Complexity
Volume 2019 (2019), Article ID 6048909, 7 pages
https://doi.org/10.1155/2019/6048909

Research Article

A Generalization of the Cauchy-Schwarz Inequality and Its Application to Stability Analysis of Nonlinear Impulsive Control Systems

Yang Peng,¹ Jiang Wu,¹ Limin Zou,² Yuming Feng,³ and Zhengwen Tu³

¹School of Statistics, Southwestern University of Finance and Economics, Chengdu 611130, China
²School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
³Key Laboratory of Intelligent Information Processing and Control, Chongqing Three Gorges University, Chongqing 404100, China

Correspondence should be addressed to Yang Peng; peng_yang2011@163.com

Received 8 October 2018; Revised 19 January 2019; Accepted 12 February 2019; Published 7 March 2019

Academic Editor: Giacomo Innocenti

Copyright © 2019 Yang Peng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

In this paper, the Euclidean norm of is defined as . We use and to denote the largest and the smallest eigenvalues of a real square matrix with real eigenvalues, respectively. Let

be a spectral decomposition with is orthogonal. Then the functional calculus for is defined as

where is a continuous real-valued function defined on a real interval and is a real symmetrical matrix with eigenvalues in [1].

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 [2–4]. For example, impulsive control method can be used in the synchronization and stabilization of chaos systems [5–11] and neural network systems [12–23].

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

where is the state variable and is output, and , , are constant matrices. The nonlinear part is a continuous function which satisfies and . If , then there will be a jump in the system and , with and . Without loss of generality, we assume that

For simplicity, we can rewrite this last system as

The stability problems of nonlinear impulsive control system (5) have been investigated extensively in the literature in the past several decades. For example, a number of sufficient conditions for the stability of nonlinear impulsive control system (5) are derived in [24–27]. Inequalities play an important role in their research, for instance, by using the Cauchy-Schwarz inequality [1] and comparison principle [27], and Yang showed a sufficient condition for the stability of nonlinear impulsive control system (5). For more results on applications of the Cauchy-Schwarz inequality to impulsive control theory, the reader is referred to [4] and the references therein.

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 (5). We end up this note with a numerical example which will show the effectiveness of our result.

2. A Generalization of the Cauchy-Schwarz Inequality

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

Lemma 1. Let be positive definite and suppose that are the largest and the smallest eigenvalues of , respectively. If satisfy

for a certain , then

where

Proof. First we assume that . Let

and then, we have

and

Small calculations show that and are the eigenvalues of . Suppose that are the largest and the smallest eigenvalues of , respectively. Then we have

and

It follows from (12) and (13) that

It can easily be seen that the function

is decreasing and so

which is equivalent to

where

Note that

and

It follows that

Meanwhile, by the Cauchy-Schwarz inequality, we have

On the other hand, the arithmetic-geometric mean inequality for scalars implies that

It follows from (21), (22), and (23) that

By using inequalities (17) and (24), we obtain

Now we consider the general situation. For arbitrary , we have

By inequality (25), we have

where

Inequality (6) implies that and so

Small calculations show that the function

is decreasing and so

It follows from (27) and (31) that

This completes the proof of our result.

Remark 2. By the Cauchy-Schwarz inequality, we know that condition (6) holds for any if we choose . And so Lemma 1 is a generalization of the Cauchy-Schwarz inequality:

Remark 3. If is orthogonal, then we can choose and Lemma 1 is the well-known Wielandt inequality:

3. An Application of Lemma 1

Let us recall the definition of the angle between two vectors :

In the course of experiment, we note that for some systems the state variable and nonlinear part have special relationships. For instance, Lü et al. [28] presented the following chaotic system:

where . Note that , and so . That is, they are orthogonal. So we want to know whether the angle between and has an effect on the stability of systems. And the results showed in [24–27] do not take into account this factor. This is the motivation for the present paper.

In this section, as an application of Lemma 1, we present a new sufficient condition for the stability of nonlinear impulsive control system (5). Compared with Theorem 3 in [27] (see also Theorem 3.1.5 in [4]), if we consider the angle factor, then we will get a larger stable region for some systems.

Lemma 4 (see [1]). Suppose that is a real symmetrical matrix and let be the largest and smallest eigenvalues of , respectively. Then

for any .

Theorem 5. Let be positive definite and suppose that are the largest and smallest eigenvalues of , respectively. Let be the largest eigenvalue of with . Suppose that is the largest eigenvalue of . If

for a certain and

where

then the origin of nonlinear impulsive control system (5) is asymptotically stable.

Proof. Let

For , we have

By Lemma 4 and noting that the matrices and have the same eigenvalues, we obtain

By Lemmas 1 and 4 and , we have

It follows from (43), (44), and (45) that

For , by using Lemma 4 again and noting that the matrices and have the same eigenvalues, we obtain

To avoid repetition, we omit the following proof because it is same as that of Theorem 3 in [27]. This completes the proof of our result.

Remark 6. If we choose , then by the Cauchy-Schwarz inequality we know that inequality (38) holds for any and condition of (40) becomes

which is the condition of Theorem 3 in [27] (see also [4]). So, our result is a generalization of Theorem 3 in [27].

Remark 7. If , condition of (40) will be replaced by

Remark 8. Let us discuss Lü’s [28] chaotic system again. Noting that and taking into consideration that we can choose , then inequality (38) holds and condition of (40) becomes

Furthermore, if we choose , then this last condition can be simplified as

which contains the condition of Theorem 3.2.1 in [4] (see also [10]).

Remark 9. Lemma 1 has some other applications in impulsive control theory; for example, by using Lemma 1 and comparison lemmas on the sufficient condition for the stability of nonlinear impulsive differential systems shown in [24–26], some results presented in [24–26] can be generalized.

4. A Numerical Example

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

In 2005, Qi and Chen et al. [29] produced a new system which is described by

where

This system is chaotic when

By definition of the Euclidean norm, we have

By Figure 1, we know that , so we can choose . By the arithmetic-geometric mean inequality for scalars we know that

So, we can choose . In this example, we choose the matrices as follows:

Simple calculations show that

and so we have

We choose , and then

Putting

the simulation results are shown in Figure 2.

Figure 1: The state trajectory of the uncontrolled chaotic system with the initial condition .

Figure 2: The state trajectory of the controlled chaotic system with the initial condition .

On the other hand, by Yang’s [27] result we know that if

then the origin of Qi’s system [29] is asymptotically stable. Figure 3 shows the stable region for different ’s.

Figure 3: The estimation of boundaries of stable region with different ’s.

From Figure 3 we know that if we consider the angle factor, then we get a larger stable region for Qi’s system.

5. Conclusion

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 1 may have other applications in related fields of control theory.

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

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

Acknowledgments

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).

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