Appendix 1: Proof of Lemma 1
The proof of inequality (6):
Choose the first three Legendre orthogonal polynomials on \([\omega ,\chi ]\) (see [28]):
$$ p_{0} (\upsilon ) = 1,p_{1} (\upsilon ) = \frac{1}{\chi - \omega }(2\upsilon - \omega - \chi ),p_{2} (\upsilon ) = \frac{1}{{(\chi - \omega )^{2} }}[6\upsilon^{2} - 6(\omega + \chi )\upsilon + (\omega^{2} + 4\omega \chi + \chi^{2} )]. $$
Simple calculation gives
$$ \begin{array}{*{20}l} {\int_{\omega }^{\chi } {p_{i} } (\upsilon )p_{j} (\upsilon ){\text{d}}\upsilon = \left\{ {\begin{array}{*{20}c} {0,\quad \;} & {i \ne j,} \\ {\frac{\chi - \omega }{{2i + 1}},} & {i = j,} \\ \end{array} } \right.} \hfill & {i,j = 0,1,2.} \hfill \\ \end{array} $$
(17)
Define \(\varsigma (\upsilon ) = {\text{col}}\)\(\{ p_{0} (\upsilon )\ell ,p_{1} (\upsilon )\ell ,p_{2} (\upsilon )\ell \} ,{{\mathbb{M}}\ominus } = [M_{1} \;M_{2} \;M_{3} ].\) For matrix \(B > {\mathbf{0}},\) it follows from the well-known Schur complement lemma that
$$ \left[ {\begin{array}{*{20}c} B & * \\ {{\mathbb{M}}^{{ \top }} } & {{\mathbb{M}}^{{ \top }} B^{ - 1} {\mathbb{M}}} \\ \end{array} } \right] \ge {\mathbf{0}} $$
which directly brings about
$$ ( * )^{{ \top }} \left[ {\begin{array}{*{20}c} B & * \\ {{\mathbb{M}}^{{ \top }} } & {{\mathbb{M}}^{{ \top }} B^{ - 1} {\mathbb{M}}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\vartheta (\upsilon )} \\ {\varsigma (\upsilon )} \\ \end{array} } \right] \ge 0 $$
or equivalently
$$ - 2\varsigma (\upsilon )^{{ \top }} {\mathbb{M}}^{{ \top }} \vartheta (\upsilon ) \le ( * )^{{ \top }} B^{ - 1} {\mathbb{M}}\varsigma (\upsilon ) + ( * )^{{ \top }} B\vartheta (\upsilon ). $$
(18)
Integrating both sides of inequality (18) from \(\omega\) to \(\chi\) yields
$$ \begin{gathered} \int_{\omega }^{\chi } {( * )^{{ \top }} B^{ - 1} {\mathbb{M}}\varsigma (\upsilon ){\text{d}}\upsilon } + \int_{\omega }^{\chi } {( * )^{{ \top }} B\vartheta (\upsilon ){\text{d}}\upsilon } \hfill \\ \ge - 2\ell^{{ \top }} M_{1}^{{ \top }} \int_{\omega }^{\chi } {p_{0} (\upsilon )\vartheta (\upsilon ){\text{d}}\upsilon } - 2\ell^{{ \top }} M_{2}^{{ \top }} \int_{\omega }^{\chi } {p_{1} (\upsilon )\vartheta (\upsilon ){\text{d}}\upsilon } - 2\ell^{{ \top }} M_{3}^{{ \top }} \int_{\omega }^{\chi } {p_{2} (\upsilon )\vartheta (\upsilon ){\text{d}}\upsilon } . \hfill \\ \end{gathered} $$
Integration by parts and applying (17) derives
$$ \begin{gathered} - 2\ell^{{ \top }} M_{1}^{{ \top }} \Theta_{1} \ell - 2\ell^{{ \top }} M_{2}^{{ \top }} \Theta_{2} \ell - 2\ell^{{ \top }} M_{3}^{{ \top }} \Theta_{3} \ell \hfill \\ \le (\chi - \omega )\ell^{{ \top }} \left( {M_{1}^{{ \top }} B^{ - 1} M_{1} + \frac{1}{3}M_{2}^{{ \top }} B^{ - 1} M_{2} + \frac{1}{5}M_{3}^{{ \top }} B^{ - 1} M_{3} } \right)\ell + \int_{\omega }^{\chi } {( * )^{{ \top }} B\vartheta (\upsilon ){\text{d}}\upsilon } , \hfill \\ \end{gathered} $$
which gives inequality (6).
