Third-order reciprocally convex approach to stability of fuzzy cellular neural networks under impulsive perturbations

Abstract

The stability is investigated for a kind of fuzzy cellular neural networks with time-varying and continuously distributed delays under impulsive perturbations. When the Wirtinger-based integral inequality is applied to partitioned integral terms in the derivation of matrix inequality conditions, a new kind of linear combination of positive functions emerges weighted by the inverses of cubic convex parameters. This paper proposes an efficient method called third-order reciprocally convex approach to manipulate such a combination by extending the reciprocal convex technique. By utilizing Briat lemma and reciprocal convex approach, this paper derives several novel sufficient conditions to ensure the global asymptotic stability of the equilibrium point of the considered networks. Based on the derived criteria, a lower-bound estimation can be obtained of the largest stability interval for a class of fuzzy cellular neural networks under impulsive perturbations. Simulation examples demonstrate that the presented method can significantly reduce the conservatism of the existing results, and lead to wider applications.

Appendix

1.1 Proof of Theorem 1

Pre- and post-multiplying the left-hand side inequality (9) by $\bigg [\begin{array}{cc} \mathbb {I} &{} 0 \\ *&{} I_{3n}\end{array}\bigg ]$ and its transpose, it follows that

$$\begin{aligned} \left[ \begin{array}{cccccc} 7P_7 &{}3P_7 &{} P_7&{} S_2 &{} S_3&{} S_4\\ *&{}3P_7 &{} 0&{} 0&{} S_3 &{} 0 \\ *&{} *&{} P_7&{} 0&{} 0&{} S_4 \\ *&{} *&{} *&{} 3P_7&{} 0&{} 0 \\ *&{} *&{} *&{} *&{} 3P_7&{} 0\\ *&{} *&{} *&{} *&{} *&{} P_7\end{array}\right] >0. \end{aligned}$$

(14)

where

$$\begin{aligned} \mathbb {I}=\left[ \begin{array}{ccc} I &{}I &{} I\\ 0 &{} I &{} 0 \\ 0 &{} 0 &{} I\end{array}\right] . \end{aligned}$$

Pre- and post-multiplying the left-hand side inequality (14) by $\bigg [\begin{array}{cc} I_{3n} &{} 0 \\ *&{} \mathbb {I}\end{array}\bigg ]$ and its transpose, it follows that

$$\begin{aligned} \left[ \begin{array}{cccccc} 7P_7 &{}3P_7 &{} P_7&{} S_2+S_3+S_4 &{} S_3&{} S_4\\ *&{}3P_7 &{} 0&{} S_3&{} S_3 &{} 0 \\ *&{} *&{} P_7&{} S_4&{} 0&{} S_4 \\ *&{} *&{} *&{} 7P_7&{} 3P_7&{} P_7 \\ *&{} *&{} *&{} *&{} 3P_7&{} 0\\ *&{} *&{} *&{} *&{} *&{} P_7\end{array}\right] >0. \end{aligned}$$

(15)

Inspired by Liu et al. (2012) and Lee and Park (2014), we consider the following Lyapunov–Krasovskii functional candidate:

$$\begin{aligned} V(t,x_t)=\sum ^8_{i=1}V_{i}(t,x_t),\nonumber \end{aligned}$$

where

$$\begin{aligned} V_{1}(t,x_t)&=x_{t}^TP_1x_{t}+\int _{t-{\tau _1}}^t {x}(s)^T{P_{2}}x(s)\mathrm{d}s \\&\quad +\int _{t-{\tau _2}}^{t-{\tau _1}} {x}(s)^T{P_{3}}x(s)\mathrm{d}s+\int _{t-\tau (t)}^t \varsigma (s)^T\tilde{Q}\varsigma (s)\mathrm{d}s,\nonumber \\ V_{2}(t,x_t)&=\tau ^2_{12}\int _{-{\tau (t)}}^{-{\tau _1}} {x}(t+s)^T{\tilde{P}(s)}x(t+s)\mathrm{d}s\\&\quad +\tau _{12}\int _{t-{\tau _2}}^{t-{\tau _1}} \int _{\theta }^{t-{\tau _1}} {x}(s)^T{P_{6}}x(s)\mathrm{d}s\mathrm{d}\theta , \nonumber \\ V_{3}(t,x_t)&=\tau _{12}^3\int _{t-{\tau _2}}^{t-{\tau _1}} \int _{\theta }^{t-{\tau _1}} {g}(x(s))^T{P_{7}}{g}(x(s))\mathrm{d}s\mathrm{d}\theta , \nonumber \\ V_{4}(t,x_t)&=\tau _{1}^3\int _{t-{\tau _1}}^t \int _{\theta }^t [{x}(s)^T{P_{8}}x(s)+\dot{x}(s)^T{P_{9}}\dot{x}(s)]\mathrm{d}s\mathrm{d}\theta ,\nonumber \\ V_{5}(t,x_t)&=\tau _{12}\int _{t-{\tau _2}}^{t-\tau _1}\int _{\theta }^t \dot{x}(s)^T{P_{10}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\quad +\int _{t-{\tau _2}}^{t-\tau _1}\left( {\int _\theta ^{t-\tau _1} {\int _\beta ^t {\dot{x}{{(s)}^T}P_{11}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta }}}\right. \\&\quad +\left. {\int _{t - \tau _2 } ^\theta {\int _\beta ^t {\dot{x}{{(s)}^T}{P_{12}}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta }}}\right) \mathrm{d}\theta ,\nonumber \\ \end{aligned}$$

$$\begin{aligned} V_{6}(t,x_t)&=\frac{\tau _{1}^2}{2}\int _{t-{\tau _1}}^{t}\left( \int _{t-{\tau _1}}^{\gamma }\int _{\theta }^t \dot{x}(s)^TP_{13}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \right. \\&\quad +\left. \int ^{t}_{\gamma }\int _{\theta }^t \dot{x}(s)^TP_{14}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \right) \mathrm{d}{\gamma }\nonumber \\&\quad +\frac{\tau _{1}^3}{6}\int _{t-{\tau _1}}^{t}\left( {\int _{\gamma }^{t}\int _\theta ^{t} {\int _\beta ^t {\dot{x}{{(s)}^T}P_{15}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta \mathrm{d}\theta }}}\right. \\&\quad +\left. {\int _{t - \tau _1} ^\gamma \int _{t - \tau _1} ^\theta {\int _\beta ^t {\dot{x}{{(s)}^T}{P_{16}}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta \mathrm{d}\theta }}}\right) \mathrm{d}{\gamma } ,\nonumber \\ V_{7}(t,x_t)&=\tau _{12}\int _{t-{\tau _2}}^{t-{\tau _1}}\left( {\int _{\gamma }^{t-{\tau _1}}\int _\theta ^{t-{\tau _1}} {\int _\beta ^t {\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta \mathrm{d}\theta }}}\right. \\&\quad +\left. {\int _{t - \tau _2} ^\gamma \int _{t - \tau _2} ^\theta {\int _\beta ^t {\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta \mathrm{d}\theta }}}\right) \mathrm{d}{\gamma } ,\nonumber \\ V_{8}(t,x_t)&=\sum \limits _{j=1}^n r_{1j}\int _0^\infty {\kappa _j}(\theta )\int _{t-\theta }^t g_j^2({x_j}(s))\mathrm{d}s\mathrm{d}\theta ,\nonumber \end{aligned}$$

with $\varsigma (s)=\mathrm{col}\{{x}(s),\ {g}(x(s))-L^+{x}(s)\}$ and

$$\begin{aligned} \tilde{P}(s)=\frac{s+\tau _2}{\tau _{12}}P_{4}-\frac{s+\tau _1}{\tau _{12}}P_{5}, \quad s\in [-\tau _2,-\tau _1]. \end{aligned}$$

It is easy to see that $\tilde{P}(s)>0$ for any $s\in [-\tau _2,-\tau _1]$ as $\tilde{P}(s)$ being intrinsically linear in s with two corresponding boundary matrices $\tilde{P}(-\tau _1)=P_{4}>0$ and $\tilde{P}(-\tau _2)=P_{5}>0.$

Calculating the upper right derivative of $V(t,x_t)$ along the solution of (1) at the continuous interval $t \in [{t_{k - 1}},{t_k}),k \in {\mathbb {Z}_ + },$ we get that

$$\begin{aligned} D^+V(t,x_t)=&\sum ^8_{i=1}D^+V_{i}(t,x_t), \end{aligned}$$

(16)

where

$$\begin{aligned}&D^+V_{1}(t,x_t) = 2{x_t}^T{P_1}\dot{x}(t)+{x_t}^T{P_{2}}x_{t}-x_{\tau _1}^T(P_{2}-P_{3})x_{\tau _1}\nonumber \\&\quad -x_{\tau _2}^T{P_{3}}x_{\tau _2}+\varsigma (t)^T\tilde{Q}\varsigma (t)-[1-\dot{\tau }(t)]\varsigma (t-\tau (t))^T\nonumber \\ {}&\quad \tilde{Q}\varsigma (t-\tau (t)),\end{aligned}$$

(17)

$$\begin{aligned}&D^+V_{2}(t,x_t) = \tau ^2_{12}x_{\tau _1}^T(P_{4}+P_{6})x_{\tau _1}-[1-\dot{\tau }(t)]\tau ^2_{12}x_{\tau }^T\nonumber \\&\quad \times {\tilde{P}(-\tau (t))}x_{\tau }-\tau _{12}\int _{t-{\tau (t)}}^{t-{\tau _1}}x(s)^T(P_{4}-P_{5})x(s)\mathrm{d}s\nonumber \\&\quad -\tau _{12}\int _{t-{\tau _2}}^{t-{\tau _1}}x(s)^TP_{6}x(s)\mathrm{d}s,\end{aligned}$$

(18)

$$\begin{aligned}&D^+V_{3}(t,x_t) = \tau _{12}^6{g}(x_{\tau _1})^T{P_{7}}{g}(x_{\tau _1})\nonumber \\&\quad -\tau _{12}^3\int _{t-{\tau _2}}^{t-{\tau _1}} {g}(x(s))^T{P_{7}}{g}(x(s))\mathrm{d}s,\end{aligned}$$

(19)

$$\begin{aligned}&D^+V_{4}(t,x_t) = \tau _{1}^4[{x_t}^T{P_{8}}x_{t}+\dot{x}(t)^T{P_{9}}\dot{x}(t)]\nonumber \\&\quad -\tau _{1}^3\int _{t-{\tau _1}}^t [{x}(s)^T{P_{8}}x(s)+\dot{x}(s)^T{P_{9}}\dot{x}(s)]\mathrm{d}s, \end{aligned}$$

(20)

$$\begin{aligned}&D^+V_{5}(t,x_t)=\tau _{12}\dot{x}(t)^T\Big [{P_{10}+\frac{\tau _{12}}{2}(P_{11}+P_{12})}\Big ]\dot{x}(t)\nonumber \\&\quad -\tau _{12}\int _{t-{\tau _2}}^{t-\tau _1}\dot{x}(s)^T{P_{10}}\dot{x}(s)\mathrm{d}s-\int _{t-{\tau _2}}^{t-\tau _1}\nonumber \\&\quad \times \left( {\int _\theta ^{t-\tau _1} { {\dot{x}{{(s)}^T}P_{11}\dot{x}(s)\mathrm{d}s}}}+ {\int _{t - \tau _2 } ^\theta {{\dot{x}{{(s)}^T}{P_{12}}\dot{x}(s)\mathrm{d}s}}}\right) \mathrm{d}\theta ,\end{aligned}$$

(21)

$$\begin{aligned}&D^+V_{6}(t,x_t)=\frac{\tau _{1}^4}{4}\dot{x}(t)^T(P_{13}+P_{14})\dot{x}(t)\nonumber \\&\quad +\frac{\tau _{1}^6}{36}\dot{x}(t)^T(P_{15}+P_{16})\dot{x}(t)\nonumber \\&\quad -\frac{\tau _{1}^2}{2}\int _{t-{\tau _1}}^{t}\left( \int _{t-{\tau _1}}^{\gamma }\dot{x}(s)^TP_{13}\dot{x}(s)\mathrm{d}s\right. \nonumber \\ {}&\quad \left. +\int ^{t}_{\gamma } \dot{x}(s)^TP_{14}\dot{x}(s)\mathrm{d}s\right) \mathrm{d}{\gamma }\nonumber \\&\quad -\frac{\tau _{1}^3}{6}\int _{t-{\tau _1}}^{t}\left( {\int _{\gamma }^{t}\int _\theta ^{t} {{\dot{x}{{(s)}^T}P_{15}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\right. \nonumber \\&\quad +\left. {\int _{t - \tau _1} ^\gamma \int _{t - \tau _1} ^\theta {{\dot{x}{{(s)}^T}{P_{16}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\right) \mathrm{d}{\gamma } ,\end{aligned}$$

(22)

$$\begin{aligned}&D^+V_{7}(t,x_t)=\frac{\tau _{12}^4}{6}\dot{x}(t)^T(P_{17}+P_{18})\dot{x}(t)\nonumber \\&\quad -\tau _{12}\int _{t-{\tau _2}}^{t-{\tau _1}}\left( {\int _{\gamma }^{t-{\tau _1}}\int _\theta ^{t-{\tau _1}} {{\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\right. \nonumber \\&\quad +\left. {\int _{t - \tau _2} ^\gamma \int _{t - \tau _2} ^\theta {{\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\right) \mathrm{d}{\gamma } ,\end{aligned}$$

(23)

$$\begin{aligned}&D^+V_{8}(t,x_t)=\sum \limits _{j=1}^n {r_{1j}\int _0^\infty {{\kappa _j}(\theta )g_j^2({x_j}(t ))} } \mathrm{d}\theta \nonumber \\&\quad -\sum \limits _{j= 1}^n {r_{1j}\int _0^\infty {{\kappa _j}(\theta )g_j^2({x_j}(t-\theta ))} } \mathrm{d}\theta . \end{aligned}$$

(24)

Setting $\vartheta =\frac{\tau (t)-\tau _{1}}{\tau _{12}},\varpi =1-\vartheta ,$ one has $\tilde{P}(-\tau (t))=(1-\vartheta )P_{4}+\vartheta P_{5}.$

