COMPLEXITY Complexity 1099-0526 1076-2787 Hindawi 10.1155/2019/6048909 6048909 Research Article A Generalization of the Cauchy-Schwarz Inequality and Its Application to Stability Analysis of Nonlinear Impulsive Control Systems http://orcid.org/0000-0002-1546-0400 Peng Yang 1 Wu Jiang 1 http://orcid.org/0000-0002-3385-6476 Zou Limin 2 http://orcid.org/0000-0003-0465-3925 Feng Yuming 3 Tu Zhengwen 3 Innocenti Giacomo 1 School of Statistics Southwestern University of Finance and Economics Chengdu 611130 China swufe.edu.cn 2 School of Mathematics and Statistics Chongqing Technology and Business University Chongqing 400067 China ctbu.edu.cn 3 Key Laboratory of Intelligent Information Processing and Control Chongqing Three Gorges University Chongqing 404100 China sanxiau.edu.cn 2019 732019 2019 08 10 2018 19 01 2019 12 02 2019 732019 2019 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.

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.

Fundamental Research Funds for the Central Universities JBK19072018278 Chongqing Research Program of Basic Research and Frontier Technology cstc2017jcyjAX0032
1. Introduction

In this paper, the Euclidean norm of xRn is defined as x=xTx. We use λmax(H) and λmin(H) to denote the largest and the smallest eigenvalues of a real square matrix H with real eigenvalues, respectively. Let(1)H=QTdiagλ1,,λnQ be a spectral decomposition with Q is orthogonal. Then the functional calculus for H is defined as(2)fH=QTdiagfλ1,,fλnQ, where ft is a continuous real-valued function defined on a real interval Ω and H is a real symmetrical matrix with eigenvalues in Ω .

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 . For example, impulsive control method can be used in the synchronization and stabilization of chaos systems  and neural network systems .

In this paper, we consider a class of nonlinear impulsive control systems as follows:(3)x˙=Ax+ϕx,y=Cx,tτk,Δx=By,t=τk,k=1,2,,where xRn is the state variable and yRm is output, and ARn×n, BRn×m, CRm×n are constant matrices. The nonlinear part ϕ:RnRn is a continuous function which satisfies ϕ(t,0)=0 and ϕ(x)Lx. If t=τk, then there will be a jump in the system and Δxτk=xτk+-xτk-=xτk+-xτk, with xτk+=limtτk+xt and xτk=xτk-=limtτk-xt. Without loss of generality, we assume that(4)t0<τ1<τ2<,limkτk=. For simplicity, we can rewrite this last system as(5)x˙=Ax+ϕx,tτk,Δx=BCx,t=τk,k=1,2,.xt0=x0.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 . Inequalities play an important role in their research, for instance, by using the Cauchy-Schwarz inequality  and comparison principle , 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  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 P be positive definite and suppose that λ2,λ1 are the largest and the smallest eigenvalues of P, respectively. If x,yRn satisfy(6)xTy2σxTxyTyfor a certain σ0,1, then(7)xTPy2λ2-gσλ1λ2+gσλ12xTPxyTPy,where(8)gσ=1-σ1+σ.

Proof.

