In this section, based on the stability analysis for an impulsive delayed system, some sufficient conditions are derived to ensure the global exponential synchronization for the master system and the slave system.

**Theorem 1** *Suppose that Assumption * 1 *holds and* {\mathrm{\Delta}}_{\mathrm{sup}}<\mathrm{\infty}. *Let* {\lambda}_{1} *be the largest eigenvalue of* {({I}_{n}+{B}_{p})}^{T}({I}_{n}+{B}_{p}) *and* {\lambda}_{2} *be the largest eigenvalue of* {B}_{l}^{T}{B}_{l}. *If there exist a positive definite matrix* *P* *such that the discrete system*

z(k+1)={J}_{k}(N+1)z(k),\phantom{\rule{1em}{0ex}}k\in Z

*is globally exponentially stable with decay rate* \sigma >0, *where*

{J}_{k}(N+1)=\left[\begin{array}{cccccc}0& 1& 0& \cdots & 0& 0\\ 0& 0& 1& \cdots & 0& 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0& 0& 0& \cdots & 1& 0\\ 0& 0& 0& \cdots & 0& 1\\ {\alpha}_{k-N}& {\alpha}_{k-N+1}& {\alpha}_{k-N+2}& \cdots & {\alpha}_{k-1}& {\tilde{\alpha}}_{k-1}\end{array}\right],

*where* {\tilde{\alpha}}_{k-1}=a{e}^{\alpha {\mathrm{\Delta}}_{k}}, {\alpha}_{k-N+i-1}=b{e}^{\alpha {\mathrm{\Delta}}_{k-N+i-1}}, i=1,2,\dots ,N, a=2{\lambda}_{1}\frac{{\lambda}_{\mathrm{max}}(P)}{{\lambda}_{\mathrm{min}}(P)}, b=2N{\lambda}_{2}\frac{{\lambda}_{\mathrm{max}}(P)}{{\lambda}_{\mathrm{min}}(P)}, \alpha =\frac{{\lambda}_{\mathrm{max}}({A}^{T}P+PA+{D}^{T}P+PD)}{{\lambda}_{\mathrm{max}}(P)}, {\lambda}_{1}={\lambda}_{\mathrm{max}}({({B}_{p}+I)}^{T}P({B}_{p}+I)), *and* {\lambda}_{2}={\lambda}_{\mathrm{max}}({B}_{l}^{T}{B}_{l}). *Then the error system* (2.4) *is globally exponentially stable with the convergence rate* -\frac{\sigma}{2{T}_{a}}, *and hence the slave system* (2.2) *can achieve global exponential synchronization with the master system* (2.1).

*Proof*

Consider a Lyapunov function in the form of

when t\in ({t}_{k-1},{t}_{k}]. The Dini derivative of V(t) along the trajectory of the error system (2.4) can be obtained as follows:

\begin{array}{rcl}\dot{V}(t)& =& {\dot{e}}^{T}(t)Pe(t)+{e}^{T}(t)P\dot{e}(t)\\ =& {e}^{T}(t)({A}^{T}P+PA)e(t)+{e}^{T}(t)P\mathrm{\Psi}(x(t),y(t))+{\mathrm{\Psi}}^{T}(x(t),y(t))Pe(t)\\ \le & {e}^{T}(t)({A}^{T}P+PA)e(t)+{e}^{T}(t)({D}^{T}P+PD)e(t)\\ =& {e}^{T}(t)({A}^{T}P+PA+{D}^{T}P+PD)e(t)\\ \le & \frac{{\lambda}_{\mathrm{max}}({A}^{T}P+PA+{D}^{T}P+PD)}{{\lambda}_{\mathrm{min}}(P)}V(t)\\ \triangleq & \alpha V(t),\end{array}

(3.1)

where the first inequality is obtained by Assumption 1 and \alpha =\frac{{\lambda}_{\mathrm{max}}({A}^{T}P+PA+{D}^{T}P+PD)}{{\lambda}_{\mathrm{min}}(P)}.

Therefore,

V(t)\le V\left({t}_{k-1}^{+}\right)exp[\alpha (t-{t}_{k-1})],\phantom{\rule{1em}{0ex}}t\in ({t}_{k-1},{t}_{k}],k=1,2,\dots .