When $\tau _1<\tau (t)\le \tau _2,$ by the condition (7), applying Lemma 1 to (18) gives

$$\begin{aligned}&-\tau _{12}\int _{t-\tau (t)}^{t-{\tau _1}}{x(s)^T{(P_{4}-P_{5}+P_{6})}x(s)}\mathrm{d}s\nonumber \\&\quad \le -\frac{\tau _{12}}{\tau (t)-{\tau _1}}\Bigg \{\bigg (\int _{t-\tau (t)}^{t-{\tau _1}}{x(s)}\mathrm{d}s\bigg )^T\nonumber \\&\qquad \times {(P_{4}-P_{5}+P_{6})}\bigg (\int _{t-\tau (t)}^{t-{\tau _1}}{x(s)}\mathrm{d}s\bigg )\nonumber \\&\qquad +3\bigg (\int _{t-\tau (t)}^{t-{\tau _1}}{x(s)}\mathrm{d}s-\frac{2}{\tau (t)-{\tau _1}}\int _{t-\tau (t)}^{t-{\tau _1}}\int _{t-\tau (t)}^{\theta } x(s) \mathrm{d}s\mathrm{d}\theta \bigg )^T\nonumber \\&\qquad \times {(P_{4}-P_{5}+P_{6})}\nonumber \\&\qquad \times \bigg (\int _{t-\tau (t)}^{t-{\tau _1}}{x(s)}\mathrm{d}s-\frac{2}{\tau (t)-{\tau _1}}\int _{t-\tau (t)}^{t-{\tau _1}}\int _{t-\tau (t)}^{\theta } x(s) \mathrm{d}s\mathrm{d}\theta \bigg )\Bigg \}\\&\quad =-\tau _{12}[\tau (t)-{\tau _1}]v_1^T{(P_{4}-P_{5}+P_{6})}v_1\nonumber \\&\qquad -\frac{3}{\vartheta }\{[\tau (t)\!-\!{\tau _1}]v_1\!-\!v_5\}^T{(P_{4}\!-\!P_{5}\!+\!P_{6})}\{[\tau (t)-{\tau _1}]v_1-v_5\}.\nonumber \end{aligned}$$

(25)

While $\tau _1\le \tau (t)<\tau _2,$ again employing Lemma 1 to (18) derives

$$\begin{aligned}&-\tau _{12}\int ^{t-\tau (t)}_{t-{\tau _2}}{x(s)^T{P_{6}}x(s)}\mathrm{d}s\nonumber \\&\quad \le \frac{\tau _{12}}{\tau _2-\tau (t)}\Bigg \{\bigg (\int ^{t-\tau (t)}_{t-{\tau _2}}{x(s)}\mathrm{d}s\bigg )^T{P_{6}}\bigg (\int ^{t-\tau (t)}_{t-{\tau _2}}{x(s)}\mathrm{d}s\bigg )\nonumber \\&\qquad +3\bigg (\int ^{t-\tau (t)}_{t-{\tau _2}}{x(s)}\mathrm{d}s-\frac{2}{\tau _2-\tau (t)}\int _{t-{\tau _2}}^{t-\tau (t)}\int _{t-\tau _2}^{\theta } x(s) \mathrm{d}s\mathrm{d}\theta \bigg )^T\nonumber \\&\qquad \times {P_{6}}\bigg (\int ^{t-\tau (t)}_{t-{\tau _2}}{x(s)}\mathrm{d}s-\frac{2}{\tau _2-\tau (t)}\int _{t-{\tau _2}}^{t-\tau (t)}\int _{t-\tau _2}^{\theta } x(s) \mathrm{d}s\mathrm{d}\theta \bigg )\Bigg \}\nonumber \\&\quad =-\tau _{12}[\tau _2-\tau (t)]v_3^T{P_{6}}v_3-\frac{3}{\varpi }\{[\tau _2-\tau (t)]v_3-v_7\}^T\nonumber \\&\qquad \times {P_{6}}\{[\tau _2-\tau (t)]v_3-v_7\}. \end{aligned}$$

(26)

Based on (4), inequalities (25) and (26) still hold for $\tau (t)=\tau _1$ or $\tau (t)=\tau _2$, respectively.

It is easy to derive that $\Xi _4>0$ from inequalities (12), based on the reciprocal convex technique of Park et al. (2011); Wu et al. 2012), one has

$$\begin{aligned}&\frac{3}{\vartheta }\{[\tau (t)-{\tau _1}]v_1-v_5\}^T{(P_{4}\!-\!P_{5}+P_{6})}\{[\tau (t)-{\tau _1}]v_1-v_5\}\\&\qquad +\frac{3}{\varpi }\{[{\tau _2}-{\tau (t)}]v_3-v_7\}^T{P_{6}}\{[\tau _2-\tau (t)]v_3-v_7\}\\&\quad \ge \bigg [\begin{array}{c} (\tau (t)-{\tau _1})v_1-v_5 \\ (\tau _2-\tau (t))v_3-v_7\end{array}\bigg ]^T\bigg [\begin{array}{cc} P_{4}-P_{5}+P_{6}&{} S_9\\ *&{}P_{6}\end{array}\bigg ]\\&\qquad \times \bigg [\begin{array}{c} (\tau (t)-{\tau _1})v_1-v_5 \\ (\tau _2-\tau (t))v_3-v_7\end{array}\bigg ]. \end{aligned}$$

When $\tau _1<\tau (t)<\tau _2,$ applying Lemma 1 to (19) gives

$$\begin{aligned}&-\tau _{12}^3\int _{t-{\tau _2}}^{t-{\tau _1}} {g}(x(s))^T{P_{7}}{g}(x(s))\mathrm{d}s\nonumber \\&\quad =-\tau _{12}^3\bigg (\int _{t-\tau (t)}^{t-{\tau _1}} {g}(x(s))^T{P_{7}}{g}(x(s))\mathrm{d}s\nonumber \\ {}&\qquad +\int _{t-{\tau _2}}^{t-\tau (t)} {g}(x(s))^T{P_{7}}{g}(x(s))\mathrm{d}s\bigg )\nonumber \\&\quad \le -\frac{\tau _{12}^3}{\tau (t)-{\tau _1}}\Bigg \{\bigg (\int _{t-\tau (t)}^{t-{\tau _1}}{g}(x(s))\mathrm{d}s\bigg )^T\nonumber \\ {}&\qquad \times {P_{7}}\bigg (\int _{t-\tau (t)}^{t-{\tau _1}}{g}(x(s))\mathrm{d}s\bigg )+3\bigg (\int _{t-\tau (t)}^{t-{\tau _1}}{{g}(x(s))}\mathrm{d}s\nonumber \\&\qquad -\frac{2}{\tau (t)-{\tau _1}}\int _{t-\tau (t)}^{t-{\tau _1}}\int _{t-\tau (t)}^{\theta } {g}(x(s)) \mathrm{d}s\mathrm{d}\theta \bigg )^T\nonumber \\&\qquad \times {P_{7}}\bigg (\int _{t-\tau (t)}^{t-{\tau _1}}{{g}(x(s))}\mathrm{d}s-\frac{2}{\tau (t)-{\tau _1}}\int _{t-\tau (t)}^{t-{\tau _1}}\nonumber \\&\qquad \times \int _{t-\tau (t)}^{\theta } {g}(x(s)) \mathrm{d}s\mathrm{d}\theta \bigg )\Bigg \}-\frac{\tau _{12}^3}{\tau _2-\tau (t)}\nonumber \\&\qquad \times \Bigg \{\bigg (\int ^{t-\tau (t)}_{t-{\tau _2}}{{g}(x(s))}\mathrm{d}s\bigg )^T{P_{7}}\bigg (\int ^{t-\tau (t)}_{t-{\tau _2}}{{g}(x(s))}\mathrm{d}s\bigg )\nonumber \\&\qquad +3\bigg (\int ^{t-\tau (t)}_{t-{\tau _2}}{{g}(x(s))}\mathrm{d}s-\frac{2}{\tau _2-\tau (t)}\int _{t-{\tau _2}}^{t-\tau (t)}\nonumber \\&\qquad \times \int _{t-\tau _2}^{\theta } {g}(x(s)) \mathrm{d}s\mathrm{d}\theta \bigg )^T {P_{7}}\bigg (\int ^{t-\tau (t)}_{t-{\tau _2}}{{g}(x(s))}\mathrm{d}s\nonumber \\&\qquad -\frac{2}{\tau _2-\tau (t)}\int _{t-{\tau _2}}^{t-\tau (t)}\int _{t-\tau _2}^{\theta } {g}(x(s)) \mathrm{d}s\mathrm{d}\theta \bigg )\Bigg \}\nonumber \\&\quad =-\tau _{12}^2\Bigg (\frac{1}{\vartheta }v_2^T{P_{7}}v_2+\frac{1}{\varpi }v_4^T{P_{7}}v_4\Bigg )\nonumber \\&\qquad -3\Bigg (\frac{1}{\vartheta ^3}\omega _1^T{P_{7}}\omega _1+\frac{1}{\varpi ^3}\omega _2^T{P_{7}}\omega _2\Bigg ), \end{aligned}$$

(27)

where $\omega _1=[\tau (t)-{\tau _1}]v_2-2v_6,\ \omega _2=[\tau _2-\tau (t)]v_4-2v_8.$

Based on the reciprocal convex technique of Park et al. (2011) and Wu et al. (2012), from inequality (8) one has

$$\begin{aligned}&\left[ \begin{array}{c} \sqrt{\frac{\varpi }{\vartheta }}v_2\\ -\sqrt{\frac{\vartheta }{\varpi }}v_4\end{array}\right] ^T\bigg [\begin{array}{cc} P_{7}&{} S_1\\ *&{}P_{7}\end{array}\bigg ]\left[ \begin{array}{c} \sqrt{\frac{\varpi }{\vartheta }}v_2\\ -\sqrt{\frac{\vartheta }{\varpi }}v_4\end{array}\right] \ge 0,\nonumber \end{aligned}$$

which implies

$$\begin{aligned} \frac{\varpi }{\vartheta }v_2^T{P_{7}}v_2+\frac{\vartheta }{\varpi }v_4^T{P_{7}}v_4\ge v_2^T{S_1}v_4+v_4^T{S_1^T}v_2.\nonumber \end{aligned}$$

Noting that $\vartheta +\varpi =1,$ one obtains

$$\begin{aligned}&\frac{1}{\vartheta }v_2^T{P_{7}}v_2+\frac{1}{\varpi }v_4^T{P_{7}}v_4 \ge \bigg [\begin{array}{c} v_2\\ v_4\end{array}\bigg ]^T\bigg [\begin{array}{cc} P_{7}&{} S_1\\ *&{}P_{7}\end{array}\bigg ]\bigg [\begin{array}{c} v_2\\ v_4\end{array}\bigg ]. \end{aligned}$$

(28)

Moreover, from inequality (9) one gets

$$\begin{aligned}&\left[ \begin{array}{c} \sqrt{\frac{\varpi }{\vartheta }}\omega _1\\ \frac{\varpi }{\vartheta }\omega _1\\ \sqrt{\frac{\varpi ^3}{\vartheta ^3}}\omega _1\\ -\sqrt{\frac{\vartheta }{\varpi }}\omega _2\\ -\frac{\vartheta }{\varpi }\omega _2\\ -\sqrt{\frac{\vartheta ^3}{\varpi ^3}}\omega _2\end{array}\right] ^T\left[ \begin{array}{cccccc} 3P_{7} &{}0&{}0&{} S_2&{}0&{}0\\ *&{} 3P_{7}&{}0&{}0&{} S_3&{}0 \\ *&{} *&{}P_{7}&{}0&{}0&{} S_4\\ *&{} *&{}*&{}3P_{7}&{}0&{}0\\ *&{} *&{}*&{}*&{}3P_{7}&{}0\\ *&{} *&{} *&{}*&{}*&{}P_{7}\end{array}\right] \left[ \begin{array}{c} \sqrt{\frac{\varpi }{\vartheta }}\omega _1\\ \frac{\varpi }{\vartheta }\omega _1\\ \sqrt{\frac{\varpi ^3}{\vartheta ^3}}\omega _1\\ -\sqrt{\frac{\vartheta }{\varpi }}\omega _2\\ -\frac{\vartheta }{\varpi }\omega _2\\ -\sqrt{\frac{\vartheta ^3}{\varpi ^3}}\omega _2\end{array}\right] \ge 0,\nonumber \end{aligned}$$

which implies

$$\begin{aligned}&\bigg [3\frac{\varpi }{\vartheta }+3\Big (\frac{\varpi }{\vartheta }\Big )^2+\Big (\frac{\varpi }{\vartheta }\Big )^3\bigg ]\omega _1^T{P_{7}}\omega _1\\&\quad \quad +\bigg [3\frac{\varpi }{\vartheta }+3\Big (\frac{\vartheta }{\varpi }\Big )^2+\Big (\frac{\vartheta }{\varpi }\Big )^3\bigg ]\omega _2^T{P_{7}}\omega _2 \nonumber \\&\quad \ge \omega _1^T{S_2}\omega _2+\omega _2^T{S_2^T}\omega _1+\omega _1^T{S_3}\omega _2+\omega _2^T{S_3^T}\omega _1\\&\quad \quad +\omega _1^T{S_4}\omega _2+\omega _2^T{S_4^T}\omega _1.\nonumber \end{aligned}$$

Noting that $(\vartheta +\varpi )^3=1,$ one has

$$\begin{aligned}&\frac{1}{\vartheta ^3}\omega _1^T{P_{7}}\omega _1+\frac{1}{\varpi ^3}\omega _2^T{P_{7}}\omega _2 \ge \bigg [\begin{array}{c} \omega _1\\ \omega _2\end{array}\bigg ]^T\nonumber \\&\quad \times \bigg [\begin{array}{cc} P_{7}&{} S_2+S_3+S_4\\ *&{}P_{7}\end{array}\bigg ]\bigg [\begin{array}{c} \omega _1\\ \omega _2\end{array}\bigg ]. \end{aligned}$$

(29)

Based on inequalities (28)–(29), when $\tau _1<\tau (t)<\tau _2,$ inequality (27) can be estimated as

$$\begin{aligned}&-\tau _{12}^3\int _{t-{\tau _2}}^{t-{\tau _1}} {g}(x(s))^T{P_{7}}{g}(x(s))\mathrm{d}s\nonumber \\&\quad \le -\tau _{12}^2\Big [\begin{array}{c} v_2\\ v_4\end{array}\Big ]^T\bigg [\begin{array}{cc} P_{7}&{} S_1\\ *&{}P_{7}\end{array}\bigg ]\Big [\begin{array}{c} v_2\\ v_4\end{array}\Big ]\nonumber \\&\qquad -3\Big [\begin{array}{c} \omega _1\\ \omega _2\end{array}\Big ]^T\bigg [\begin{array}{cc} P_{7}&{} S_2+S_3+S_4\\ *&{}P_{7}\end{array}\bigg ]\Big [\begin{array}{c} \omega _1\\ \omega _2\end{array}\Big ]. \end{aligned}$$

(30)

For $\tau (t)=\tau _1$ or $\tau (t)=\tau _2,$ it is easy to see that inequality (30) still holds according to Lemma 1.

