This section addresses the exponential stability problem of the switched system (2.4). The main result is stated as follows.

**Theorem 3.1** *If there exist positive definite matrices* P>0, Q>0 *and scalar constants* {\epsilon}_{ij}>0 (i=1,2, j=1,2,3), {\epsilon}_{14}>0, w>0, *such that the following matrix inequalities hold*:

*where*

\begin{array}{rcl}{\mathrm{\Omega}}_{1}& =& {A}^{T}P+PA+{K}^{T}{B}^{T}P+PBK+wP+Q+{\epsilon}_{11}PP+{\epsilon}_{11}^{-1}{Q}_{1}+{\epsilon}_{12}^{-1}{E}_{1}^{T}{E}_{1}+{\epsilon}_{12}P{D}_{1}{D}_{1}^{T}P\\ +{\epsilon}_{14}^{-1}{K}^{T}{E}_{3}^{T}{E}_{3}K+{\epsilon}_{14}P{D}_{3}{D}_{3}^{T}P+{\epsilon}_{13}^{-1}P{D}_{2}{D}_{2}^{T}P,\end{array}

*and*

{\mathrm{\Omega}}_{2}={A}^{T}P+PA+wP+Q+{\epsilon}_{21}PP+{\epsilon}_{21}^{-1}{Q}_{1}+{\epsilon}_{22}^{-1}{E}_{1}^{T}{E}_{1}+{\epsilon}_{22}P{D}_{1}{D}_{1}^{T}P+{\epsilon}_{23}^{-1}P{D}_{2}{D}_{2}^{T}P,

*then the system* (2.4) *is exponentially stable*. *Moreover*, *the solution* x(t) *satisfies the condition*

\parallel x(t)\parallel \le \parallel \varphi \parallel \sqrt{\frac{{\lambda}_{max}(P)+d{\lambda}_{max}(Q)}{{\lambda}_{min}(P)}}{e}^{-\frac{w(t-\tau )}{2}},\phantom{\rule{1em}{0ex}}\mathrm{\forall}t>0.

*Proof* Consider the following candidate Lyapunov-Krasovskii functional:

V(x(t))={x}^{T}(t)Px(t)+{e}^{-wt}{\int}_{t-d}^{t}{e}^{w\theta}{x}^{T}(\theta )Qx(\theta )\phantom{\rule{0.2em}{0ex}}d\theta .

(3.3)

When nT\le t<nT+\tau, the derivative of (3.3) with respect to time *t* along the trajectories of the first subsystem of the system (2.4) is calculated and estimated as follows:

