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

be a spectral decomposition with Q is orthogonal. Then the functional calculus forH is defined as f (H) = QTdiag (f (λ1) , ⋅ ⋅ ⋅ , f (λn)) Q, (2) where f(t) 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: ?̇? = Ax + φ (x) , y = Cx, t ̸ = τk, Δx = By, t = τk, k = 1, 2, . . . , (3)


Introduction
In this paper, the Euclidean norm of  ∈   is defined as ‖‖ = √   .We use  max () and  min () 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 Ω and  is a real symmetrical matrix with eigenvalues in Ω [1].
In this paper, we consider a class of nonlinear impulsive control systems as follows: ẋ =  +  () ,  = ,  ̸ =   , Δ = ,  =   ,  = 1, 2, . . ., where  ∈   is the state variable and  ∈   is output, and  ∈  × ,  ∈  × ,  ∈  × are constant matrices.The nonlinear part  :   →   is a continuous function which satisfies (, 0) = 0 and ‖()‖ ≤ ‖‖.If  =   , then there will be a jump in the system and Δ( ().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][25][26][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.

A Generalization of the Cauchy-Schwarz Inequality
In this section, we will give a generalized Cauchy-Schwarz inequality. Lemma for a certain  ∈ [0, 1], then where Proof.First we assume that ‖‖ = ‖‖ = 1.
and then, we have and Small calculations show that 1 + |  | and 1 − |  | are the eigenvalues of   .Suppose that  2 ,  1 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 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 .By the Cauchy-Schwarz inequality, we know that condition (6) holds for any ,  ∈   if we choose  = 1.
And so Lemma 1 is a generalization of the Cauchy-Schwarz inequality: Remark .If ,  ∈   is orthogonal, then we can choose  = 0 and Lemma 1 is the well-known Wielandt inequality:

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  ∈ [0,1].Note that  = [,,]  , () = [0, −, ]  and so   () = 0.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][25][26][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  2 ,  1 be the largest and smallest eigenvalues of , respectively.en for a certain  ∈ [0, 1] and where then the origin of nonlinear impulsive control system ( ) is asymptotically stable.
Remark .If we choose  = 1, 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 .If  = , condition of (40) will be replaced by Remark .Let us discuss Lü's [28] chaotic system again.Noting that   () = 0 and taking into consideration that we can choose  = 0, 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 .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][25][26], some results presented in [24][25][26] can be generalized.

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 ] . (53 This system is chaotic when So, we can choose  = 1/9.In this example, we choose the matrices , ,  as follows: then the origin of Qi's system [29] is asymptotically stable.Figure 3 shows the stable region for 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.

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.

Figure 3 :
Figure 3: The estimation of boundaries of stable region with different 's.
1. Let  be positive definite and suppose that  2 ,  1 are the largest and the smallest eigenvalues of , respectively.