When $\tau _1>0,$ utilizing Lemma 1 and the Leibniz–Newton formula to (20) derives

$$\begin{aligned}&-\tau _1^3\int ^{t}_{t-{\tau _1}}{x(s)^T{P_{8}}x(s)}\mathrm{d}s\nonumber \\&\quad \le -\tau _1^2\Bigg \{\bigg (\int ^{t}_{t-{\tau _1}}{x(s)}\mathrm{d}s\bigg )^T{P_{8}}\bigg (\int ^{t}_{t-{\tau _1}}{x(s)}\mathrm{d}s\bigg )\nonumber \\&\qquad +3\bigg (\int ^{t}_{t-{\tau _1}}{x(s)}\mathrm{d}s-\frac{2}{\tau _1}\int ^{t}_{t-{\tau _1}}\int _{t-\tau _1}^{\theta } x(s) \mathrm{d}s\mathrm{d}\theta \bigg )^T\nonumber \\&\qquad \times {P_{8}}\bigg (\int ^{t}_{t-{\tau _1}}{x(s)}\mathrm{d}s-\frac{2}{\tau _1}\int ^{t}_{t-{\tau _1}}\int _{t-\tau _1}^{\theta } x(s) \mathrm{d}s\mathrm{d}\theta \bigg )\Bigg \}\nonumber \\&\quad =-\tau _1^2v_{10}^T{P_{8}}v_{10}-3(\tau _1v_{10}-2v_{12})^T{P_{8}}(\tau _1v_{10}-2v_{12}),\end{aligned}$$

(31)

$$\begin{aligned}&-\tau _1^3\int ^{t}_{t-{\tau _1}}{\dot{x}(s)^T{P_{9}}\dot{x}(s)}\mathrm{d}s\nonumber \\&\quad \le -\tau _1^2\Bigg \{\bigg (\int ^{t}_{t-{\tau _1}}{\dot{x}(s)}\mathrm{d}s\bigg )^T{P_{9}}\bigg (\int ^{t}_{t-{\tau _1}}{\dot{x}(s)}\mathrm{d}s\bigg )\nonumber \\&\qquad +3\bigg (\int ^{t}_{t-{\tau _1}}{\dot{x}(s)}\mathrm{d}s-\frac{2}{\tau _1}\int ^{t}_{t-{\tau _1}}\int _{t-\tau _1}^{\theta } \dot{x}(s) \mathrm{d}s\mathrm{d}\theta \bigg )^T\nonumber \\&\qquad \times {P_{9}}\bigg (\int ^{t}_{t-{\tau _1}}{\dot{x}(s)}\mathrm{d}s-\frac{2}{\tau _1}\int ^{t}_{t-{\tau _1}}\int _{t-\tau _1}^{\theta } \dot{x}(s) \mathrm{d}s\mathrm{d}\theta \bigg )\Bigg \}\nonumber \\&\qquad =-\tau _1^2(x_{t}-x_{\tau _1})^T{P_{9}}(x_{t}-x_{\tau _1})\nonumber \\&\qquad -3(\tau _1x_{t}+\tau _1x_{\tau _1}-2v_{10})^T\nonumber \\&\qquad \times {P_{9}}(\tau _1x_{t}+\tau _1x_{\tau _1}-2v_{10}). \end{aligned}$$

(32)

For $\tau _1=0,$ we have $v_{10}=v_{12}=0.$ Thus inequalities (31) and (32) still hold.

When $\tau _1<\tau (t)<\tau _2,$ applying Lemma 1 and the Leibniz–Newton formula to (21) gives

$$\begin{aligned}&-\tau _{12}\int ^{t-{\tau _1}}_{t-\tau (t)}{\dot{x}(s)^T{P_{10}}\dot{x}(s)}\mathrm{d}s\nonumber \\&\quad \le -\frac{\tau _{12}}{\tau (t)-\tau _1}\Bigg \{\bigg (\int ^{t-{\tau _1}}_{t-\tau (t)}{\dot{x}(s)}\mathrm{d}s\bigg )^T{P_{10}}\bigg (\int ^{t-{\tau _1}}_{t-\tau (t)}{\dot{x}(s)}\mathrm{d}s\bigg )\nonumber \\&\qquad +3\bigg (\int ^{t-{\tau _1}}_{t-\tau (t)}{\dot{x}(s)}\mathrm{d}s-\frac{2}{\tau (t)-\tau _1}\int ^{t-{\tau _1}}_{t-\tau (t)}\int _{t-\tau (t)}^{\theta } \dot{x}(s) \mathrm{d}s\mathrm{d}\theta \bigg )^T\nonumber \\&\qquad \times {P_{10}}\bigg (\int ^{t-{\tau _1}}_{t-\tau (t)}{\dot{x}(s)}\mathrm{d}s-\frac{2}{\tau (t)-\tau _1}\int ^{t-{\tau _1}}_{t-\tau (t)}\int _{t-\tau (t)}^{\theta } \dot{x}(s) \mathrm{d}s\mathrm{d}\theta \bigg )\Bigg \}\nonumber \\&\quad =-\frac{1}{\vartheta }\big \{(x_{\tau _1}-x_{\tau })^T{P_{10}}(x_{\tau _1}-x_{\tau })\\&\qquad +3(x_{\tau _1}+x_{\tau }-2v_{1})^T{P_{10}}(x_{\tau _1}+x_{\tau }-2v_{1})\big \},\nonumber \\&-\tau _{12}\int ^{t-\tau (t)}_{t-{\tau _2}}{\dot{x}(s)^T{P_{10}}\dot{x}(s)}\mathrm{d}s\nonumber \\&\quad \le -\frac{\tau _{12}}{\tau _2-\tau (t)}\Bigg \{\bigg (\int ^{t-\tau (t)}_{t-{\tau _2}}{\dot{x}(s)}\mathrm{d}s\bigg )^T{P_{10}}\bigg (\int ^{t-\tau (t)}_{t-{\tau _2}}{\dot{x}(s)}\mathrm{d}s\bigg )\nonumber \\&\qquad +3\bigg (\int ^{t-\tau (t)}_{t-{\tau _2}}{\dot{x}(s)}\mathrm{d}s-\frac{2}{\tau _2-\tau (t)}\int ^{t-\tau (t)}_{t-{\tau _2}}\int _{t-\tau _2}^{\theta } \dot{x}(s) \mathrm{d}s\mathrm{d}\theta \bigg )^T\nonumber \\&\qquad \times {P_{10}}\bigg (\int ^{t-\tau (t)}_{t-{\tau _2}}{\dot{x}(s)}\mathrm{d}s-\frac{2}{\tau _2-\tau (t)}\int ^{t-\tau (t)}_{t-{\tau _2}}\int _{t-\tau _2}^{\theta } \dot{x}(s) \mathrm{d}s\mathrm{d}\theta \bigg )\Bigg \}\nonumber \\&\quad =-\frac{1}{\varpi }\{(x_{\tau }-x_{\tau _2})^T{P_{10}}(x_{\tau }-x_{\tau _2})\end{aligned}$$

(33)

$$\begin{aligned}&\qquad +3(x_{\tau }+x_{\tau _2}-2v_{3})^T{P_{10}}(x_{\tau }+x_{\tau _2}-2v_{3})\}.\nonumber \end{aligned}$$

(34)

Obviously the following equalities hold for any $t>0$:

$$\begin{aligned}&\int _{t-\tau _2}^{t-\tau _1}\int _\theta ^{t-\tau _1} \dot{x}(s)^TP_{11}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\quad =\int _{t-\tau (t)}^{t-\tau _1}\int _\theta ^{t-\tau _1} \dot{x}(s)^TP_{11}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\quad \quad +[\tau _2-\tau (t)]\int _{t-\tau (t)}^{t-\tau _1} \dot{x}(s)^TP_{11}\dot{x}(s)\mathrm{d}s\nonumber \\&\quad \quad +\int _{t-\tau _2}^{t-\tau (t)}\int _\theta ^{t-\tau (t)} \dot{x}(s)^TP_{11}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta ,\end{aligned}$$

(35)

$$\begin{aligned}&\int _{t-\tau _2}^{t-\tau _1}\int _{t - \tau _2 } ^\theta \dot{x}(s)^TP_{12}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\quad =\int _{t-\tau (t)}^{t-\tau _1}\int _{t-\tau (t)}^\theta \dot{x}(s)^TP_{12}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\quad \quad +[\tau (t)-\tau _1]\int _{t - \tau _2 }^{t-\tau (t)} \dot{x}(s)^TP_{12}\dot{x}(s)\mathrm{d}s\nonumber \\&\quad \quad +\int _{t-\tau _2}^{t-\tau (t)}\int _{t-\tau _2} ^\theta \dot{x}(s)^TP_{12}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta . \end{aligned}$$

(36)

When $\tau _1<\tau (t)<\tau _2,$ from Lemma 2 and the Leibniz–Newton formula one gets

$$\begin{aligned}&\int _{t-\tau (t)}^{t-\tau _1}\int _\theta ^{t-\tau _1} \dot{x}(s)^TP_{11}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\quad \ge \frac{2}{[\tau (t)-\tau _1]^2}\bigg (\int _{t-\tau (t)}^{t-\tau _1}\int _\theta ^{t-\tau _1} \dot{x}(s)\mathrm{d}s\mathrm{d}\theta \bigg )^T\nonumber \\&\quad \quad \times P_{11}\bigg (\int _{t-\tau (t)}^{t-\tau _1}\int _\theta ^{t-\tau _1} \dot{x}(s)\mathrm{d}s\mathrm{d}\theta \bigg )\nonumber \\&\quad =2(x_{\tau _1}-v_1)^TP_{11}(x_{\tau _1}-v_1),\end{aligned}$$

(37)

$$\begin{aligned}&\int _{t-\tau _2}^{t-\tau (t)}\int _\theta ^{t-\tau (t)} \dot{x}(s)^TP_{11}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\quad \ge \frac{2}{[\tau _2-\tau (t)]^2}\bigg (\int _{t-\tau _2}^{t-\tau (t)}\int _\theta ^{t-\tau (t)} \dot{x}(s)\mathrm{d}s\mathrm{d}\theta \bigg )^T\nonumber \\&\quad \quad \times P_{11}\bigg (\int _{t-\tau _2}^{t-\tau (t)}\int _\theta ^{t-\tau (t)} \dot{x}(s)\mathrm{d}s\mathrm{d}\theta \bigg )\nonumber \\&\quad =2(x_{\tau }-v_3)^TP_{11}(x_{\tau }-v_3),\end{aligned}$$

(38)

$$\begin{aligned}&\int _{t-\tau (t)}^{t-\tau _1}\int _{t-\tau (t)}^\theta \dot{x}(s)^TP_{12}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\quad \ge \frac{2}{[\tau (t)-\tau _1]^2}\bigg (\int _{t-\tau (t)}^{t-\tau _1}\int _{t-\tau (t)}^\theta \dot{x}(s)\mathrm{d}s\mathrm{d}\theta \bigg )^T\nonumber \\&\quad \quad \times P_{12}\bigg (\int _{t-\tau (t)}^{t-\tau _1}\int _{t-\tau (t)}^\theta \dot{x}(s)\mathrm{d}s\mathrm{d}\theta \bigg )\nonumber \\&\quad =2(v_1-x_{\tau })^TP_{12}(v_1-x_{\tau }),\end{aligned}$$

(39)

$$\begin{aligned}&\int _{t-\tau _2}^{t-\tau (t)}\int _{t-\tau _2} ^\theta \dot{x}(s)^TP_{12}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\quad \ge \frac{2}{[\tau _2-\tau (t)]^2}\bigg (\int _{t-\tau _2}^{t-\tau (t)}\int _{t-\tau _2} ^\theta \dot{x}(s)\mathrm{d}s\mathrm{d}\theta \bigg )^T\nonumber \\&\quad \quad \times P_{12}\bigg (\int _{t-\tau _2}^{t-\tau (t)}\int _{t-\tau _2} ^\theta \dot{x}(s)\mathrm{d}s\mathrm{d}\theta \bigg )\nonumber \\&\quad =2(v_3-x_{\tau _2})^TP_{12}(v_3-x_{\tau _2}). \end{aligned}$$

(40)

Based on (4), inequalities (37)–(40) still hold for any $t>0$ with $\tau (t)=\tau _1$ and $\tau (t)=\tau _2.$

Similar to inequalities (33) and (34), when $\tau _1<\tau (t)<\tau _2,$ from Lemma 1 and the Leibniz–Newton formula, one derives

$$\begin{aligned}&[\tau _2-\tau (t)]\int ^{t-{\tau _1}}_{t-\tau (t)}{\dot{x}(s)^T{P_{11}}\dot{x}(s)}\mathrm{d}s\nonumber \\&\quad \ge \frac{\varpi }{\vartheta }\big \{(x_{\tau _1}-x_{\tau })^T{P_{11}}(x_{\tau _1}-x_{\tau })\nonumber \\&\quad \quad +3(x_{\tau _1}+x_{\tau }-2v_{1})^T{P_{11}}(x_{\tau _1}+x_{\tau }-2v_{1})\big \},\end{aligned}$$

(41)

$$\begin{aligned}&[\tau (t)-\tau _1]\int ^{t-\tau (t)}_{t-{\tau _2}}{\dot{x}(s)^T{P_{12}}\dot{x}(s)}\mathrm{d}s\nonumber \\&\quad \ge \frac{\vartheta }{\varpi }\big \{(x_{\tau }-x_{\tau _2})^T{P_{12}}(x_{\tau }-x_{\tau _2}) +3(x_{\tau }+x_{\tau _2}-2v_{3})^T\nonumber \\&\quad \quad \times {P_{12}}(x_{\tau }+x_{\tau _2}-2v_{3})\big \}. \end{aligned}$$

(42)

