Abstract
This paper focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Itô formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.
1. Introduction
The nondeterministic (i.e., stochastic) phenomena are frequently encountered in many practical systems. These systems should be described by stochastic differential equations (SDEs for short) instead of ordinary ones. On the other hand, time delays are included in many practical systems, such as networks control systems, traffic systems, production process control systems, and population and economic dynamic systems, that is, the current and future states of the systems dependent on their departed states. In current years, the study of analysis and synthesis of stochastic time delay systems, which are described by stochastic delayed differential equations (SDDEs for short), is a popular topic in the field of control theory [1–7]. Because the existence of time delay is often the reason of instability and deteriorates the control performance, the studies on time delay systems stability and control have important theoretical and practical values.
A real dynamic system is influenced by both stochastic disturbances and time delays, so when we consider the behavior of a dynamic system, we use the stochastic delayed differential equation and the stochastic functional differential equation (SFDE for short) as modeling tools to investigate stability of stochastic dynamic systems with discrete delays or distributed delays. So far, these topics have received a lot of attention and there are so many references about them. For example, Cong [1] and Li et al. [2] obtained exponential stability conditions of linear stochastic neutral delay systems. Mao [8], Mao and Shah [9], Zhu and Hu [10], Zhu and Hu [11], S. Xie and L. Xie [12], and Zhu et al. [13] established some stability criteria of the stochastic system with discrete delays. An improved delay-dependent stability criterion is derived for stochastic delay systems by a strict LMI in [14]. Hu and Wu [15], Wu et al. [16], Yin et al. [17], and Zhou et al. [18] established some stability criteria of the stochastic system with distributed delays. However, discrete delays and distributed delays always coexist in real dynamic systems; thus, it is reasonable to consider them together and it leads us to investigate stochastic differential equations with mixed delays (SMDDEs for short).
Although stochastic differential systems with mixed delays received increasing attention recently, there is a little previous literature, as systematic research on such system has not been developed yet. For example, Zhu and Song [19] obtained some exponential stability results for a class of impulsive nonlinear stochastic differential equations with mixed delays by Razumikhin technique, but these sufficient conditions only ensure the exponential stability of the trivial solution in the mean square and did not give a bound for the time delay . Deng et al. [20] and L. Xu and D. Xu [21] focused on the corresponding study of exponential stability of neural network model. Thus, this paper aims to fill the gap in a sense. In this paper, we investigate not only the exponential stability in the mean square but also the almost surely exponential stability for a class of SMDDEs based on Lyapunov stability theory, Itô formula, stochastic analysis, inequality technique, and so on. We first consider the existence and uniqueness of the adapted solution by employing fixed point theorem. Next, some sufficient conditions of exponential stability and corollaries for stochastic differential systems with mixed delays are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of SMDDE implies the almost surely exponentially stable. The obtained results generalize and improve some recent results (for instance, [19–21]). In particular, our theoretical results show that if SDE is exponentially stable and the time delay is sufficiently small, then the corresponding SMDDE will remain exponentially stable. Moreover, the time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice. It should be mentioned that the approach provided here is different from those used in [19–21]. Finally, we present a simple example to illustrate the effectiveness of our stable results.
The rest of this paper is organized as follows. In Section 2, we give the preliminary results about SMDDEs. Main results and proofs for SMDDEs are provided in Section 3. Section 4 presents a simple example to illustrate our stable results. Section 5 lists some concluding remarks.
2. Preliminaries
Throughout this paper and unless specified, we let be an m-dimensional Brownian motion defined on a complete probability space with a natural filtration (i.e., and augmented by all the P-null sets in ). Denote by the Euclidean norm. If is a vector or matrix, its transpose is denoted by . If is a matrix, denote by the operator norm of ; that is, . is the initial path of , where is a given finite time delay and is the set of continuous functions from into . Moreover, denote by the family of -valued adapted stochastic processes , such that is -measurable and ().
We also use the notation which is -valued adapted stochastic processes s.t. .
We consider the following stochastic differential equations with mixed delays: where and represent the nonlinear uncertainties and represent given functional of the path segment of ; is the given averaging parameter. Furthermore, we always assume that for the stability purpose of this paper.
