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We study the input-to-state stability of singularly perturbed control systems with delays. By using the generalized Halanay inequality and Lyapunov functions, we derive the input-to-state stability of some classes of linear and nonlinear singularly perturbed control systems with delays.

The stability properties of control systems are an important research field. The concept of input-to-state stability (ISS) of the control systems was proposed by Sontag [

Singularly perturbed control systems are a special class of control systems which is characterized by small parameters multiplying the highest derivates. Recently, many attentions have been devoted to the study of singularly perturbed systems, in particular, to their stability properties. Saberi and Khalil [

The previous studies have mainly focused on the exponential stability of singularly perturbed systems with or without delays and the ISS of singularly perturbed control systems without delay. There are no results about the ISS of delay singulary perturbed control systems. In this paper, we study the ISS of some classes of delay singularly perturbed control systems. By using the generalized Halanay inequality and the Lyapunov functions, we obtain the sufficient conditions under which these delay singularly perturbed control systems are input-to-state stable.

We introduce the following symbols (cf. [

The matrix

A real

For any measurable locally essentially bounded function

A function

A function

Let

Then there exists a positive-definitive matrix

The following generalized Halanay inequality will play a key role in studying the ISS for the system (

Suppose

where

Let

Let

Moreover, by the conditions (i)–(iii), the estimate (

Consider the delay singularly perturbed control systems

The delay singularly perturbed control system (

In this section, we are concerned with ISS of the following linear delay singularly perturbed control systems as a special class of (

There exist positive constants

From Assumption

There exist bounded functions

(1) There exists a positive number

(2)

(3)

(4)

where

If Assumptions

Let

From

In this section, we are concerned with ISS of the following nonlinear delay singularly perturbed control systems as a special class of (

There exist positive constants

If Assumption

There exist bounded functions

(1) There exist a positive number

(2)

(3)

(4)

where

If Assumptions

Let

From

By the definitions of

Consider the following linear delay system as an application of Theorem

Consider the following nonlinear delay system as an application of Theorem

Let

In this paper, we have studied the input-to-state stability of two classes of the linear and nonlinear delay singularly perturbed control systems. The generalized Halanay inequality and the Lyapunov function play important roles in obtaining the stability results. The sufficient conditions of input-to-state stability for delay singularly perturbed control systems are given.

This work is supported by projects NSF of China (11126329, 11271311, 11201510), NSF of Hunan Province (09JJ3002), Projects Board of Education of Chongqing City (KJ121110). The authors express their sincere thanks to the referees for their useful comments and suggestions, which led to improvements of the presentation.