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The

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 [

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 [

With respect to the stochastic differential systems, we just mention the following representative works. The practical stability in the

In this paper, we are concerned with the problem of the practical stability in the

Consider the stochastic system described by the following

(Lipschitz condition) for all

(Linear growth condition) for all

where

Note that

System (

In Definition

Noticing the notations of

System (

If the conditions of Definition

In Section

In this section, the practical stability in the

If the following conditions are met:

there exists a function

For all

By the notations of

In Theorem

If the following conditions are met:

there exists a function

Let

Noticing the notation of

In Theorem

If the following conditions are met:

there exists a function

Let

Noticing the (

In the theorems above, some sufficient conditions that guarantee the

In this section, one numerical example is given to demonstrate the result in Theorem

Consider the one-dimensional stochastic differential equation as follow:

Let

Now, we investigate the practical stability in the 1st mean for (

We define a Lyapunov-like function as

So, when

Now, we have the fact that

Illustration of the practical stability in the 1st mean.

This paper mainly establishes the sufficient conditions of practical stability in the

For further studies, we can extend practical stability in the

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).