(3.2)

On the other hand, it follows from (2.4) for t={t}_{k}^{+}, k=1,2,\dots , that we obtain

\begin{array}{rcl}V\left(e\left({t}_{k}^{+}\right)\right)& =& [{e}^{T}({t}_{k}){({B}_{p}+I)}^{T}+\sum _{i=k-N}^{k-1}{e}^{T}({t}_{i}){B}_{l}^{T}]P[({B}_{p}+I)e({t}_{k})+{B}_{l}\sum _{i=k-N}^{k-1}e({t}_{i})]\\ =& {e}^{T}({t}_{k}){({B}_{p}+I)}^{T}P({B}_{p}+I)e({t}_{k})+\sum _{i=k-N}^{k-1}{e}^{T}({t}_{i}){B}_{l}^{T}P{B}_{l}\sum _{i=k-N}^{k-1}e({t}_{i})\\ +2{e}^{T}({t}_{k}){({B}_{p}+I)}^{T}P{B}_{l}\sum _{i=k-N}^{k-1}e({t}_{i}).\end{array}

(3.3)

By Lemmas 1 and 2, we can obtain that

where {\lambda}_{1}={\lambda}_{\mathrm{max}}({({B}_{p}+I)}^{T}P({B}_{p}+I)), {\lambda}_{2}={\lambda}_{\mathrm{max}}({B}_{l}^{T}{B}_{l}), a=2{\lambda}_{1}\frac{{\lambda}_{\mathrm{max}}(P)}{{\lambda}_{\mathrm{min}}(P)} and b=2N{\lambda}_{2}\frac{{\lambda}_{\mathrm{max}}(P)}{{\lambda}_{\mathrm{min}}(P)} are utilized.

From (3.4)-(3.6), we have

\begin{array}{rcl}V\left(e\left({t}_{k}^{+}\right)\right)& \le & aV(e({t}_{k}))+b\sum _{i=k-N}^{k-1}V(e({t}_{i}))\\ \le & aV\left(e\left({t}_{k-1}^{+}\right)\right){e}^{\alpha ({t}_{k}-{t}_{k-1})}+b\sum _{i=k-N}^{k-1}V\left(e\left({t}_{i-1}^{+}\right)\right){e}^{\alpha ({t}_{i}-{t}_{i-1})}\\ =& a{e}^{\alpha {\mathrm{\Delta}}_{k}}V\left(e\left({t}_{k-1}^{+}\right)\right)+b\sum _{i=k-N}^{k-1}{e}^{\alpha {\mathrm{\Delta}}_{i}}V\left(e\left({t}_{i-1}^{+}\right)\right)\\ \triangleq & \sum _{i=1}^{N}{\alpha}_{k-N+i-1}V\left({t}_{k-N+i-2}^{+}\right)+{\tilde{\alpha}}_{k-1}V\left({t}_{k-1}^{+}\right),\end{array}

(3.7)

where {\tilde{\alpha}}_{k-1}=a{e}^{\alpha {\mathrm{\Delta}}_{k}}, {\alpha}_{k-N+i-1}=b{e}^{\alpha {\mathrm{\Delta}}_{k-N+i-1}}, i=1,2,\dots ,N.

Similar to the proof of Theorem 4.2 in [22], by (3.7), for k\in Z, let

\{\begin{array}{c}{\omega}_{1}(k)=V(e({t}_{k+1}^{+})),\hfill \\ {\omega}_{2}(k)=V(e({t}_{k+2}^{+})),\hfill \\ \vdots \hfill \\ {\omega}_{N+1}(k)=V(e({t}_{k+N+1}^{+}))\hfill \end{array}

(3.8)

and \omega (k)={({\omega}_{1}(k),{\omega}_{2}(k),\dots ,{\omega}_{N+1}(k))}^{T}. Then the system of difference equations obtained above together with (3.7) and (3.8) can be expressed as

\omega (k-N)\le {J}_{k}(N+1)\omega (k-N-1),

where

{J}_{k}(N+1)=\left[\begin{array}{cccccc}0& 1& 0& \cdots & 0& 0\\ 0& 0& 1& \cdots & 0& 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0& 0& 0& \cdots & 1& 0\\ 0& 0& 0& \cdots & 0& 1\\ {\alpha}_{k-N}& {\alpha}_{k-N+1}& {\alpha}_{k-N+2}& \cdots & {\alpha}_{k-1}& {\tilde{\alpha}}_{k-1}\end{array}\right].

