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Practical Stability in the th Mean for Itô Stochastic Differential Equations
The th mean practical stability problem is studied for a general class of Itô-type stochastic differential equations over both finite and infinite time horizons. Instead of the comparison principle, a function which is nonnegative, nondecreasing, and differentiable is cooperated with the Lyapunov-like functions to analyze the practical stability. By using this technique, the difficulty in finding an auxiliary deterministic stable system is avoided. Then, some sufficient conditions are established that guarantee the th moment practical stability of the considered equations. Moreover, the practical stability is compared with traditional Lyapunov stability; some differences between them are given. Finally, the results derived in this paper are demonstrated by an illustrative example.
Lyapunov stability is one of the most important conceptions of stability and has been widely applied to many fields involving nearly all aspects of reality. As we all know, however, the Lyapunov stability is usually employed to study the steady-state property over an infinite horizon and cannot cope with the transient behavior of the trajectory. Therefore, even a stable system in the sense of Lyapunov cannot be applied in the practice since the trajectory exhibits undesirable transient behaviors such as exceeding certain boundary imposed on the trajectory. Moreover, for a Lyapunov stable system, the domain of the desired attractor may be too small to control the initial perturbation in it, which also limits the uses of the Lyapunov stability. On the other hand, for an unstable system in the sense of Lyapunov, it is often the case that its trajectory oscillates sufficiently near by the desired state, which is absolutely acceptable in the practical engineering. As such, we are more interested in the transient behavior over a finite or infinite horizon rather than the steady-state property over an infinite horizon. For this purpose, a new notion of stability, that is, the practical stability has first been proposed in , where it has been shown that the Lyapunov stability may not assure the practical stability and vice versa. Subsequently, the theory on the practical stability has been developed in [2–4].
Up to now, the practical stability problem has been well investigated for deterministic differential equations and many desirable results have been achieved. For example, in , a concept of finite time stability, as one special case of practical stability proposed in , has been introduced to examine the behavior of systems contained within prespecified bounds during a fixed time interval. The practical stability with respect to a set rather than the particular state has been extended. In [7, 8], some results on the practical stability have been obtained for discontinuous systems and some differences between the practical stability and the Lyapunov stability have been given. In , by using the method of Lyapunov function and Dini derivative, some sufficient conditions have been derived for various types of practical stability. In , a new definition of generalized practical stability is introduced. By making use of Lyapunov-like functions, some sufficient conditions are established.
With respect to the stochastic differential systems, we just mention the following representative works. The practical stability in the th mean has been proposed for discontinuous systems in . In , by using the Lyapunov-like functions and the comparison principle, a unified approach is developed to deal with the problems of both the th mean Lyapunov stability and the th mean practical stability for the delayed stochastic systems. In , some criteria of practical stability in probability have been established in terms of deterministic auxiliary systems with initial conditions. The results obtained in [11, 13] have been further extended to a class of large-scale Itô-type stochastic systems in , where the initial conditions of the resulting auxiliary systems are random. In all papers mentioned above, the practical stability of the stochastic systems is determined through testing one corresponding auxiliary deterministic system, whereas, in , the sufficient conditions for practical stability in the mean square for a class of stochastic dynamical systems are established by using an integrable function and Lyapunov-like functions instead of the comparison principle.
In this paper, we are concerned with the problem of the practical stability in the th mean for a general class of Itô-type stochastic differential equations over both finite and infinite time intervals. By using Lyapunov-like functions and a nonnegative, nondecreasing, and differentiable function , some criteria are established to ensure the th mean practical stability for the considered stochastic system. This technique avoids the difficulty in finding an auxiliary deterministic stable system. Moreover, the practical stability is compared with traditional Lyapunov stability and some differences between them are presented. Finally, an illustrative example is provided to demonstrate the results derived in this paper.
Notation. denotes the -dimensional Euclidean space. denotes the interval , where (in this paper, can be finite or infinite). represents the family of nonnegative, nondecreasing, and differentiable functions on . represents the family of all real-valued functions defined on which are continuously twice differentiable in and once differentiable in . Let be a complete probability space. For a random variable , means the th mean of . The followings are the other notions in this paper: where are given.
2. Preliminaries and Definitions
Consider the stochastic system described by the following -dimensional stochastic differential equation: where is the stochastic increment in the sense of Itô and is an -dimensional Brownian motion. and are and matrix functions, respectively. And is the initial value. Then, we let be any solution process of (2.1) with the initial value . Furthermore, we assume that (2.1) satisfies the theorem of the existence and uniqueness of solutions  as follows.(i)(Lipschitz condition) for all and , (ii)(Linear growth condition) for all , and ,
where and are two positive constants.
Note that and satisfy the conditions By using Itô formula, The derivative of the Lyapunov-like function with respect to along the solution of (2.1) is given by where Now, we give the definitions on the practical stability in the th mean for (2.1).
Definition 2.1. System (2.1) is said to be practically stable in the th mean (PSM) with respect to , ; if there exist , then one has ; one implies that
Remark 2.2. In Definition 2.1, for , if the is finite time, then the system (2.1) is called finite time practically stable, which is one special case of practical stability.
Noticing the notations of and above, we can see that is a subset of the initial-state set when the initial time is , and is a subset of the state space at time . Therefore, it is easy to see that ; one implies that , , and . Thus, we give the following definition which is equal to Definition 2.1.
