Practical Stability in the 𝑝th Mean for Itô Stochastic Differential Equations

Abstract

The 𝑝th 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 𝜂(𝑡) 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 𝑝th 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.

1. Introduction

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 [1], 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 [5], a concept of finite time stability, as one special case of practical stability proposed in [6], 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 𝑥=0 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 [9], by using the method of Lyapunov function and Dini derivative, some sufficient conditions have been derived for various types of practical stability. In [10], 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 𝑝th mean has been proposed for discontinuous systems in [11]. In [12], by using the Lyapunov-like functions and the comparison principle, a unified approach is developed to deal with the problems of both the 𝑝th mean Lyapunov stability and the 𝑝th mean practical stability for the delayed stochastic systems. In [13], 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 [14], 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 [15], 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 𝑝th 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 𝜂(𝑡), some criteria are established to ensure the 𝑝th 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. 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space. 𝑇₀ denotes the interval [𝑡₀,𝑇), where 𝑡₀,𝑇∈𝑅₊ (in this paper, 𝑇 can be finite or infinite). 𝑀[𝑡₀,𝑇) represents the family of nonnegative, nondecreasing, and differentiable functions on [𝑡₀,𝑇). 𝐶^1,2(𝑇₀×𝑅^𝑛,𝑅₊) represents the family of all real-valued functions 𝑉(𝑡,𝑥(𝑡)) defined on 𝑇₀×𝑅^𝑛 which are continuously twice differentiable in 𝑥(𝑡)∈𝑅^𝑛 and once differentiable in 𝑡∈𝑅₊. Let (Ω,ℱ,𝑃) be a complete probability space. For a random variable 𝜉, 𝐸‖𝜉‖^𝑝 means the 𝑝th mean of 𝜉. The followings are the other notions in this paper: 𝑆₀(𝑡)=𝑥(𝑡)∈𝑅^𝑛∶𝐸‖𝑥(𝑡)‖^𝑝,𝑆(𝑡)=𝑥(𝑡)<𝜆∈𝑅^𝑛∶𝐸‖𝑥(𝑡)‖^𝑝,𝑉≤𝐴_𝑆0𝑀(𝑡)=sup𝐸𝑉(𝑡,𝑥(𝑡))∶𝑥(𝑡)∈𝑆₀(𝑡,𝑉𝑆_𝑚)(𝑡)=inf{𝐸𝑉(𝑡,𝑥(𝑡))∶𝑥(𝑡)∈𝜕𝑆(𝑡)}, (1.1) where 𝜆,𝐴(𝜆<𝐴) are given.

2. Preliminaries and Definitions

Consider the stochastic system described by the following 𝑛-dimensional stochastic differential equation:d𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))d𝑡+𝑔(𝑡,𝑥(𝑡))d𝐵(𝑡)on𝑡∈𝑇₀,𝑥𝑡₀=𝑥₀, (2.1) where d𝑥(𝑡) is the stochastic increment in the sense of Itô and 𝐵(𝑡) is an 𝑚-dimensional Brownian motion. 𝑓(𝑡,𝑥(𝑡)) and 𝑔(𝑡,𝑥(𝑡)) are 𝑛×1 and 𝑛×𝑚 matrix functions, respectively. And 𝑥(𝑡₀)=𝑥₀ is the initial value. Then, we let 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝑥₀) be any solution process of (2.1) with the initial value 𝑥(𝑡₀)=𝑥₀. Furthermore, we assume that (2.1) satisfies the theorem of the existence and uniqueness of solutions [16] as follows.(i)(Lipschitz condition) for all 𝑥(𝑡),𝑦(𝑡)∈𝑅^𝑛,and 𝑡∈𝑇₀, 𝑓(𝑡,𝑥(𝑡))−𝑓𝑡,𝑦(𝑡)²𝑔(𝑡,𝑥(𝑡))−𝑔𝑡,𝑦(𝑡)²≤𝐾𝑥(𝑡)−𝑦(𝑡)². (2.2)(ii)(Linear growth condition) for all 𝑥(𝑡),𝑦(𝑡)∈𝑅^𝑛, and 𝑡∈𝑇₀, 𝑓(𝑡,𝑥(𝑡))²𝑔(𝑡,𝑥(𝑡))²≤𝐾^∗1+‖𝑥(𝑡)‖², (2.3)

where 𝐾 and 𝐾^∗ are two positive constants.

