In this section, new stability criteria for system (1) are derived by using the Lyapunov method and LMI framework. For the sake of simplicity of matrix and vector representation, \(e_{i}\in\mathbb{R}^{9n\times n}\) are defined as block entry matrices, for example, \(e^{T}_{4}=[0\ 0 \ 0 \ I \ 0 \ 0 \ 0 \ 0 \ 0]\). The other notations are the following:
$$\begin{aligned}& \Gamma=[\textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}}-C& 0 & 0 & A & B & 0 & 0 & 0 & -I\end{array}\displaystyle ], \\& \varpi_{1}(t)=\left[\textstyle\begin{array}{@{}c@{\quad}c@{}}x^{T}(t-h(t))& \int^{t-h(t)}_{t-h}\frac {x^{T}(s)}{h-h(t)}\,ds\end{array}\displaystyle \right]^{T}, \qquad \varpi_{2}(t)=\left[\textstyle\begin{array}{@{}c@{\quad}c@{}} x^{T}(t) & \int^{t}_{t-h(t)}\frac{x^{T}(s)}{h(t)}\,ds\end{array}\displaystyle \right]^{T}, \\& \varpi_{3}(t)=\left[\textstyle\begin{array}{@{}c@{\quad}c@{}}x^{T}(t-h(t)) & \int^{t}_{t-h(t)}\frac {x^{T}(s)}{h(t)}\,ds\end{array}\displaystyle \right]^{T}, \qquad \varpi_{4}(t)=\left[\textstyle\begin{array}{@{}c@{\quad}c@{}}x^{T}(t-h)& \int^{t-h(t)}_{t-h}\frac{x^{T}(s)}{h-h(t)}\,ds\end{array}\displaystyle \right]^{T}, \\& \xi^{T}(t)=\bigl[x^{T}(t),x^{T}\bigl(t-h(t) \bigr),x^{T}(t-h),f^{T}\bigl(x(t)\bigr), f^{T} \bigl(x\bigl(t-h(t)\bigr)\bigr), \\& \hphantom{\xi^{T}(t)={}}f^{T}\bigl(x(t-h)\bigr), \eta_{1}^{T}(t), \eta_{2}^{T}(t), \dot{x}^{T}(t)\bigr], \\& \eta_{1}(t)= \int^{t}_{t-h(t)}\frac{x(s)}{h(t)}\,ds, \qquad \eta_{2}(t)= \int ^{t-h(t)}_{t-h}\frac{x(s)}{h-h(t)}\,ds, \\& \lambda^{\delta}=\operatorname{diag}\bigl\{ \lambda_{1}^{\delta}, \ldots,\lambda_{n}^{\delta }\bigr\} =\lambda_{m}+\delta( \lambda_{M}-\lambda_{m}), \\& \lambda_{m}= \operatorname{diag}\bigl\{ \lambda_{1}^{-},\ldots, \lambda_{n}^{-}\bigr\} , \qquad \lambda_{M}= \operatorname{diag}\bigl\{ \lambda _{1}^{+},\ldots, \lambda_{n}^{+}\bigr\} , \\& \Xi_{[h(t)]}=-h(t)e_{7}U_{2}e^{T}_{7}- \bigl(h-h(t)\bigr)e_{8}U_{2}e^{T}_{8} \\& \hphantom{\Xi_{[h(t)]}={}}{}+[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2}& e_{8}\end{array}\displaystyle ] \bigl(h-h(t)\bigr)\operatorname{sym}\bigl\{ Y_{1}[\textstyle\begin{array}{@{}c@{\quad}c@{}}I& -I\end{array}\displaystyle ]\bigr\} [\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2} & e_{8}\end{array}\displaystyle ]^{T} \\& \hphantom{\Xi_{[h(t)]}={}}{}+[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{1} & e_{7}\end{array}\displaystyle ]h(t)\operatorname{sym}\bigl\{ Y_{2}[\textstyle\begin{array}{@{}c@{\quad}c@{}}I& -I\end{array}\displaystyle ] \bigr\} [\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{1} & e_{7}\end{array}\displaystyle ]^{T} \\& \hphantom{\Xi_{[h(t)]}={}}{}+[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2} & e_{7}\end{array}\displaystyle ]h(t)\operatorname{sym}\bigl\{ Y_{3}[\textstyle\begin{array}{@{}c@{\quad}c@{}}-I& I\end{array}\displaystyle ]\bigr\} [\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2}& e_{7}\end{array}\displaystyle ]^{T} \\& \hphantom{\Xi_{[h(t)]}={}}{}+[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{3} & e_{8}\end{array}\displaystyle ]\bigl(h-h(t)\bigr)\operatorname{sym}\bigl\{ Y_{4}[\textstyle\begin{array}{@{}c@{\quad}c@{}}-I& I\end{array}\displaystyle ]\bigr\} [\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{3} & e_{8}\end{array}\displaystyle ]^{T}, \\& \Phi_{a}=\Phi_{1}+\Phi_{2}+\Phi_{3}, \\& \Phi_{b}=\Phi_{1}+\Phi_{2}+ \Phi_{3}^{*}, \\& \Phi_{1}=\operatorname{sym}\bigl(e_{1}Pe_{9}^{T} \bigr)+\operatorname{sym}\bigl((e_{4}-e_{1}\lambda _{m})K_{1}e_{9}^{T}\bigr)+ \operatorname{sym}\bigl((e_{1}\lambda_{M}-e_{4})K_{2}e_{9}^{T} \bigr) \\& \hphantom{\Phi_{1}={}}{}+[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{1} & e_{4}\end{array}\displaystyle ](Q_{1}+Q_{2})[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{1}& e_{4}\end{array}\displaystyle ]^{T}-(1- \mu)[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2}& e_{5}\end{array}\displaystyle ]Q_{1}[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2} & e_{5}\end{array}\displaystyle ]^{T} \\& \hphantom{\Phi_{1}={}}{}-[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{3} & e_{6}\end{array}\displaystyle ]Q_{2}[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{3} & e_{6}\end{array}\displaystyle ]^{T}, \\& \Phi_{2}=e_{9}\biggl(h^{2}U_{1}+ \frac {h^{2}}{2}(R_{1}+R_{2})\biggr)e^{T}_{9}+he_{1}U_{2}e^{T}_{1} \\& \hphantom{\Phi_{2}={}}{}-[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{1}-e_{2} & e_{2}-e_{3}\end{array}\displaystyle ] \left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} U_{1} & S_{1} \\ * & U_{1} \end{array}\displaystyle \right ] [\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{1}-e_{2}& e_{2}-e_{3}\end{array}\displaystyle ]^{T} \\& \hphantom{\Phi_{2}={}}{}-[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{1}+e_{2}-2e_{7} & e_{2}+e_{3}-2e_{8}\end{array}\displaystyle ]\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} 3U_{1} & S_{2} \\ * & 3U_{1} \end{array}\displaystyle \right ] [\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{1}+e_{2}-2e_{7} & e_{2}+e_{3}-2e_{8}\end{array}\displaystyle ]^{T} \\& \hphantom{\Phi_{2}={}}{}+\operatorname{sym}\bigl((e_{1}F_{1}+e_{9}F_{2}) \Gamma\bigr), \\& \Phi_{3}=-2\bigl[e_{4}-e_{5}-(e_{1}-e_{2}) \lambda _{m}\bigr]H_{1}\bigl[e_{4}-e_{5}-(e_{1}-e_{2}) \lambda^{\delta }\bigr]^{T}-2\bigl[e_{5}-e_{6}-(e_{2}-e_{3}) \lambda_{m}\bigr] \\& \hphantom{\Phi_{3}={}}{}\times H_{2}\bigl[e_{5}-e_{6}-(e_{2}-e_{3}) \lambda^{\delta}\bigr]^{T}-\operatorname{sym}\bigl(e_{1} \Pi _{1}\bigl(\lambda_{m}\lambda^{\delta} \bigr)e^{T}_{1}\bigr)+\operatorname{sym}\bigl(e_{1} \Pi_{1}\bigl(\lambda _{m}+\lambda^{\delta} \bigr)e^{T}_{4}\bigr) \\& \hphantom{\Phi_{3}={}}{}-\operatorname{sym}\bigl(e_{4}\Pi_{1}e^{T}_{4} \bigr)-\operatorname{sym}\bigl(e_{2}\Pi_{2}\bigl( \lambda_{m}\lambda^{\delta }\bigr)e^{T}_{2} \bigr)+\operatorname{sym}\bigl(e_{2}\Pi_{2}\bigl( \lambda_{m}+\lambda^{\delta}\bigr)e^{T}_{5} \bigr) \\& \hphantom{\Phi_{3}={}}{}-\operatorname{sym}\bigl(e_{5}\Pi_{2}e^{T}_{5} \bigr)-\operatorname{sym}\bigl(e_{3}\Pi_{3}\bigl( \lambda_{m}\lambda^{\delta }\bigr)e^{T}_{3} \bigr) \\& \hphantom{\Phi_{3}={}}{}+\operatorname{sym}\bigl(e_{3}\Pi_{3}\bigl( \lambda_{m}+\lambda^{\delta }\bigr)e^{T}_{6} \bigr)-\operatorname{sym}\bigl(e_{6}\Pi_{3}e^{T}_{6} \bigr), \\& \Phi^{*}_{3}=-2\bigl[e_{4}-e_{5}-(e_{1}-e_{2}) \lambda^{\delta }\bigr]H_{3}\bigl[e_{4}-e_{5}-(e_{1}-e_{2}) \lambda _{M}\bigr]^{T}-2\bigl[e_{5}-e_{6}-(e_{2}-e_{3}) \lambda^{\delta}\bigr] \\& \hphantom{\Phi^{*}_{3}={}}{}\times H_{4}\bigl[e_{5}-e_{6}-(e_{2}-e_{3}) \lambda_{M}\bigr]^{T}-\operatorname{sym}\bigl(e_{1} \Pi _{4}\bigl(\lambda^{\delta}\lambda_{M} \bigr)e^{T}_{1}\bigr)+\operatorname{sym}\bigl(e_{1} \Pi_{4}\bigl(\lambda ^{\delta}+\lambda_{M} \bigr)e^{T}_{4}\bigr) \\& \hphantom{\Phi^{*}_{3}={}}{}-\operatorname{sym}\bigl(e_{4}\Pi_{4}e^{T}_{4} \bigr)-\operatorname{sym}\bigl(e_{2}\Pi_{5}\bigl( \lambda^{\delta}\lambda _{M}\bigr)e^{T}_{2} \bigr)+\operatorname{sym}\bigl(e_{2}\Pi_{5}\bigl( \lambda^{\delta}+\lambda _{M}\bigr)e^{T}_{5} \bigr) \\& \hphantom{\Phi^{*}_{3}={}}{}-\operatorname{sym}\bigl(e_{5}\Pi_{5}e^{T}_{5} \bigr)-\operatorname{sym}\bigl(e_{3}\Pi_{6}\bigl( \lambda^{\delta}\lambda _{M}\bigr)e^{T}_{3} \bigr) \\& \hphantom{\Phi^{*}_{3}={}}{}+\operatorname{sym}\bigl(e_{3}\Pi_{6}\bigl( \lambda^{\delta}+\lambda _{M}\bigr)e^{T}_{6} \bigr)-\operatorname{sym}\bigl(e_{6}\Pi_{6}e^{T}_{6} \bigr), \\& \Sigma_{a}=\frac{h^{2}}{2}[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{1}& e_{7}\end{array}\displaystyle ]X_{2}[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{1} & e_{7}\end{array}\displaystyle ]^{T}+ \frac{h^{2}}{2}[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2} & e_{7}\end{array}\displaystyle ]X_{3}[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2}& e_{7}\end{array}\displaystyle ]^{T}, \\& \Sigma_{b}=\frac{h^{2}}{2}[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2}& e_{8}\end{array}\displaystyle ]X_{1}[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2} & e_{8}\end{array}\displaystyle ]^{T}+ \frac{h^{2}}{2}[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{3} & e_{8}\end{array}\displaystyle ]X_{4}[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{3} & e_{8}\end{array}\displaystyle ]^{T}. \end{aligned}$$
3.1 Stability analysis
The following theorem is given for system (1) with \(\omega(t)=0\) as the first result.