The proof of inequality (7):
Choosing the first three weighted Legendre orthogonal polynomials on \([\omega ,\chi ]\)
$$ q_{0} (\upsilon ) = 1,q_{1} (\upsilon ) = \frac{1}{\chi - \omega }(3\upsilon - \omega - 2\chi ),q_{2} (\upsilon ) = \frac{1}{{(\chi - \omega )^{2} }}[10(\upsilon - \omega )^{2} - 12(\upsilon - \omega )(\chi - \omega ) + 3(\chi - \omega )^{2} ], $$
simple calculation gives
$$ \begin{array}{*{20}l} {\int_{\omega }^{\chi } {(\upsilon - \omega )q_{i} } (\upsilon )q_{j} (\upsilon ){\text{d}}\upsilon = \left\{ {\begin{array}{*{20}c} {0,\quad \quad \;} & {i \ne j,} \\ {\frac{{(\chi - \omega )^{2} }}{2(i + 1)},} & {i = j,} \\ \end{array} } \right.} \hfill & {i,j = 0,1,2.} \hfill \\ \end{array} $$
(19)
Define \(\mu (\upsilon ) = {\text{col}}\)\(\{ q_{0} (\upsilon )\ell ,q_{1} (\upsilon )\ell ,q_{2} (\upsilon )\ell \} ,{{\mathbb{M}}\ominus } = [M_{1} \;M_{2} \;M_{3} ].\) For matrix \(B > {\mathbf{0}},\) it follows from the Schur complement Lemma that
$$ \left[ {\begin{array}{*{20}c} B & * \\ {{\mathbb{M}}^{{ \top }} } & {{\mathbb{M}}^{{ \top }} B^{ - 1} {\mathbb{M}}} \\ \end{array} } \right] \ge {\mathbf{0}} $$
which directly brings about
$$ ( * )^{{ \top }} \left[ {\begin{array}{*{20}c} B & * \\ {{\mathbb{M}}^{{ \top }} } & {{\mathbb{M}}^{{ \top }} B^{ - 1} {\mathbb{M}}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\vartheta (\upsilon )} \\ {\mu (\upsilon )} \\ \end{array} } \right] \ge 0 $$
or equivalently
$$ - 2\mu (\upsilon )^{{ \top }} {\mathbb{M}}^{{ \top }} \vartheta (\upsilon ) \le ( * )^{{ \top }} B^{ - 1} {\mathbb{M}}\mu (\upsilon ) + ( * )^{{ \top }} B\vartheta (\upsilon ). $$
(20)
For \(\upsilon \in [\omega ,\chi ],\) multiplying \(\upsilon - \omega\) and integrating both sides of inequality (20) from \(\omega\) to \(\chi\) yields
$$ \begin{gathered} \int_{\omega }^{\chi } {(\upsilon - \omega )( * )^{{ \top }} B^{ - 1} {\mathbb{M}}\mu (\upsilon ){\text{d}}\upsilon } + \int_{\omega }^{\chi } {(\upsilon - \omega )( * )^{{ \top }} B\vartheta (\upsilon ){\text{d}}\upsilon } \hfill \\ \ge - 2\ell^{{ \top }} M_{1}^{{ \top }} \int_{\omega }^{\chi } {(\upsilon - \omega )\vartheta (\upsilon ){\text{d}}\upsilon } - 2\ell^{{ \top }} M_{2}^{{ \top }} \int_{\omega }^{\chi } {(\upsilon - \omega )q_{1} (\upsilon )\vartheta (\upsilon ){\text{d}}\upsilon } - 2\ell^{{ \top }} M_{3}^{{ \top }} \int_{\omega }^{\chi } {(\upsilon - \omega )q_{2} (\upsilon )\vartheta (\upsilon ){\text{d}}\upsilon } . \hfill \\ \end{gathered} $$
Integration by parts and applying (19) derives
$$ \begin{gathered} - 2\ell^{{ \top }} M_{1}^{{ \top }} \Theta_{1} \ell - 2\ell^{{ \top }} M_{2}^{{ \top }} \Theta_{2} \ell - 2\ell^{{ \top }} M_{3}^{{ \top }} \Theta_{3} \ell \hfill \\ \le \frac{1}{2}(\chi - \omega )^{2} \ell^{{ \top }} \left( {M_{1}^{{ \top }} B^{ - 1} M_{1} + \frac{1}{2}M_{2}^{{ \top }} B^{ - 1} M_{2} + \frac{1}{3}M_{3}^{{ \top }} B^{ - 1} M_{3} } \right)\ell + \int_{\omega }^{\chi } {(\upsilon - \omega )( * )^{{ \top }} B\vartheta (\upsilon ){\text{d}}\upsilon } , \hfill \\ \end{gathered} $$
which gives inequality (7).