Note that 𝑆₀(𝑡) and 𝑆(𝑡) satisfy the conditions 𝑆₀(𝑡)⊂𝑆(𝑡),𝜕𝑆₀(𝑡)∩𝜕𝑆(𝑡)=∅. (2.4) By using Itô formula, The derivative of the Lyapunov-like function 𝑉(𝑡,𝑥(𝑡))∈𝐶^1,2(𝑇₀×𝑅^𝑛,𝑅₊) with respect to 𝑡 along the solution 𝑥(𝑡) of (2.1) is given by d𝑉(𝑡,𝑥(𝑡))=𝐿𝑉(𝑡,𝑥(𝑡))d𝑡+𝑉_𝑥(𝑡,𝑥(𝑡))𝑔(𝑡,𝑥(𝑡))d𝐵(𝑡), (2.5) where𝐿𝑉(𝑡,𝑥(𝑡))=𝑉_𝑡(𝑡,𝑥(𝑡))+𝑉_𝑥((𝑡𝑓((𝑡+1𝑡,𝑥))𝑡,𝑥))2trace𝑔^𝑇(𝑡,𝑥(𝑡))𝑉_𝑥𝑥(𝑡,𝑥(𝑡))𝑔(𝑡,𝑥(𝑡)). (2.6) Now, we give the definitions on the practical stability in the 𝑝th mean for (2.1).

Definition 2.1. System (2.1) is said to be practically stable in the 𝑝th mean (PSM) with respect to (𝜆,𝐴), 0<𝜆<𝐴; if there exist (𝜆,𝐴), then one has 𝐸‖𝑥₀‖^𝑝<𝜆; one implies that 𝐸𝑥(𝑡;𝑡₀,𝑥₀)^𝑝<𝐴∀𝑡∈𝑇₀. (2.7)

Remark 2.2. In Definition 2.1, for 𝑇₀=[𝑡₀,𝑇), if the 𝑇 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 𝑆₀(𝑡) and 𝑆(𝑡) above, we can see that 𝑆₀(𝑡₀) is a subset of the initial-state set when the initial time is 𝑡₀, and 𝑆(𝑡) is a subset of the state space at time 𝑡. Therefore, it is easy to see that 𝐸‖𝑥₀‖^𝑝<𝜆; one implies that 𝑥(𝑡₀)∈𝑆₀(𝑡₀), 𝐸‖𝑥(𝑡;𝑡₀,𝑥₀)‖^𝑝<𝐴, and 𝑥(𝑡;𝑡₀,𝑥₀)∈int𝑆(𝑡). 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 (𝜆,𝐴), 0<𝜆<𝐴, if, for given 𝑆₀(𝑡),𝑆(𝑡) with 𝑆₀(𝑡)⊂𝑆(𝑡) and 𝜕𝑆₀(𝑡)∩𝜕𝑆(𝑡)=∅, one has 𝑥(𝑡₀)∈𝑆₀(𝑡₀)  then it is implied that 𝑥𝑡;𝑡₀,𝑥₀∈int𝑆(𝑡)∀𝑡∈𝑇₀. (2.8)

Remark 2.4. If the conditions of Definition 2.3 are satisfied, then the system (2.1) is also said to be practically stable in the 𝑝th mean with respect to (𝑆₀(𝑡₀),𝑆(𝑡)).

In Section 3, the criteria for practical stability in the 𝑝th mean will be established for (2.1).

3. Practical Stability Criteria

In this section, the practical stability in the 𝑝th 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 𝜂(𝑡).