Theorem 3.1
For given scalars
\(0<\delta\leq1\), \(h>0\), and
μ
and diagonal matrices
\(\lambda_{m}=\operatorname{diag}\{\lambda_{1}^{-},\ldots,\lambda_{n}^{-}\}\)
and
\(\lambda_{M}=\operatorname{diag}\{\lambda_{1}^{+},\ldots,\lambda_{n}^{+}\}\), system (1) with
\(\omega(t)=0\)
is asymptotically stable if there exist positive definite matrices
P, \(Q_{i}\), \(U_{i}\), \(R_{i}\) (\(i=1,2\)) and positive diagonal matrices
\(K_{i}=\operatorname{diag}(k_{i1},\ldots,k_{in})\) (\(i=1,2\)), \(H_{i}=\operatorname{diag}(h_{i1},\ldots,h_{in})\) (\(i=1,\ldots,4\)), and
\(\Pi _{i}=\operatorname{diag}(\pi_{i1},\ldots,\pi_{in})\) (\(i=1,\ldots,6\)) for any matrices
\(Y_{i}\) (\(k=1,\ldots,4\)), \(S_{i}\) (\(i=1,2\)), \(F_{i}\) (\(i=1,2\)), and
\(X_{i}\) (\(i=1,\ldots,4\)) of appropriate dimensions such that the following conditions hold:
$$\begin{aligned}& \bigl(\Gamma^{\perp}\bigr)^{T}(\Xi_{[h(t)=0]}+ \Phi_{i}+\Sigma_{j}) \bigl(\Gamma^{\perp }\bigr)< 0 \quad (\forall i,j=a,b), \end{aligned}$$
(7)
$$\begin{aligned}& \bigl(\Gamma^{\perp}\bigr)^{T}(\Xi_{[h(t)=h]}+ \Phi_{i}+\Sigma_{j}) \bigl(\Gamma^{\perp }\bigr)< 0 \quad (\forall i,j=a,b), \end{aligned}$$
(8)
$$\begin{aligned}& \left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} U_{1}+R_{1} & S_{1} \\ {*} & U_{1}+R_{2} \end{array}\displaystyle \right ]\geq0, \qquad \left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} 3(U_{1}+R_{1}) & S_{2} \\ {*} & 3(U_{1}+R_{2}) \end{array}\displaystyle \right ]\geq0. \end{aligned}$$
(9)
Proof
Let us consider the Lyapunov-Krasovskii functional candidate
$$ V(t,x_{t})=\sum_{i=1}^{5}V_{i}(t,x_{t}), $$
(10)
where
$$\begin{aligned}& V_{1}(t,x_{t})= x^{T}(t)Px(t) +2\sum ^{n}_{i=1} \int^{x_{i}(t)}_{0}\bigl[k_{1i} \bigl(f_{i}(s)-\lambda _{i}^{-}s \bigr)+k_{2i}\bigl(\lambda_{i}^{+}s-f_{i}(s) \bigr)\bigr]\,ds, \\& V_{2}(t,x_{t})= \int^{t}_{t-h(t)}\varepsilon^{T}(s)Q_{1} \varepsilon(s)\,ds+ \int^{t}_{t-h}\varepsilon^{T}(s)Q_{2} \varepsilon(s)\,ds, \\& V_{3}(t,x_{t})=h \int^{0}_{-h} \int^{t}_{t+\theta}\dot{x}^{T}(s)U_{1} \dot {x}(s)\,ds\,d\theta+ \int^{0}_{-h} \int^{t}_{t+\theta}x^{T}(s)U_{2}x(s) \,ds\,d\theta, \\& V_{4}(t,x_{t})= \int^{0}_{-h} \int^{0}_{\theta} \int^{t}_{t+\vartheta}\dot {x}^{T}(s)R_{1} \dot{x}(s)\, ds\, d\vartheta\, d\theta, \\& V_{5}(t,x_{t})= \int^{0}_{-h} \int^{\theta}_{-h} \int^{t}_{t+\vartheta}\dot {x}^{T}(s)R_{2} \dot{x}(s)\, ds\, d\vartheta\, d\theta \end{aligned}$$
and
$$ \varepsilon(t)=\left[\textstyle\begin{array}{@{}c@{\quad}c@{}}x^{T}(t) & f^{T}(x(t))\end{array}\displaystyle \right]^{T}. $$
Then, calculating the time derivative of \(V(t,x_{t})\) along the trajectory of system (1) yields
$$\begin{aligned}& \dot{V}_{1}(t,x_{t})= 2x^{T}(t)P\dot{x}(t) +2 \sum^{n}_{i=1}\bigl[k_{1i} \bigl(f_{i}\bigl(x_{i}(t)\bigr)-\lambda _{i}^{-}x_{i}(t) \bigr)+k_{2i}\bigl(\lambda_{i}^{+}x_{i}(t)-f_{i} \bigl(x_{i}(t)\bigr)\bigr)\bigr] \dot{x}_{i}(t) \\& \hphantom{\dot{V}_{1}(t,x_{t})}= 2x^{T}(t)P\dot{x}(t)+2\bigl[f\bigl(x(t)\bigr)- \lambda_{m}x(t)\bigr]^{T}K_{1}\dot {x}(t)+2\bigl[ \lambda_{M}x(t)-f\bigl(x(t)\bigr)\bigr]^{T}K_{2} \dot{x}(t) \\& \hphantom{\dot{V}_{1}(t,x_{t})}= \xi^{T}(t) \bigl(\operatorname{sym} \bigl(e_{1}Pe_{9}^{T}+(e_{4}-e_{1} \lambda _{m})K_{1}e_{9}^{T}+(e_{1} \lambda_{M}-e_{4})K_{2}e_{9}^{T} \bigr)\bigr)\xi(t), \end{aligned}$$
(11)