Appendix 2
Denote \(e_{i} = [\begin{array}{*{20}c} {\underbrace {{{\mathbf{0}}\,\;{\kern 1pt} ...\;\,{\kern 1pt} {\mathbf{0}}}}_{i - 1}} & {I_{n} } & {\underbrace {{{\mathbf{0}}\,\;{\kern 1pt} ...\;\,{\kern 1pt} {\mathbf{0}}}}_{17 - i}} \\ \end{array} ],{ }i = 1,2,...,17,\)
$$ e_{u} = - \Omega e_{1} + \Pi e_{8} ,e_{v} = - \Delta e_{4} + \Sigma e_{2} ,{\mathbf{\mathbb{Q}}}_{j} = {\text{diag}}\{ Q_{j} ,3Q_{j} ,5Q_{j} \} ,j = 3,4,...,8, $$
$$ \begin{gathered} e_{a} = {\text{col}}\{ e_{1} - e_{2} ,e_{1} + e_{2} - 2e_{10} ,e_{1} - e_{2} + 6e_{10} - 6e_{14} \} , \hfill \\ e_{b} = {\text{col}}\{ e_{2} - e_{3} ,e_{2} + e_{3} - 2e_{11} ,e_{2} - e_{3} + 6e_{11} - 6e_{15} \} , \hfill \\ e_{c} = {\text{col}}\{ e_{4} - e_{5} ,e_{4} + e_{5} - 2e_{12} ,e_{4} - e_{5} + 6e_{12} - 6e_{16} \} , \hfill \\ e_{d} = {\text{col}}\{ e_{5} - e_{6} ,e_{5} + e_{6} - 2e_{13} ,e_{5} - e_{6} + 6e_{13} - 6e_{17} \} , \hfill \\ \Phi_{k} = \left[ {\begin{array}{*{20}c} {\overline{\mu }S_{1} } & {\overline{\mu }S_{2} + R_{k} } \\ * & {\overline{\mu }S_{3} } \\ \end{array} } \right],\Phi_{k + 2} = \left[ {\begin{array}{*{20}c} {\overline{\kappa }U_{1} } & {\overline{\kappa }U_{2} + R_{k + 2} } \\ * & {\overline{\kappa }U_{3} } \\ \end{array} } \right],\;k = 1,2, \hfill \\ \Xi_{0} = {\text{sym}}\left\{ {\left[ {\begin{array}{*{20}c} {e_{1} } \\ {\mathbf{0}} \\ \end{array} } \right]^{{ \top }} P_{1} \left[ {\begin{array}{*{20}c} {e_{u} } \\ {e_{1} - e_{3} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {e_{4} } \\ {\mathbf{0}} \\ \end{array} } \right]^{{ \top }} P_{2} \left[ {\begin{array}{*{20}c} {e_{v} } \\ {e_{4} - e_{6} } \\ \end{array} } \right]} \right\}, \hfill \\ \end{gathered} $$
$$ \begin{gathered} \Xi_{1} = {\text{sym}}\left\{ {\left[ {\begin{array}{*{20}c} {e_{1} } \\ {e_{10} } \\ \end{array} } \right]^{{ \top }} P_{5} \left[ {\begin{array}{*{20}c} {\mathbf{0}} \\ {e_{1} - e_{2} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {e_{1} } \\ {e_{11} } \\ \end{array} } \right]^{{ \top }} P_{6} \left[ {\begin{array}{*{20}c} {\mathbf{0}} \\ {e_{2} - e_{3} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {e_{4} } \\ {e_{12} } \\ \end{array} } \right]^{{ \top }} P_{7} \left[ {\begin{array}{*{20}c} {\mathbf{0}} \\ {e_{4} - e_{5} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {e_{4} } \\ {e_{13} } \\ \end{array} } \right]^{{ \top }} P_{8} \left[ {\begin{array}{*{20}c} {\mathbf{0}} \\ {e_{5} - e_{6} } \\ \end{array} } \right]} \right\} \hfill \\ \quad \quad \; + {\text{sym}}\left\{ {[(\Theta_{1} e_{4} - e_{7} )^{{ \top }} \Lambda_{1} + (e_{7} - \Theta_{2} e_{4} )^{{ \top }} \Lambda_{2} ]e_{v} + (\Theta_{1} e_{4} - e_{7} )^{{ \top }} Y_{1} (e_{7} - \Theta_{2} e_{4} )} \right\} \hfill \\ \quad \quad \; + {\text{sym}}\left\{ {(\Theta_{1} e_{5} - e_{8} )^{{ \top }} Y_{2} (e_{8} - \Theta_{2} e_{5} ) + (\Theta_{1} e_{6} - e_{9} )^{{ \top }} Y_{3} (e_{9} - \Theta_{2} e_{6} )} \right\} \hfill \\ \quad \quad \; + {\text{sym}}\left\{ {[\Theta_{1} (e_{4} - e_{5} ) - (e_{7} - e_{8} )]^{{ \top }} Y_{4} [(e_{7} - e_{8} ) - \Theta_{2} (e_{4} - e_{5} )]} \right\} \hfill \\ \quad \quad \; + {\text{sym}}\left\{ {[\Theta_{1} (e_{5} - e_{6} ) - (e_{8} - e_{9} )]^{{ \top }} Y_{5} [(e_{8} - e_{9} ) - \Theta_{2} (e_{5} - e_{6} )]} \right\} \hfill \\ \quad \quad \; + {\text{sym}}\left\{ {[\Theta_{1} (e_{4} - e_{6} ) - (e_{7} - e_{9} )]^{{ \top }} Y_{6} [(e_{7} - e_{9} ) - \Theta_{2} (e_{4} - e_{6} )]} \right\} \hfill \\ \end{gathered} $$
$$ \begin{gathered} \Xi_{\mu } = e_{2}^{{ \top }} (Q_{1} - Q_{2} )e_{2} + ( * )^{{ \top }} P_{5} \left[ {\begin{array}{*{20}c} {e_{1} } \\ {e_{10} } \\ \end{array} } \right] - ( * )^{{ \top }} P_{6} \left[ {\begin{array}{*{20}c} {e_{1} } \\ {e_{11} } \\ \end{array} } \right] + {\text{sym}}\left\{ {\left[ {\begin{array}{*{20}c} {e_{1} } \\ {e_{10} } \\ \end{array} } \right]^{{ \top }} P_{5} \left[ {\begin{array}{*{20}c} {\mathbf{0}} \\ {e_{2} - e_{10} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {e_{1} } \\ {e_{11} } \\ \end{array} } \right]^{{ \top }} P_{6} \left[ {\begin{array}{*{20}c} {\mathbf{0}} \\ {e_{11} - e_{2} } \\ \end{array} } \right]} \right\}, \hfill \\ \Xi_{\kappa } = ( * )^{{ \top }} (P_{3} - P_{4} )\left[ {\begin{array}{*{20}c} {e_{5} } \\ {e_{8} } \\ \end{array} } \right] + ( * )^{{ \top }} P_{7} \left[ {\begin{array}{*{20}c} {e_{4} } \\ {e_{12} } \\ \end{array} } \right] - ( * )^{{ \top }} P_{8} \left[ {\begin{array}{*{20}c} {e_{4} } \\ {e_{13} } \\ \end{array} } \right] + {\text{sym}}\left\{ {\left[ {\begin{array}{*{20}c} {e_{4} } \\ {e_{12} } \\ \end{array} } \right]^{{ \top }} P_{7} \left[ {\begin{array}{*{20}c} {\mathbf{0}} \\ {e_{5} - e_{12} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {e_{4} } \\ {e_{13} } \\ \end{array} } \right]^{{ \top }} P_{8} \left[ {\begin{array}{*{20}c} {\mathbf{0}} \\ {e_{13} - e_{5} } \\ \end{array} } \right]} \right\}, \hfill \\ \end{gathered} $$