Theorem 3.1. If the following conditions are met:(1)𝑆₀(𝑡)⊂𝑆(𝑡) and 𝜕𝑆₀(𝑡)∩𝜕𝑆(𝑡)=∅for all 𝑡∈𝑇₀,(2)there exists a function 𝑉(𝑡,𝑥(𝑡))∈𝐶^1,2(𝑇₀×𝑅^𝑛,𝑅₊), which is satisfying the following conditions:(a)𝐸𝐿𝑉(𝑡,𝑥(𝑡))≤0𝑡∈𝑇₀,𝑥(𝑡)∈𝑆(𝑡), (3.1)(b)𝑉^𝑆0_𝑀𝑡₀<𝑉_𝑚𝑆(𝑡)∀𝑡∈𝑇₀, (3.2)then (2.1) is PSM with respect to (𝜆,𝐴).

Proof. For all 𝑥₀∈𝑆₀(𝑡₀), let 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝑥₀) be a solution of (2.1) with the initial value 𝑥₀. For contradiction, we assume that there exists a first time 𝑡₁∈𝑇₀ such that 𝐸‖𝑥(𝑡)‖^𝑝<𝐴 for 𝑡₀≤𝑡<𝑡₁ and 𝐸‖𝑥(𝑡₁)‖^𝑝=𝐴.
By the notations of 𝑉^𝑆0_𝑀(𝑡), 𝑉𝑆_𝑚(𝑡), and (2)-(b), we have 𝑉^𝑆0_𝑀𝑡₀<𝑉_𝑚𝑆𝑡₁≤𝐸𝑉𝑡₁,𝑥𝑡₁. (3.3) Noticing the 𝑉(𝑡,𝑥(𝑡)) and (2.5), (2.6), it can be obtained that 𝑉𝑡₁,𝑥𝑡₁−𝑉𝑡₀,𝑥𝑡₀=^𝑡1_𝑡0𝐿𝑉(𝑠,𝑥(𝑠))d𝑠+^𝑡1_𝑡0𝑉_𝑥(𝑠,𝑥(𝑠))𝑔(𝑠,𝑥(𝑠))d𝐵(𝑠). (3.4) By the assumption (2)-(a) and taking the expected value on the both sides of (3.4), we have 𝐸𝑉𝑡₁,𝑥𝑡₁−𝑉𝑡₀,𝑥𝑡₀=𝐸^𝑡1_𝑡0𝐿𝑉(𝑠,𝑥(𝑠))d𝑠=^𝑡1_𝑡0𝐸𝐿𝑉(𝑠,𝑥(𝑠))d𝑠≤0 (3.5) because 𝐸^𝑡1_𝑡0𝑉_𝑥(𝑠,𝑥(𝑠))𝑔(𝑠,𝑥(𝑠))d𝐵(𝑠)=0, (3.6) then we have 𝑉^𝑆0_𝑀𝑡₀<𝑉𝑆_𝑚𝑡₁≤𝐸𝑉𝑡₁,𝑥𝑡₁≤𝐸𝑉𝑡₀,𝑥𝑡₀≤𝑉^𝑆0_𝑀𝑡₀. (3.7) This is a contradiction, so the proof is complete.

Remark 3.2. In Theorem 3.1, if the 𝑇₀ is a finite time interval, then (2.1) is practically stable on finite time. Furthermore, it should be pointed out that the condition 𝐸𝐿𝑉(𝑡,𝑥(𝑡))≤0 is necessary to guarantee the 𝑝th moment stability for (2.1) in the sense of Lyapunov. However, it would be too strict for the 𝑝th mean practical stability of (2.1). In the following theorem, this condition is replaced by 𝐸𝐿𝑉(𝑡,𝑥(𝑡))≤d𝜂(𝑡)d𝑡, (3.8) where 𝜂(𝑡)∈𝑀[𝑡₀,𝑇).