$$\begin{aligned}& \dot{V}_{2}(t,x_{t})\leq \varepsilon^{T}(t) (Q_{1}+Q_{2})\varepsilon (t)-(1-\mu)\varepsilon^{T} \bigl(t-h(t)\bigr)Q_{1}\varepsilon\bigl(t-h(t)\bigr) - \varepsilon^{T}(t-h)Q_{2}\varepsilon(t-h) \\& \hphantom{\dot{V}_{2}(t,x_{t})}= \xi^{T}(t) \bigl([\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{1} & e_{4}\end{array}\displaystyle ](Q_{1}+Q_{2})[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{1} & e_{4}\end{array}\displaystyle ]^{T}-(1- \mu )[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2} & e_{5}\end{array}\displaystyle ]Q_{1}[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2}& e_{5}\end{array}\displaystyle ]^{T} \\& \hphantom{\dot{V}_{2}(t,x_{t})\leq{}}{} -[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{3} & e_{6}\end{array}\displaystyle ]Q_{2}[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{3} & e_{6}\end{array}\displaystyle ]^{T} \bigr)\xi(t), \end{aligned}$$
(12)
$$\begin{aligned}& \dot{V}_{3}(t,x_{t})= h^{2} \dot{x}^{T}(t)U_{1}\dot{x}(t)-h \int ^{t}_{t-h}\dot{x}^{T}(s)U_{1} \dot{x}(s)\,ds +hx^{T}(t)U_{2}x(t) \\& \hphantom{\dot{V}_{3}(t,x_{t})={}}{}- \int^{t}_{t-h}x^{T}(s)U_{2}x(s) \,ds. \end{aligned}$$
(13)
By using Lemma 2.1 we can obtain
$$\begin{aligned}& -h \int^{t}_{t-h}\dot{x} ^{T}(s)U_{1} \dot{x}(s)\,ds \\& \quad = -h \int^{t}_{t-h(t)}\dot{x}^{T}(s)U_{1} \dot{x}(s)\,ds-h \int ^{t-h(t)}_{t-h}\dot{x}^{T}(s)U_{1} \dot{x}(s)\,ds \\& \quad \leq -\frac{h}{h(t)}\bigl(x(t)-x\bigl(t-h(t)\bigr)\bigr)^{T}U_{1} \bigl(x(t)-x\bigl(t-h(t)\bigr)\bigr) \\& \qquad {}-\frac {h}{h-h(t)}\bigl(x\bigl(t-h(t) \bigr)-x(t-h)\bigr)^{T} \\& \qquad {}\times U_{1}\bigl(x \bigl(t-h(t)\bigr)-x(t-h)\bigr)-\frac{3h}{h(t)} \bigl(x(t)+x\bigl(t-h(t)\bigr)-2\eta _{1}(t)\bigr)^{T} \\& \qquad {}\times U_{1} \bigl(x(t)+x\bigl(t-h(t)\bigr)-2\eta_{1}(t)\bigr) \\& \qquad {}- \frac{3h}{h-h(t)}\bigl(x\bigl(t-h(t)\bigr)+x(t-h)-2\eta _{2}(t)\bigr)^{T} \\& \qquad {}\times U_{1}\bigl(x\bigl(t-h(t) \bigr)+x(t-h)-2\eta_{2}(t)\bigr). \end{aligned}$$
Using Jensen’ inequality to estimate the \(U_{2}\)-dependent integral term in (13) yields
$$\begin{aligned}& - \int^{t}_{t-h}x ^{T}(s)U_{2}x(s) \,ds=- \int ^{t}_{t-h(t)}x^{T}(s)U_{2}x(s) \,ds- \int ^{t-h(t)}_{t-h}x^{T}(s)U_{2}x(s) \,ds \\& \hphantom{\int^{t}_{t-h}x ^{T}(s)U_{2}x(s) \,ds}\leq -\frac{1}{h(t)}\biggl( \int^{t}_{t-h(t)}x(s)\,ds\biggr)^{T}U_{2} \biggl( \int ^{t}_{t-h(t)}x(s)\,ds\biggr) \\& \hphantom{\int^{t}_{t-h}x ^{T}(s)U_{2}x(s) \,ds={}}{}-\frac{1}{h-h(t)} \biggl( \int^{t-h(t)}_{t-h}x(s)\,ds\biggr)^{T} U_{2}\biggl( \int^{t-h(t)}_{t-h}x(s)\,ds\biggr) \\& \hphantom{\int^{t}_{t-h}x ^{T}(s)U_{2}x(s) \,ds}= -h(t)\eta_{1}^{T}(t)U_{2} \eta_{1}(t)-\bigl(h-h(t)\bigr)\eta_{2}^{T}(t)U_{2} \eta_{2}(t), \\& \dot{V}_{4}(t,x_{t})= \frac{h^{2}}{2} \dot{x}^{T}(t)R_{1}\dot{x}(t)- \int _{-h}^{0} \int_{t+\theta}^{t} \dot{x}^{T}(s)R_{1} \dot{x}(s)\,ds\,d\theta \\& \hphantom{\dot{V}_{4}(t,x_{t})}= \frac{h^{2}}{2}\dot{x}^{T}(t)R_{1} \dot{x}(t) -\bigl(h-h(t)\bigr) \int_{t-h(t)}^{t}\dot{x}^{T}(s)R_{1} \dot{x}(s)\,ds\,d\theta \\& \hphantom{\dot{V}_{4}(t,x_{t})={}}{}- \int_{-h}^{-h(t)} \int_{t+\theta}^{t-h(t)} \dot{x}^{T}(s)R_{1} \dot{x}(s)\,ds\,d\theta - \int_{-h(t)}^{0} \int_{t+\theta}^{t} \dot{x}^{T}(s)R_{1} \dot{x}(s)\,ds\,d\theta \\& \hphantom{\dot{V}_{4}(t,x_{t})}\leq \frac{h^{2}}{2}\dot{x}^{T}(t)R_{1} \dot{x}(t)-\biggl(\frac{h-h(t)}{h(t)}\biggr) \bigl[\bigl(x(t)-x\bigl(t-h(t)\bigr) \bigr)^{T}R_{1}\bigl(x(t)-x\bigl(t-h(t)\bigr)\bigr) \\& \hphantom{\dot{V}_{4}(t,x_{t})={}}{}+3\bigl(x(t)+x\bigl(t-h(t)\bigr)-2\eta_{1}(t) \bigr)^{T}R_{1}\bigl(x(t)+x\bigl(t-h(t)\bigr)-2\eta _{1}(t)\bigr)\bigr] \\& \hphantom{\dot{V}_{4}(t,x_{t})={}}{}- \int_{-h}^{-h(t)} \int_{t+\theta}^{t-h(t)} \dot{x}^{T}(s)R_{1} \dot{x}(s)\,ds\,d\theta - \int_{-h(t)}^{0} \int_{t+\theta}^{t} \dot{x}^{T}(s)R_{1} \dot{x}(s)\,ds\,d\theta, \end{aligned}$$