According to inequalities (10) and (11), the following inequalities hold:

$$\begin{aligned}&\left[ \begin{array}{c} \sqrt{\frac{\varpi }{\vartheta }}(x_{\tau _1}-x_{\tau })\\ -\sqrt{\frac{\vartheta }{\varpi }}(x_{\tau }-x_{\tau _2})\end{array}\right] ^T\bigg [\begin{array}{cc} P_{10}+P_{11} &{} S_5\\ *&{} P_{10}+P_{12} \end{array}\bigg ]\\&\quad \times \left[ \begin{array}{c} \sqrt{\frac{\varpi }{\vartheta }}(x_{\tau _1}-x_{\tau })\\ -\sqrt{\frac{\vartheta }{\varpi }}(x_{\tau }-x_{\tau _2})\end{array}\right] \ge 0, \nonumber \\&\left[ \begin{array}{c} \sqrt{\frac{\varpi }{\vartheta }}(x_{\tau _1}+x_{\tau }-2v_{1})\\ -\sqrt{\frac{\vartheta }{\varpi }}(x_{\tau }+x_{\tau _2}-2v_{3})\end{array}\right] ^T\bigg [\begin{array}{cc} P_{10}+P_{11} &{} S_6\\ *&{} P_{10}+P_{12} \end{array}\bigg ]\\&\quad \times \left[ \begin{array}{c} \sqrt{\frac{\varpi }{\vartheta }}(x_{\tau _1}+x_{\tau }-2v_{1})\\ -\sqrt{\frac{\vartheta }{\varpi }}(x_{\tau }+x_{\tau _2}-2v_{3})\end{array}\right] \ge 0,\nonumber \end{aligned}$$

which implies

$$\begin{aligned}&\frac{\varpi }{\vartheta }(x_{\tau _1}-x_{\tau })^T(P_{10}+P_{11})(x_{\tau _1}-x_{\tau }) +\frac{\vartheta }{\varpi }(x_{\tau }-x_{\tau _2})^T\nonumber \\&\quad \quad \times (P_{10}+P_{12})(x_{\tau }-x_{\tau _2})\nonumber \\&\quad \ge (x_{\tau _1}-x_{\tau })^T{S_5}(x_{\tau }-x_{\tau _2})+(x_{\tau }-x_{\tau _2})^T{S_5^T}(x_{\tau _1}-x_{\tau }),\nonumber \\&\frac{\varpi }{\vartheta }(x_{\tau _1}+x_{\tau }-2v_{1})^T(P_{10}+P_{11})(x_{\tau _1}+x_{\tau }-2v_{1})\\&\quad \quad +\frac{\vartheta }{\varpi }(x_{\tau }+x_{\tau _2}-2v_{3})^T(P_{10}+P_{12})(x_{\tau }+x_{\tau _2}-2v_{3})\nonumber \\&\quad \ge (x_{\tau _1}+x_{\tau }-2v_{1})^T{S_6}(x_{\tau }+x_{\tau _2}-2v_{3})\\&\quad \quad +(x_{\tau }+x_{\tau _2}-2v_{3})^T{S_6^T}(x_{\tau _1}+x_{\tau }-2v_{1}).\nonumber \end{aligned}$$

Noting that $\vartheta +\varpi =1,$ one obtains

$$\begin{aligned}&\frac{1}{\vartheta }(x_{\tau _1}-x_{\tau })^TP_{10}(x_{\tau _1}-x_{\tau }) +\frac{\varpi }{\vartheta }(x_{\tau _1}-x_{\tau })^T\nonumber \\&\qquad \times P_{11}(x_{\tau _1}-x_{\tau })+\frac{1}{\varpi }(x_{\tau }-x_{\tau _2})^TP_{10}(x_{\tau }-x_{\tau _2})\nonumber \\&\quad \quad +\frac{\vartheta }{\varpi }(x_{\tau }-x_{\tau _2})^TP_{12}(x_{\tau }-x_{\tau _2})\nonumber \\&\quad \ge \bigg [\begin{array}{c} x_{\tau _1}-x_{\tau }\\ x_{\tau }-x_{\tau _2}\end{array}\bigg ]^T\bigg [\begin{array}{cc} P_{10}&{} S_5\\ *&{}P_{10}\end{array}\bigg ]\bigg [\begin{array}{c} x_{\tau _1}-x_{\tau }\\ x_{\tau }-x_{\tau _2}\end{array}\bigg ],\\&\frac{1}{\vartheta }(x_{\tau _1}+x_{\tau }-2v_{1})^TP_{10}(x_{\tau _1}+x_{\tau }-2v_{1})\nonumber \\&\quad \quad +\frac{\varpi }{\vartheta }(x_{\tau _1}+x_{\tau }-2v_{1})^TP_{11}(x_{\tau _1}+x_{\tau }-2v_{1})\nonumber \\&\quad \quad +\frac{1}{\varpi }(x_{\tau }+x_{\tau _2}-2v_{3})^TP_{10}(x_{\tau }+x_{\tau _2}-2v_{3})\nonumber \\&\quad \quad +\frac{\vartheta }{\varpi }(x_{\tau }+x_{\tau _2}-2v_{3})^TP_{12}(x_{\tau }+x_{\tau _2}-2v_{3})\nonumber \end{aligned}$$

(43)

$$\begin{aligned} \ge&\bigg [\begin{array}{c} x_{\tau _1}+x_{\tau }-2v_{1}\\ x_{\tau }+x_{\tau _2}-2v_{3}\end{array}\bigg ]^T\bigg [\begin{array}{cc} P_{10}&{} S_6\\ *&{}P_{10}\end{array}\bigg ]\bigg [\begin{array}{c} x_{\tau _1}+x_{\tau }-2v_{1}\\ x_{\tau }+x_{\tau _2}-2v_{3}\end{array}\bigg ]. \end{aligned}$$

(44)

Substituting (33)–(44) into (21) derives

$$\begin{aligned}&D^+V_{5}(t,x_t)\le \tau _{12}\dot{x}(t)^T\Big [{P_{10}+\frac{\tau _{12}}{2}(P_{11}+P_{12})}\Big ]\dot{x}(t)\nonumber \\&\quad -\bigg [\begin{array}{c} x_{\tau _1}-x_{\tau }\\ x_{\tau }-x_{\tau _2}\end{array}\bigg ]^T\bigg [\begin{array}{cc} P_{10}&{} S_5\\ *&{}P_{10}\end{array}\bigg ]\bigg [\begin{array}{c} x_{\tau _1}-x_{\tau }\\ x_{\tau }-x_{\tau _2}\end{array}\bigg ]\nonumber \\&\quad -3\bigg [\begin{array}{c} x_{\tau _1}+x_{\tau }-2v_{1}\\ x_{\tau }+x_{\tau _2}-2v_{3}\end{array}\bigg ]^T\bigg [\begin{array}{cc} P_{10}&{} S_6\\ *&{}P_{10}\end{array}\bigg ]\bigg [\begin{array}{c} x_{\tau _1}+x_{\tau }-2v_{1}\\ x_{\tau }+x_{\tau _2}-2v_{3}\end{array}\bigg ]\nonumber \\&\quad -2(x_{\tau _1}-v_1)^TP_{11}(x_{\tau _1}-v_1)-2(x_{\tau }-v_3)^TP_{11}(x_{\tau }-v_3)\nonumber \\&\quad -2(v_1-x_{\tau })^TP_{12}(v_1-x_{\tau })-2(v_3-x_{\tau _2})^TP_{12}(v_3-x_{\tau _2}). \end{aligned}$$

(45)

It is easily proved that inequality (45) holds for any $t>0$ with $\tau (t)=\tau _1$ or $\tau (t)=\tau _2.$

On the other hand, from Lemma 2 and the Leibniz–Newton formula one derives

$$\begin{aligned}&-\frac{\tau _{1}^2}{2}\int _{t-{\tau _1}}^{t}\left( \int _{t-{\tau _1}}^{\gamma }\dot{x}(s)^TP_{13}\dot{x}(s)\mathrm{d}s+\int ^{t}_{\gamma } \dot{x}(s)^TP_{14}\dot{x}(s)\mathrm{d}s\right) \mathrm{d}{\gamma }\nonumber \\&\quad \le -\left( \int _{t-{\tau _1}}^{t}\int _{t-{\tau _1}}^{\gamma }\dot{x}(s)\mathrm{d}s\mathrm{d}{\gamma }\right) ^TP_{13}\left( \int _{t-{\tau _1}}^{t}\int _{t-{\tau _1}}^{\gamma }\dot{x}(s)\mathrm{d}s\mathrm{d}{\gamma }\right) \nonumber \\&\qquad -\left( \int _{t-{\tau _1}}^{t}\int ^{t}_{\gamma } \dot{x}(s)\mathrm{d}s\mathrm{d}{\gamma }\right) ^TP_{14}\left( \int _{t-{\tau _1}}^{t}\int ^{t}_{\gamma } \dot{x}(s)\mathrm{d}s\mathrm{d}{\gamma }\right) \nonumber \\&\quad =\qquad -(v_{10}-\tau _1x_{\tau _1})^TP_{13}(v_{10}-\tau _1x_{\tau _1})-(\tau _1x_{t}-v_{10})^T\nonumber \\&\qquad \times P_{14}(\tau _1x_{t}-v_{10}),\end{aligned}$$

(46)

$$\begin{aligned}&\qquad -\frac{\tau _{1}^3}{6}\int _{t-{\tau _1}}^{t}\left( {\int _{\gamma }^{t}\int _\theta ^{t} {{\dot{x}{{(s)}^T}P_{15}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\right. \nonumber \\&+\left. {\int _{t - \tau _1} ^\gamma \int _{t - \tau _1} ^\theta {{\dot{x}{{(s)}^T}{P_{16}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\right) \mathrm{d}{\gamma }\nonumber \\&\quad \le -\left( \int _{t-{\tau _1}}^{t}{\int _{\gamma }^{t}\int _\theta ^{t} {{\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\right) ^T\nonumber \\&\qquad \times P_{15}\left( \int _{t-{\tau _1}}^{t}{\int _{\gamma }^{t}\int _\theta ^{t} {{\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\right) \nonumber \\&\qquad -\left( \int _{t-{\tau _1}}^{t}{\int _{t - \tau _1} ^\gamma \int _{t - \tau _1} ^\theta {{\dot{x}{(s)}\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\right) ^T\nonumber \\&\qquad \times {P_{16}}\left( \int _{t-{\tau _1}}^{t}{\int _{t - \tau _1} ^\gamma \int _{t - \tau _1} ^\theta {{\dot{x}{(s)}\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\right) \nonumber \\&\quad =\qquad -\Big (\frac{1}{2}\tau _1^2x_{t}-v_{13}\Big )^TP_{15}\Big (\frac{1}{2}\tau _1^2x_{t}-v_{13}\Big )\nonumber \\&\qquad -\Big (v_{12}-\frac{1}{2}\tau _1^2x_{\tau _1}\Big )^TP_{16}\Big (v_{12}-\frac{1}{2}\tau _1^2x_{\tau _1}\Big ). \end{aligned}$$

(47)

Obviously the following equalities hold for any $t>0$

$$\begin{aligned}&\int _{t-{\tau _2}}^{t-{\tau _1}}{\int _{\gamma }^{t-{\tau _1}}\int _\theta ^{t-{\tau _1}} {{\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad =\int _{t-{\tau (t)}}^{t-{\tau _1}}{\int _{\gamma }^{t-{\tau _1}}\int _\theta ^{t-{\tau _1}} {{\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad \quad +\int ^{t-{\tau (t)}}_{t-{\tau _2}}{\int _{\gamma }^{t-{\tau (t)}}\int _\theta ^{t-{\tau (t)}} {{\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad \quad +\int _{t-{\tau _2}}^{t-{\tau (t)}}{\int _{\gamma }^{t-{\tau (t)}}\int _{t-{\tau (t)}}^{t-{\tau _1}} {{\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad \quad +\int _{t-{\tau _2}}^{t-{\tau (t)}}{\int _{t-{\tau (t)}}^{t-{\tau _1}}\int _\theta ^{t-{\tau _1}} {{\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }. \end{aligned}$$

(48)

When $\tau _1<\tau (t)<\tau _2,$ from Lemma 2 and the Leibniz–Newton formula one gets

$$\begin{aligned}&\tau _{12}\int _{t-{\tau (t)}}^{t-{\tau _1}}{\int _{\gamma }^{t-{\tau _1}}\int _\theta ^{t-{\tau _1}} {{\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad \ge \frac{6\tau _{12}}{[\tau (t)-{\tau _1}]^3}\bigg (\int _{t-{\tau (t)}}^{t-{\tau _1}}{\int _{\gamma }^{t-{\tau _1}}\int _\theta ^{t-{\tau _1}} {{\dot{x}{{(s)}}\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\bigg )^T\nonumber \\&\quad \quad \times P_{17}\bigg (\int _{t-{\tau (t)}}^{t-{\tau _1}}{\int _{\gamma }^{t-{\tau _1}}\int _\theta ^{t-{\tau _1}} {{\dot{x}{{(s)}}\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\bigg )\nonumber \\&\quad =\frac{3}{2\vartheta }\{[\tau (t)-{\tau _1}]x_{\tau _1}-v_{9}\}^TP_{17}\{[\tau (t)-{\tau _1}]x_{\tau _1}-v_{9}\},\end{aligned}$$

(49)

$$\begin{aligned}&\tau _{12}\int ^{t-{\tau (t)}}_{t-{\tau _2}}{\int _{\gamma }^{t-{\tau (t)}}\int _\theta ^{t-{\tau (t)}} {{\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad \ge \frac{6\tau _{12}}{[{\tau _2}-\tau (t)]^3}\bigg (\int ^{t-{\tau (t)}}_{t-{\tau _2}}{\int _{\gamma }^{t-{\tau (t)}}\int _\theta ^{t-{\tau (t)}} {{\dot{x}{{(s)}}\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\bigg )^T\nonumber \\&\quad \quad \times P_{17}\bigg (\int ^{t-{\tau (t)}}_{t-{\tau _2}}{\int _{\gamma }^{t-{\tau (t)}}\int _\theta ^{t-{\tau (t)}} {{\dot{x}{{(s)}}\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\bigg )\nonumber \\&\quad =\frac{3}{2\varpi }\{[{\tau _2}-\tau (t)]x_{\tau }-v_{11}\}^TP_{17}\{[{\tau _2}-\tau (t)]x_{\tau }-v_{11}\},\end{aligned}$$

(50)

$$\begin{aligned}&\int _{t-{\tau _2}}^{t-{\tau (t)}}{\int _{t-{\tau (t)}}^{t-{\tau _1}}\int _\theta ^{t-{\tau _1}} {{\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad =[{\tau _2}-\tau (t)]{\int _{t-{\tau (t)}}^{t-{\tau _1}}\int _\theta ^{t-{\tau _1}} {{\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\nonumber \\&\quad \ge \frac{2[{\tau _2}-\tau (t)]}{[\tau (t)-{\tau _1}]^2}\bigg ({\int _{t-{\tau (t)}}^{t-{\tau _1}}\int _\theta ^{t-{\tau _1}} {{\dot{x}{{(s)}}\mathrm{d}s\mathrm{d}\theta }}}\bigg )^T\nonumber \\&\quad \quad \times P_{17}\bigg ({\int _{t-{\tau (t)}}^{t-{\tau _1}}\int _\theta ^{t-{\tau _1}} {{\dot{x}{{(s)}}\mathrm{d}s\mathrm{d}\theta }}}\bigg )\nonumber \\&\quad =2[{\tau _2}-\tau (t)](x_{\tau _1}-v_{1})^TP_{17}(x_{\tau _1}-v_{1}). \end{aligned}$$

