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The Mean Stability Criteria in terms of Two Measures for Stochastic Differential Equations with Coefficient’s Uncertainty
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.
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  for demonstrating stability of ODEs. In the 1960s, Movchan  studied the stability with two measures; such works were also developed and can be seen in . In the past decades, Lyapunov’s method is modified to the study of stability of Markovian processes , stochastic differential systems based on Brownian motions , semimartingales , or Lévy processes  and is also developed with the form of exponential stability  or LaSalle theorem , and so forth. Recently, the stability for systems with unknown parameters is also discussed , and the theorems of stability are widely applied in aerospace , state-feedback control , automatic control , neural networks , 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  does not imply that . So, we extend Lyapunov’s methods used by  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 , 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 , or Theorems and in ). 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 ):
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 ). 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 equationThen, for every solution of (1) , implies
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 bemean -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
Example 15. Consider the following 2-dimensional Itô’s system: Let ; suppose has the form ThenIn order to make -positive, we let Let , ; in order to find to satisfywe 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 functionalwhere , 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 haveThis 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.
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.
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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