(14)
$$\begin{aligned}& \dot{V}_{5}(t,x_{t})= \frac{h^{2}}{2} \dot{x}^{T}(t)R_{2}\dot{x}(t)- \int _{-h}^{0} \int_{t-h}^{t+\theta} \dot{x}^{T}(s)R_{2} \dot{x}(s)\,ds\,d\theta \\& \hphantom{\dot{V}_{5}(t,x_{t})}= \frac{h^{2}}{2}\dot{x}^{T}(t)R_{2} \dot{x}(t) -h(t) \int_{t-h}^{t-h(t)}\dot{x}^{T}(s)R_{2} \dot{x}(s)\,ds\,d\theta \\& \hphantom{\dot{V}_{5}(t,x_{t})={}}{}- \int_{-h(t)}^{0} \int_{t-h(t)}^{t+\theta} \dot{x}^{T}(s)R_{2} \dot{x}(s)\,ds\,d\theta - \int_{-h}^{-h(t)} \int_{t-h}^{t+\theta} \dot{x}^{T}(s)R_{2} \dot{x}(s)\,ds\,d\theta \\& \hphantom{\dot{V}_{5}(t,x_{t})}\leq \frac{h^{2}}{2}\dot{x}^{T}(t)R_{2} \dot{x}(t)-\biggl(\frac{h(t)}{h-h(t)}\biggr) \bigl[\bigl(x\bigl(t-h(t)\bigr)-x(t-h) \bigr)^{T} \\& \hphantom{\dot{V}_{5}(t,x_{t})={}}{}\times R_{2}\bigl(x\bigl(t-h(t)\bigr)-x(t-h)\bigr) +3\bigl(x\bigl(t-h(t)\bigr)+x(t-h)-2\eta_{2}(t) \bigr)^{T} \\& \hphantom{\dot{V}_{5}(t,x_{t})={}}{}\times R_{2}\bigl(x\bigl(t-h(t)\bigr)+x(t-h)-2\eta _{2}(t)\bigr)\bigr] \\& \hphantom{\dot{V}_{5}(t,x_{t})={}}{}- \int_{-h(t)}^{0} \int_{t-h(t)}^{t+\theta} \dot{x}^{T}(s)R_{2} \dot{x}(s)\,ds\,d\theta - \int_{-h}^{-h(t)} \int_{t-h}^{t+\theta} \dot{x}^{T}(s)R_{2} \dot{x}(s)\,ds\,d\theta. \end{aligned}$$
(15)
On one hand, from Lemma 2.4 it is clear that if there exist matrices \(S_{1}\) and \(S_{2}\) satisfying (9), then the estimation of the \(U_{1}\)-dependent integral term in (13), the \(R_{1}\)-dependent integral term in (14), and the \(R_{2}\)-dependent integral term in (15) can be obtained as follows:
$$\begin{aligned}& -\xi^{T}(t)\biggl\{ \frac{1}{\alpha}(e_{1}-e_{2})U_{1}(e_{1}-e_{2})^{T}+ \frac {1}{\beta}(e_{2}-e_{3})U_{1}(e_{2}-e_{3})^{T} \\& \qquad {}+\frac{\beta}{\alpha}(e_{1}-e_{2})R_{1}(e_{1}-e_{2})^{T}+\frac{\alpha}{\beta}(e_{2}-e_{3})R_{2}(e_{2}-e_{3})^{T} \biggr\} \xi(t) \\& \quad \leq-\xi^{T}(t)\left [ \textstyle\begin{array}{@{}c@{}} e^{T}_{1}-e^{T}_{2} \\ e^{T}_{2}-e^{T}_{3} \end{array}\displaystyle \right ]^{T} \left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} U_{1} & S_{1} \\ * & U_{1} \end{array}\displaystyle \right ] \left [ \textstyle\begin{array}{@{}c@{}} e^{T}_{1}-e^{T}_{2} \\ e^{T}_{2}-e^{T}_{3} \end{array}\displaystyle \right ] \xi(t), \end{aligned}$$
(16)
$$\begin{aligned}& -\xi^{T}(t)\biggl\{ \frac{1}{\alpha }(e_{1}+e_{2}-2e_{7})3U_{1}(e_{1}+e_{2}-2e_{7})^{T}+ \frac{1}{\beta }(e_{2}+e_{3}-2e_{8}) 3U_{1}(e_{2}+e_{3}-2e_{8})^{T} \\& \qquad {}+\frac{\beta}{\alpha}(e_{1}+e_{2}-2e_{7})3R_{1}(e_{1}+e_{2}-2e_{7})^{T} +\frac{\alpha}{\beta }(e_{2}+e_{3}-2e_{8})3R_{2}(e_{2}+e_{3}-2e_{8})^{T} \biggr\} \xi(t) \\& \quad \leq-\xi^{T}(t)\left [ \textstyle\begin{array}{@{}c@{}} e^{T}_{1}+e^{T}_{2}-2e^{T}_{7} \\ e^{T}_{2}+e^{T}_{3}-2e^{T}_{8} \end{array}\displaystyle \right ]^{T} \left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} 3U_{1} & S_{2} \\ {*} & 3U_{1} \end{array}\displaystyle \right ] \left [ \textstyle\begin{array}{@{}c@{}} e^{T}_{1}+e^{T}_{2}-2e^{T}_{7} \\ e^{T}_{2}+e^{T}_{3}-2e^{T}_{8} \end{array}\displaystyle \right ] \xi(t), \end{aligned}$$
(17)
where \(\alpha=\frac{h(t)}{h}\) and \(\beta=\frac{h-h(t)}{h}\).