Theorem 3.3. If the following conditions are met:(1)𝑆₀(𝑡)⊂𝑆(𝑡) and 𝜕𝑆₀(𝑡)∩𝜕𝑆(𝑡)=∅ for all 𝑡∈𝑇₀,(2)there exists a function 𝜂(𝑡)∈𝑀[𝑡₀,𝑇), which satisfies the following conditions:(a)𝐸𝐿𝑉(𝑡,𝑥(𝑡))≤d𝜂(𝑡)d𝑡𝑡∈𝑇₀,𝑥(𝑡)∈𝑆(𝑡), (3.9)(b)𝜂𝑡₀=𝑉^𝑆0_𝑀𝑡₀, (3.10)(c)𝜂(𝑡)<𝑉𝑆_𝑚(𝑡)∀𝑡∈𝑇₀, (3.11)then (2.1) is PSM with respect to (𝜆,𝐴).

Proof. Let 𝑥(𝑡) be a solution of (2.1) with the initial value 𝑥₀∈𝑆₀(𝑡₀). For contradiction, we assume that the result is not true, which means that there exists a first time 𝑡₁∈𝑇₀ such that 𝐸‖𝑥(t)‖^𝑝<𝐴 for 𝑡₀≤𝑡<𝑡₁ and 𝐸‖𝑥(𝑡₁)‖^𝑝=𝐴.
Noticing the notation of 𝑉𝑆_𝑚(𝑡), we have 𝑉𝑆_𝑚𝑡₁≤𝐸𝑉𝑡₁,𝑥𝑡₁. (3.12) By using (2.5), (2.6), it follows that 𝑉𝑡₁,𝑥𝑡₁−𝑉𝑡₀,𝑥𝑡₀=^𝑡1_𝑡0𝐿𝑉(𝑠,𝑥(𝑠))d𝑠+^𝑡1_𝑡0𝑉_𝑥(𝑠,𝑥(𝑠))𝑔(𝑠,𝑥(𝑠))d𝐵(𝑠). (3.13) Taking the expectation on the both sides of (3.13), considering 𝐸^𝑡1_𝑡0𝑉_𝑥(𝑠,𝑥(𝑠))𝑔(𝑠,𝑥(𝑠))d𝐵(𝑠)=0 (3.14) and the assumption (2)-(a), we obtain 𝐸𝑉𝑡₁,𝑥𝑡₁=𝐸𝑉𝑡₀,𝑥𝑡₀+^𝑡1_𝑡0𝐸𝐿𝑉(𝑠,𝑥(𝑠))d𝑠≤𝐸𝑉𝑡₀,𝑥𝑡₀+^𝑡1_𝑡0d𝜂(𝑠)=𝐸𝑉𝑡₀,𝑥𝑡₀+𝜂𝑡₁−𝜂𝑡₀=𝜂𝑡₁−𝑉^𝑆0_𝑀𝑡₀−𝐸𝑉𝑡₀,𝑥𝑡₀≤𝜂𝑡₁. (3.15) Then, it follows from (3.12) and (3.15) that 𝑉𝑆_𝑚𝑡₁≤𝐸𝑉𝑡₁,𝑥𝑡₁≤𝜂𝑡₁ (3.16) 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 𝜂(𝑡) 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 𝐸𝐿𝑉(𝑡,𝑥(𝑡))≤d𝜂(𝑡)d𝑡 (3.17) holds on 𝑥(𝑡)∈𝑆(𝑡). Next, the condition 𝑥(𝑡)∈𝑆(𝑡) will be replaced by a weaker one, that is, 𝑥(𝑡)∈𝑆(𝑡)/𝑆₀(𝑡).

Theorem 3.5. If the following conditions are met:(1)𝑆₀(𝑡)⊂𝑆(𝑡) and 𝜕𝑆₀(𝑡)∩𝜕𝑆(𝑡)=∅ for all 𝑡∈𝑇₀,(2)there exists a function 𝜂(𝑡)∈𝑀[𝑡₀,𝑇), which satisfies the following conditions:(a)𝐸𝐿𝑉(𝑡,𝑥(𝑡))≤d𝜂(𝑡)d𝑡𝑡∈𝑇₀(𝑡)∈𝑆(𝑡),𝑥𝑆₀(𝑡), (3.18)(b)𝜂(𝑡)=𝑉^𝑆0_𝑀(𝑡),𝑥(𝑡)∈𝑆₀(𝑡), (3.19)(c)𝜂(𝑡)<𝑉𝑆_𝑚(𝑡)∀𝑡∈𝑇₀, (3.20)then (2.1) is PSM with respect to (𝜆,𝐴).