For simplicity, in what follows, we write sometimes, where .
To develop our theories and results, we need to introduce the following concepts and important inequalities. For stochastic system, exponential stability in mean square and almost surely exponential stability are generally used [13].
Definition 1. The trivial solution of (1) is said to be th moment exponentially stable, if there exists a positive constant such that for any .
Especially, , and it is called mean square exponentially stable.
Definition 2. The trivial solution of (1) is said to be almost surely exponentially stable. If there exists a positive constant such that for any .
Lemma 3 (see [22]). For any real matrices and a constant , the following matrix inequality holds:
Lemma 4 (Cauchy-Schwarz inequality). Let and be real functions which are continuous on the closed interval . Then,
3. Main Results
3.1. Existence and Uniqueness Result of the Solution for SMDDEs
We make the following assumptions for the coefficients of (1).(H3.1)Let be fixed time duration, , , and there exists a constant such that (H3.2).
Theorem 5. Let (H3.1) and (H3.2) hold. Then for any , (1) has a unique t-continuous adapted solution, denoted by and . So (1) has a trivial solution .
Inspired by the literature [23], we present the proof of the Theorem 5 as follows.
Proof. Let us define a norm in Banach space as follows: Clearly it is equivalent to the original norm of . We consider where , . Define a mapping such that . We desire to prove that is a contraction mapping under the norm . For arbitrary , set and , . Then, satisfies Applying Itô’s formula to , we have Integrating from to and taking the expectation in the above, we get Lemma 3 yields Then by (H3.1), we obtain Thus, where Lemma 4 yields It then follows from (17) that Then, Let ; then the above yields That is, This implies that is a strict contraction mapping. Then it follows from the fixed point theorem that (1) has a unique solution in . Since and satisfy (H3.1) and (H3.2), we can easily derive that and is continuous with respect to . Furthermore, by , (1) has a trivial solution .
For simplicity, in what follows we write .
3.2. Exponential Stability for SMDDEs
We make the following assumptions for the coefficients of (1).
(H3.3) There exist nonnegative constants , , for any such that and for any such that (H3.4) By (H3.1), one has .
In the study of mean square exponential stability, it is often to use a quadratic function as the Lyapunov function; that is, , where is a symmetric positive definite matrix.
Theorem 6. Let (H3.3) and (H3.4) hold; then the trivial solution of (1) is exponentially stable in the mean square. Assume that there exist symmetric positive definite matrices and a constant such that
In order to prove Theorem 6, we need three lemmas, proofs of which are left in appendix.
Lemma 7. Fix the initial data arbitrarily. Then, for any , where is a constant larger than .
Lemma 8. Fix the initial data arbitrarily. Then, for any .
Lemma 9. Let (H3.3) and (H3.4) hold. Then, for any , where is a constant larger than .
Based on the above Lemmas 7–9, we now carry out a proof for Theorem 6.
Proof of Theorem 6. Fix the initial data arbitrarily. Applying Itô’s formula to , we have
Combining Lemmas 3 and (24) as well as (H3.4), we can estimate the first item of (32) as follows:
where .
By (24), the last item of (32) yields
Substituting the above two into (32), we get
For small enough , we derive
If (25) holds, then we can choose small enough such that
Applying Itô’s formula to , we derive
Substituting (35) into the above yields
Taking the expectation in the above, we have
Now we apply Lemmas 7–9 to the last three terms on the right-hand side of (40) to get an estimate of as follows:
for , where
Since is positive definite,
where is the smallest eigenvalue of .
Then,
It then follows from (41) that
Hence,
This easily yields
Then (1) is exponentially stable in the mean square.
Theorem 10. Let , under the same assumption as Theorem 6, if Then,
Proof. let , under the same assumption as Theorem 6. It follows from (48) that
for all , . Then
For , , we have
Let be arbitrary. By Doob’s martingale inequality, it follows from (52) that
Thus, it follows from the Borel-Cantelli lemma that, for almost all , there exists , and ,
So, for , ,
This easily yields
Since is arbitrary, we must have
Remark 11. The exponentially stable in the mean square of SMDDE (1) implies the almost surely exponentially stable. In general, Theorem 10 is still true for th moment exponential stability.