On the other hand, according to Lemma 2.3, we obtain
$$\begin{aligned}& -\biggl( \int_{-h}^{-h(t)} \int_{t+\theta}^{t-h(t)} \dot{x}^{T}(s)R_{1} \dot{x}(s)\,ds\,d\theta + \int_{-h(t)}^{0} \int_{t+\theta}^{t} \dot{x}^{T}(s)R_{1} \dot{x}(s)\,ds\,d\theta \\& \quad {}+ \int_{-h(t)}^{0} \int _{t-h(t)}^{t+\theta} \dot{x}^{T}(s)R_{2} \dot{x}(s)\,ds\,d\theta+ \int_{-h}^{-h(t)} \int_{t-h}^{t+\theta} \dot{x}^{T}(s)R_{2} \dot{x}(s)\,ds\,d\theta\biggr) \\& \quad \leq\xi^{T}(t)[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2} & e_{8}\end{array}\displaystyle ]\bigl(h-h(t)\bigr)\operatorname{sym} \bigl\{ Y_{1}[\textstyle\begin{array}{@{}c@{\quad}c@{}}I& -I\end{array}\displaystyle ]\bigr\} [\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2} & e_{8}\end{array}\displaystyle ]^{T}\xi(t) ^{T}(t) \\& \qquad {}+\xi[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{1}& e_{7}\end{array}\displaystyle ]h(t)\operatorname{sym}\bigl\{ Y_{2}[\textstyle\begin{array}{@{}c@{\quad}c@{}}I& -I\end{array}\displaystyle ] \bigr\} [\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{1} & e_{7}\end{array}\displaystyle ]^{T}\xi(t) \\& \qquad {}+\xi^{T}(t)[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2} & e_{7}\end{array}\displaystyle ]h(t) \operatorname{sym}\bigl\{ Y_{3}[\textstyle\begin{array}{@{}c@{\quad}c@{}}-I& I\end{array}\displaystyle ]\bigr\} [\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{2} & e_{7}\end{array}\displaystyle ]^{T} \xi(t) \\& \qquad {}+\xi^{T}(t)[\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{3}& e_{8}\end{array}\displaystyle ]\bigl(h-h(t)\bigr)\operatorname{sym} \bigl\{ Y_{4}[\textstyle\begin{array}{@{}c@{\quad}c@{}}-I& I\end{array}\displaystyle ]\bigr\} [\textstyle\begin{array}{@{}c@{\quad}c@{}}e_{3} & e_{8}\end{array}\displaystyle ]^{T}\xi(t) \\& \qquad {}+ \frac{(h-h(t))^{2}}{2}\varpi_{1}^{T}(t)X_{1} \varpi_{1}(t)+\frac {h(t)^{2}}{2}\varpi_{2}^{T}(t)X_{2} \varpi_{2}(t) \\& \qquad {}+\frac{h(t)^{2}}{2}\varpi_{3}^{T}(t)X_{3} \varpi_{3}(t)+\frac {(h-h(t))^{2}}{2}\varpi_{4}^{T}(t)X_{4} \varpi_{4}(t). \end{aligned}$$
(18)
Now, letting \(M=\varpi_{1}^{T}(t)X_{1}\varpi_{1}(t)+\varpi _{4}^{T}(t)X_{4}\varpi_{4}(t)\) and \(Z=\varpi_{2}^{T}(t)X_{2}\varpi _{2}(t)+\varpi_{3}^{T}(t)X_{3}\varpi_{3}(t)\), define the vector-valued function
$$\begin{aligned} g\bigl(h(t)\bigr) =&\frac{(h-h(t))^{2}}{2}\varpi_{1}^{T}(t)X_{1} \varpi_{1}(t)+\frac {h(t)^{2}}{2}\varpi_{2}^{T}(t)X_{2} \varpi_{2}(t) \\ &{}+\frac{h(t)^{2}}{2}\varpi_{3}^{T}(t)X_{3} \varpi_{3}(t)+\frac{(h-h(t))^{2}}{2}\varpi_{4}^{T}(t)X_{4} \varpi_{4}(t) \\ =&\frac{(h-h(t))^{2}}{2}M+\frac{h(t)^{2}}{2}Z. \end{aligned}$$
(19)
When \(h(t)=\frac{h}{M+Z}\), we have \(\dot{g}(h(t))=0\), and in this case, we can obtain a minimum value. So, it is clear that we can get a maximum value at the endpoints.
Case I: when \(M\geq Z\),
$$ g\bigl(h(t)\bigr)=\frac{(h-h(t))^{2}}{2}M+\frac{h(t)^{2}}{2}Z\leq g(0)=\frac{h^{2}}{2}M. $$
(20)
Case II: when \(M< Z\),
$$ g\bigl(h(t)\bigr)=\frac{(h-h(t))^{2}}{2}M+\frac{h(t)^{2}}{2}Z\leq g(h)=\frac{h^{2}}{2}Z. $$
(21)
In addition, for any matrices \(F_{1}\) and \(F_{2}\) with appropriate dimension, the following zero equation holds:
$$ 2\bigl[x^{T}(t)F_{1}+\dot{x}^{T}(t)F_{2} \bigr] \bigl[-\dot {x}(t)-Cx(t)+Af\bigl(x(t)\bigr)+Bf\bigl(x\bigl(t-h(t)\bigr) \bigr)\bigr]=0. $$
(22)
Furthermore, by introducing a parameter δ for the bound of the activation function we will consider two subintervals, \(\lambda_{i}^{-}\leq(f_{i}(a)-f_{i}(b))/(a-b)\leq\lambda_{i}^{\delta}\) and \(\lambda_{i}^{\delta}\leq(f_{i}(a)-f_{i}(b))/(a-b)\leq\lambda_{i}^{+}\), where \(\lambda_{i}^{\delta}=\lambda_{i}^{-}+\delta(\lambda _{i}^{+}-\lambda_{i}^{-})\).
Case I: \(\lambda_{i}^{-}\leq\frac{f_{i}(a)-f_{i}(b)}{a-b}\leq\lambda _{i}^{\delta}\).
For Case I, the following conditions hold:
$$ \lambda_{i}^{-}\leq\frac {f_{i}(x_{i}(t))-f_{i}(x_{i}(t-h(t)))}{x_{i}(t)-x_{i}(t-h(t))}\leq \lambda_{i}^{\delta}, \quad i=1,2,\ldots,n $$
and
$$ \lambda_{i}^{-}\leq\frac {f_{i}(x_{i}(t-h(t)))-f_{i}(x_{i}(t-h))}{x_{i}(t-h(t))-x_{i}(t-h)}\leq \lambda_{i}^{\delta}, \quad i=1,2,\ldots,n. $$
Then, for any appropriate diagonal matrices \(H_{i}=\operatorname{diag}\{h_{i1},\ldots ,h_{in}\}>0\), \(i=1,2\), we have:
