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We address the problem of the globally asymptotic stability for a class of stochastic nonlinear systems with the output feedback control. By using the backstepping design method, a novel dynamic output feedback controller is designed to ensure that the stochastic nonlinear closed-loop system is globally asymptotically stable in probability. Our way is different from the traditional mathematical induction method. Indeed, we develop a new method to study the globally asymptotic stability by introducing a series of specific inequalities. Moreover, an example and its simulations are given to illustrate the theoretical result.

As is well known, the stability problem of nonlinear systems with the state feedback or output feedback control has received much attention since it can be extensively applied in many fields such as engineering and finance. In the practical application, nonlinear systems with the feedback control can model many kinds of stochastic influences either natural or man-made. The output feedback control especially has been used more widely for the reason that a system by the output feedback is more flexible to respond to the information of control systems than the state feedback.

In recent years, there has been a larger number of research works on the global stability for nonlinear systems with the output feedback control [

It is worth pointing out that the problem of global output feedback stability for a class of deterministic lower-triangular systems has been solved in [

In the spirit of stochastic stability theorem of Khasminskii [

The rest of this paper is arranged as follows. In Section

In this section, we mainly give the definition of the globally asymptotic stability in probability and introduce several preliminary lemmas.

Consider the following stochastic nonlinear systems:

The function

The equilibrium

For any given

Consider system (

for (

the equilibrium

Let

For any given real numbers

For any constants

We first prove (

For a series of numbers

Let the vector

For vectors

In this section, we design a novel linear observer system (

Consider the following nonlinear stochastic system:

The nonlinear functions

The linear observer system is designed as

Assume that Assumption

Consider the following Lyapunov function

It follows from Assumption

Substituting (

Now, we take the Lyapunov function as follows:

Letting

Letting

In [

In this section, we will use an example to illustrate our main result.

Consider the following stochastic nonlinear system:

Now, taking

The first state response in Example

The second state response in Example

The control input in Example

Figures

In this paper, we have studied the problem of globally asymptotic stability of stochastic nonlinear systems by the output feedback with a novel method. It is worth pointing out that the design of the dynamic output feedback controller plays an important role in the proof of our main result, especially that the Young inequality is a key tool. We believe that our formulation and approach can be used to analyse the stabilization problem of stochastic nonlinear systems with input delays, in which the feedback domination design will be a more complex structure.

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

This work was jointly supported by the National Natural Science Foundation of China (61374080), the Natural Science Foundation of Zhejiang Province (LY12F03010), and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.