching as follows: on with initial data and , which is regarded as a -valued stochastic process. Moreover, , , .
Let () be a right-continuous Markov chain on the probability space taking values in a finite state space with generator given by where . Here, is the transition rate from to if , while .
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 Then, always has an eigenvalue . The algebraic interpretation of irreducibility is rank . Under this condition, the Markov chain has a unique stationary distribution , subject to and for all . For a real-valued function defined on , we define for each .
Let denote the family of all nonnegative functions on , which are continuously twice differentiable in and once differentiable in . If , define an operator from to by where
For a parameter , we rewrite the Markov chain as and the generator as . is given by where represents the fast varying motions and represents the slowly changing dynamics. Set , , and . For the sake of simplicity, suppose that with , , and where is a generator of a Markov chain taking values in , for every .
We give the first assumption as follows.
Assumption 1. For each , is irreducible.
In order to emphasize the effect of the fast switching, (1) can be given by
To assure the existence and uniqueness of the solution, we give the following standard assumptions.
Assumption 2 (local Lipschitz condition). For each integer , there exists a constant such that for all , and those with , and , .
Assumption 3 (linear growth condition). There is an , for any , , such that
Assumption 4. For all and those , there is a constant such that
Under Assumptions 2, 3, and 4, (10) has a unique solution denoted by on , where is dependent on the initial value (see ). Moreover, for every and any compact subset of , there is a positive constant which is independent of such that
Since the state space of the Markov chain is large, it is too complicated to deal with directly. We need to analyse the limit equation of (10). To continue, make all 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 (see ). Define for each with and . It is easy to know that , , and are the limits with respect to the stationary distribution of the Markov chain. Consider that, for any , are nonnegative definite matrices, so we denote its “square root” of by . For degenerate diffusions, we can see the argument in .
The limit equation of (10) is defined as follows:
3. Exponential Stability of NSFDE with Two-Time-Scale Markovian Switching
In this section, we establish the Razumikhin-type theorem on the exponential stability for (10). Denote by the family of nonnegative real-valued functions defined on that are -times continuously differentiable with respect to . At the same time, we need another assumption and a lemma with respect to for some .
Assumption 5. For each , as . Moreover, , for , where denotes the th derivative of with respect to and denotes the function of satisfying .
Lemma 6. Suppose that ; there is a positive constant such that Then, for any , the solution for (10) satisfies
Proof. Note the following elementary inequality: We have from condition (20) that, for any , Then, Therefore, the desired result holds.
Theorem 7. Let Assumptions 1–4 hold and let , , , be all positive numbers and . Assume that there exists a function satisfying Assumption 5, such that for all , , . Consider the following: for all , , , and provided , satisfying for all . Then, for all , where being the root of the following equation:
In other words, the trivial solution of (10) is th moment exponentially stable and the th moment Lyapunov exponent is not greater than .
By the definition of , we know that
Extend to by setting . Recalling the facts that is continuous for all and is right continuous, it is easy to see that is right continuous on . Let be arbitrary and define
for all . We claim that
Note that, for each , either or .
If , because is right continuous on , it is easy to obtain that, for all sufficiently small, ; hence, and .
If , we have for all .
Then, for all .
On the other hand, by Lemma 6, we derive Then, where ; that is, .
Consequently, there exists a sufficiently small , such that, for any , for all . Thus, which implies that By the condition of , we get
Next, we consider By the definition of operator , we have
Therefore, By the definition of , This implies that By the argument of Lemma 7.14 in , the right side of the above inequality is equivalent to ; that is, . Similarly, we can show that By the definition of and , we have Hence, By the argument of Lemma 7.14 in , the right side of the above inequality is equivalent to ; that is, . Therefore, That is So, for all sufficiently small, and hence . Inequality (34) holds.
It follows from (34) that for all . By the definition of , By Lemma 6, we derive That is,
4. Neutral Stochastic Functional System with Pure Jump
In this section, we discuss the stability of the following neutral stochastic functional system with pure jump: where , , . We assume that each column of the matrix depends on only through the th coordinate ; that is, is an -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 limit system of (56) is defined as follows: where