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We study the stabilization problem of discrete-time planar switched linear systems with impulse. When all subsystems are controllable, based on an explicit estimation on the state transition matrix, we establish a sufficient condition such that the switched impulsive system is stabilizable under arbitrary switching signal with given switching frequency. When there exists at least one uncontrollable subsystem, a sufficient condition is also given to guarantee the stabilization of the switched impulsive system under appropriate switching signal.

Recent years have witnessed a rapid progress for switched systems, for example, see monographs [

During the last three decades, there is an increasing interest on the stability analysis for switched systems. For stability issues; one important problem is to find conditions that guarantee asymptotic stability of the switched system for arbitrary switching signal. Such a problem is usually studied by using a common Lyapunov functional approach, especially by using a common quadratic Lyapunov functional approach [

For systems that switch among a finite set of controllable linear systems, the stabilization problem of continuous-time switched systems with arbitrary switching frequency was studied in [

In this paper, motivated by the work in [

This paper is organized as follows. In Section

Consider the following planar discrete-time linear system:

Under the following linear feedback law:

When the system (

Let

First, we consider the case of single input, that is,

In particular, for any

Set

Let

We have that

It is not difficult to see that

that is,

It implies that

Consequently,

Let

Second, noting that

So,

Since

For the multiple-input case, one sees that for any

The proof of Lemma

When

Define the following Lyapunov function:

It is easy to see that

For system (

Let the Lyapunov function be defined by (

By (

By (

Thus,

By induction, we have

This completes the proof of Lemma

Now, we study the stabilization of the following discrete-time switched linear system:

Throughout this paper, we assume that

Under the linear feedback law

Denote the frequency of the switching signal by

Assume that (H1) and (H2) hold and

For any

Since

By the analysis and (H1), we obtain

Noting that

Next, we consider the case when there exist both controllable subsystems and uncontrollable subsystems for system (

Denote the switching frequency of those controllable subsystems by

Similar to the above analysis, for any given feedback matrices

Set

Assume that (H2), (

For any

For any

If

If

Based on the same analysis, there exists a feedback matrix

We now choose

Noting that

In order to illustrate the theoretical result, we consider two examples.

Consider the switched systems (

It is not difficult to verify that

By Lemma

Let

We can get

Consider the switched systems (

It is not difficult to verify that

By Lemma

Let

By Theorem

By Theorem

In this paper, the stabilization problem of discrete-time planar switched linear systems with impulse is investigated. When all the subsystems are controllable, we first establish an estimation on the transition matrix for each controllable subsystem, which is a discrete analogue of the corresponding result in [

The authors thank the reviewers for their helpful and valuable comments on this paper. This work was supported by the Natural Science Foundation of Shandong Province under Grant nos. JQ201119 and ZR2010AL002 and the National Natural Science Foundation of China under Grant no. 61174217.