(3.9)

Let the comparison system be

\{\begin{array}{c}z(k+1)={J}_{k}(N+1)z(k),\hfill \\ z(N)=\omega (-1).\hfill \end{array}

(3.10)

Then, by the comparison principle, we can get

\omega (k-N-1)\le z(k),\phantom{\rule{1em}{0ex}}k\ge N,k\in Z.

Thus, by the condition in the theorem, there exists a constant K>0 such that

\parallel \omega (k-N-1)\parallel \le K{e}^{-\sigma (k-N)}\parallel \omega (-1)\parallel ,\phantom{\rule{1em}{0ex}}k\ge N,k\in Z,

where \parallel \omega (-1)\parallel ={\sum}_{i=0}^{N}{V}^{2}(e({t}_{i}^{+})).

From (3.8), for t={t}_{k}, k\in Z, we can get that

V\left(e\left({t}_{k}^{+}\right)\right)={\omega}_{N+1}(k-N-1)\le \parallel \omega (k-N-1)\parallel \le K\parallel \omega (-1)\parallel {e}^{-\sigma (k-N)}.

(3.11)

Hence, by Lemma 2, (3.2) and (3.11), and for any t\in ({t}_{k-1},{t}_{k}], k\in Z, we get

\begin{array}{rcl}{\parallel e(t)\parallel}^{2}& \le & \frac{1}{{\lambda}_{\mathrm{min}}(P)}V(e(t))\le \frac{1}{{\lambda}_{\mathrm{min}}(P)}V\left({t}_{k-1}^{+}\right){e}^{\alpha (t-{t}_{k-1})}\\ \le & \frac{K\parallel \omega (-1)\parallel {e}^{\alpha {\mathrm{\Delta}}_{\mathrm{sup}}}}{{\lambda}_{\mathrm{min}}(P)}{e}^{\sigma (N+1)}{e}^{-\sigma k}\triangleq \tilde{{K}^{2}}{e}^{-\sigma k},\end{array}

where \tilde{{K}^{2}}=\frac{K\parallel \omega (-1)\parallel {e}^{\alpha {\mathrm{\Delta}}_{\mathrm{sup}}}}{{\lambda}_{\mathrm{min}}(P)}{e}^{\sigma (N+1)}.

Let {N}_{\zeta}(t,{t}_{0}) be the number of impulsive times of the impulsive sequence *ζ* in the interval ({t}_{0},t). Hence, we can obtain

{\parallel e(t)\parallel}^{2}\le \tilde{{K}^{2}}{e}^{-\sigma {N}_{\zeta}(t,{t}_{0})}.

(3.12)

Since the average impulsive interval of the impulsive sequence \zeta =\{{t}_{1},{t}_{2},\dots \} is equal to {T}_{a}, we have

{N}_{\zeta}(t,{t}_{0})\ge \frac{t-{t}_{0}}{{T}_{a}}-{N}_{0},\phantom{\rule{1em}{0ex}}\mathrm{\forall}T\ge t\ge 0.

Hence, by (3.12), we get

\parallel e(t)\parallel \le \tilde{K}{e}^{\frac{\sigma {N}_{0}}{2}}{e}^{-\frac{\sigma}{2{T}_{a}}(t-{t}_{0})}.

Thus, the trivial solution e=0 of the error system (2.4) is globally exponentially stable with the convergence rate -\frac{\sigma}{2{T}_{a}}, and hence the slave system (2.2) can achieve global exponential synchronization with the master system (2.1). □

**Remark 3** In this paper, a modified impulsive control system is adopted to provide the basis for developing global exponential synchronization between the master system and the slave system, which can reduce the impulsive times and the control cost effectively. In addition, to stabilize the error system (2.4) more effectively, we can also consider that the error at the current time instant and the previous time instants play different roles in the impulsive control system. For example, we can suppose that {B}_{p}=\frac{\eta {b}_{p}}{{b}_{p}+{b}_{l}}I and {B}_{l}=\frac{\eta {b}_{l}}{N({b}_{p}+{b}_{l})}I, where *η*, {b}_{p} and {b}_{l} are constants, and |{b}_{p}|\ge |{b}_{l}|. Obviously, it is a special case of Theorem 1.

**Remark 4** Note that in the proof of Theorem 1, the concept of an average impulsive interval is employed to prove the global exponential stability for the error system under Assumption 1. By this approach, the requirement on the lower bound and upper bound of impulsive interval is removed in Theorem 1, which is different from the conventional ones in the literature.