$$ \begin{gathered} \Xi_{2} = {\text{sym}}\left\{ {\left[ {\begin{array}{*{20}c} {\mathbf{0}} \\ {e_{10} } \\ \end{array} } \right]^{{ \top }} P_{1} \left[ {\begin{array}{*{20}c} {e_{u} } \\ {e_{1} - e_{3} } \\ \end{array} } \right]} \right\},\quad \Xi_{3} = {\text{sym}}\left\{ {\left[ {\begin{array}{*{20}c} {e_{1} } \\ {e_{10} } \\ \end{array} } \right]^{{ \top }} P_{5} \left[ {\begin{array}{*{20}c} {e_{u} } \\ {\mathbf{0}} \\ \end{array} } \right]} \right\} - e_{10}^{{ \top }} (\overline{\mu }S_{1} )e_{10} \hfill \\ \quad \;\;\;\quad \;\;\; + {\text{sym\{ }}6(e_{10} - e_{14} )^{{ \top }} (\overline{\mu }S_{2} + R_{1} )e_{10} \} - 3( * )^{{ \top }} (\overline{\mu }S_{1} )(e_{10} - e_{14} ) - 12e_{10}^{{ \top }} (\overline{\mu }S_{3} )e_{10} , \hfill \\ \end{gathered} $$
$$ \begin{gathered} \Xi_{4} = {\text{sym}}\left\{ {\left[ {\begin{array}{*{20}c} {\mathbf{0}} \\ {e_{11} } \\ \end{array} } \right]^{{ \top }} P_{1} \left[ {\begin{array}{*{20}c} {e_{u} } \\ {e_{1} - e_{3} } \\ \end{array} } \right]} \right\},\quad \Xi_{5} = {\text{sym}}\left\{ {\left[ {\begin{array}{*{20}c} {e_{1} } \\ {e_{11} } \\ \end{array} } \right]^{{ \top }} P_{6} \left[ {\begin{array}{*{20}c} {e_{u} } \\ {\mathbf{0}} \\ \end{array} } \right]} \right\} - e_{11}^{{ \top }} (\overline{\mu }S_{1} )e_{11} \hfill \\ \quad \;\;\;\,\,\quad \;\;\; + {\text{sym}}\{ 6(e_{11} - e_{15} )^{{ \top }} (\overline{\mu }S_{2} + R_{2} )e_{11} \} - 3( * )^{{ \top }} (\overline{\mu }S_{1} )(e_{11} - e_{15} ) - 12e_{11}^{{ \top }} (\overline{\mu }S_{3} )e_{11} , \hfill \\ \end{gathered} $$
$$ \begin{gathered} \Xi_{6} = {\text{sym}}\left\{ {\left[ {\begin{array}{*{20}c} {\mathbf{0}} \\ {e_{12} } \\ \end{array} } \right]^{{ \top }} P_{2} \left[ {\begin{array}{*{20}c} {e_{v} } \\ {e_{4} - e_{6} } \\ \end{array} } \right]} \right\},\quad \Xi_{7} = \left\{ {\left[ {\begin{array}{*{20}c} {e_{4} } \\ {e_{12} } \\ \end{array} } \right]^{{ \top }} P_{7} \left[ {\begin{array}{*{20}c} {e_{v} } \\ {\mathbf{0}} \\ \end{array} } \right]} \right\} - e_{12}^{{ \top }} (\overline{\kappa }U_{1} )e_{12} \hfill \\ \quad \;\;\;\,\,\quad \;\;\;\,\,{\text{ + sym\{ }}6(e_{12} - e_{16} )^{{ \top }} (\overline{\kappa }U_{2} + R_{3} )e_{12} - 3( * )^{{ \top }} (\overline{\kappa }U_{1} )(e_{12} - e_{16} ) - 12e_{12}^{{ \top }} (\overline{\kappa }U_{3} )e_{12} , \hfill \\ \end{gathered} $$