\begin{array}{rcl}\dot{V}(x(t))& =& {\dot{x}}^{T}(t)Px(t)+{x}^{T}(t)P\dot{x}(t)-w{e}^{-wt}{\int}_{t-d}^{t}{e}^{w\theta}{x}^{T}(\theta )Qx(\theta )\phantom{\rule{0.2em}{0ex}}d\theta \\ +{e}^{-wt}[{e}^{wt}{x}^{T}(t)Qx(t)-{e}^{w(t-d)}{x}^{T}(t-d)Qx(t-d)]\\ =& {\dot{x}}^{T}(t)Px(t)+{x}^{T}(t)P\dot{x}(t)-w{e}^{-wt}{\int}_{t-d}^{t}{e}^{w\theta}{x}^{T}(\theta )Qx(\theta )\phantom{\rule{0.2em}{0ex}}d\theta \\ +{x}^{T}(t)Qx(t)-{e}^{-wd}{x}^{T}(t-d)Qx(t-d)\\ =& [(A+\mathrm{\Delta}A(t))x(t)+({A}_{d}+\mathrm{\Delta}{A}_{d}(t))x(t-d)\\ {+(B+\mathrm{\Delta}B(t))Kx(t)+f(x(t),x(t-d))]}^{T}Px(t)\\ +{x}^{T}(t)P[(A+\mathrm{\Delta}A(t))x(t)+({A}_{d}+\mathrm{\Delta}{A}_{d}(t))x(t-d)\\ +(B+\mathrm{\Delta}B(t))Kx(t)+f(x(t),x(t-d))]\\ -w{e}^{-wt}{\int}_{t-d}^{t}{e}^{w\theta}{x}^{T}(\theta )Qx(\theta )\phantom{\rule{0.2em}{0ex}}d\theta +{x}^{T}(t)Qx(t)-{e}^{-wd}{x}^{T}(t-d)Qx(t-d)\\ =& {x}^{T}(t)[{A}^{T}P+PA+{K}^{T}{B}^{T}P+PBK+Q]x(t)\\ +{x}^{T}(t-d){A}_{d}^{T}Px(t)+{x}^{T}(t)P{A}_{d}x(t-d)\\ +2{x}^{T}(t)Pf(x(t),x(t-d))+{x}^{T}(t)[\mathrm{\Delta}{A}^{T}(t)P+P\mathrm{\Delta}A(t)\\ +{K}^{T}\mathrm{\Delta}{B}^{T}(t)P+P\mathrm{\Delta}B(t)K]x(t)\\ +{x}^{T}(t-d)\mathrm{\Delta}{A}_{d}^{T}(t)Px(t)+{x}^{T}(t)P\mathrm{\Delta}{A}_{d}(t)x(t-d)\\ -wV(x(t))+w{x}^{T}(t)Px(t)-{e}^{-wd}{x}^{T}(t-d)Qx(t-d)\\ \le & {x}^{T}(t)[{A}^{T}P+PA+{K}^{T}{B}^{T}P+PBK+wP+Q]x(t)\\ +{x}^{T}(t-d){A}_{d}^{T}Px(t)+{x}^{T}(t)P{A}_{d}x(t-d)+2{x}^{T}(t)Pf(x(t),x(t-d))\\ +{x}^{T}(t)[{E}_{1}^{T}{F}^{T}(t){D}_{1}^{T}P+P{D}_{1}F(t){E}_{1}+{K}^{T}{E}_{3}^{T}{F}^{T}(t){D}_{3}^{T}P\\ +P{D}_{3}F(t){E}_{3}K]x(t)+{x}^{T}(t-d){E}_{2}^{T}{F}^{T}(t){D}_{2}^{T}Px(t)+{x}^{T}(t)P{D}_{2}F(t){E}_{2}x(t-d)\\ -wV(x(t))-{e}^{-wd}{x}^{T}(t-d)Qx(t-d).\end{array}