(51)

Based on (4), inequalities (49)–(51) still hold for any $t>0$ with $\tau (t)=\tau _1$ and $\tau (t)=\tau _2.$

It is easy to derive that $\Xi _5>0$ from inequalities (12), based on the reciprocal convex technique of Park et al. (2011); Wu et al. 2012), one has

$$\begin{aligned}&\quad \frac{1}{\vartheta }\{[\tau (t)-{\tau _1}]x_{\tau _1}-v_{9}\}^TP_{17}\{[\tau (t)-{\tau _1}]x_{\tau _1}-v_{9}\}\\&\quad +\frac{1}{\varpi }\{[{\tau _2}-\tau (t)]x_{\tau }-v_{11}\}^TP_{17}\{[{\tau _2}-\tau (t)]x_{\tau }-v_{11}\}\\&\ge \bigg [\begin{array}{c}(\tau (t)-{\tau _1})x_{\tau _1}-v_{9} \\ ({\tau _2}-\tau (t))x_{\tau }-v_{11} \end{array}\bigg ]^T\bigg [\begin{array}{cc} P_{17}&{} S_{10}\\ *&{}P_{17}\end{array}\bigg ]\\&\qquad \bigg [\begin{array}{c}(\tau (t)-{\tau _1})x_{\tau _1}-v_{9} \\ ({\tau _2}-\tau (t))x_{\tau }-v_{11} \end{array}\bigg ]. \end{aligned}$$

Similar to inequalities (33) and (34), when $\tau _1<\tau (t)<\tau _2,$ from Lemma 1 nd the Leibniz–Newton formula one gets

$$\begin{aligned}&\tau _{12}\int _{t-{\tau _2}}^{t-{\tau (t)}}{\int _{\gamma }^{t-{\tau (t)}}\int _{t-{\tau (t)}}^{t-{\tau _1}} {{\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad =\frac{\tau _{12}}{2}[{\tau _2}-{\tau (t)}]^2\int _{t-{\tau (t)}}^{t-{\tau _1}}\dot{x}(s)^TP_{17}\dot{x}(s)\mathrm{d}s\nonumber \\&\quad \ge \frac{1}{2\vartheta }\{[({\tau _2}-{\tau (t)})(x_{\tau _1}-x_{\tau })]^T{P_{17}}[({\tau _2}-{\tau (t)})(x_{\tau _1} -x_{\tau })]\nonumber \\&\quad \quad +3[({\tau _2}-{\tau (t)})(x_{\tau _1}+x_{\tau }-2v_{1})]^T\nonumber \\&\quad \quad \times {P_{17}}\big [({\tau _2}-{\tau (t)})(x_{\tau _1}+x_{\tau }-2v_{1})\big ]\}. \end{aligned}$$

(52)

Obviously the following equalities hold for any $t>0$

$$\begin{aligned}&\int _{t-{\tau _2}}^{t-{\tau _1}}{\int _{t - \tau _2} ^\gamma \int _{t - \tau _2} ^\theta {{\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad =\int _{t-{\tau (t)}}^{t-{\tau _1}}{\int _{t - \tau _2} ^{t-{\tau (t)}}\int _{t - \tau _2} ^\theta {{\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad \quad +\int _{t-{\tau (t)}}^{t-{\tau _1}}{\int _{t-{\tau (t)}}^\gamma \int _{t - \tau _2} ^{t-{\tau (t)}}{{\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad \quad +\int _{t-{\tau (t)}}^{t-{\tau _1}}{\int _{t-{\tau (t)}}^\gamma \int _{t-{\tau (t)}}^\theta {{\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad \quad +\int _{t-{\tau _2}}^{t-{\tau (t)}}{\int _{t - \tau _2} ^\gamma \int _{t - \tau _2} ^\theta {{\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }. \end{aligned}$$

(53)

When $\tau _1<\tau (t)<\tau _2,$ from Lemma 2 and the Leibniz–Newton formula one gets

$$\begin{aligned}&\int _{t-{\tau (t)}}^{t-{\tau _1}}{\int _{t - \tau _2} ^{t-{\tau (t)}}\int _{t - \tau _2} ^\theta {{\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad =[\tau (t)-\tau _1]{\int _{t - \tau _2} ^{t-{\tau (t)}}\int _{t - \tau _2} ^\theta {{\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\nonumber \\&\quad \ge \frac{2[\tau (t)-\tau _1]}{[\tau _2-\tau (t)]^2}\bigg ({\int _{t - \tau _2} ^{t-{\tau (t)}}\int _{t - \tau _2} ^\theta {{\dot{x}{{(s)}}\mathrm{d}s\mathrm{d}\theta }}}\bigg )^T\nonumber \\&\quad \quad \times {P_{18}}\bigg ({\int _{t - \tau _2} ^{t-{\tau (t)}}\int _{t - \tau _2} ^\theta {{\dot{x}{{(s)}}\mathrm{d}s\mathrm{d}\theta }}}\bigg )\nonumber \\&\quad =2[\tau (t)-\tau _1](v_3-x_{\tau _2})^T{P_{18}}(v_3-x_{\tau _2}),\end{aligned}$$

(54)

$$\begin{aligned}&\tau _{12}\int _{t-{\tau (t)}}^{t-{\tau _1}}{\int _{t-{\tau (t)}}^\gamma \int _{t-{\tau (t)}}^\theta {{\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad \ge \frac{6\tau _{12}}{[\tau (t)-\tau _1]^3}\bigg (\int _{t-{\tau (t)}}^{t-{\tau _1}}{\int _{t-{\tau (t)}}^\gamma \int _{t-{\tau (t)}}^\theta {{\dot{x}{{(s)}}\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\bigg )^T\nonumber \\&\quad \quad \times {P_{18}}\bigg (\int _{t-{\tau (t)}}^{t-{\tau _1}}{\int _{t-{\tau (t)}}^\gamma \int _{t-{\tau (t)}}^\theta {{\dot{x}{{(s)}}\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\bigg )\nonumber \\&\quad =\frac{3}{2\vartheta }\{[\tau (t)-\tau _1]x_{\tau }-v_5\}^T{P_{18}}\{[\tau (t)-\tau _1]x_{\tau }-v_5\},\end{aligned}$$

(55)

$$\begin{aligned}&\tau _{12}\int _{t-{\tau _2}}^{t-{\tau (t)}}{\int _{t - \tau _2} ^\gamma \int _{t - \tau _2} ^\theta {{\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad \ge \frac{6\tau _{12}}{[\tau _2-\tau (t)]^3}\bigg (\int _{t-{\tau _2}}^{t-{\tau (t)}}{\int _{t - \tau _2} ^\gamma \int _{t - \tau _2} ^\theta {{\dot{x}{{(s)}}\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\bigg )^T\nonumber \\&\quad \quad \times {P_{18}}\bigg (\int _{t-{\tau _2}}^{t-{\tau (t)}}{\int _{t - \tau _2} ^\gamma \int _{t - \tau _2} ^\theta {{\dot{x}{{(s)}}\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\bigg )\nonumber \\&\quad =\frac{3}{2\varpi }\{[\tau _2-\tau (t)]x_{\tau _2}-v_7\}^T{P_{18}}\{[\tau _2-\tau (t)]x_{\tau _2}-v_7\}, \end{aligned}$$

(56)

Based on (4), inequalities (54)–(56) still hold for any $t>0$ with $\tau (t)=\tau _1$ and $\tau (t)=\tau _2.$

It is easy to derive that $\Xi _6>0$ from inequalities (12), based on the reciprocal convex technique of Park et al. (2011); Wu et al. 2012), one has

$$\begin{aligned}&\quad \frac{1}{\vartheta }\{[\tau (t)-\tau _1]x_{\tau }-v_5\}^T{P_{18}}\{[\tau (t)-\tau _1]x_{\tau }-v_5\}\\&\quad +\frac{1}{\varpi }\{[\tau _2-\tau (t)]x_{\tau _2}-v_7\}^T{P_{18}}\{[\tau _2-\tau (t)]x_{\tau _2}-v_7\}\\&\ge \bigg [\begin{array}{c}(\tau (t)-\tau _1)x_{\tau }-v_5 \\ ({\tau _2}-\tau (t))x_{\tau _2}-v_7 \end{array}\bigg ]^T\bigg [\begin{array}{cc} P_{18}&{} S_{11}\\ *&{}P_{18}\end{array}\bigg ]\\&\quad \times \bigg [\begin{array}{c}(\tau (t)-\tau _1)x_{\tau }-v_5 \\ ({\tau _2}-\tau (t))x_{\tau _2}-v_7 \end{array}\bigg ]. \end{aligned}$$

Similar to inequalities (33) and (34), when $\tau _1<\tau (t)<\tau _2,$ from Lemma 1 and the Leibniz–Newton formula one gets

$$\begin{aligned}&\tau _{12}\int _{t-{\tau (t)}}^{t-{\tau _1}}{\int _{t-{\tau (t)}}^\gamma \int _{t - \tau _2} ^{t-{\tau (t)}}{{\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad =\frac{\tau _{12}}{2}[{\tau (t)}-{\tau _1}]^2\int _{t - \tau _2} ^{t-{\tau (t)}}\dot{x}(s)^TP_{18}\dot{x}(s)\mathrm{d}s\nonumber \\&\quad \ge \frac{1}{2\varpi }\Big \{\big [({\tau (t)}-{\tau _1})(x_{\tau }-x_{\tau _2})\big ]^T{P_{18}} \big [({\tau (t)}-{\tau _1})(x_{\tau }-x_{\tau _2})\big ]\nonumber \\&\quad \quad +3\big [({\tau (t)}-{\tau _1})(x_{\tau }+x_{\tau _2}-2v_{3})\big ]^T\nonumber \\&\quad \quad \times {P_{18}}\big [({\tau (t)}-{\tau _1})(x_{\tau }+x_{\tau _2}-2v_{3})\big ]\Big \}. \end{aligned}$$

(57)

Applying inequalities (12) derives $\Xi _2>0,\Xi _3>0$; thus one has the following inequalities

$$\begin{aligned}&\left[ \begin{array}{c} \sqrt{\frac{\varpi }{\vartheta }}[{\tau _2}-\tau (t)](x_{\tau _1}-x_{\tau })\\ -\sqrt{\frac{\vartheta }{\varpi }}[\tau (t)-{\tau _1}](x_{\tau }-x_{\tau _2})\end{array}\right] ^T\bigg [\begin{array}{cc} P_{17} &{} S_7\\ *&{} P_{18} \end{array}\bigg ]\nonumber \\&\qquad \times \left[ \begin{array}{c} \sqrt{\frac{\varpi }{\vartheta }}[{\tau _2}-\tau (t)](x_{\tau _1}-x_{\tau })\\ -\sqrt{\frac{\vartheta }{\varpi }}[\tau (t)-{\tau _1}](x_{\tau }-x_{\tau _2})\end{array}\right] \ge 0, \nonumber \end{aligned}$$

$$\begin{aligned}&\left[ \begin{array}{c} \sqrt{\frac{\varpi }{\vartheta }}[{\tau _2}-\tau (t)](x_{\tau _1}+x_{\tau }-2v_{1})\\ -\sqrt{\frac{\vartheta }{\varpi }}[\tau (t)-{\tau _1}](x_{\tau }+x_{\tau _2}-2v_{3})\end{array}\right] ^T\bigg [\begin{array}{cc} P_{17} &{} S_8\\ *&{} P_{18} \end{array}\bigg ]\nonumber \\&\qquad \times \left[ \begin{array}{c} \sqrt{\frac{\varpi }{\vartheta }}[{\tau _2}-\tau (t)](x_{\tau _1}+x_{\tau }-2v_{1})\\ -\sqrt{\frac{\vartheta }{\varpi }}[\tau (t)-{\tau _1}](x_{\tau }+x_{\tau _2}-2v_{3})\end{array}\right] \ge 0.\nonumber \end{aligned}$$

Noting that $\vartheta +\varpi =1,$ similar to inequality (28), from the reciprocal convex technique one has

$$\begin{aligned}&\frac{1}{\vartheta }[({\tau _2}-{\tau (t)})(x_{\tau _1}-x_{\tau })]^T{P_{17}}[({\tau _2}-{\tau (t)})(x_{\tau _1} -x_{\tau })]\nonumber \\&\quad \quad +\frac{1}{\varpi }[({\tau (t)}-{\tau _1})(x_{\tau }-x_{\tau _2})]^T{P_{18}} [({\tau (t)}-{\tau _1})(x_{\tau }-x_{\tau _2})]\nonumber \\&\quad \ge \bigg [\begin{array}{c} [{\tau _2}-\tau (t)](x_{\tau _1}-x_{\tau })\\ \left[ \tau (t)-{\tau _1}\right] (x_{\tau }-x_{\tau _2})\end{array}\bigg ]^T\bigg [\begin{array}{cc} P_{17} &{} S_7\\ *&{} P_{18}\end{array}\bigg ]\nonumber \\&\qquad \times \bigg [\begin{array}{c} [{\tau _2}-\tau (t)](x_{\tau _1}-x_{\tau })\\ \left[ \tau (t)-{\tau _1}\right] (x_{\tau }-x_{\tau _2})\end{array}\bigg ],\end{aligned}$$

(58)

$$\begin{aligned}&\frac{1}{\vartheta }[({\tau _2}-{\tau (t)})(x_{\tau _1}+x_{\tau }-2v_{1})]^T{P_{17}}[({\tau _2}-{\tau (t)})\nonumber \\&\qquad \times (x_{\tau _1} +x_{\tau }-2v_{1})]+\frac{1}{\varpi }[({\tau (t)}-{\tau _1})\nonumber \\&\quad \quad (x_{\tau }+x_{\tau _2}-2v_{3})]^T{P_{18}}[({\tau (t)}-{\tau _1})(x_{\tau } +x_{\tau _2}-2v_{3})]\nonumber \\&\quad \ge \bigg [\begin{array}{c} [{\tau _2}-\tau (t)](x_{\tau _1}+x_{\tau }-2v_{1})\\ \left[ \tau (t)-{\tau _1}\right] (x_{\tau }+x_{\tau _2}-2v_{3})\end{array}\bigg ]^T\bigg [\begin{array}{cc} P_{17} &{} S_8\\ *&{} P_{18} \end{array}\bigg ]\nonumber \\&\quad \quad \times \bigg [\begin{array}{c} [{\tau _2}-\tau (t)](x_{\tau _1}+x_{\tau }-2v_{1})\\ \left[ \tau (t)-{\tau _1}\right] (x_{\tau }+x_{\tau _2}-2v_{3})\end{array}\bigg ]. \end{aligned}$$