Let us single out two important special cases.
Case 1. If , then (1) reduces to nonlinear deterministic differential equation with mixed delay
Applying Theorem 6 to (58), we obtain the following useful result.
Corollary 12. Let (H3.4) and condition (22) hold, then the trivial solution of (58) is exponentially stable in the mean square. Assume that there exists a symmetric positive definite matrices and a constant such that
Remark 13. The bound for the time delay when SMDDE (1) is exponentially stable follows from (25), and the bound for the corresponding deterministic case follows from (60).
Case 2. If the time delay , then (1) reduces to nonlinear SDE
Corollary 14. If SDE (61) is exponentially stable and the time delay is sufficiently small, then the corresponding SMDDE (1) will remain exponentially stable.
Proof. SMDDE (1) can be rewritten as It is clear that if the time delay is sufficiently small, then SMDDE (1) is regarded as the perturbed system of the corresponding SDE (61) (without delay). On the other hand, the perturbation term could be so small that the perturbed equation (1) would behave in a similar way as (61) asymptotically. Applying Theorem 5 and Remark 12 in [15], we derive SMDDE (1) which will remain exponentially stable.
4. Example
Let us now present a simple example to illustrate our results which can help us find the time delay upper limit.
Example 1. Let us now consider a two-dimensional SMDDE where for , in , and : represent the nonlinear uncertainties. is a given two-dimensional Brownian motion, represent given functionals of the path segment of , is the given averaging parameter, and is a given finite time delay.
For convenience, let us choose to be the second-order identity matrix, and . We assume that, for any , Via a simple calculation, it is easy to estimate that that is, , , , , , and .
By plugging these into (25), it is easy to find ; that is, if , then (64) is exponentially stable in the mean square and is also almost surely exponentially stable.
If , by plugging , , , , and into (60), then, we would conclude that (64) is exponentially stable provided .
Remark 15. Figures 1 and 2 give the simulation results of Example 1 when , (Figure 1) and , (Figure 2). The parameter values used in the calculations are , , and .
5. Concluding Remarks
In this paper, we investigate not only the exponential stability in the mean square but also the almost surely exponential stability for a class of SMDDEs based on Lyapunov stability theory, Itô formula, stochastic analysis, inequality technique, and so on. We first consider the existence and uniqueness of the adapted solution by employing fixed point theorem. Next, some sufficient conditions of exponential stability and corollaries for stochastic differential systems with mixed delays are obtained by using Lyapunov function. Theorem 10 shows that the exponentially stable in the mean square of SMDDE implies the almost surely exponentially stable. The obtained results generalize and improve some recent results (for instance, [19–21]). In particular, our theoretical results show that if SDE is exponentially stable and the time delay is sufficiently small, then the corresponding SMDDE will remain exponentially stable. Moreover, the time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice. It should be mentioned that the approach provided here is different from those used in [19–21]. Finally, we present a simple example to illustrate the effectiveness of our stable results. Another challenging problem is to study a class of stochastic differential equations with mixed variable delays or a class of stochastic control systems with correlated state and observation noises (for instance, [24]). We hope to study these problems in forthcoming papers.
Appendix
We now present proofs of Lemmas 7–9.
Proof of Lemma 7. For any , we easily get for any , where .
Proof of Lemma 8. Similar to (26), for any , we have
For , it then follows from (17) that
Similar to (26) and substituting (26) into the above, we obtain (29); that is
where .
Substituting (29) into (A.2) yields
The relation (28) in Lemma 8 is then proved.
Proof of Lemma 9. Similar to (26), for any , we have
Substituting (27) into the above inequality yields
The relation (30) in Lemma 9 is then proved.
On the other hand, for , we have
By (22) and (23), it follows from Lemmas 3-4 that
Similar to (27), for , we have
where .
Substituting (26), (28), and (30) into (A.10), for , we get
where .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by the Fundamental Research Funds for the Central Universities (JBK130401).