Using Lemma 2.1 and Lemma 2.2, we get

\begin{array}{rcl}\dot{V}(x(t))& \le & {x}^{T}(t)[{A}^{T}P+PA+{K}^{T}{B}^{T}P+PBK+wP+Q]x(t)\\ +{x}^{T}(t-d){A}_{d}^{T}Px(t)+{x}^{T}(t)P{A}_{d}x(t-d)\\ +{\epsilon}_{11}{x}^{T}(t)PPx(t)+{\epsilon}_{11}^{-1}{\parallel f(x(t),x(t-d))\parallel}^{2}\\ +{x}^{T}(t)[{\epsilon}_{12}^{-1}{E}_{1}^{T}{E}_{1}+{\epsilon}_{12}P{D}_{1}{D}_{1}^{T}P+{\epsilon}_{14}^{-1}{K}^{T}{E}_{3}^{T}{E}_{3}K\\ +{\epsilon}_{14}P{D}_{3}{D}_{3}^{T}P]x(t)+{\epsilon}_{13}{x}^{T}(t-d){E}_{2}^{T}{E}_{2}x(t-d)\\ +{\epsilon}_{13}^{-1}{x}^{T}(t)P{D}_{2}{D}_{2}^{T}Px(t)-wV(x(t))\\ -{e}^{-wd}{x}^{T}(t-d)Qx(t-d)\\ \le & {x}^{T}(t)[{A}^{T}P+PA+{K}^{T}{B}^{T}P+PBK+wP+Q]x(t)\\ +{x}^{T}(t-d){A}_{d}^{T}Px(t)+{x}^{T}(t)P{A}_{d}x(t-d)\\ +{\epsilon}_{11}{x}^{T}(t)PPx(t)+{\epsilon}_{11}^{-1}[{x}^{T}(t){Q}_{1}x(t)+{x}^{T}(t-d){Q}_{2}x(t-d)]\\ +{x}^{T}(t)[{\epsilon}_{12}^{-1}{E}_{1}^{T}{E}_{1}+{\epsilon}_{12}P{D}_{1}{D}_{1}^{T}P+{\epsilon}_{14}^{-1}{K}^{T}{E}_{3}^{T}{E}_{3}K+{\epsilon}_{14}P{D}_{3}{D}_{3}^{T}P]x(t)\\ +{\epsilon}_{13}^{-1}{x}^{T}(t)P{D}_{2}{D}_{2}^{T}Px(t)+{\epsilon}_{13}{x}^{T}(t-d){E}_{2}^{T}{E}_{2}x(t-d)\\ -wV(x(t))-{e}^{-wd}{x}^{T}(t-d)Qx(t-d)\\ =& {x}^{T}(t)[{A}^{T}P+PA+{K}^{T}{B}^{T}P+PBK+wP+Q\\ +{\epsilon}_{11}PP+{\epsilon}_{11}^{-1}{Q}_{1}+{\epsilon}_{12}^{-1}{E}_{1}^{T}{E}_{1}+{\epsilon}_{12}P{D}_{1}{D}_{1}^{T}P\\ +{\epsilon}_{14}^{-1}{K}^{T}{E}_{3}^{T}{E}_{3}K+{\epsilon}_{14}P{D}_{3}{D}_{3}^{T}P+{\epsilon}_{13}^{-1}P{D}_{2}{D}_{2}^{T}P]x(t)\\ +{x}^{T}(t-d){A}_{d}^{T}Px(t)+{x}^{T}(t)P{A}_{d}x(t-d)\\ +{x}^{T}(t-d)[{\epsilon}_{11}^{-1}{Q}_{2}+{\epsilon}_{13}{E}_{2}^{T}{E}_{2}-{e}^{-wd}Q]x(t-d)-wV(x(t))\\ =& {\left[\begin{array}{c}x(t)\\ x(t-d)\end{array}\right]}^{T}\left[\begin{array}{cc}{\mathrm{\Omega}}_{1}& P{A}_{d}\\ {A}_{d}^{T}P& {\epsilon}_{11}^{-1}{Q}_{2}+{\epsilon}_{13}{E}_{2}^{T}{E}_{2}-{e}^{-wd}Q\end{array}\right]\left[\begin{array}{c}x(t)\\ x(t-d)\end{array}\right]-wV(x(t)),\end{array}

where

\begin{array}{rcl}{\mathrm{\Omega}}_{1}& =& {A}^{T}P+PA+{K}^{T}{B}^{T}P+PBK+wP+Q+{\epsilon}_{11}PP+{\epsilon}_{11}^{-1}{Q}_{1}+{\epsilon}_{12}^{-1}{E}_{1}^{T}{E}_{1}+{\epsilon}_{12}P{D}_{1}{D}_{1}^{T}P\\ +{\epsilon}_{14}^{-1}{K}^{T}{E}_{3}^{T}{E}_{3}K+{\epsilon}_{14}P{D}_{3}{D}_{3}^{T}P+{\epsilon}_{13}^{-1}P{D}_{2}{D}_{2}^{T}P.\end{array}

From (3.1), we have

\dot{V}(x(t))\le -wV(x(t)),\phantom{\rule{1em}{0ex}}nT\le t<nT+\tau ,

(3.4)

which implies that when nT\le t<nT+\tau,

V(x(t))\le V(x(nT)){e}^{-w(t-nT)}.

(3.5)