**Remark 5** If N=0, the modified impulsive control scheme is the normal impulsive one, such as in [15–17]. Hence, by Theorem 1, we only need a positive definite matrix *P* such that |{\tilde{\alpha}}_{k-1}|<1, \mathrm{\forall}k\in Z, where {\tilde{\alpha}}_{k-1}=a{e}^{\alpha {\mathrm{\Delta}}_{k}}, i=1,2,\dots ,N, are the same as in Theorem 1. Then the slave system (2.2) can achieve global exponential synchronization with the master system (2.1). In fact, it can be seen from (3.7) that {\tilde{\alpha}}_{k-1} is the impulsive strength of the impulsive signal if N=0. If |{\tilde{\alpha}}_{k-1}|<1, \mathrm{\forall}k\in Z, the impulse is beneficial for the error system since the difference is reduced. Thus, the error system can be stable easily with the impulsive control system.

In the following, by using Theorem 1, we give some simple corollaries of Theorem 1.

**Corollary 1** *Suppose the impulsive interval is a positive constant* Δ, *and the impulsive gain matrix* {B}_{p}={b}_{p}I, *and* {B}_{l}={b}_{l}I. *If there exists a positive definite matrix* *P* *such that*

\rho (J(N+1))<{e}^{-\sigma},

*where*

J(N+1)=\left[\begin{array}{cccccc}0& 1& 0& \cdots & 0& 0\\ 0& 0& 1& \cdots & 0& 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0& 0& 0& \cdots & 1& 0\\ 0& 0& 0& \cdots & 0& 1\\ b{e}^{\alpha \mathrm{\Delta}}& b{e}^{\alpha \mathrm{\Delta}}& b{e}^{\alpha \mathrm{\Delta}}& \cdots & b{e}^{\alpha \mathrm{\Delta}}& a{e}^{\alpha \mathrm{\Delta}}\end{array}\right],

*where* a=2{(1+{b}_{p})}^{2}\frac{{\lambda}_{\mathrm{max}}(P)}{{\lambda}_{\mathrm{min}}(P)}, b=2N{b}_{l}^{2}\frac{{\lambda}_{\mathrm{max}}(P)}{{\lambda}_{\mathrm{min}}(P)} *and* *α* *is the same as in Theorem * 1. *Then the error system* (2.4) *is globally exponentially stable with the convergence rate* -\frac{\sigma}{2{T}_{a}}, *and hence the slave system* (2.2) *can achieve global exponential synchronization with the master system* (2.1).

*Proof* The proof is similar to Theorem 1. □

**Corollary 2** *If there exists a positive constant* 0<\gamma <1 *such that every root* {\lambda}_{j} (j=1,2,\dots ,N+1) *of the characteristic polynomial*

{F}_{k}(\lambda )\triangleq {\lambda}^{N+1}-{\tilde{\alpha}}_{k-1}{\lambda}^{N}-{\alpha}_{k-1}{\lambda}^{N}-\cdots -{\alpha}_{k-N+1}\lambda -{\alpha}_{k-N}

*satisfies* |{\lambda}_{j}|\le \gamma <1, j=1,2,\dots ,N+1, *where* {\tilde{\alpha}}_{k-1}=a{e}^{\alpha {\mathrm{\Delta}}_{k}}, {\alpha}_{k-N+i-1}=b{e}^{\alpha {\mathrm{\Delta}}_{k-N+i-1}}, i=1,2,\dots ,N, *are the same as in Theorem * 1. *Then the error system* (2.4) *is globally exponentially stable with the convergence rate* -\frac{\sigma}{2{T}_{a}}, *and hence the slave system* (2.2) *can achieve global exponential synchronization with the master system* (2.1).

*Proof* In fact, {F}_{k}(\lambda ) is the characteristic polynomial of {J}_{k}(n+1) in Theorem 1. Hence, if every root satisfies |{\lambda}_{j}|\le \gamma <1, j=1,2,\dots ,N+1, there exists a constant \sigma >0 such that |{\lambda}_{j}|\le \gamma \le {e}^{-\sigma}<1, j=1,2,\dots ,N+1, then the spectral radius of {J}_{k}(n+1) satisfies \rho ({J}_{k}(n+1))\le \gamma <1. Thus, we conclude that this corollary is true. □