$$\begin{aligned}& 0 \leq-2 {\sum }_{i=1}^{n}h_{1i} \bigl[f_{i}\bigl(x_{i}(t)\bigr)-f_{i} \bigl(x_{i}\bigl(t-h(t)\bigr)\bigr)-\lambda _{i}^{-} \bigl(x_{i}(t)-x_{i}\bigl(t-h(t)\bigr)\bigr)\bigr] \\& \hphantom{0\leq{}}{}\times \bigl[f_{i}\bigl(x_{i}(t) \bigr)-f_{i}\bigl(x_{i}\bigl(t-h(t)\bigr)\bigr)- \lambda_{i}^{\delta }\bigl(x_{i}(t)-x_{i} \bigl(t-h(t)\bigr)\bigr)\bigr] \\& \hphantom{0}=-2\xi^{T}(t)\bigl[e_{4}-e_{5}-(e_{1}-e_{2}) \lambda _{m}\bigr]H_{1}\bigl[e_{4}-e_{5}-(e_{1}-e_{2}) \lambda^{\delta}\bigr]^{T}\xi(t), \end{aligned}$$
(23)
$$\begin{aligned}& 0 \leq-2 {\sum }_{i=1}^{n}h_{2i} \bigl[f_{i}\bigl(x_{i}\bigl(t-h(t)\bigr) \bigr)-f_{i}\bigl(x_{i}(t-h)\bigr)-\lambda _{i}^{-}\bigl(x_{i}\bigl(t-h(t) \bigr)-x_{i}(t-h)\bigr)\bigr] \\& \hphantom{0\leq{}}{}\times\bigl[f_{i}\bigl(x_{i}\bigl(t-h(t) \bigr)\bigr)-f_{i}\bigl(x_{i}(t-h)\bigr)- \lambda_{i}^{\delta }\bigl(x_{i}\bigl(t-h(t) \bigr)-x_{i}(t-h)\bigr)\bigr] \\& \hphantom{0}=-2\xi^{T}(t)\bigl[e_{5}-e_{6}-(e_{2}-e_{3}) \lambda _{m}\bigr]H_{2}\bigl[e_{5}-e_{6}-(e_{2}-e_{3}) \lambda^{\delta}\bigr]^{T}\xi(t). \end{aligned}$$
(24)
When \(b=0\), we have \(\lambda_{i}^{-}\leq\frac{f_{i}(a)}{a}\leq\lambda _{i}^{\delta}\) and, for any scalars \(\pi_{1i}>0\), \(i=1,2,\ldots,n\),
$$ 2\sum_{i=1}^{n}\pi_{1i} \bigl(f_{i}\bigl(x_{i}(t)\bigr)-\lambda _{i}^{-}x_{i}(t) \bigr) \bigl(f_{i}\bigl(x_{i}(t)\bigr)-\lambda_{i}^{\delta}x_{i}(t) \bigr)\leq0, $$
which is equivalent to
$$\begin{aligned}& 2\varepsilon^{T}(t)\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} \Pi_{1}\lambda_{m}\lambda^{\delta} & -\frac{\Pi_{1}}{2}(\lambda _{m}+\lambda^{\delta})\\ * & \Pi_{1} \end{array}\displaystyle \right ]\varepsilon(t) \\& \quad =\xi^{T}(t) \bigl(\operatorname{sym}\bigl(e_{1} \Pi_{1}\bigl(\lambda_{m}\lambda^{\delta } \bigr)e^{T}_{1}\bigr)-\operatorname{sym}\bigl(e_{1} \Pi_{1}\bigl(\lambda_{m}+\lambda^{\delta} \bigr)e^{T}_{4}\bigr) +\operatorname{sym} \bigl(e_{4}\Pi_{1}e^{T}_{4}\bigr)\bigr) \xi(t) \\& \quad \leq0, \end{aligned}$$
(25)
where \(\Pi_{1}=\operatorname{diag}\{\pi_{11},\ldots,\pi_{1n}\}\).
Similarly, for any appropriately diagonal matrices \(\Pi_{i}=\operatorname{diag}\{\pi _{i1},\ldots,\pi_{in}\}>0\), \(i=2,3\), we have:
$$\begin{aligned}& 2\varepsilon^{T}\bigl(t-h(t)\bigr)\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} \Pi_{2}\lambda_{m}\lambda^{\delta} & -\frac{\Pi_{2}}{2}(\lambda _{m}+\lambda^{\delta}) \\ {*} & \Pi_{2} \end{array}\displaystyle \right ]\varepsilon\bigl(t-h(t)\bigr) \\& \quad =\xi^{T}(t) \bigl(\operatorname{sym}\bigl(e_{2} \Pi_{2}\bigl(\lambda_{m}\lambda^{\delta } \bigr)e^{T}_{2}\bigr)-\operatorname{sym}\bigl(e_{2} \Pi_{2}\bigl(\lambda_{m}+\lambda^{\delta} \bigr)e^{T}_{5}\bigr) +\operatorname{sym} \bigl(e_{5}\Pi_{2}e^{T}_{5}\bigr)\bigr) \xi(t) \\& \quad \leq0, \end{aligned}$$
(26)
$$\begin{aligned}& 2\varepsilon^{T}(t-h)\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} \Pi_{3}\lambda_{m}\lambda^{\delta} & -\frac{\Pi_{3}}{2}(\lambda _{m}+\lambda^{\delta}) \\ {*} & \Pi_{3} \end{array}\displaystyle \right ]\varepsilon(t-h) \\& \quad =\xi^{T}(t) \bigl(\operatorname{sym}\bigl(e_{3} \Pi_{3}\bigl(\lambda_{m}\lambda^{\delta } \bigr)e^{T}_{3}\bigr)-\operatorname{sym}\bigl(e_{3} \Pi_{3}\bigl(\lambda_{m}+\lambda^{\delta} \bigr)e^{T}_{6}\bigr) +\operatorname{sym} \bigl(e_{6}\Pi_{3}e^{T}_{6}\bigr)\bigr) \xi(t) \\& \quad \leq0. \end{aligned}$$
(27)
Combining the inequalities from (11) to (27) together gives the upper bound of \(\dot{V}(t,x_{t})\):
$$ \dot{V}(t,x_{t})\leq\xi^{T}(t) ( \Xi_{[h(t)]}+\Phi_{a}+\Sigma_{j})\xi(t) \quad (\forall j=a,b). $$
(28)
Case II: \(\lambda_{i}^{\delta}\leq\frac{f_{i}(a)-f_{i}(b)}{a-b}\leq \lambda_{i}^{+}\).
Case II can be discussed similarly as the procedure in Case I. Then we obtain:
$$ 0\leq\xi^{T}(t)\Phi^{*}_{4}\xi(t), $$
(29)
where \(H_{3}\), \(H_{4}\), and \(\Pi_{i}\) (\(i=4,\ldots,6\)) are defined in Theorem 3.1.
Combining the inequalities from (11) to (22), together with (29), gives the upper bound of \(\dot{V}(t,x_{t})\):
$$ \dot{V}(t,x_{t})\leq\xi^{T}(t) ( \Xi_{[h(t)]}+\Phi_{b}+\Sigma_{j})\xi(t) \quad (\forall j=a,b). $$
(30)
Using the fact that \(\Xi_{[h(t)]}\) is dependent on \(h(t)\) and applying Lemma 2.5 with \(\Gamma\xi(t)=0\), it follows that if LMIs (7), (8) hold, then system (1) with \(\omega(t)=0\) is asymptotically stable. This ends the proof. □
3.2 Extended dissipative analysis
In this section, by assuming zero initial conditions we establish the extended dissipativity condition for all nonzero \(\omega(t)\in\ell _{2}[0,\infty]\).
