Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 348235, 7 pages
http://dx.doi.org/10.1155/2015/348235

Research Article

The Mean Stability Criteria in terms of Two Measures for Stochastic Differential Equations with Coefficient’s Uncertainty

Rui Zhang,¹ Yinjing Guo,¹ Xiangrong Wang,² and Xueqing Zhang³

¹College of Electronic Communication and Physics, Shandong University of Science and Technology, Qingdao 266590, China
²College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
³College of Environmental Science and Engineering, Ocean University of China, Qingdao 266100, China

Received 21 April 2015; Revised 23 July 2015; Accepted 9 August 2015

Academic Editor: Son Nguyen

Copyright © 2015 Rui Zhang 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.

Abstract.

This paper extends the stochastic stability criteria of two measures to the mean stability and proves the stability criteria for a kind of stochastic Itô’s systems. Moreover, by applying optimal control approaches, the mean stability criteria in terms of two measures are also obtained for the stochastic systems with coefficient’s uncertainty.

1. Introduction

Lyapunov’s method, which makes an essential use of auxiliary functions (also called Lyapunov functions), is an important approach to study the stability of differential systems including ordinary differential equations (ODEs) and stochastic differential equations (SDEs). This method started in Lyapunov’s original work in 1892 [1] for demonstrating stability of ODEs. In the 1960s, Movchan [2] studied the stability with two measures; such works were also developed and can be seen in [3]. In the past decades, Lyapunov’s method is modified to the study of stability of Markovian processes [4], stochastic differential systems based on Brownian motions [5], semimartingales [6], or Lévy processes [7] and is also developed with the form of exponential stability [8] or LaSalle theorem [9], and so forth. Recently, the stability for systems with unknown parameters is also discussed [10], and the theorems of stability are widely applied in aerospace [11], state-feedback control [12], automatic control [13], neural networks [14], and other fields.

In this paper, we will discuss the following stochastic Itô’s systems:

where satisfy the usual Lipschitzian conditions and is -dimensional standard Brownian motion. It is well known that, for a stochastic process and a given positive function , almost surely discussed in [15] does not imply that . So, we extend Lyapunov’s methods used by [3] for ODEs to the stochastic cases and study the mean stability criteria in terms of two measures for system (1).

This paper is organized as follows: In Section 2, we first introduce Lyapunov’s derivatives for (1) and deduce the basic comparison results in terms of Lyapunov’s function. In Section 3, we prove the stochastic two-measure stability criteria for Itô systems, which can be seen as the extension of that of the ODEs. As described in [16], stability, robustness, and optimality can be considered systematically and simultaneously. In Section 4, the optimal control approach is extended to the stochastic systems with coefficient’s uncertainty.

2. Basic Comparison Results for Stochastic Differential Equations

Let be a given completed probability space, and is a standard Brownian motion with filtration:

Let , be deterministic functions and satisfy the following Lipschitz condition and linear growth condtion: there exists , for every , such that

For a given function , , we denote

where represents a -dimensional random variable with standard normal distribution; that is, and is a -order identity matrix.

Remark 1. We use the notation to emphasize the definition with respect to system (1). For convenience, we use the shortened form to substitute .

Remark 2. If , is Lyapunov’s operator associated with (1); that is,

where is the partial derivative for , is the gradient of for , and is the Hessian matrix of for .

Since (4) is dependent on expectation calculating, it is not easy to check whether exists or not. The following lemma gives a condition for the existence of .

Lemma 3. Let be continuous one-order differentiable for and also satisfy the following condition:

where is continuous on and is locally bounded for . Then exists at .

Proof. Denote ; then, for fixed ,

By (6) and distribution of , we have the first item of the right side of (7) that is bounded:

Since is differentiable for , so the last item of the right side of (7) is also bounded. Therefore, the supremum limit of (7) exists; that is, exist exactly.

The following lemmas will be used later.

Lemma 4 (see Theorem   in [6], or Theorems and   in [17]). Suppose satisfy (3). Then, stochastic differential equation (1) admits a unique strong solution such that, for any , there exists ( is a constant dependent only on , , and ):

Lemma 5. Suppose satisfies (6), and is the solution of (1); let ; then

where and is the usual right upper Dini derivative defined by

Proof. For small , we have

We now prove that the first two items of the right side are the higher infinitesimal of . By (1), we know that

For convenience, we denote , similar meaning for . We have

By Lemma 4 and inequality (10), we have

Let ; then ( replaces for shortening)

Now we estimate the order of

Since and , so

where are continuous positive functions. By (19), we have

Since

by (17) and (20), we see that the first two items of right side are higher infinitesimal of . So we have

For the last two items of (13), since is independent of with normal distribution , so we have

This proves (11).

The following lemma will be used in the proof of Theorem 7.

Lemma 6 (see Theorem   in [3]). Let and be the maximal solution of

existing on . Suppose and , , where is any fixed Dini derivative. Then implies , .

Now we formulate the basic comparison results in terms of Lyapunov function .

Theorem 7. Assume satisfies

where is concave for . Let be the maximal solution of the differential equation

Then, for every solution of (1) , implies

Proof. Denote . By Lemma 5 and the concave of we have

By Lemma 6, we can obtain the result (27).