Proof. Let 𝑥(𝑡) be a solution of (2.1) with the initial value 𝑥₀∈𝑆₀(𝑡₀). For contradiction, we assume that there exists a first time 𝑡₂∈𝑇₀ such that 𝐸‖𝑥(𝑡)‖^𝑝<𝐴 for 𝑡₀≤𝑡<𝑡₂ and 𝐸‖𝑥(𝑡₂)‖^𝑝=𝐴. Due to the continuity of 𝐸‖𝑥(𝑡)‖^𝑝 and the connectivity of 𝑆(𝑡),𝑆₀(𝑡), there exists such a time 𝑡₁ and 𝐸‖𝑥(𝑡₁)‖^𝑝=𝜆 holds for the last time before the time 𝑡₂. So, we get that 𝑥(𝑡)∈𝑆(𝑡)/𝑆₀(𝑡) when 𝑡∈[𝑡₁,𝑡₂].
Noticing the (2.5), (2.6), it can be obtained that, when 𝑡∈[𝑡₁,𝑡₂], 𝑉𝑡₂,𝑥𝑡₂−𝑉𝑡₁,𝑥𝑡₁=^𝑡2_𝑡1𝐿𝑉(𝑠,𝑥(𝑠))d𝑠+^𝑡2_𝑡1𝑉_𝑥(𝑠,𝑥(𝑠))𝑔(𝑠,𝑥(𝑠))d𝐵(𝑠). (3.21) By virtue of 𝐸^𝑡2_𝑡1𝑉_𝑥(𝑠,𝑥(𝑠))𝑔(𝑠,𝑥(𝑠))d𝐵(𝑠)=0, (3.22) we take the conditional expectation of (3.21) conditioning on the initial value 𝑥(𝑡₀)=𝑥₀; it can be seen from condition (2)-(a) that 𝐸𝑉𝑡₂,𝑥𝑡₂−𝑉𝑡₁,𝑥𝑡₁∣𝑥𝑡₀=𝑥₀=𝐸^𝑡2_𝑡1𝐿𝑉(𝑠,𝑥(𝑠))d𝑠∣𝑥𝑡₀=𝑥₀=^𝑡2_𝑡1𝐸𝐿𝑉(𝑠,𝑥(𝑠))d𝑠≤^𝑡2_𝑡1d𝜂(𝑠)=𝜂𝑡₂−𝜂𝑡₁. (3.23) Taking the expectation on the both sides of (3.23) and using the assumption (2)-(b), we obtain 𝐸𝑉𝑡₂,𝑥𝑡₂=𝐸𝑉𝑡₁,𝑥𝑡₁+𝜂𝑡₂−𝜂𝑡₁=𝜂𝑡₂−𝜂𝑡₁−𝐸𝑉𝑡₁,𝑥𝑡₁≤𝜂𝑡₂−𝜂𝑡₁−𝑉^𝑆0_𝑀𝑡₁=𝜂𝑡₂. (3.24) Then, 𝑉𝑆_𝑚𝑡₂≤𝐸𝑉𝑡₂,𝑥𝑡₂≤𝜂𝑡₂. (3.25) Noticing the assumption (2)-(c), this is a contradiction, Then, the proof is complete.

In the theorems above, some sufficient conditions that guarantee the 𝑝th 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 𝑝th mean practically stable. In Section 4, an example will be employed to demonstrate the obtained results.

4. Example

In this section, one numerical example is given to demonstrate the result in Theorem 3.3. The results obtained in Theorems 3.1 and 3.5 can be verified in the same way.

Example 4.1. Consider the one-dimensional stochastic differential equation as follow: d𝑥(𝑡)=𝑥(𝑡)sin(𝑡)d𝑡+d𝐵(𝑡)on𝑡∈𝑡₀,𝑥𝑡,𝑇₀=𝑥₀, (4.1) where 𝐵(𝑡) is a one-dimensional Brownian motion.