(59)

Substituting (58)–(59) into (52) and (57) derives

$$\begin{aligned}&\tau _{12}\int _{t-{\tau _2}}^{t-{\tau (t)}}{\int _{\gamma }^{t-{\tau (t)}}\int _{t-{\tau (t)}}^{t-{\tau _1}} {{\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad \quad +\tau _{12}\int _{t-{\tau (t)}}^{t-{\tau _1}}{\int _{t-{\tau (t)}}^\gamma \int _{t - \tau _2} ^{t-{\tau (t)}}{{\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\quad \ge \bigg [\begin{array}{c} [{\tau _2}-\tau (t)](x_{\tau _1}-x_{\tau })\\ \left[ \tau (t)-{\tau _1}\right] (x_{\tau }-x_{\tau _2})\end{array}\bigg ]^T\Xi _2\bigg [\begin{array}{c} [{\tau _2}-\tau (t)](x_{\tau _1}-x_{\tau })\\ \left[ \tau (t)-{\tau _1}\right] (x_{\tau }-x_{\tau _2})\end{array}\bigg ]\nonumber \\&\quad \quad +\bigg [\begin{array}{c} [{\tau _2}-\tau (t)](x_{\tau _1}+x_{\tau }-2v_{1})\\ \left[ \tau (t)-{\tau _1}\right] (x_{\tau }+x_{\tau _2}-2v_{3})\end{array}\bigg ]^T\nonumber \\&\quad \quad \times \Xi _3\bigg [\begin{array}{c} [{\tau _2}-\tau (t)](x_{\tau _1}+x_{\tau }-2v_{1})\\ \left[ \tau (t)-{\tau _1}\right] (x_{\tau }+x_{\tau _2}-2v_{3})\end{array}\bigg ]. \end{aligned}$$

(60)

Based on (4), it is easily proved that inequality (60) holds for any $t>0$ with $\tau (t)=\tau _1$ or $\tau (t)=\tau _2.$

From Cauchy–Schwarz inequality and equality (2), the following inequality holds

$$\begin{aligned}&\sum \limits _{j=1}^n {r_{1j}\int _0^\infty {{\kappa _j}(\theta )g_j^2({x_j}(t ))} } \mathrm{d}\theta \nonumber \\ {}&\quad -\sum \limits _{j= 1}^n {r_{1j}\int _0^\infty {{\kappa _j}(\theta )g_j^2({x_j}(t-\theta ))} } \mathrm{d}\theta \nonumber \\ {}&\quad = {g}(x_{t})^TR_{1}g(x_{t})-\sum \limits _{j=1}^n r_{1j}\int _0^\infty {{\kappa _j}(\theta )\mathrm{d}\theta }\nonumber \\&\qquad \times \int _0^\infty {{\kappa _j}(\theta )g_j^2({x_j}(t- \theta ))} \mathrm{d}\theta \nonumber \\&\quad \le {g}(x_{t})^TR_{1}g(x_{t})-\sum \limits _{j=1}^n {r_{1j}{{\left( {\int _0^\infty {{\kappa _j}(\theta ){g_j}({x_j}(t-\theta ))\mathrm{d}\theta } } \right) }^2}}\nonumber \\&\quad = {g}(x_{t})^TR_{1}g(x_{t})-\left( {\int _{-\infty }^t {\kappa (s)g(x(s))\mathrm{d}s} } \right) \nonumber \\&\quad \quad \times R_{1}\left( {\int _{-\infty }^t {\kappa (s)g(x(s))\mathrm{d}s} } \right) , \end{aligned}$$

(61)

where $R_1=\mathrm{diag}\{r_{11},\ r_{12}, \ldots ,\ r_{1n} \}.$

Moreover, based on (H2), the following matrix inequalities hold for any positive diagonal matrices $T_i(i=1,2,3)$:

$$\begin{aligned} 0&\le -g(x_t)^TT_1g(x_t)+2x_t^TT_1L_2g(x_t)-x_t^TT_1L_1x_t,\end{aligned}$$

(62)

$$\begin{aligned} 0&\le -g(x_\tau )^TT_2g(x_\tau )+2x_\tau ^TT_2L_2g(x_\tau )-x_\tau ^TT_2L_1x_\tau ,\end{aligned}$$

(63)

$$\begin{aligned} 0&\le -g(x_{\tau _1})^TT_3g(x_{\tau _1})+2x_{\tau _1}^TT_3L_2g(x_{\tau _1})-x_{\tau _1}^TT_3L_1x_{\tau _1}. \end{aligned}$$

(64)

From Eq. (3), the following inequality holds for any positive diagonal matrix $R_{2}=\mathrm{diag}\{r_{21},\ r_{22}, \ldots ,\ r_{2n} \}$

$$\begin{aligned} 0&= 2\sum \limits _{i=1}^n \dot{x}_{i}(t)^Tr_{2i}\bigg [-\dot{x}_{i}(t)-{d_i}{x_i}(t)+\sum \limits _{j=1}^n a_{ij}{g_j}({x_j}(t)) \nonumber \\&\qquad +\sum \limits _{j=1}^n b_{ij}{g_j}({x_j}(t-\tau (t))) \nonumber \\&\qquad + \bigwedge \limits _{j=1}^n \alpha _{ij}\int _{-\infty }^t {\kappa _j}(t- s){f_j}({x_j}(s)+z_j^*) \mathrm{d}s\nonumber \\&\quad - \bigwedge \limits _{j= 1}^n \alpha _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}(z_j^*) \mathrm{d}s \nonumber \\&\quad +\bigvee \limits _{j=1}^n \beta _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}({x_j}(s)+z_j^*)\mathrm{d}s\nonumber \\&\quad -\bigvee \limits _{j=1}^n \beta _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}(z_j^*)\mathrm{d}s\bigg ]. \end{aligned}$$

(65)

Based on Lemma 3, one obtains the following inequalities:

$$\begin{aligned}&\left| \bigwedge \limits _{j=1}^n \alpha _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}({x_j}(s)+z_j^*) \mathrm{d}s\right. \nonumber \\&\quad \quad -\left. \bigwedge \limits _{j=1}^n \alpha _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}(z_j^*) \mathrm{d}s \right| \nonumber \\&\quad \le \sum \limits _{j=1}^n \left| \alpha _{ij} \right| \times \left| \int _{-\infty }^t {\kappa _j}(t- s){f_j}({x_j}(s)+z_j^*) \mathrm{d}s \right. \\&\quad \quad \left. -\int _{-\infty }^t {\kappa _j}(t-s){f_j}(z_j^*) \mathrm{d}s \right| \nonumber \\ {}&\quad =\sum \limits _{j=1}^n \left| \alpha _{ij} \right| \times \left| \int _{-\infty }^t {\kappa _j}(t- s){g_j}({x_j}(s)) \mathrm{d}s\right| ,\nonumber \\&\left| \bigvee \limits _{j=1}^n \beta _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}({x_j}(s)+z_j^*) \mathrm{d}s\right. \\&\quad \quad -\left. \bigvee \limits _{j=1}^n \beta _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}(z_j^*) \mathrm{d}s \right| \nonumber \\&\quad \le \sum \limits _{j=1}^n \left| \beta _{ij} \right| \times \left| \int _{-\infty }^t {\kappa _j}(t- s){f_j}({x_j}(s)+z_j^*) \mathrm{d}s\right. \nonumber \\&\quad \quad \left. -\int _{-\infty }^t {\kappa _j}(t-s){f_j}(z_j^*) \mathrm{d}s \right| \nonumber \\&\quad =\sum \limits _{j=1}^n \left| \beta _{ij} \right| \times \left| \int _{-\infty }^t {\kappa _j}(t- s){g_j}({x_j}(s)) \mathrm{d}s\right| .\nonumber \end{aligned}$$

By applying Lemma 4 and the well-known Cauchy inequality $X^TY+Y^TX\le X^TP^{-1}X+Y^TPY$, the following inequality holds for any positive diagonal matrix $T_4$:

$$\begin{aligned}&2\sum ^n_{i=1}r_{2i}\dot{x}_i(t)\left( \bigwedge \limits _{j=1}^n \alpha _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}({x_j}(s)+z_j^*) \mathrm{d}s\right. \\&\quad \quad -\bigwedge \limits _{j=1}^n \alpha _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}(z_j^*) \mathrm{d}s \nonumber \\&\quad \quad + \bigvee \limits _{j=1}^n \beta _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}({x_j}(s)+z_j^*) \mathrm{d}s\nonumber \\&\quad \quad -\left. \bigvee \limits _{j=1}^n \beta _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}(z_j^*) \mathrm{d}s \right) \nonumber \\&\quad \le 2\sum ^n_{i=1}r_{2i}|{\dot{x}_i}(t)|\left( \left| \bigwedge \limits _{j=1}^n \alpha _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}({x_j}(s)+z_j^*) \mathrm{d}s\right. \right. \\&\quad \quad -\left. \left. \bigwedge \limits _{j=1}^n \alpha _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}(z_j^*) \mathrm{d}s \right| \right. \nonumber \end{aligned}$$

$$\begin{aligned}&\left| \bigvee \limits _{j=1}^n \beta _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}({x_j}(s)+z_j^*) \mathrm{d}s\right. \nonumber \\&\quad \quad -\left. \left. \bigvee \limits _{j=1}^n \beta _{ij}\int _{-\infty }^t {\kappa _j}(t-s){f_j}(z_j^*) \mathrm{d}s\right| \right) \nonumber \\&\quad \le 2|\dot{x}(t)|^TR_{2}\left( \left| \alpha \right| +\left| \beta \right| \right) \left| \int _{-\infty }^t {\kappa }(t- s){g}({x}(s)) \mathrm{d}s\right| \nonumber \\&\quad \le |\dot{x}(t)|^TR_{2}\left( \left| \alpha \right| +\left| \beta \right| \right) T_4^{-1}\left( \left| \alpha \right| +\left| \beta \right| \right) R_{2}|\dot{x}(t)|\nonumber \\&\quad \quad +\left| \int _{-\infty }^t {\kappa }(t- s){g}({x}(s)) \mathrm{d}s\right| ^T T_4\left| \int _{-\infty }^t {\kappa }(t- s){g}({x}(s)) \mathrm{d}s\right| \nonumber \\&\quad \le n\dot{x}(t)^TR_{2}\Upsilon T_4^{-1}\Upsilon R_{2}\dot{x}(t)+ \left( \int _{-\infty }^t {\kappa }(t- s){g}({x}(s)) \mathrm{d}s\right) ^T \nonumber \\&\quad \quad \times T_4\left( \int _{-\infty }^t {\kappa }(t- s){g}({x}(s)) \mathrm{d}s\right) . \end{aligned}$$

(66)

Substituting (17)–(66) into (16) yields

$$\begin{aligned}&\mathrm{D}^+{V}(t,x_t)\le \xi (t)^T\nonumber \\&\quad \times \bigg \{\Omega +\Omega _{\tau }+ne_{22}^TR_{2}\Upsilon T_4^{-1}\Upsilon R_{2}e_{22}-\Psi _{1\tau }^T\widetilde{\Xi }_1\Psi _{1\tau }\!-\!\sum _{i=2}^{6}\Psi _{i\tau }^T\Xi _i\Psi _{i\tau }\bigg \} \xi (t),\nonumber \\&\quad t \in [{t_{k- 1}},{t_k}),k \in {\mathbb {Z}_+}, \end{aligned}$$

(67)

where

$$\begin{aligned} \Psi _{1\tau }&=\bigg [\begin{array}{c} [\tau (t)-{\tau _1}] e_{9}-2e_{13}\\ \left[ {\tau _2}-\tau (t)\right] e_{11}-2e_{15}\end{array}\bigg ],\nonumber \\ \Psi _{2\tau }&=\bigg [\begin{array}{c} [{\tau _2}-\tau (t)](e_{3}-e_{2})\\ \left[ \tau (t)-{\tau _1}\right] (e_{2}-e_{4})\end{array}\bigg ],\nonumber \\ \Psi _{3\tau }&=\bigg [\begin{array}{c} [{\tau _2}-\tau (t)](e_{3}+e_{2}-2e_{8})\\ \left[ \tau (t)-{\tau _1}\right] (e_{2}+e_{4}-2e_{10})\end{array}\bigg ],\\ \Psi _{4\tau }&=\bigg [\begin{array}{c} [\tau (t)-{\tau _1}]e_{8}-e_{12}\\ \left[ {\tau _2}-\tau (t)\right] e_{10}-e_{14}\end{array}\bigg ],\\ \Psi _{5\tau }&=\bigg [\begin{array}{c} [\tau (t)-{\tau _1}]e_{3}-e_{16}\\ \left[ {\tau _2}-\tau (t)\right] e_{2}-e_{18}\end{array}\bigg ],\\ \Psi _{6\tau }&=\bigg [\begin{array}{c} [\tau (t)-{\tau _1}]e_{2}-e_{12}\\ \left[ {\tau _2}-\tau (t)\right] e_{4}-e_{14}\end{array}\bigg ],\\ \widetilde{\Xi }_1&=3\bigg [\begin{array}{cc} P_7&{} S_2+S_3+S_4 \\ *&{}P_7\end{array}\bigg ],\nonumber \end{aligned}$$

with

$$\begin{aligned} \Omega _{\tau }&=-[{\tau _2}-\tau (t)]\big [(1-\tau _d)\tau _{12}{e_{2}^T}P_{4}e_{2}+e_{10}^T{P_{6}}e_{10}\\&\quad \, +2\tau _{12}(e_{3}-e_{8})^TP_{17}(e_{3}-e_{8})\big ]\\&\quad \, -[\tau (t)-{\tau _1}]\left[ (1-\tau _d)\tau _{12}{e_{2}^T}P_{5}e_{2}+e_{8}^T\right. \\&\quad \, \left. \times {(P_{4}-P_{5}+P_{6})}e_{8} +2\tau _{12}(e_{10}-e_{4})^T{P_{18}}(e_{10}-e_{4})\right] . \end{aligned}$$

In order for applying Lemma 5 to get a less conservative result without increasing restrictive conditions, one adds $\pm 18\Psi _{1\tau }^T\overline{P}\Psi _{1\tau }$ to the left side of the following inequality:

$$\begin{aligned}&\Omega +\Omega _{\tau }+ne_{22}^TR_{2}\Upsilon T_4^{-1}\Upsilon R_{2}e_{22}-\Psi _{1\tau }^T\widetilde{\Xi }_1\Psi _{1\tau }\nonumber \\&\quad - \sum _{i=2}^{6}\Psi _{i\tau }^T\Xi _i\Psi _{i\tau }<0, \end{aligned}$$

