This paper studies the asymptotic stability problem for a class of uncertain impulsive switched systems with discrete and distributed delays. Based on Lyapunov functional theory, delay-dependent sufficient LMI conditions are established for the asymptotic stability of the considered systems. Moreover, an appropriate feedback controller is constructed for stabilizing the corresponding closed-loop system. The results are illustrated to be efficient through an example.

A switched system is a type of hybrid system which is a combination of discrete and continuous dynamical systems. These systems arise as models for phenomena which cannot be described by exclusively continuous or exclusively discrete processes. Recently, on the basis of Lyapunov functions and other analysis tools, the stability and stabilization for switched systems have been investigated and many variable results have been obtained; see [

On the other hand, time delays and uncertainties happen frequently in various engineering, biological, and economical systems, and they many result in instability. Many stability criteria have been derived for continuous dynamical systems with time delays or uncertainties; see [

In this paper, the problem of delay-dependent stability analysis and synthesis for impulsive switched system with discrete and distributed delays is studied. The uncertainties under consideration are norm bounded. Based on Lyapunov functional approach and linear matrix inequality technology, some new delay-dependent stability and stabilization conditions are derived. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.

Consider the following impulsive switched system with mixed delays:

Let

For any constant matrix

Suppose that there exist symmetric positive definite matrices

When

The stability condition

Define

From conditions (

Next, for the impulsive switching time point

This completes the proof.

In this section, we focus on designing a memoryless state feedback controller in the form of

Suppose that there exist symmetric positive definite matrices

Then the trivial solution of the impulsive switched system (

Substitute

Replacing

Similar to the proof of Theorem

Define

This completes the proof.

As an illustration, we consider a system in the form of (

Letting

In this paper, the asymptotic stability problem for a class of uncertain impulsive switched systems with discrete and distributed delays is discussed. Firstly, delay-dependent stability criteria have been obtained by choosing proper Lyapunov function. Furthermore, some appropriate feedback controllers have been constructed to ensure the asymptotic stability of the closed-loop systems. A numerical example is solved by MATLAB Toolbox to illustrate that the results obtained are effective.