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The problem of robust exponential stabilization for dynamical nonlinear systems with uncertainties and time-varying delay is considered in the paper. By constructing the proposed Lyapunov-Krasovskii functional approach, continuous state feedback controllers are put forward, and the criteria which guarantee the exponential stabilization of the nonlinear systems with uncertainties and time-varying delay are established in terms of solutions to the standard Riccati differential equations. Furthermore, based on the Lyapunov method and the linear matrix inequality approach, the sufficient conditions of exponential stability for a class of uncertain systems with time-varying delays and nonlinear perturbations are derived. Finally, two numerical examples are given to demonstrate the validity of the results.

The stability problem and stabilization problem of time-delay systems are important problems not yet completely solved and continuously investigated by many people. For control systems with delayed state, existing stability criteria can be classified into two categories, that is, delay-independent ones and delay-dependent ones, and delay-independent ones are usually more conservative than the delay-dependent ones. On the other hand, it is unavoidable to include uncertain parameters and perturbations in practical control systems due to modeling errors, measurement errors, approximations, and so on. Therefore, the robust stabilization problem of uncertain dynamical systems with time-varying delays has attracted considerable attention of many researchers in recent years, and most of these papers have always adopted linear matrix equalities to guarantee the exponential stability of dynamical systems by employing Leibniz-Newton formula or different transformations. For instance, by employing a descriptor model transformation and a decomposition technique of the delay term matrix, the robust stability of uncertain linear systems with a single time-varying delay and nonlinear perturbations is investigated in [

Being different from them, this paper chooses the Riccati differential equation to solve the stabilization of uncertain time-delay systems. The sufficient conditions which guarantee that the uncertain systems with time-varying delay and nonlinear perturbations are exponentially stabilizable are presented by employing Lyapunov-Krasovskii functional, and the controllers are constructed.

The rest of this paper is organized as follows. Section

For convenience, we now introduce the following notations that will be employed throughout the paper. The notation

Now, let us consider a class of uncertain systems with time-varying delay and nonlinear perturbations of the form

The purpose of this paper is to design a state feedback controller

Before proposing our theorems, we introduce for (

For all

Assumption

For any real vectors

Given constant symmetric matrices

In this section, we will present our main results on the robust exponential stabilization of system (

Given positive numbers

We need the following assumption.

For any

Suppose that condition (H1) and Assumptions

Let

Therefore, we get

Hence we have

Note that from the proof of Theorem

In addition, we consider a class of uncertain systems with time-varying delays and simple nonlinear perturbations as follows:

Given positive numbers

We have the following theorem.

Suppose that condition (H1) holds. If there exist positive numbers

Moreover, the solution

From (

For system (

Noticing

Noticing

Hence we have

In this section, we will give numerical examples to demonstrate the effective of the proposed methods.

Consider system (

The systems in the examples that are dealt by [

Consider system (

Taking

We can verify that all the conditions of Theorem

By Theorem

For

The state

The problem of robust stabilization for a class of dynamical nonlinear systems with uncertainties and time-varying delays has been considered. On condition that the derivative of time-varying delays has restriction, a novel stability criterion which can guarantee the exponential stabilization of the uncertain systems with time-varying delay and nonlinear perturbations has been established by using the Riccati differential equation. The continuous state feedback controller has been proposed. Furthermore, based on the Lyapunov method, a linear matrix inequality approach to robust exponentially stabilization for a class of uncertain systems with time-varying delays and nonlinear perturbations via linear memoryless state feedback has been proposed. Finally, two illustrative examples have been given to demonstrate the utilization of the results.

This work was supported by the National Nature Science Foundation of China under Grant no. 50977047 and the Natural Science Foundation of Tianjin under Grant no. 11JCYBJC06800.