$$ \begin{gathered} \Xi_{8} = {\text{sym}}\left\{ {\left[ {\begin{array}{*{20}c} {\mathbf{0}} \\ {e_{13} } \\ \end{array} } \right]^{{ \top }} P_{2} \left[ {\begin{array}{*{20}c} {e_{v} } \\ {e_{4} - e_{6} } \\ \end{array} } \right]} \right\},\quad \Xi_{9} = {\text{sym}}\left\{ {\left[ {\begin{array}{*{20}c} {e_{4} } \\ {e_{13} } \\ \end{array} } \right]^{{ \top }} P_{8} \left[ {\begin{array}{*{20}c} {e_{v} } \\ {\mathbf{0}} \\ \end{array} } \right]} \right\} - e_{13}^{{ \top }} (\overline{\kappa }U_{1} )e_{13} \hfill \\ \quad \;\;\;\,\,\quad \;\;\;\,\,{\text{ + sym}}\{ 6(e_{13} - e_{17} )^{{ \top }} (\overline{\kappa }U_{2} + R_{4} )e_{13} \} - 3( * )^{{ \top }} (\overline{\kappa }U_{1} )(e_{13} - e_{17} ) - 12e_{13}^{{ \top }} (\overline{\kappa }U_{3} )e_{13} , \hfill \\ \end{gathered} $$
$$ \begin{gathered} \Xi_{a} = \Xi_{0} + \Xi^{\prime}_{1} + \overline{\mu }(\Xi_{4} + \Xi_{5} ) + \overline{\kappa }(\Xi_{8} + \Xi_{9} ),\;\;\Xi_{b} = \Xi_{0} + \Xi^{\prime}_{1} + \overline{\mu }(\Xi_{4} + \Xi_{5} ) + \overline{\kappa }(\Xi_{6} + \Xi_{7} ), \hfill \\ \Xi_{c} = \Xi_{0} + \Xi^{\prime}_{1} + \overline{\mu }(\Xi_{2} + \Xi_{3} ) + \overline{\kappa }(\Xi_{8} + \Xi_{9} ),\;\;\Xi_{d} = \Xi_{0} + \Xi^{\prime}_{1} + \overline{\mu }(\Xi_{2} + \Xi_{3} ) + \overline{\kappa }(\Xi_{6} + \Xi_{7} ),\; \hfill \\ \Xi^{\prime}_{1} = \Xi_{1} + \mu_{d} \Xi_{\mu } + \kappa_{d} \Xi_{\kappa } . \hfill \\ \end{gathered} $$
Appendix 3: Proof of Theorem 1
Consider the following LKF:
$$ V(t) = \sum\limits_{j = 1}^{7} {V_{j} } (t), $$
where
$$\begin{aligned} V_{l} (t) & = ( * )^{ { \top }} P_{l} \eta _{{lt}} ,\;l = 1,2, \hfill \\ V_{3} (t) & = 2\sum\limits_{{i = 1}}^{n} {\int_{0}^{{w_{i} (t)}} {\{ \lambda _{{1i}} [\theta _{i}^{ + } \upsilon - \rho _{i} (} } \upsilon )] + \lambda _{{2i}} [\rho _{i} (\upsilon ) - \theta _{i}^{ - } \upsilon ]\} {\text{d}}\upsilon , \hfill \\ \end{aligned}$$
$$\begin{gathered} V_{4} (t) = \int_{{t - \mu (t)}}^{t} {u_{\upsilon }^{ { \top }} } Q_{1} u_{\upsilon } {\text{d}}\upsilon + \int_{{t - \bar{\mu }}}^{{t - \mu (t)}} {u_{\upsilon }^{ { \top }} } Q_{2} u_{\upsilon } {\text{d}}\upsilon + \int_{{t - \kappa (t)}}^{t} {\eta _{{3\upsilon }}^{ { \top }} } P_{3} \eta _{{3\upsilon }} {\text{d}}\upsilon + \int_{{t - \bar{\kappa }}}^{{t - \kappa (t)}} {\eta _{{3\upsilon }}^{ { \top }} } P_{4} \eta _{{3\upsilon }} {\text{d}}\upsilon , \hfill \\ V_{5} (t) = \bar{\mu }\int_{{t - \bar{\mu }}}^{t} {(\bar{\mu } - t + \upsilon )\eta _{{4\upsilon }}^{ { \top }} S\eta _{{4\upsilon }} {\text{d}}\upsilon } + \bar{\kappa }\int_{{t - \bar{\kappa }}}^{t} {(\bar{\kappa } - t + \upsilon )\eta _{{5\upsilon }}^{ { \top }} U\eta _{{5\upsilon }} {\text{d}}\upsilon } , \hfill \\ \end{gathered}$$