Similarly, when nT+\tau \le t<(n+1)T, we have

\begin{array}{rcl}\dot{V}(x(t))& =& {\dot{x}}^{T}(t)Px(t)+{x}^{T}(t)P\dot{x}(t)-w{e}^{-wt}{\int}_{t-d}^{t}{e}^{w\theta}{x}^{T}(\theta )Qx(\theta )\phantom{\rule{0.2em}{0ex}}d\theta \\ +{x}^{T}(t)Qx(t)-{e}^{-wd}{x}^{T}(t-d)Qx(t-d)\\ =& {[(A+\mathrm{\Delta}A(t))x(t)+({A}_{d}+\mathrm{\Delta}{A}_{d}(t))x(t-d)+f(x(t),x(t-d))]}^{T}Px(t)\\ +{x}^{T}(t)P[(A+\mathrm{\Delta}A(t))x(t)+({A}_{d}+\mathrm{\Delta}{A}_{d}(t))x(t-d)+f(x(t),x(t-d))]\\ -w{e}^{-wt}{\int}_{t-d}^{t}{e}^{w\theta}{x}^{T}(\theta )Qx(\theta )\phantom{\rule{0.2em}{0ex}}d\theta +{x}^{T}(t)Qx(t)-{e}^{-wd}{x}^{T}(t-d)Qx(t-d)\\ =& {x}^{T}(t)[{A}^{T}P+PA+Q]x(t)+{x}^{T}(t-d){A}_{d}^{T}Px(t)+{x}^{T}(t)P{A}_{d}x(t-d)\\ +2{x}^{T}(t)Pf(x(t),x(t-d))+{x}^{T}(t)[\mathrm{\Delta}{A}^{T}(t)P+P\mathrm{\Delta}A(t)]x(t)\\ +{x}^{T}(t-d)\mathrm{\Delta}{A}_{d}^{T}(t)Px(t)+{x}^{T}(t)P\mathrm{\Delta}{A}_{d}(t)x(t-d)\\ -wV(x(t))+w{x}^{T}(t)Px(t)-{e}^{-wd}{x}^{T}(t-\tau )Qx(t-\tau )\\ \le & {x}^{T}(t)[{A}^{T}P+PA+wP+Q]x(t)+{x}^{T}(t-d){A}_{d}^{T}Px(t)+{x}^{T}(t)P{A}_{d}x(t-d)\\ +2{x}^{T}(t)Pf(x(t),x(t-d))+{x}^{T}(t)[{E}_{1}^{T}{F}^{T}(t){D}_{1}^{T}P+P{D}_{1}F(t){E}_{1}]x(t)\\ +{x}^{T}(t-d){E}_{2}^{T}{F}^{T}(t){D}_{2}^{T}Px(t)+{x}^{T}(t)P{D}_{2}F(t){E}_{2}x(t-d)\\ -wV(x(t))-{e}^{-wd}{x}^{T}(t-d)Qx(t-d)\\ \le & {x}^{T}(t)[{A}^{T}P+PA+wP+Q]x(t)+{x}^{T}(t-d){A}_{d}^{T}Px(t)+{x}^{T}(t)P{A}_{d}x(t-d)\\ +{\epsilon}_{21}{x}^{T}(t)PPx(t)+{\epsilon}_{21}^{-1}{\parallel f(x(t),x(t-d))\parallel}^{2}\\ +{x}^{T}(t)[{\epsilon}_{22}^{-1}{E}_{1}^{T}{E}_{1}+{\epsilon}_{22}P{D}_{1}{D}_{1}^{T}P]x(t)\\ +{\epsilon}_{23}{x}^{T}(t-d){E}_{2}^{T}{E}_{2}x(t-d)+{\epsilon}_{23}^{-1}{x}^{T}(t)P{D}_{2}{D}_{2}^{T}Px(t)-wV(x(t))\\ -{e}^{-wd}{x}^{T}(t-d)Qx(t-d)\\ \le & {x}^{T}(t)[{A}^{T}P+PA+wP+Q]x(t)+{x}^{T}(t-d){A}_{d}^{T}Px(t)+{x}^{T}(t)P{A}_{d}x(t-d)\\ +{\epsilon}_{21}{x}^{T}(t)PPx(t)+{\epsilon}_{21}^{-1}[{x}^{T}(t){Q}_{1}x(t)+{x}^{T}(t-d){Q}_{2}x(t-d)]\\ +{\epsilon}_{23}^{-1}{x}^{T}(t)P{D}_{2}{D}_{2}^{T}Px(t)+{x}^{T}(t)[{\epsilon}_{22}^{-1}{E}_{1}^{T}{E}_{1}+{\epsilon}_{22}P{D}_{1}{D}_{1}^{T}P]x(t)\\ +{\epsilon}_{23}{x}^{T}(t-d){E}_{2}^{T}{E}_{2}x(t-d)-wV(x(t))\\ -{e}^{-wd}{x}^{T}(t-d)Qx(t-d)\\ =& {x}^{T}(t)[{A}^{T}P+PA+wP+Q+{\epsilon}_{21}PP+{\epsilon}_{21}^{-1}{Q}_{1}\\ +{\epsilon}_{22}^{-1}{E}_{1}^{T}{E}_{1}+{\epsilon}_{22}P{D}_{1}{D}_{1}^{T}P+{\epsilon}_{23}^{-1}P{D}_{2}{D}_{2}^{T}P]x(t)\\ +{x}^{T}(t-d)[{\epsilon}_{21}^{-1}{Q}_{2}+{\epsilon}_{23}{E}_{2}^{T}{E}_{2}-{e}^{-wd}Q]x(t-d)-wV(x(t))\\ +{x}^{T}(t-d){A}_{d}^{T}Px(t)+{x}^{T}(t)P{A}_{d}x(t-d)\\ =& {\left[\begin{array}{c}x(t)\\ x(t-d)\end{array}\right]}^{T}\left[\begin{array}{cc}{\mathrm{\Omega}}_{2}& P{A}_{d}\\ {A}_{d}^{T}P& {\epsilon}_{21}^{-1}{Q}_{2}+{\epsilon}_{23}{E}_{2}^{T}{E}_{2}-{e}^{-wd}Q\end{array}\right]\left[\begin{array}{c}x(t)\\ x(t-d)\end{array}\right]-wV(x(t)),\end{array}