Definition 2.3. System (2.1) is said to be PSM with respect to , , if, for given with and , one has then it is implied that
3. Practical Stability Criteria
In this section, the practical stability in the th mean will be investigated in detail, and some stability criteria will be derived for (2.1) by using a Lyapunov-like function and a nonnegative, nondecreasing, and differentiable function .
Theorem 3.1. If the following conditions are met:(1) and for all ,(2)there exists a function , which is satisfying the following conditions:(a)(b)then (2.1) is PSM with respect to .
Proof. For all , let be a solution of (2.1) with the initial value . For contradiction, we assume that there exists a first time such that for and .
By the notations of , , and (2)-(b), we have Noticing the and (2.5), (2.6), it can be obtained that By the assumption (2)-(a) and taking the expected value on the both sides of (3.4), we have because then we have This is a contradiction, so the proof is complete.
Remark 3.2. In Theorem 3.1, if the is a finite time interval, then (2.1) is practically stable on finite time. Furthermore, it should be pointed out that the condition is necessary to guarantee the th moment stability for (2.1) in the sense of Lyapunov. However, it would be too strict for the th mean practical stability of (2.1). In the following theorem, this condition is replaced by where .
Theorem 3.3. If the following conditions are met:(1) and for all ,(2)there exists a function , which satisfies the following conditions:(a)(b)(c)then (2.1) is PSM with respect to .
Proof. Let be a solution of (2.1) with the initial value . For contradiction, we assume that the result is not true, which means that there exists a first time such that for and .
Noticing the notation of , we have By using (2.5), (2.6), it follows that Taking the expectation on the both sides of (3.13), considering and the assumption (2)-(a), we obtain Then, it follows from (3.12) and (3.15) that which contradicts with the condition (2)-(c) of Theorem 3.3, and, hence, the proof is complete.
Remark 3.4. In Theorem 3.3, we mainly use the function and the Lyapunov-like functions but not the comparison principle in [11–14] to achieve the result, which avoids the difficulty in finding an auxiliary deterministic stable system. Here, we assume that holds on . Next, the condition will be replaced by a weaker one, that is, .
Theorem 3.5. If the following conditions are met:(1) and for all ,(2)there exists a function , which satisfies the following conditions:(a)(b)(c)then (2.1) is PSM with respect to .
Proof. Let be a solution of (2.1) with the initial value . For contradiction, we assume that there exists a first time such that for and . Due to the continuity of and the connectivity of , there exists such a time and holds for the last time before the time . So, we get that when .
Noticing the (2.5), (2.6), it can be obtained that, when , By virtue of we take the conditional expectation of (3.21) conditioning on the initial value ; it can be seen from condition (2)-(a) that Taking the expectation on the both sides of (3.23) and using the assumption (2)-(b), we obtain Then, Noticing the assumption (2)-(c), this is a contradiction, Then, the proof is complete.
In the theorems above, some sufficient conditions that guarantee the th mean practical stability are derived for (2.1). It is worth mentioning that the establishment of the practical stability criteria here avoids introducing other auxiliary stable systems, which make it convenient to determine whether an Itô-type stochastic differential system is the th mean practically stable. In Section 4, an example will be employed to demonstrate the obtained results.
Example 4.1. Consider the one-dimensional stochastic differential equation as follow: where is a one-dimensional Brownian motion.
Now, we investigate the practical stability in the 1st mean for (4.1) with respect to and . One assumes that the initial value satisfies the conditions and for . Then, we approximate the value of and .
We define a Lyapunov-like function as Due to the fact that is a positive-definite function, one can easily get when .
So, when , it is obvious that By using the Itô formula, we calculate the derivative of the Lyapunov-like function along the solution of (4.1), and noticing the (2.6), we have Taking the expectation on both sides of (4.4), one obtains so we define From (4.4)–(4.6), it can be easily verified that the condition (2)-(a) of Theorem 3.3 is satisfied. Then, by the condition (2)-(b) of Theorem 3.3, we have and hence, it can be obtained from (4.6) that On the other hand, from the condition (2)-(c) of Theorem 3.3, we have So, we have
Now, we have the fact that and . According to Theorem 3.3, (4.1) is practically stable in the 1st mean with respect to and on . In the simulation, we take 50 initial values satisfying . For every initial value, the 1st mean orbit and the maximum of for are computed numerically. The simulation result is depicted in Figure 1.
This paper mainly establishes the sufficient conditions of practical stability in the th mean for the Itô-type stochastic differential equation over finite or infinite time interval. By using Lyapunov-like functions and a nonnegative, nondecreasing, and differentiable function instead of the comparison principle, the difficulty in finding an auxiliary deterministic stable system is avoided. Moreover, this paper indicates that the practical stability can be examined over finite or infinite time interval and it can be used to depict the transient behavior of the trajectory.
For further studies, we can extend practical stability in the th mean to uniformly practical stability and strict practical stability in the th mean by the same methods in this paper. And, we can also consider other techniques to establish the sufficient conditions for the practical stability in probability and the almost sure practical stability instead of the comparison principle. Other future research topics include the investigation on the filtering and control problems for uncertain nonlinear stochastic systems; see, for example, [17–26].
This paper is supported by the National Natural Science Foundation of China (No. 60974030) and the Science and Technology Project of Education Department in Fujian Province, China (No. JA11211).
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