$$\begin{gathered} V_{6} (t) = \frac{1}{2}\int_{{t - \bar{\mu }}}^{t} {(\bar{\mu } - t + \upsilon )\dot{u}_{\upsilon }^{ { \top }} [2Q_{3} + (\bar{\mu } - t + \upsilon )Q_{4} + (\bar{\mu } + t - \upsilon )Q_{5} ]\dot{u}_{\upsilon } {\text{d}}\upsilon } \hfill \\ \quad \quad + \frac{1}{2}\int_{{t - \bar{\kappa }}}^{t} {(\bar{\kappa } - t + \upsilon )\dot{w}_{\upsilon }^{ { \top }} [2Q_{6} + (\bar{\kappa } - t + \upsilon )Q_{7} + (\bar{\kappa } + t - \upsilon )Q_{8} ]\dot{w}_{\upsilon } {\text{d}}\upsilon } , \hfill \\ V_{7} (t) = \mu (t)\eta _{{6t}}^{ { \top }} P_{5} \eta _{{6t}} + [\bar{\mu } - \mu (t)]\eta _{{7t}}^{ { \top }} P_{6} \eta _{{7t}} + \kappa (t)\eta _{{8t}}^{ { \top }} P_{7} \eta _{{8t}} + [\bar{\kappa } - \kappa (t)]\eta _{{9t}}^{ { \top }} P_{8} \eta _{{9t}} , \hfill \\ \end{gathered}$$
with
$$ \begin{gathered} \eta_{1t} = {\text{col}}\left\{ {u_{t} ,\int_{{t - \overline{\mu }}}^{t} {u_{\upsilon } } {\text{d}}\upsilon } \right\},\;\eta_{2t} = {\text{col}}\left\{ {w_{t} ,\int_{{t - \overline{\kappa }}}^{t} {w_{\upsilon } } {\text{d}}\upsilon } \right\},\;\eta_{3\upsilon } = \left[ {\begin{array}{*{20}c} {w_{\upsilon } } \\ {\rho (w_{\upsilon } )} \\ \end{array} } \right], \hfill \\ \eta_{4\upsilon } = \left[ {\begin{array}{*{20}c} {u_{\upsilon } } \\ {\dot{u}_{\upsilon } } \\ \end{array} } \right],\,\eta_{5\upsilon } = \left[ {\begin{array}{*{20}c} {w_{\upsilon } } \\ {\dot{w}_{\upsilon } } \\ \end{array} } \right],\;\eta_{6t} = \left[ {\begin{array}{*{20}c} {u_{t} } \\ {v_{1t} } \\ \end{array} } \right],\;\eta_{7t} = \left[ {\begin{array}{*{20}c} {u_{t} } \\ {v_{2t} } \\ \end{array} } \right],\;\eta_{8t} = \left[ {\begin{array}{*{20}c} {w_{t} } \\ {v_{3t} } \\ \end{array} } \right],\;\eta_{9t} = \left[ {\begin{array}{*{20}c} {w_{t} } \\ {v_{4t} } \\ \end{array} } \right].\; \hfill \\ \end{gathered} $$
Denote \(\Lambda_{k} = {\text{diag\{ }}\lambda_{k1}^{ + } ,\lambda_{k2}^{ + } , \cdots ,\lambda_{kn}^{ + } ,\;k = 1,2,\) the derivative of \(V(t,u_{t} ,w_{t} )\) with respect to time \(t \in [0, + \infty )\) along the solution of (3) is
$$ \dot{V}(t) = \sum\limits_{j = 1}^{7} {\dot{V}_{j} } (t) $$
(21)
where
$$ \dot{V}_{1} (t) = 2\left[ {\begin{array}{*{20}c} {u_{t} } \\ {\mu (t)v_{1t} + [\overline{\mu } - \mu (t)]v_{2t} } \\ \end{array} } \right]^{{ \top }} P_{1} \left[ {\begin{array}{*{20}c} {\dot{u}_{t} } \\ {u_{t} - u_{{\overline{\mu }}} } \\ \end{array} } \right], $$