(68)

and gives

$$\begin{aligned} \widetilde{\Omega }_{\tau }- \sum _{i=1}^{6}\Psi _{i\tau }^T\Xi _i\Psi _{i\tau }<0, \end{aligned}$$

(69)

where

$$\begin{aligned} \widetilde{\Omega }_{\tau }&\quad =\Omega +\Omega _{\tau }+18\Psi _{1\tau }^T\overline{P}\Psi _{1\tau }+ne_{22}^TR_{2}\Upsilon T_4^{-1}\Upsilon R_{2}e_{22}.\nonumber \end{aligned}$$

For convenience, we denote

$$\begin{aligned} \Lambda _{j\tau }&=- \sum _{i=j+1}^{6}\Psi _{i\tau }^T\Xi _i\Psi _{i\tau }+\sum _{i=1}^{j-1}\mathrm{sym}\big \{\Pi _i\Psi _{i\tau }\big \},\\ \overline{\Lambda }_{j\tau }&=\Pi _1+\sum \limits _{i=2}^{j}\Psi _{i\tau }^T\Pi _{ i+5}^T,\ \ j=2,3,4,5,\\ \widetilde{\Lambda }_{k\tau }&=\Pi _2+\sum \limits _{i=3}^{k}\Psi _{i\tau }^T\Pi _{i+9}^T,k=3,4,5,6;\\ \widehat{\Lambda }_{l\tau }&=\Pi _3+\sum \limits ^l_{i=4}\Psi _{i\tau }^T\Pi _{i+1 2}^T,l=4,5,6;\\ \check{\Lambda }_{p\tau }&=\Pi _4+\sum \limits ^p_{i=5}\Psi _{i\tau }^T\Pi _{i+14 }^T,p=5,6. \end{aligned}$$

Applying inequality (15) and the definition of a positive definite matrix, one derives that $\Xi _1>0.$ Moreover, from Lemma 5 one gets that the inequality (69) is true if and only if for any appropriate dimensional matrix $\Pi _1$ the following inequality holds:

$$\begin{aligned}&\left[ \begin{array}{cc} \widetilde{\Omega }_{\tau }-\sum \limits _{i=2}^{6}\Psi _{i\tau }^T\Xi _i\Psi _{i\tau }+\mathrm{sym}\big \{\Pi _1\Psi _{1\tau }\big \} &{} \Pi _1\\ *&{} -\Xi _1\end{array}\right] \nonumber \\&\quad =\left[ \begin{array}{cc} \widetilde{\Omega }_{\tau }+\Lambda _{2\tau } &{} \Pi _1\\ *&{} -\Xi _1\end{array}\right] \nonumber \\&\quad \quad -\left[ \begin{array}{c} \Psi _{2\tau }^T\\ 0\end{array}\right] \Xi _2\left[ \begin{array}{c} \Psi _{2\tau }^T\\ 0\end{array}\right] ^T<0. \end{aligned}$$

(70)

Utilizing (12) yields $\Xi _2>0.$ Again from Lemma 5, one has that inequality (70) is true if and only if the following inequality holds for any appropriate dimensional matrix $\overline{\Pi }=\mathrm{col}\big \{\Pi _2,\ \Pi _7\big \}$

$$\begin{aligned}&\left[ \begin{array}{cc} \bigg [\begin{array}{cc} \widetilde{\Omega }_{\tau }+\Lambda _{2\tau } &{} \Pi _1\\ *&{} -\Xi _1\end{array}\bigg ]+\mathrm{sym}\bigg \{\overline{\Pi }\bigg [\begin{array}{c} \Psi _{2\tau }^T\\ 0\end{array}\bigg ]^T\bigg \} &{} \overline{\Pi }\\ *&{} -\Xi _2\end{array}\right] \nonumber \\&\quad =\left[ \begin{array}{ccc} \widetilde{\Omega }_{\tau }+\Lambda _{2\tau }+\mathrm{sym}\big \{\Pi _2\Psi _{2\tau }\big \} &{} \overline{\Lambda }_{2\tau } &{} \Pi _2\\ *&{} -\Xi _1 &{} \Pi _7\\ *&{} *&{} -\Xi _2\end{array}\right] \nonumber \\&\quad =\left[ \begin{array}{ccc} \widetilde{\Omega }_{\tau }+\Lambda _{3\tau } &{} \overline{\Lambda }_{2\tau } &{} \Pi _2\\ *&{} -\Xi _1 &{} \Pi _7\\ *&{} *&{} -\Xi _2\end{array}\right] \nonumber \\&\quad \quad -\left[ \begin{array}{c} \Psi _{3\tau }^T\\ 0\\ 0\end{array}\right] \Xi _3\left[ \begin{array}{c} \Psi _{3\tau }^T\\ 0\\ 0\end{array}\right] ^T<0. \end{aligned}$$

(71)

Employing (12) yields $\Xi _3>0.$ Again from Lemma 5, one obtains that inequality (71) is true if and only if the following inequality holds for any appropriate dimensional matrix $\widehat{\Pi }=\mathrm{col}\big \{\Pi _3,\ \Pi _8,\ \Pi _{12}\big \}$

$$\begin{aligned}&\left[ \begin{array}{cc} \left[ \begin{array}{ccc} \widetilde{\Omega }_{\tau }+\Lambda _{3\tau } &{} \overline{\Lambda }_{2\tau } &{} \Pi _2\\ *&{} -\Xi _1 &{} \Pi _7\\ *&{} *&{} - \Xi _2\end{array}\right] +\mathrm{sym}\left\{ \left[ \begin{array}{c} \Psi _{3\tau }^T\\ 0\\ 0\end{array}\right] \widehat{\Pi }^T\right\} &{} \widehat{\Pi }\\ *&{} -\Xi _3\end{array}\right] \nonumber \\&\quad =\left[ \begin{array}{cccc} \widetilde{\Omega }_{\tau }+\Lambda _{3\tau }+\mathrm{sym}\big \{\Pi _3\Psi _{3\tau }\big \} &{} \overline{\Lambda }_{3\tau } &{} \widetilde{\Lambda }_{3\tau }&{}\Pi _3 \\ *&{} -\Xi _1 &{} \Pi _7&{} \Pi _8\\ *&{} *&{} -\Xi _2&{} \Pi _{12} \\ *&{} *&{} *&{}- \Xi _3\end{array}\right] \nonumber \\&\quad =\left[ \begin{array}{cccc} \widetilde{\Omega }_{\tau }+\Lambda _{4\tau } &{} \overline{\Lambda }_{3\tau } &{} \widetilde{\Lambda }_{3\tau }&{} \Pi _3\\ *&{} -\Xi _1 &{} \Pi _7&{} \Pi _8\\ *&{} *&{} -\Xi _2&{} \Pi _{12} \\ *&{} *&{} *&{}-\Xi _3\end{array}\right] -\left[ \begin{array}{c} \Psi _{4\tau }^T\\ 0\\ 0\\ 0\end{array}\right] \Xi _4\left[ \begin{array}{c} \Psi _{4\tau }^T\\ 0\\ 0\\ 0\end{array}\right] ^T<0. \end{aligned}$$

(72)

It ia easy to derive from inequalities (12) that $\Xi _4>0.$ Again from Lemma 5, one obtains that inequality (72) is true if and only if the following inequality holds for any appropriate dimensional matrix $\widetilde{\Pi }=\mathrm{col}\big \{\Pi _4,\ \Pi _9,\ \Pi _{13},\ \Pi _{16}\big \}$

$$\begin{aligned}&\left[ \begin{array}{cc} \left[ \begin{array}{cccc} \widetilde{\Omega }_{\tau }+\Lambda _{4\tau } &{} \overline{\Lambda }_{3\tau } &{} \widetilde{\Lambda }_{3\tau }&{} \Pi _3\\ *&{} -\Xi _1 &{} \Pi _7&{} \Pi _8\\ *&{} *&{} -\Xi _2&{} \Pi _{12} \\ *&{} *&{} *&{}- \Xi _3\end{array}\right] +\mathrm{sym}\left\{ \left[ \begin{array}{c} \Psi _{4\tau }^T\\ 0\\ 0\\ 0\end{array}\right] \widetilde{\Pi }^T\right\} &{} \widetilde{\Pi }\\ *&{} -\Xi _4\end{array}\right] \nonumber \\&\quad =\left[ \begin{array}{ccccc} \widetilde{\Omega }_{\tau }+\Lambda _{4\tau }+\mathrm{sym}\big \{\Pi _4\Psi _{4\tau }\big \} &{} \overline{\Lambda }_{4\tau } &{} \widetilde{\Lambda }_{4\tau }&{} \widehat{\Lambda }_{4\tau }&{} \Pi _4\\ *&{} - \Xi _1 &{} \Pi _7&{} \Pi _8&{} \Pi _9\\ *&{} *&{} -\Xi _2&{} \Pi _{12}&{} \Pi _{13} \\ *&{} *&{} *&{}-\Xi _3&{} \Pi _{16}\\ *&{} *&{} *&{}*&{}-\Xi _4 \end{array}\right] \nonumber \\&\quad =\left[ \begin{array}{ccccc} \widetilde{\Omega }_{\tau }+\Lambda _{5\tau } &{} \overline{\Lambda }_{4\tau } &{} \widetilde{\Lambda }_{4\tau }&{} \widehat{\Lambda }_{4\tau }&{} \Pi _4\\ *&{} - \Xi _1 &{} \Pi _7&{} \Pi _8&{} \Pi _9\\ *&{} *&{} -\Xi _2&{} \Pi _{12}&{} \Pi _{13} \\ *&{} *&{} *&{}-\Xi _3&{} \Pi _{16}\\ *&{} *&{} *&{}*&{}-\Xi _4 \end{array}\right] \!-\!\left[ \begin{array}{c} \Psi _{5\tau }^T\\ 0\\ 0\\ 0\\ 0\end{array}\right] \Xi _5\left[ \begin{array}{c} \Psi _{5\tau }^T\\ 0\\ 0\\ 0\\ 0\end{array}\right] ^T\!<\!0, \end{aligned}$$

(73)

Based on inequalities (12) that $\Xi _5>0.$ Again from Lemma 5, one obtains that inequality (73) is true if and only if the following inequality holds for any appropriate dimensional matrix $\check{\Pi }=\mathrm{col}\big \{\Pi _5,\ \Pi _{10},\ \Pi _{14},\ \Pi _{17},\ \Pi _{19}\big \}$

$$\begin{aligned}&\left[ \begin{array}{cc} \left[ \begin{array}{ccccc} \widetilde{\Omega }_{\tau }+\Lambda _{5\tau } &{} \overline{\Lambda }_{4\tau } &{} \widetilde{\Lambda }_{4\tau }&{} \widehat{\Lambda }_{4\tau }&{} \Pi _4\\ *&{} - \Xi _1 &{} \Pi _7&{} \Pi _8&{} \Pi _9\\ *&{} *&{} -\Xi _2&{} \Pi _{12}&{} \Pi _{13} \\ *&{} *&{} *&{}-\Xi _3&{} \Pi _{16}\\ *&{} *&{} *&{}*&{}-\Xi _4 \end{array}\right] +\mathrm{sym}\left\{ \left[ \begin{array}{c} \Psi _{5\tau }^T\\ 0\\ 0\\ 0\\ 0\end{array}\right] \check{\Pi }^T\right\} &{} \check{\Pi }\\ *&{} -\Xi _5\end{array}\right] \nonumber \\&\quad =\left[ \begin{array}{cccccc} \widetilde{\Omega }_{\tau }+\Lambda _{5\tau }+\mathrm{sym}\big \{\Pi _5\Psi _{5\tau }\big \} &{} \overline{\Lambda }_{5\tau } &{} \widetilde{\Lambda }_{5\tau }&{} \widehat{\Lambda }_{5\tau }&{} \check{\Lambda }_{5\tau }&{} \Pi _{5}\\ *&{} -\Xi _1 &{} \Pi _7&{} \Pi _8&{} \Pi _9&{} \Pi _{10}\\ *&{} *&{} -\Xi _2&{} \Pi _{12}&{} \Pi _{13} &{} \Pi _{14}\\ *&{} *&{} *&{}-\Xi _3&{} \Pi _{16}&{} \Pi _{17}\\ *&{} *&{} *&{}*&{}- \Xi _4&{} \Pi _{19} \\ *&{} *&{} *&{} *&{}*&{}- \Xi _5\end{array}\right] \nonumber \\&\quad \!=\!\left[ \!\begin{array}{cccccc} \widetilde{\Omega }_{\tau }+\Lambda _{6\tau } &{} \overline{\Lambda }_{5\tau } &{} \widetilde{\Lambda }_{5\tau }&{} \widehat{\Lambda }_{5\tau }&{} \check{\Lambda }_{5\tau }&{} \Pi _{5}\\ *&{} - \Xi _1 &{} \Pi _7&{} \Pi _8&{} \Pi _9&{} \Pi _{10}\\ *&{} *&{} -\Xi _2&{} \Pi _{12}&{} \Pi _{13} &{} \Pi _{14}\\ *&{} *&{} *&{}-\Xi _3&{} \Pi _{16}&{} \Pi _{17}\\ *&{} *&{} *&{}*&{}-\Xi _4&{} \Pi _{19} \\ *&{} *&{} *&{} *&{}*&{}-\Xi _5\end{array}\!\right] \!-\!\left[ \!\begin{array}{c} \Psi _{6\tau }^T\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\!\right] \Xi _6\left[ \!\begin{array}{c} \Psi _{6\tau }^T\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\!\right] ^T\!<\!0, \end{aligned}$$

(74)