Remark 8. If , the inequality (25) became

3. Stability Criteria in terms of Two Measures

Now we discuss the two-measure stability criteria for the stochastic differential system (1). We assume for all . Firstly, we give some definitions for stochastic stability.

Definition 9. The stochastic differential system (1) is said to be

mean -equistable, if for each and , there exists a function which is continuous in for each such that

 where is any solution of (1);mean -uniformly stable, if holds with being independent of ;mean -quasiequiasymptotically stable, if for each and , there exist positive number and such that

mean -quasiuniform asymptotically stable if holds with and being independent of ;mean -asymptotically stable if holds and, given , there exists a such that

mean -uniformly equiasymptotically stable, if and hold together;mean -uniformly asymptotically stable if and hold simultaneously;mean -unstable if fails to hold.

The following classes of functions will be used in this paper:

 , , 

Definition 10. Let . Then, we say that is finer than if there exists a function such that . Furthermore, if is independent of then we call uniformly finer than .

Definition 11. Let . If there exists a function convex such that , then we call -positive definite. If there exists a concave function such that , then we call -decrescent.

Theorem 12. Assume that

and is uniformly finer than ,, satisfies (6), and is -positive definite and -decrescent, and , for .

Then, the stability properties of the trivial solution of (26) imply the corresponding -stability properties of (1).

Proof. Since is -positive definite, so there exists a convex such that

Suppose the trivial solution of (26) is equistable and    is its maximal solutions with initial time and initial value , then, for every , there exists , when ,

Let ; then, by Theorem 7, we have

Since is -decrescent, there exists a concave such that

So

Since is continuous and strictly increasing, so let ; then when , inequality (34) holds. Combining (33), (34), and (35) and using the strictly increase of , we can gain

which implies (1) -equistability.

Remark 13. If , then condition can be replaced by

Remark 14. The stabilities of auxiliary ordinary differential equation (26) are defined by Definition   in [3].

Example 15. Consider the following 2-dimensional Itô’s system:

Let ; suppose has the form

Then

In order to make -positive, we let

Let , ; in order to find to satisfy

we set

When , combining (43) and (45), we have that, when

there exists which satisfies (44). Moreover, the trivial solution of

is uniformly asymptotically stable; by Theorem 12, the stochastic differential is mean--asymptotically stable.

However, in practice, the coefficients maybe have some uncertainty properties; that is, we only know the range of the parameters . Then how to discuss the stability of such systems with uncertainty is still a very interesting problem. The following section will introduce an optimal control approach to discuss the stability of such systems with uncertainty.

4. The Stability for Systems with Uncertainty

Consider the following stochastic differential equations with uncertainty:

where is an uncertainty function, and , that is, the trivial solution, is of (48). Furthermore, we also suppose there exists a function such that

Now we discuss how to determine the asymptotic stability of system (48) for all uncertainty function . Similar to the methods applied by [18, 19], we can translate this stability problem into an optimal problem.

For the nominal system

suppose we can find a state-feedback control that minimizes the cost functional

where , is the initial value of (50), , and is admissible on for each , and satisfies

Let the value function

Theorem 16. Suppose is an optimal control of problem (53), and there exists satisfying

and the value function , is also positive, and is decrescent, then system (48) is uniformly mean uniformly asymptotically stable for all uncertainties .

Proof. The values function satisfies the Hamilton-Jacobi-Bellman equation:

So, the optimal control satisfies

Then, the Lyapunov generator of (48) for is given as

So, for the solutions of (48) with uncertainty , applying Itô’s formula to , we have, when ,

Hence

So is decreasing on . Now we show when . Suppose . Since is decrescent, so there exists and , such that () combining with (59); we have

This contradicts the fact that . This implies that system (48) is uniformly mean uniformly asymptotically stable.

Corollary 17. Specially, let

The value function can be given by and the optimal control , where satisfies the following Riccati equation:

Let be the maximal eigenvalue of . If , then the corresponding system with uncertainty is uniformly mean -equistable for all uncertainties (which is also mean square asymptotically stable for all uncertainties ).

Example 18. Consider (41) with uncertainty coefficients. In (41), replace by and , respectively, and takes values in , in , , and . Then the system with uncertainty is obtained:

Let

And let be the 2-order identical matrix. Let ; then

We can take and the auxiliary optimal problem is

with the cost functional

, solving (62) with ; we have

By Corollary 17, we can determine that the stochastic system (63) is uniformly mean uniformly stable for all uncertainties.

5. Conclusion

In this paper, we extend the stability criteria of two measures to the mean stability situations for the stochastic systems with uncertainty. For the usual SDE, we give the results of mean stability criteria which are the basic criteria for such systems. As far as the systems with uncertainty, in order to resolve the difficulties coming from the coefficient uncertainty, we use the optimal control results as an auxiliary method to determine the mean stability. Furthermore, the stability criteria in terms of two measures for other stochastic systems, such as systems with Markovian jumps or Poisson jumps, are worth further studying.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the NSF of China (61471224, 61402265, and 11271007), the NSF of Shandong Province (ZR2011FM018), and the Doctoral Fund of Ministry of Education (20123718110010).

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