First we assume that x=y=1. Let(9)X=x,y, and then, we have(10)XTX=1xTyyTx1 and(11)XTPX=xTPxxTPyyTPxyTPy.Small calculations show that 1+xTy and 1-xTy are the eigenvalues of XTX. Suppose that μ2,μ1 are the largest and the smallest eigenvalues of XTPX, respectively. Then we have(12)μ2=λmaxXTPXλ2λmaxXTX=1+xTyλ2and(13)μ1=λminXTPXλ1λminXTX=1-xTyλ1.It follows from (12) and (13) that(14)1-xTyλ11+xTyλ2μ1μ2. It can easily be seen that the function(15)gt=1-t1+t,0t1, is decreasing and so(16)gμ1μ2g1-qλ11+qλ2, which is equivalent to(17)μ2-μ1μ2+μ1λ2-gqλ1λ2+gqλ1,where(18)q=xTy0,1. Note that(19)μ2+μ12=xTPx+yTPy2 and(20)μ2-μ12=μ2+μ12-4μ1μ2=xTPx+yTPy2-4xTPxyTPy+4xTPy2. It follows that(21)μ2-μ1μ2+μ12=1-4xTPxyTPy-xTPy2xTPx+yTPy2.Meanwhile, by the Cauchy-Schwarz inequality, we have(22)xTPy2=P1/2xTP1/2y2xTPxyTPy.On the other hand, the arithmetic-geometric mean inequality for scalars implies that(23)xTPx+yTPy24xTPxyTPy.It follows from (21), (22), and (23) that(24)xTPy2μ2-μ1μ2+μ12xTPxyTPy.By using inequalities (17) and (24), we obtain(25)xTPy2λ2-gqλ1λ2+gqλ12xTPxyTPy.Now we consider the general situation. For arbitrary x,yRn, we have(26)xx=yy=1 By inequality (25), we have(27)xTPy2λ2-gqλ1λ2+gqλ12xTPxyTPywhere(28)gq=1-q1+q,q=xTyxy. Inequality (6) implies that qσ and so(29)gσgq. Small calculations show that the function(30)ft=a-ta+t,a>1,0t1, is decreasing and so(31)λ2/λ1-gqλ2/λ1+gqλ2/λ1-gσλ2/λ1+gσ.It follows from (27) and (31) that(32)xTPy2λ2-gσλ1λ2+gσλ12xTPxyTPy. This completes the proof of our result.

Remark 2.

By the Cauchy-Schwarz inequality, we know that condition (6) holds for any x,yRn if we choose σ=1. And so Lemma 1 is a generalization of the Cauchy-Schwarz inequality:(33)xTPy2=P1/2xTP1/2y2xTPxyTPy.

Remark 3.

If x,yRn is orthogonal, then we can choose σ=0 and Lemma 1 is the well-known Wielandt inequality:(34)xTPy2λ2-λ1λ2+λ12xTPxyTPy.

3. An Application of Lemma <xref ref-type="statement" rid="lem2.1">1</xref>

Let us recall the definition of the angle between two vectors y,zRn:(35)θ=arccosyTzyz,θ0,π. In the course of experiment, we note that for some systems the state variable x and nonlinear part ϕ(x) have special relationships. For instance, Lü et al.  presented the following chaotic system:(36)x˙=25α+10y-x,y˙=28-35αx+29α-1y-xz,z˙=-α+83z+xy, where α0,1. Note that x=x,y,zT, ϕ(x)=[0,-xz,xy]T and so xTϕ(x)=0. That is, they are orthogonal. So we want to know whether the angle between x and ϕ(x) has an effect on the stability of systems. And the results showed in  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  (see also Theorem 3.1.5 in ), if we consider the angle factor, then we will get a larger stable region for some systems.

Lemma 4 (see [<xref ref-type="bibr" rid="B8">1</xref>]).

Suppose that H is a real symmetrical matrix and let λ2,λ1 be the largest and smallest eigenvalues of H, respectively. Then(37)λ1yTyyTHyλ2yTy,for any yRn.

Theorem 5.

Let P be positive definite and suppose that λ2,λ1 are the largest and smallest eigenvalues of P, respectively. Let λ3 be the largest eigenvalue of P-1Q with Q=PA+ATP. Suppose that λ4 is the largest eigenvalue of P-1I+BCTPI+BC. If(38)xTϕx2σxTxϕxTϕxfor a certain σ0,1 and(39)λ3+2Lλ2-gσλ1λ2+gσλ1λ2λ10,(40)λ3+2Lλ2-gσλ1λ2+gσλ1λ2λ1τk+1-τk-lnγλ4,where(41)gσ=1-σ1+σ,γ>1,then the origin of nonlinear impulsive control system (5) is asymptotically stable.

Proof.

