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

Authors: Yang Peng; Jiang Wu; Limin Zou; Yuming Feng; Zhengwen Tu
Journal: Complexity
 (2019)

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## 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.


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## Body

## 1. Introduction

In this paper, the Euclidean norm ofx∈Rn 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 [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:(3)x˙=Ax+ϕx,y=Cx,t≠τk,Δx=By,t=τk,k=1,2,…,where x∈Rn is the state variable and y∈Rm is output, and A∈Rn×n, B∈Rn×m, C∈Rm×n are constant matrices. The nonlinear part ϕ:Rn→Rn 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+=limt→τk+xt and xτk=xτk-=limt→τk-xt. Without loss of generality, we assume that(4)t0<τ1<τ2<⋯,limk→∞⁡τk=∞. For simplicity, we can rewrite this last system as(5)x˙=Ax+ϕx,t≠τk,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 [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.
LetP be positive definite and suppose that λ2,λ1 are the largest and the smallest eigenvalues of P, respectively. If x,y∈Rn satisfy(6)xTy2≤σxTxyTyfor a certain σ∈0,1, then(7)xTPy2≤λ2-gσλ1λ2+gσλ12xTPxyTPy,where(8)gσ=1-σ1+σ.Proof.
First we assume thatx=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=λmaxXTPX≤λ2λmaxXTX=1+xTyλ2and(13)μ1=λminXTPX≥λ1λminXTX=1-xTyλ1.It follows from (12) and (13) that(14)1-xTyλ11+xTyλ2≤μ1μ2. It can easily be seen that the function(15)gt=1-t1+t,0≤t≤1, is decreasing and so(16)gμ1μ2≤g1-qλ11+qλ2, which is equivalent to(17)μ2-μ1μ2+μ1≤λ2-gqλ1λ2+gqλ1,where(18)q=xTy∈0,1. 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=P1/2xTP1/2y2≤xTPxyTPy.On the other hand, the arithmetic-geometric mean inequality for scalars implies that(23)xTPx+yTPy2≥4xTPxyTPy.It follows from (21), (22), and (23) that(24)xTPy2≤μ2-μ1μ2+μ12xTPxyTPy.By using inequalities (17) and (24), we obtain(25)xTPy2≤λ2-gqλ1λ2+gqλ12xTPxyTPy.Now we consider the general situation. For arbitrary x,y∈Rn, we have(26)xx=yy=1 By inequality (25), we have(27)xTPy2≤λ2-gqλ1λ2+gqλ12xTPxyTPywhere(28)gq=1-q1+q,q=xTyxy. Inequality (6) implies that q≤σ and so(29)gσ≤gq. Small calculations show that the function(30)ft=a-ta+t,a>1,0≤t≤1, is decreasing and so(31)λ2/λ1-gqλ2/λ1+gq≤λ2/λ1-gσλ2/λ1+gσ.It follows from (27) and (31) that(32)xTPy2≤λ2-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,y∈Rn if we choose σ=1. And so Lemma 1 is a generalization of the Cauchy-Schwarz inequality:(33)xTPy2=P1/2xTP1/2y2≤xTPxyTPy.Remark 3.
Ifx,y∈Rn is orthogonal, then we can choose σ=0 and Lemma 1 is the well-known Wielandt inequality:(34)xTPy2≤λ2-λ1λ2+λ12xTPxyTPy.


## 3. An Application of Lemma1

Let us recall the definition of the angle between two vectorsy,z∈Rn:(35)θ=arccos⁡yTzyz,θ∈0,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. [28] presented the following chaotic system:(36)x˙=25α+10y-x,y˙=28-35αx+29α-1y-xz,z˙=-α+83z+xy, where α∈0,1. 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 [24–27] do not take into account this factor. This is the motivation for the present paper.In this section, as an application of Lemma1, 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 thatH is a real symmetrical matrix and let λ2,λ1 be the largest and smallest eigenvalues of H, respectively. Then(37)λ1yTy≤yTHy≤λ2yTy,for any y∈Rn.Theorem 5.
LetP 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)xTϕx2≤σxTxϕxTϕxfor a certain σ∈0,1 and(39)λ3+2Lλ2-gσλ1λ2+gσλ1λ2λ1≥0,(40)λ3+2Lλ2-gσλ1λ2+gσλ1λ2λ1τk+1-τk≤-ln⁡γλ4,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 t≠τk, 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×P1/2x≤λ3xTP1/2P1/2x=λ3Vx.By Lemmas 1 and 4 and ϕ(x)≤Lx, we have(45)2xTPϕx≤2λ2-gσλ1λ2+gσλ1xTPxϕxTPϕx≤2λ2-gσλ1λ2+gσλ1λ2xTPxϕxTϕx≤2Lλ2-gσλ1λ2+gσλ1λ2xTPxxTx=2Lλ2-gσλ1λ2+gσλ1×λ2xTPxxTP1/2P-1P1/2x≤2Lλ2-gσλ1λ2+gσλ1×λ2λ1xTPxxTP1/2P1/2x=2Lλ2-gσλ1λ2+gσλ1λ2λ1Vx.It follows from (43), (44), and (45) that(46)D+Vxt≤λ3+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=τk≤λ4xTP1/2P1/2xt=τk=λ4Vxt=τk. 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 chooseσ=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≤-ln⁡γλ4,γ>1, 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.
IfP=I, condition of (40) will be replaced by(49)λ3+2σLτk+1-τk≤-ln⁡γλ4,γ>1.Remark 8.
Let us discuss Lü’s [28] 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≤-ln⁡γλ4,γ>1.Furthermore, if we choose P=I, then this last condition can be simplified as(51)λ3τk+1-τk≤-ln⁡γλ4,γ>1, which contains the condition of Theorem 3.2.1 in [4] (see also [10]).Remark 9.
Lemma1 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(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+x2y2≤max⁡x,y,zx2+y2+z2=max⁡x,y,zx By Figure 1, we know that maxx,y,z≤45, so we can choose L=45. By the arithmetic-geometric mean inequality for scalars we know that(56)xTϕx2=x2y2z2=x2y2z23×x2y2y2z2x2z23≤19x2+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≤-ln⁡0.1715γ62.9638. We choose γ=1.1, and then(60)τk+1-τk≤0.0265. Putting(61)τk+1-τk=0.0260, the simulation results are shown in Figure 2.Figure 1
The state trajectory of the uncontrolled chaotic system with the initial conditionx(0)=(3,4,5)T.Figure 2
The state trajectory of the controlled chaotic system with the initial conditionx(0)=(3,4,5)T.On the other hand, by Yang’s [27] result we know that if(62)τk+1-τk≤-ln⁡0.1715γ122.9638, then the origin of Qi’s system [29] is asymptotically stable. Figure 3 shows the stable region for different γ’s.Figure 3
The estimation of boundaries of stable region with differentγ’s.From Figure3 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 Lemma1 may have other applications in related fields of control theory.



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