Let 𝐾=𝐾^∗=1; it is obvious that (4.1) satisfies both the Lipschitz condition and the Linear growth condition, so the existence and uniqueness of the solution 𝑥(𝑡) of (4.1) is guaranteed.

Now, we investigate the practical stability in the 1st mean for (4.1) with respect to 𝜆=1 and 𝐴=2. One assumes that the initial value 𝑥(𝑡₀) satisfies the conditions 𝐸|𝑥(𝑡₀)|<1 and 𝐸|𝑥(𝑡;𝑡₀,𝑥₀)|<2 for 𝑡∈𝑇₀. Then, we approximate the value of 𝑡₀ and 𝑇.

We define a Lyapunov-like function as𝑉(𝑡,𝑥(𝑡))=|𝑥(𝑡)|. (4.2) Due to the fact that 𝑉(𝑡,𝑥(𝑡)) is a positive-definite function, one can easily get 𝑉(𝑡,𝑥(𝑡))>0 when 𝑥(𝑡)≠0.

So, when 𝑥(𝑡)≠0, it is obvious that 𝑉_𝑡(𝑡,𝑥(𝑡))=0,𝑉_𝑥(𝑡,𝑥(𝑡))=𝑉1,𝑥>0,−1,𝑥<0,^𝑥𝑥(𝑡,𝑥(𝑡))=0. (4.3) By using the Itô formula, we calculate the derivative of the Lyapunov-like function 𝑉(𝑡,𝑥(𝑡)) along the solution 𝑥(𝑡) of (4.1), and noticing the (2.6), we have 𝐿𝑉(𝑡,𝑥(𝑡))=𝑉_𝑡(𝑡,𝑥(𝑡))+𝑉_𝑥((𝑡𝑓((𝑡+1𝑡,𝑥))𝑡,𝑥))2trace𝑔^𝑇(𝑡,𝑥(𝑡))𝑉_𝑥𝑥((𝑡𝑔((𝑡(𝑡)(𝑡)≤|𝑥(𝑡.𝑡,𝑥))𝑡,𝑥))=0±𝑥sin+0)| (4.4) Taking the expectation on both sides of (4.4), one obtains𝐸[𝐿𝑉(𝑡,𝑥(𝑡))]≤𝐸|𝑥(𝑡)|<𝐴=2, (4.5) so we define𝜂(𝑡)=2𝑡. (4.6) 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 𝜂𝑡₀=𝑉^𝑆0_𝑀𝑡₀=sup𝐸𝑥𝑡₀;𝑥𝑡₀∈𝑆₀𝑡₀=𝜆=1 (4.7) and hence, it can be obtained from (4.6) that 𝑡₀=12. (4.8) On the other hand, from the condition (2)-(c) of Theorem 3.3, we have 𝜂(𝑡)<𝑉𝑆_𝑚(𝑡)=inf{𝐸|𝑥(𝑡)|∶𝑥(𝑡)∈𝜕𝑆(𝑡)}=𝐴=2. (4.9) So, we have 𝑡<1. (4.10)

Now, we have the fact that 𝑡₀=1/2 and 𝑇=1. According to Theorem 3.3, (4.1) is practically stable in the 1st mean with respect to 𝜆=1 and 𝐴=2 on 𝑡∈[1/2,1). In the simulation, we take 50 initial values satisfying 𝐸|𝑥(1/2)|<1. For every initial value, the 1st mean orbit and the maximum of 𝐸|𝑥(𝑡)| for 𝑡∈[1/2,1) are computed numerically. The simulation result is depicted in Figure 1.

Figure 1 

Illustration of the practical stability in the 1st mean.

5. Conclusion

This paper mainly establishes the sufficient conditions of practical stability in the 𝑝th 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 𝜂(𝑡) 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 𝑝th mean to uniformly practical stability and strict practical stability in the 𝑝th 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].

Acknowledgment

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).