Let(42)Vxt=xTPx. For tτk, we have(43)D+Vxt=xTPA+ATPx+2xTPϕx.By Lemma 4 and noting that the matrices P-1/2PA+ATPP-1/2 and P-1PA+ATP have the same eigenvalues, we obtain(44)xTPA+ATPx=xTP1/2×P-1/2PA+ATPP-1/2×P1/2xλ3xTP1/2P1/2x=λ3Vx.By Lemmas 1 and 4 and ϕ(x)Lx, we have(45)2xTPϕx2λ2-gσλ1λ2+gσλ1xTPxϕxTPϕx2λ2-gσλ1λ2+gσλ1λ2xTPxϕxTϕx2Lλ2-gσλ1λ2+gσλ1λ2xTPxxTx=2Lλ2-gσλ1λ2+gσλ1×λ2xTPxxTP1/2P-1P1/2x2Lλ2-gσλ1λ2+gσλ1×λ2λ1xTPxxTP1/2P1/2x=2Lλ2-gσλ1λ2+gσλ1λ2λ1Vx.It follows from (43), (44), and (45) that(46)D+Vxtλ3+2Lλ2-gσλ1λ2+gσλ1λ2λ1Vx. For t=τk, by using Lemma 4 again and noting that the matrices P-1I+BCTPI+BC and P-1/2I+BCTPI+BCP-1/2 have the same eigenvalues, we obtain(47)Vx+BCxt=τk=x+BCxTPx+BCxt=τk=xTI+BCTPI+BCxt=τk=xTP1/2P-1/2×I+BCTPI+BC×P-1/2P1/2xt=τkλ4xTP1/2P1/2xt=τk=λ4Vxt=τk. To avoid repetition, we omit the following proof because it is same as that of Theorem 3 in . This completes the proof of our result.

Remark 6.

If we choose σ=1, then by the Cauchy-Schwarz inequality we know that inequality (38) holds for any x,ϕ(x) and condition of (40) becomes(48)λ3+2Lλ2λ1τk+1-τk-lnγλ4,γ>1, which is the condition of Theorem 3 in  (see also ). So, our result is a generalization of Theorem 3 in .

Remark 7.

If P=I, condition of (40) will be replaced by(49)λ3+2σLτk+1-τk-lnγλ4,γ>1.

Remark 8.

Let us discuss Lü’s  chaotic system again. Noting that xTϕ(x)=0 and taking into consideration that we can choose σ=0, then inequality (38) holds and condition of (40) becomes(50)λ3+2Lλ2-λ1λ2+λ1λ2λ1τk+1-τk-lnγλ4,γ>1.Furthermore, if we choose P=I, then this last condition can be simplified as(51)λ3τk+1-τk-lnγλ4,γ>1, which contains the condition of Theorem 3.2.1 in  (see also ).

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 , some results presented in  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.  produced a new system which is described by(52)x˙=Ax+ϕx, where(53)A=-aa0c-1000-b,x=xyz,ϕx=yz-xzxy. This system is chaotic when(54)a=35,b=83,c=25.By definition of the Euclidean norm, we have(55)ϕx=y2z2+x2z2+x2y2maxx,y,zx2+y2+z2=maxx,y,zx By Figure 1, we know that maxx,y,z45, so we can choose L=45. By the arithmetic-geometric mean inequality for scalars we know that(56)xTϕx2=x2y2z2=x2y2z23×x2y2y2z2x2z2319x2+y2+z2x2y2+y2z2+x2z2=19xTxϕxTϕx. So, we can choose σ=1/9. In this example, we choose the matrices B,P,C as follows:(57)B=-0.60-0.010.01-0.01-0.6000.010-0.60,P=C=100010001. Simple calculations show that(58)λ3=32.9638,λ4=0.1715, and so we have(59)τk+1-τk-ln0.1715γ62.9638. We choose γ=1.1, and then(60)τk+1-τk0.0265. Putting(61)τk+1-τk=0.0260, the simulation results are shown in Figure 2.

The state trajectory of the uncontrolled chaotic system with the initial condition x(0)=(3,4,5)T.

The state trajectory of the controlled chaotic system with the initial condition x(0)=(3,4,5)T.

On the other hand, by Yang’s  result we know that if(62)τk+1-τk-ln0.1715γ122.9638, then the origin of Qi’s system  is asymptotically stable. Figure 3 shows the stable region for different γ’s.

The estimation of boundaries of stable region with different γ’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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