From inequalities (12) one gets $\Xi _6>0.$ Again from Lemma 5, one obtains that inequality (74) is true if and only if the following inequality holds for any appropriate dimensional matrix $\underline{\Pi }=\mathrm{col}\big \{\Pi _6,\ \Pi _{11},\ \Pi _{15},\ \Pi _{18},\ \Pi _{20},\ \Pi _{21}\big \}$

$$\begin{aligned}&\left[ \begin{array}{cc} \left[ \begin{array}{cccccc} \widetilde{\Omega }_{\tau }+\Lambda _{6\tau } &{} \overline{\Lambda }_{5\tau } &{} \widetilde{\Lambda }_{5\tau }&{} \widehat{\Lambda }_{5\tau }&{} \check{\Lambda }_{5\tau }&{} \Pi _{5}\\ *&{} -\Xi _1 &{} \Pi _7&{} \Pi _8&{} \Pi _9&{} \Pi _{10}\\ *&{} *&{} -\Xi _2&{} \Pi _{12}&{} \Pi _{13} &{} \Pi _{14}\\ *&{} *&{} *&{}-\Xi _3&{} \Pi _{16}&{} \Pi _{17}\\ *&{} *&{} *&{}*&{}- \Xi _4&{} \Pi _{19} \\ *&{} *&{} *&{} *&{}*&{}- \Xi _5\end{array}\right] +\mathrm{sym}\left\{ \left[ \begin{array}{c} \Psi _{6\tau }^T\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right] \underline{\Pi }^T\right\} &{} \underline{\Pi }\\ *&{} - \Xi _6\end{array}\right] \nonumber \\&\quad \!=\!\left[ \begin{array}{ccccccc} \widetilde{\Omega }_{\tau }+\sum \limits _{i=1}^6\mathrm{sym}\big \{\Pi _i\Psi _{i\tau }\big \} &{} \overline{\Lambda }_{6\tau } &{} \widetilde{\Lambda }_{6\tau }&{} \widehat{\Lambda }_{6\tau }&{} \check{\Lambda }_{6\tau }&{} \Pi _{5}+\Psi _{6\tau }^T\Pi _{21}^T&{} \Pi _{6}\\ *&{} -\Xi _1 &{} \Pi _7&{} \Pi _8&{} \Pi _9&{} \Pi _{10}&{} \Pi _{11}\\ *&{} *&{} - \Xi _2&{} \Pi _{12}&{} \Pi _{13} &{} \Pi _{14}&{} \Pi _{15}\\ *&{} *&{} *&{}- \Xi _3&{} \Pi _{16}&{} \Pi _{17}&{} \Pi _{18}\\ *&{} *&{} *&{}*&{}-\Xi _4&{} \Pi _{19} &{} \Pi _{20}\\ *&{} *&{} *&{} *&{}*&{}-\Xi _5&{} \Pi _{21}\\ *&{} *&{} *&{} *&{} *&{}*&{}-\Xi _6\end{array}\right] \!<\!0. \end{aligned}$$

(75)

From the well-known Schur complement, ones deduces that inequalities (75) are equivalent to the following inequality:

$$\begin{aligned}&\left[ \begin{array}{ccccccccc} \Omega +\Omega _{\tau }+\sum \limits _{i=1}^6\mathrm{sym}\big \{\Pi _i\Psi _{i\tau }\big \} &{} \overline{\Lambda }_{6\tau } &{} \widetilde{\Lambda }_{6\tau }&{} \widehat{\Lambda }_{6\tau }&{} \check{\Lambda }_{6\tau }&{} \Pi _{5}+\Psi _{6\tau }^T\Pi _{21}^T&{} \Pi _{6}&{} 6\Psi _{1\tau }^T\overline{P} &{} ne_{22}^TR_{2}\Upsilon \\ *&{} -\Xi _1 &{} \Pi _7&{} \Pi _8&{} \Pi _9&{} \Pi _{10}&{} \Pi _{11}&{}0 &{} 0\\ *&{} *&{} -\Xi _2&{} \Pi _{12}&{} \Pi _{13} &{} \Pi _{14}&{} \Pi _{15}&{}0 &{} 0\\ *&{} *&{} *&{}- \Xi _3&{} \Pi _{16}&{} \Pi _{17}&{} \Pi _{18}&{}0 &{} 0\\ *&{} *&{} *&{}*&{}- \Xi _4&{} \Pi _{19} &{} \Pi _{20}&{}0 &{} 0\\ *&{} *&{} *&{} *&{}*&{}- \Xi _5&{} \Pi _{21}&{}0 &{} 0\\ *&{} *&{} *&{} *&{} *&{}*&{}-\Xi _6&{}0 &{} 0\\ *&{}0 &{} 0&{}0 &{} 0&{}0 &{} 0&{}-2\overline{P}&{} 0\\ *&{} 0 &{} 0 &{} 0&{}0 &{} 0&{}0 &{}0 &{} -nT_4\end{array}\right] <0. \end{aligned}$$

(76)

The condition (76) is intrinsically linear in $\tau (t)$ and thus can be treated non-conservatively by two corresponding boundary LMIs (12): one for $\tau (t)=\tau _1,$ and the other for $\tau (t)=\tau _2,$ which imply $\mathrm{D}^+{V}(t,x_t)<0$ for any $t\in [t_{k - 1},t_k),\ k \in \mathbb {Z}_+.$

When $t = t_k,\ k \in \mathbb {Z}_ +,$ from the condition (H5), one has

$$\begin{aligned}&V(t_k,x(t_k))\nonumber \\&\quad =x(t_k)^TP_1x(t_k)+\int _{t_k-{\tau _1}}^{t_k} {x}(s)^T{P_{2}}x(s)\mathrm{d}s \nonumber \\&\qquad +\int _{t_k-{\tau _2}}^{t_k-{\tau _1}} {x}(s)^T{P_{3}}x(s)\mathrm{d}s +\int _{t_k-\tau ({t_k})}^{t_k} \varsigma (s)^T\tilde{Q}\varsigma (s)\mathrm{d}s\nonumber \\&\qquad +\int _{t_k-{\tau ({t_k})}}^{t_k-{\tau _1}} {x}(s)^T{\tilde{P}(s-t_k)}x(s)\mathrm{d}s\nonumber \\&\qquad +\int _{t_k-{\tau _2}}^{t_k-{\tau _1}} \int _{\theta }^{t_k-{\tau _1}} {x}(s)^T{P_{6}}x(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\qquad +\tau _{12}^3\int _{t_k-{\tau _2}}^{t_k-{\tau _1}} \int _{\theta }^{t_k-{\tau _1}} {g}(x(s))^T{P_{7}}{g}(x(s))\mathrm{d}s\mathrm{d}\theta \nonumber \\&\qquad +\tau _{1}^3\int _{t_k-{\tau _1}}^{t_k} \int _{\theta }^{t_k} [{x}(s)^T{P_{8}}x(s)+\dot{x}(s)^T{P_{9}}\dot{x}(s)]\mathrm{d}s\mathrm{d}\theta \nonumber \\&\qquad +\tau _{12}\int _{t_k-{\tau _2}}^{t_k-\tau _1}\int _{\theta }^{t_k} \dot{x}(s)^T{P_{10}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\qquad +\int _{t_k-{\tau _2}}^{t_k-\tau _1}\left( {\int _\theta ^{t_k-\tau _1} {\int _\beta ^{t_k} {\dot{x}{{(s)}^T}P_{11}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta }}}\right. \nonumber \\&\qquad \left. + {\int _{t_k - \tau _2 } ^\theta {\int _\beta ^{t_k} {\dot{x}{{(s)}^T}{P_{12}}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta }}}\right) \mathrm{d}\theta \nonumber \\&\qquad +\frac{\tau _{1}^2}{2}\int _{t_k-{\tau _1}}^{t_k}\left( \int _{t_k-{\tau _1}}^{\gamma }\int _{\theta }^{t_k} \dot{x}(s)^TP_{13}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \right. \nonumber \\&\qquad \left. +\int ^{t_k}_{\gamma }\int _{\theta }^{t_k} \dot{x}(s)^TP_{14}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \right) \mathrm{d}{\gamma }\nonumber \\&\qquad +\frac{\tau _{1}^3}{6}\int _{t_k-{\tau _1}}^{t_k}{\int _{\gamma }^{t_k}\int _\theta ^{t_k} {\int _\beta ^{t_k} {\dot{x}{{(s)}^T}P_{15}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta \mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\qquad +\frac{\tau _{1}^3}{6}\int _{t_k-{\tau _1}}^{t_k}{\int _{t_k - \tau _1} ^\gamma \int _{t_k - \tau _1} ^\theta {\int _\beta ^{t_k} {\dot{x}{{(s)}^T}{P_{16}}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta \mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\qquad +\tau _{12}\int _{t_k-{\tau _2}}^{t_k-{\tau _1}}{\int _{\gamma }^{t_k-{\tau _1}}\int _\theta ^{t_k-{\tau _1}} {\int _\beta ^{t_k} {\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta \mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\ \end{aligned}$$

$$\begin{aligned}&\qquad +\tau _{12}\int _{t_k-{\tau _2}}^{t_k-{\tau _1}} {\int _{t_k - \tau _2} ^\gamma \int _{t_k - \tau _2} ^\theta {\int _\beta ^{t_k} {\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta \mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\qquad +\sum \limits _{j=1}^n r_{1j}\int _0^\infty {\kappa _j}(\theta )\int _{t_k-\theta }^{t_k} g_j^2({x_j}(s))\mathrm{d}s\mathrm{d}\theta \nonumber \\&\quad = [x({t_k^-})+\Delta x({t_k}) ]^T{P_1}[x({t_k^-})+\Delta x({t_k})]\nonumber \\&\qquad +\int _{t_k^--{\tau _1}}^{t_k^-} {x}(s)^T{P_{2}}x(s)\mathrm{d}s +\int _{t_k^--{\tau _2}}^{t_k^--{\tau _1}}{x}(s)^T{P_{3}}x(s)\mathrm{d}s\nonumber \\&\qquad +\int _{t_k^--\tau ({t_k^-})}^{t_k^-} \varsigma (s)^T\tilde{Q}\varsigma (s)\mathrm{d}s\nonumber \\&\qquad +\int _{t_k^--{\tau ({t_k^-})}}^{t_k^--{\tau _1}} {x}(s)^T{\tilde{P}(s-t_k^-)}x(s)\mathrm{d}s\nonumber \\&\qquad +\int _{t_k^--{\tau _2}}^{t_k^--{\tau _1}} \int _{\theta }^{t_k^--{\tau _1}} {x}(s)^T{P_{6}}x(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\qquad +\tau _{12}^3\int _{t_k^--{\tau _2}}^{t_k^--{\tau _1}} \int _{\theta }^{t_k^--{\tau _1}} {g}(x(s))^T{P_{7}}{g}(x(s))\mathrm{d}s\mathrm{d}\theta \end{aligned}$$

$$\begin{aligned}&\qquad +\tau _{1}^3\int _{t_k^--{\tau _1}}^{t_k^-} \int _{\theta }^{t_k^-} [{x}(s)^T{P_{8}}x(s)+\dot{x}(s)^T{P_{9}}\dot{x}(s)]\mathrm{d}s\mathrm{d}\theta \nonumber \\&\qquad +\tau _{12}\int _{t_k^--{\tau _2}}^{t_k^--\tau _1}\int _{\theta }^{t_k^-} \dot{x}(s)^T{P_{10}}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\qquad +\int _{t_k^--{\tau _2}}^{t_k^--\tau _1}\left( {\int _\theta ^{t_k^--\tau _1} {\int _\beta ^{t_k^-} {\dot{x}{{(s)}^T}P_{11}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta }}}\right. \nonumber \\&\qquad \left. + {\int _{t_k^- - \tau _2 } ^\theta {\int _\beta ^{t_k^-} {\dot{x}{{(s)}^T}{P_{12}}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta }}}\right) \mathrm{d}\theta \nonumber \\&\qquad +\frac{\tau _{1}^2}{2}\int _{t_k^--{\tau _1}}^{t_k^-}\left( \int _{t_k^--{\tau _1}}^{\gamma }\int _{\theta }^{t_k^-} \dot{x}(s)^TP_{13}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \right. \nonumber \\&\qquad \left. +\int ^{t_k^-}_{\gamma }\int _{\theta }^{t_k^-} \dot{x}(s)^TP_{14}\dot{x}(s)\mathrm{d}s\mathrm{d}\theta \right) \mathrm{d}{\gamma }\nonumber \\&\qquad +\frac{\tau _{1}^3}{6}\int _{t_k^--{\tau _1}}^{t_k^-}{\int _{\gamma }^{t_k^-}\int _\theta ^{t_k^-} {\int _\beta ^{t_k^-} {\dot{x}{{(s)}^T}P_{15}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta \mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\qquad +\frac{\tau _{1}^3}{6}\int _{t_k^--{\tau _1}}^{t_k^-}{\int _{t_k^- - \tau _1} ^\gamma \int _{t_k^- - \tau _1} ^\theta {\int _\beta ^{t_k^-} {\dot{x}{{(s)}^T}{P_{16}}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta \mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\qquad +\tau _{12}\int _{t_k^--{\tau _2}}^{t_k^--{\tau _1}}{\int _{\gamma }^{t_k^--{\tau _1}}\int _\theta ^{t_k^--{\tau _1}} {\int _\beta ^{t_k^-} {\dot{x}{{(s)}^T}P_{17}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta \mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\qquad +\tau _{12}\int _{t_k^--{\tau _2}}^{t_k^--{\tau _1}} {\int _{t_k^- - \tau _2} ^\gamma \int _{t_k^- - \tau _2} ^\theta {\int _\beta ^{t_k^-} {\dot{x}{{(s)}^T}{P_{18}}\dot{x}(s)\mathrm{d}s\mathrm{d}\beta \mathrm{d}\theta }}}\mathrm{d}{\gamma }\nonumber \\&\qquad +\sum \limits _{j=1}^n r_{1j}\int _0^\infty {\kappa _j}(\theta )\int _{t_k^--\theta }^{t_k^-} g_j^2({x_j}(s))\mathrm{d}s\mathrm{d}\theta \nonumber \\&={V}({t_k^-},x({t_k^-}))+ x({t_k^-})^T[{(I - {\Gamma _k})^T}P_1(I - {\Gamma _k})-P_1]x({t_k^-}). \end{aligned}$$

(77)

On the other hand, it follows from (5) that

$$\begin{aligned} \left( {\begin{array}{l@{\quad }l} I &{} 0 \\ 0 &{} {P_1}^{ - 1} \\ \end{array}} \right) \left( {\begin{array}{l@{\quad }l} {P_1} &{} {(I - {\Gamma _k}){P_1}} \\ * &{} {P_1} \\ \end{array}} \right) \left( {\begin{array}{l@{\quad }l} I &{} 0 \\ 0 &{} {P_1}^{ - 1} \\ \end{array}} \right) \ge 0,\nonumber \end{aligned}$$

that is

$$\begin{aligned} \left( {\begin{array}{l@{\quad }l} {P_1} &{} {I - {\Gamma _k}} \\ * &{} {P_1}^{ - 1} \\ \end{array}} \right) \ge 0.\nonumber \end{aligned}$$

From the Schur complement, one gets

$$\begin{aligned} {P_1} - {(I - {\Gamma _k})^T}{P_1}(I - {\Gamma _k}) \ge 0. \end{aligned}$$

(78)

By combining (78) with (77), one obtains

$$\begin{aligned} {V}({t_k},x({t_k}))\le {V}({t_k^-},x({t_k^-})),\quad k \in {\mathbb {Z}_ + }.\nonumber \end{aligned}$$

Therefore, the system (3) is asymptotically stable, which implies that the equilibrium point of model (1) is globally asymptotically stable. This ends the proof of Theorem 1.