Theorem 3.2
For given scalars
\(0<\delta\leq1\), \(h>0\), and
μ, diagonal matrices
\(\lambda_{m}=\operatorname{diag}\{\lambda_{1}^{-},\ldots, \lambda_{n}^{-}\}\)
and
\(\lambda_{M}=\operatorname{diag}\{\lambda_{1}^{+},\ldots,\lambda_{n}^{+}\}\), and matrices
\(\Psi_{i}\) (\(i=1,\ldots,4\)) satisfying Assumption
2.2, system (1) is asymptotically stable and extended dissipative if there exist positive definite matrices
P, \(Q_{i}\), \(U_{i}\), \(R_{i}\) (\(i=1,2\)) and positive diagonal matrices
\(K_{i}=\operatorname{diag}(k_{i1},\ldots,k_{in})\) (\(i=1,2\)), \(H_{i}=\operatorname{diag}(h_{i1},\ldots,h_{in})\) (\(i=1,\ldots,4\)), and
\(\Pi _{i}=\operatorname{diag}(\pi_{i1},\ldots,\pi_{in})\) (\(i=1,\ldots,6\)) for any matrices
\(Y_{i}\) (\(k=1,\ldots,4\)), \(S_{i}\) (\(i=1,2\)), \(F_{i}\) (\(i=1,2\)), and
\(X_{i}\) (\(i=1,\ldots,4\)) of appropriate dimensions such that LMIs (9) and the following conditions hold:
$$\begin{aligned}& \bigl(\bar{\Gamma}^{\perp}\bigr)^{T}(\bar{\Xi}_{[h(t)=0]}+ \bar{\Phi}_{i}+\bar {\Sigma}_{j}) \bigl(\bar{ \Gamma}^{\perp}\bigr)< 0 \quad (\forall i,j=a,b), \end{aligned}$$
(31)
$$\begin{aligned}& \bigl(\bar{\Gamma}^{\perp}\bigr)^{T}(\bar{\Xi}_{[h(t)=h]}+ \bar{\Phi}_{i}+\bar {\Sigma}_{j}) \bigl(\bar{ \Gamma}^{\perp}\bigr)< 0 \quad (\forall i,j=a,b), \end{aligned}$$
(32)
$$\begin{aligned}& P-D^{T}\Psi_{4}D\geq0, \end{aligned}$$
(33)
where
$$\begin{aligned}& \bar{\Gamma}=[\textstyle\begin{array}{@{}c@{\quad}c@{}}\Gamma & I\end{array}\displaystyle ], \qquad \bar{\Xi}_{[h(t)]}=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} \Xi_{[h(t)]} & 0 \\ {*} & 0 \end{array}\displaystyle \right ], \\& \bar{\Phi}_{i}= \left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} \bar{\Phi}_{1} & \bar{\Phi}_{2} \\ * & -\Psi_{3} \end{array}\displaystyle \right ], \qquad \bar{ \Sigma}_{j}=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} \Sigma_{j} & 0 \\ * & 0 \end{array}\displaystyle \right ], \\& \bar{\Phi}_{1}=\Phi_{i}-e_{1}D^{T} \Psi_{1}De_{1}^{T},\qquad \bar{ \Phi}_{2}=\bigl[\textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}}-\Psi^{T}_{2}D & 0& 0& 0 & 0 & 0 & 0 & 0 & 0\end{array}\displaystyle \bigr]^{T}. \end{aligned}$$
Proof
From (28) and (30) we have \(\dot{V}(t,x_{t})\leq\xi^{T}(t)(\Xi_{[h(t)]}+\Phi_{i}+\Sigma_{j})\xi(t)\) (\(\forall i,j=a,b\)), and it is clear that
$$ \bar{\xi}^{T}(t) (\bar{\Xi}_{[h(t)]}+\bar{\Phi}_{i}+ \bar{\Sigma}_{j})\bar {\xi}(t) =\xi^{T}(t) ( \Xi_{[h(t)]}+\Phi_{i}+\Sigma_{j})\xi(t)-J(t), $$
where \(\bar{\xi}^{T}(t)=[\xi^{T}(t)\ \omega^{T}(t)]^{T}\) and \(J(t)\) are defined in Definition 2.1. By Lemma 2.5, (31) and (32) are equivalent to \(\bar{\xi}^{T}(t)(\bar{\Xi }_{[h(t)]}+\bar{\Phi}_{i}+\bar{\Sigma}_{j})\bar{\xi}(t)<0\) (\(\forall i,j=a,b\)). Therefore, we can obtain
$$ \dot{V}(t)\leq\bar{\xi}^{T}(t) (\bar{\Xi}_{[h(t)]}+\bar{ \Phi}_{i}+\bar {\Sigma}_{j})\bar{\xi}(t) +J(t)\leq J(t). $$
By integrating both sides of this inequality from 0 to \(t\geq0\) we can obtain
$$ \int_{0}^{t}J(s)\,ds\geq V(t)-V(0)\geq x^{T}(t)Px(t). $$
(34)
Considering the two cases of \(\Psi_{4}=0\) and \(\Psi_{4}>0\), due to the extended dissipativity condition, we can represent the strictly \((\mathcal{Q},\mathcal{S},\mathcal{R})\)-dissipativity condition, the \(H_{\infty}\) performance, and the passivity when \(\Psi _{4}=0\) or the \(\ell_{2}-\ell_{\infty}\) performance criterion when \(\Psi_{4}>0\).
On one hand, we consider \(\Psi_{4}=0\) and from (34) we can get that
$$ \int_{0}^{t_{f}}J(s)\,ds\geq0. $$
(35)
This implies Assumption 2.2 with \(\Psi_{4}=0\).
On the other hand, when \(\Psi_{4}>0\), as mentioned in Assumption 2.2, we have the matrices \(\Psi_{1}=0\), \(\Psi_{2}=0\), and \(\Psi_{3}>0\) in this case. Then, for any \(0\leq t\leq t_{f}\), (34) leads to \(\int _{0}^{t_{f}}J(s)\,ds\geq\int_{0}^{t}J(s)\,ds\geq x^{T}(t)Px(t)\). Therefore, according to (33), we have
$$ y^{T}(t)\Psi_{4}y(t)=x^{T}(t)D^{T} \Phi_{4}Dx(t)\leq x^{T}(t)Px(t)\leq \int_{0}^{t_{f}}J(s)\,ds. $$
(36)
From (35) and (36) we get that system (1) is extended dissipative. This completes the proof. □