^{1}

^{1}

^{2}

^{3}

^{3}

^{1}

^{2}

^{3}

In this paper, we first present a generalization of the Cauchy-Schwarz inequality. As an application of our result, we obtain a new sufficient condition for the stability of a class of nonlinear impulsive control systems. We end up this note with a numerical example which shows the effectiveness of our method.

In this paper, the Euclidean norm of

During the last three decades, many people have studied impulsive control method because it is an efficient way in dealing with the stability of complex systems [

In this paper, we consider a class of nonlinear impulsive control systems as follows:

In this paper, we first present a generalization of the Cauchy-Schwarz inequality by using some results of matrix analysis and techniques of inequalities. As an application of our result, we obtain a new sufficient condition for the stability of nonlinear impulsive control system (

In this section, we will give a generalized Cauchy-Schwarz inequality.

Let

First we assume that

By the Cauchy-Schwarz inequality, we know that condition (

If

Let us recall the definition of the angle between two vectors

In this section, as an application of Lemma

Suppose that

Let

Let

If we choose

If

Let us discuss Lü’s [

Lemma

We end up this paper with a numerical example which shows the effectiveness of our method.

In 2005, Qi and Chen et al. [

The state trajectory of the uncontrolled chaotic system with the initial condition

The state trajectory of the controlled chaotic system with the initial condition

On the other hand, by Yang’s [

The estimation of boundaries of stable region with different

From Figure

In this paper, a generalization of the Cauchy-Schwarz inequality is presented. Then we use this inequality to analyze asymptotic stability for a class of nonlinear impulsive control systems. We think that Lemma

The Matlab code data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

All authors contributed equally to the writing of this paper. All authors read and approved the final version of this paper.

The authors wish to express their heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript. This work was supported by the Fundamental Research Funds for the Central Universities (No. JBK19072018278) and the Chongqing Research Program of Basic Research and Frontier Technology (No. cstc2017jcyjAX0032).