where {\mathrm{\Omega}}_{2}={A}^{T}P+PA+wP+Q+{\epsilon}_{21}PP+{\epsilon}_{21}^{-1}{Q}_{1}+{\epsilon}_{22}^{-1}{E}_{1}^{T}{E}_{1}+{\epsilon}_{22}P{D}_{1}{D}_{1}^{T}P+{\epsilon}_{23}^{-1}P{D}_{2}{D}_{2}^{T}P.

From (3.2), we have

\dot{V}(x(t))\le -wV(x(t)),\phantom{\rule{1em}{0ex}}nT+\tau \le t<(n+1)T,

(3.6)

which implies that when nT+\tau \le t<(n+1)T,

V(x(t))\le V(x(nT+\tau )){e}^{-w(t-nT-\tau )}.

(3.7)

From (3.5) and (3.7), it follows that:

When 0\le t<\tau, V(x(t))\le V({x}_{0}){e}^{-wt}, and

V(x(\tau ))\le V({x}_{0}){e}^{-w\tau}.

When \tau \le t<T,

When T\le t<T+\tau,

When T+\tau \le t<2T,

When 2T\le t<2T+\tau,

When nT\le t<nT+\tau,

When nT+\tau \le t<(n+1)T,

\begin{array}{rcl}V(x(t))& \le & V(x(nT+\tau )){e}^{-w(t-nT-\tau )}\\ \le & V({x}_{0}){e}^{-wnT}{e}^{-w(t-nT-\tau )}\\ =& V({x}_{0}){e}^{-w(t-\tau )}.\end{array}

(3.9)

From (3.8) and (3.9), it follows that for any t>0, we can obtain

V(x(t))\le V({x}_{0}){e}^{-w(t-\tau )},\phantom{\rule{1em}{0ex}}\mathrm{\forall}t>0.

(3.10)

From (3.3) and (3.10), it follows that for any t>0,

\begin{array}{rcl}{x}^{T}(t)x(t)& \le & \frac{V(x(t))}{{\lambda}_{min}(P)}\\ \le & \frac{1}{{\lambda}_{min}(P)}V({x}_{0}){e}^{-w(t-\tau )}.\end{array}

Hence, we get

\parallel x(t)\parallel \le \sqrt{\frac{V({x}_{0})}{{\lambda}_{min}(P)}}{e}^{-\frac{w(t-\tau )}{2}},\phantom{\rule{1em}{0ex}}\mathrm{\forall}t>0,

Noticing

\begin{array}{rcl}V({x}_{0})& =& {x}^{T}(0)Px(0)+{\int}_{-d}^{0}{e}^{w\theta}{x}^{T}(\theta )Qx(\theta )\phantom{\rule{0.2em}{0ex}}d\theta \\ \le & ({\lambda}_{max}(P)+d{\lambda}_{max}(Q)){\parallel \varphi \parallel}^{2},\end{array}

we get

\parallel x(t)\parallel \le \parallel \varphi \parallel \sqrt{\frac{{\lambda}_{max}(P)+d{\lambda}_{max}(Q)}{{\lambda}_{min}(P)}}{e}^{-\frac{w(t-\tau )}{2}},\phantom{\rule{1em}{0ex}}\mathrm{\forall}t>0,

which completes the proof. □

**Corollary 3.1** *If there exist positive definite matrices* P>0, Q>0, *and scalar constants* {\epsilon}_{j}>0 (j=1,2,3,4), w>0, *such that the following LMIs hold*:

*were*

*then the system* (2.4) *is exponentially stable*. *Moreover*, *the solution* x(t) *satisfies the condition*

\parallel x(t)\parallel \le \parallel \varphi \parallel \sqrt{\frac{{\lambda}_{max}(P)+d{\lambda}_{max}(Q)}{{\lambda}_{min}(P)}}{e}^{-\frac{w(t-\tau )}{2}},\phantom{\rule{1em}{0ex}}\mathrm{\forall}t>0.

*Proof* Set

According to Lemma 2.3 and (3.11) and (3.12), we have

Taking {\epsilon}_{11}={\epsilon}_{21}={\epsilon}_{1}, {\epsilon}_{12}={\epsilon}_{22}={\epsilon}_{2}, {\epsilon}_{13}={\epsilon}_{23}={\epsilon}_{3}, {\epsilon}_{14}={\epsilon}_{4}, we have

\begin{array}{rcl}{\mathrm{\Omega}}_{2}& =& {A}^{T}P+PA+wP+Q+{\epsilon}_{21}PP+{\epsilon}_{21}^{-1}{Q}_{1}+{\epsilon}_{22}^{-1}{E}_{1}^{T}{E}_{1}+{\epsilon}_{22}P{D}_{1}{D}_{1}^{T}P+{\epsilon}_{23}^{-1}P{D}_{2}{D}_{2}^{T}P\\ =& {A}^{T}P+PA+wP+Q+{\epsilon}_{1}PP+{\epsilon}_{1}^{-1}{Q}_{1}+{\epsilon}_{2}^{-1}{E}_{1}^{T}{E}_{1}+{\epsilon}_{2}P{D}_{1}{D}_{1}^{T}P+{\epsilon}_{3}^{-1}P{D}_{2}{D}_{2}^{T}P\\ =& {\mathrm{\Omega}}_{4},\end{array}

Hence, (3.14) implies that (3.2) holds and {\mathrm{\Omega}}_{4}<0. From (3.13), we have

\left[\begin{array}{cc}{\mathrm{\Omega}}_{3}& P{A}_{d}\\ {A}_{d}^{T}P& {\epsilon}_{1}^{-1}{Q}_{2}+{\epsilon}_{3}{E}_{2}^{T}{E}_{2}-{e}^{-wd}Q\end{array}\right]+\left[\begin{array}{cc}{\mathrm{\Omega}}_{4}& 0\\ 0& 0\end{array}\right]<0.

(3.15)

Since

\begin{array}{rcl}{\mathrm{\Omega}}_{1}& =& {A}^{T}P+PA+{K}^{T}{B}^{T}P+PBK+wP+Q+{\epsilon}_{11}PP+{\epsilon}_{11}^{-1}{Q}_{1}+{\epsilon}_{12}^{-1}{E}_{1}^{T}{E}_{1}+{\epsilon}_{12}P{D}_{1}{D}_{1}^{T}P\\ +{\epsilon}_{14}^{-1}{K}^{T}{E}_{3}^{T}{E}_{3}K+{\epsilon}_{14}P{D}_{3}{D}_{3}^{T}P+{\epsilon}_{13}^{-1}P{D}_{2}{D}_{2}^{T}P\\ =& {A}^{T}P+PA+{K}^{T}{B}^{T}P+PBK+wP+Q+{\epsilon}_{1}PP+{\epsilon}_{1}^{-1}{Q}_{1}+{\epsilon}_{2}^{-1}{E}_{1}^{T}{E}_{1}+{\epsilon}_{2}P{D}_{1}{D}_{1}^{T}P\\ +{\epsilon}_{4}^{-1}{K}^{T}{E}_{3}^{T}{E}_{3}K+{\epsilon}_{4}P{D}_{3}{D}_{3}^{T}P+{\epsilon}_{3}^{-1}P{D}_{2}{D}_{2}^{T}P\\ =& {\mathrm{\Omega}}_{4}+{K}^{T}{B}^{T}P+PBK+{\epsilon}_{4}^{-1}{K}^{T}{E}_{3}^{T}{E}_{3}K+{\epsilon}_{4}P{D}_{3}{D}_{3}^{T}P={\mathrm{\Omega}}_{4}+{\mathrm{\Omega}}_{3},\end{array}

