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We investigate the state-feedback stabilization problem for a class of stochastic feedforward nonlinear time-delay systems. By using the homogeneous domination approach and choosing an appropriate Lyapunov-Krasovskii functional, the delay-independent state-feedback controller is explicitly constructed such that the closed-loop system is globally asymptotically stable in probability. A simulation example is provided to demonstrate the effectiveness of the proposed design method.

In recent years, the study on stochastic lower-triangular nonlinear systems has received considerable attention from both theoretical and practical point of views see, for instance, [

Feedforward (also called upper-triangular) system is another important class of nonlinear systems. Firstly, from a theoretical point of view, since they are not feedback linearizable and maybe not stabilized by applying the conventional backstepping method, the stabilization problem of these systems is more difficult than that of lower-triangular systems. Secondly, many physical devices, such as the cart-pendulum system in [

However, all these above-mentioned results are limited to deterministic systems. There are fewer results on stochastic feedforward nonlinear systems until now, due to the special characteristics of this system. To the best of the authors’ knowledge, [

The purpose of this paper is to further weaken the assumptions on the drift and diffusion terms of system (

The paper is organized as follows. Section

The following notations, definitions, and lemmas are to be used throughout the paper.

Consider the following stochastic time-delay system:

For any given

The equilibrium

For fixed coordinates

The dilation

A function

A vector field

A homogeneous

For system (

Given a dilation weight

Suppose that

Let

The objective of this paper is to design a state-feedback controller for system (

For system (

For

The time-varying delay

When

Firstly, we introduce the following coordinate transformation:

We construct a state-feedback controller for system (

Introducing

The first virtual controller

In this step, we can get the following lemma.

Suppose that at step

See the Appendix.

At step

We state the main result in this paper.

If Assumptions

the closed-loop system has a unique solution on

the equilibrium at the origin of the closed-loop system is GAS in probability.

We prove Theorem

By Lemma 4.3 in [

From

By Assumption

Since

Equation (

By Steps 1–3 and Lemma

In this paper, the homogeneous domination idea is generalized to stochastic feedforward nonlinear time-delay systems (

Due to the special upper-triangular structure and the appearance of time-varying delay, there is no efficient method to solve the stabilization problem of system (

One of the main obstacles in the stability analysis is how to deal with the effect of time-varying delay. In this paper, by constructing an appropriate Lyapunov-Krasovskii functional (

It is worth pointing out that the rigorous proof of Theorem

Consider the following stochastic nonlinear system:

By Lemma

Choosing

By (

In simulation, we choose the initial values

(a) The response of the closed-loop system (

By using the homogeneous domination approach, this paper further studied the state-feedback stabilization problem for a class of stochastic feedforward nonlinear time-delay systems (

There still exist some problems to be investigated. One is to consider the output-feedback control of switched stochastic system (

According to (

Using (

Choosing

The authors declare that there is no conflict of interests.

The authors would like to express sincere gratitude to editor and reviewers for their helpful suggestions in improving the quality of this paper. This work was partially supported by the National Natural Science Foundation of China (nos. 61304002, 61304003, 61203123, and 61304054), the Fundamental Research Funds for the Central Universities of China (no. 11CX04044A), the Shandong Provincial Natural Science Foundation of China (no. ZR2012FQ019), and the Polish-Norwegian Research Programme operated by the National Center for Research and 24 Development under the Norwegian Financial Mechanism 2009–2014 in the frame of Project Contract (no. Pol-Nor/200957/47/2013).