(3.15) implies that (3.1) holds. According to Theorem 3.1, the system (2.4) is exponentially stable, and

\parallel x(t)\parallel \le \parallel \varphi \parallel \sqrt{\frac{{\lambda}_{max}(P)+d{\lambda}_{max}(Q)}{{\lambda}_{min}(P)}}{e}^{-\frac{w(t-\tau )}{2}},\phantom{\rule{1em}{0ex}}\mathrm{\forall}t>0,

which completes the proof. □

The following theorem can be directly deduced from Theorem 3.1 by applying Lemma 2.3.

**Theorem 3.2** *If there exist positive definite matrices* P>0, Q>0 *and scalar constants* {\epsilon}_{ij}>0 (i=1,2, j=1,2,3), {\epsilon}_{14}>0, w>0, *such that the following LMIs hold*:

*where*

{\overline{\mathrm{\Omega}}}_{1}={A}^{T}P+PA+{K}^{T}{B}^{T}P+PBK+wP+Q+{\epsilon}_{11}{Q}_{1}+{\epsilon}_{12}{E}_{1}^{T}{E}_{1}+{\epsilon}_{14}{K}^{T}{E}_{3}^{T}{E}_{3}K,

*and*

{\overline{\mathrm{\Omega}}}_{2}={A}^{T}P+PA+wP+Q+{\epsilon}_{21}{Q}_{1}+{\epsilon}_{22}{E}_{1}^{T}{E}_{1},

*then the system* (2.4) *is exponentially stable*. *Moreover*, *the solution* x(t) *satisfies the condition*

\parallel x(t)\parallel \le \parallel \varphi \parallel \sqrt{\frac{{\lambda}_{max}(P)+d{\lambda}_{max}(Q)}{{\lambda}_{min}(P)}}{e}^{-\frac{w(t-\tau )}{2}},\phantom{\rule{1em}{0ex}}\mathrm{\forall}t>0.

Consider the following time-delay systems:

\begin{array}{r}\dot{x}(t)=Ax(t)+{A}_{d}x(t-d)+Bu(t)+f(x(t),x(t-d)),\\ x(t)=\varphi (t),\phantom{\rule{1em}{0ex}}t\in [-d,0],\end{array}

(3.16)

where x\in {R}^{n} is a state vector, and u\in {R}^{m} is the external input of the system (3.16). f:{R}^{n}\to {R}^{n} is a continuous nonlinear function satisfying f(0,0)=0, and there exist positive definite matrices {Q}_{1}, {Q}_{2} such that {\parallel f(x(t),x(t-d))\parallel}^{2}\le {x}^{T}(t){Q}_{1}x(t)+{x}^{T}(t-d){Q}_{2}x(t-d) for x(t),x(t-d)\in {R}^{n}.

With the control law (2.3), the system (3.16) can be rewritten as

\{\begin{array}{cc}\dot{x}(t)=Ax(t)+{A}_{d}x(t-d)+BKx(t)+f(x(t),x(t-d)),\hfill & nT\le t<nT+\tau ,\hfill \\ \dot{x}(t)=Ax(t)+{A}_{d}x(t-d)+f(x(t),x(t-d)),\hfill & nT+\tau \le t<(n+1)T.\hfill \end{array}

(3.17)

From Corollary 3.1, the following corollary can be immediately obtained.

**Corollary 3.2** *If there exist positive definite matrices* P>0, Q>0, *and scalar constants* {\epsilon}_{1}>0, w>0, *such that the following LMIs hold*:

*then the system* (3.17) *is exponentially stable*. *Moreover*, *the solution* x(t) *satisfies the condition*

\parallel x(t)\parallel \le \parallel \varphi \parallel \sqrt{\frac{{\lambda}_{max}(P)+d{\lambda}_{max}(Q)}{{\lambda}_{min}(P)}}{e}^{-\frac{w(t-\tau )}{2}},\phantom{\rule{1em}{0ex}}\mathrm{\forall}t>0.