(22)
$$ \dot{V}_{2} (t) = 2\left[ {\begin{array}{*{20}c} {w_{t} } \\ {\kappa (t)v_{3t} + [\overline{\kappa } - \kappa (t)]v_{4t} } \\ \end{array} } \right]^{{ \top }} P_{2} \left[ {\begin{array}{*{20}c} {\dot{w}_{t} } \\ {w_{t} - w_{{\overline{\kappa }}} } \\ \end{array} } \right], $$
(23)
$$ \dot{V}_{3} (t) = 2\left\{ {[\Theta_{1} w_{t} - \rho (w_{t} )]^{{ \top }} \Lambda_{1} + [\rho (w_{t} ) - \Theta_{2} w_{t} ]^{{ \top }} \Lambda_{2} } \right\}\dot{w}_{t} , $$
(24)
$$\begin{gathered} \dot{V}_{4} (t) = u_{t}^{ { \top }} Q_{1} u_{t} - [1 - \dot{\mu }(t)](Q_{1} - Q_{2} )u_{\mu }^{ { \top }} u_{\mu } - u_{{\bar{\mu }}}^{ { \top }} Q_{2} u_{{\bar{\mu }}} \hfill \\ \quad \quad \quad + ( * )^{ { \top }} P_{3} \eta _{{3t}} - [1 - \dot{\kappa }(t)]( * )^{ { \top }} (P_{3} - P_{4} )\eta _{{3(t - \kappa (t))}} - ( * )^{ { \top }} P_{4} \eta _{{3(t - \bar{\kappa })}} , \hfill \\ \end{gathered}$$
(25)
$$\dot{V}_{5} (t) = \bar{\mu }^{2} \eta _{{4t}}^{ { \top }} S\eta _{{4t}} - \bar{\mu }\int_{{t - \bar{\mu }}}^{t} {\eta _{{4\upsilon }}^{ { \top }} } S\eta _{{4\upsilon }} {\text{d}}\upsilon + \bar{\kappa }^{2} \eta _{{5t}}^{ { \top }} U\eta _{{5t}} - \bar{\kappa }\int_{{t - \bar{\kappa }}}^{t} {\eta _{{5\upsilon }}^{ { \top }} } U\eta _{{5\upsilon }} {\text{d}}\upsilon ,$$
(26)
$$ \begin{gathered} \dot{V}_{6} (t) = \frac{1}{2}\overline{\mu }\dot{u}_{t}^{{ \top }} (2Q_{3} + \overline{\mu }Q_{4} + \overline{\mu }Q_{5} )\dot{u}_{t} + V_{a} (t) - V_{b} (t) - V_{c} (t) \hfill \\ \quad \quad \quad + \frac{1}{2}\overline{\kappa }\dot{w}_{t}^{{ \top }} (2Q_{6} + \overline{\kappa }Q_{7} + \overline{\kappa }Q_{8} )\dot{w}_{t} + V_{d} (t) - V_{e} (t) - V_{f} (t), \hfill \\ \end{gathered} $$
(27)
$$ \begin{gathered} \dot{V}_{7} (t) = \dot{\mu }(t)\left[ {( * )^{{ \top }} P_{5} \eta_{6t} - ( * )^{{ \top }} P_{6} \eta_{7t} } \right] + 2\mu (t)\eta_{6t}^{{ \top }} P_{5} \left[ {\begin{array}{*{20}c} {\dot{u}_{t} } \\ {\dot{v}_{1t} } \\ \end{array} } \right] + 2[\overline{\mu } - \mu (t)]\eta_{7t}^{{ \top }} P_{6} \left[ {\begin{array}{*{20}c} {\dot{u}_{t} } \\ {\dot{v}_{2t} } \\ \end{array} } \right] \hfill \\ \quad \quad \quad + \dot{\kappa }(t)\left[ {( * )^{{ \top }} P_{7} \eta_{8t} - ( * )^{T} P_{8} \eta_{9t} } \right] + 2\kappa (t)\eta_{8t}^{{ \top }} P_{7} \left[ {\begin{array}{*{20}c} {\dot{w}_{t} } \\ {\dot{v}_{3t} } \\ \end{array} } \right] + 2[\overline{\kappa } - \kappa (t)]\eta_{9t}^{{ \top }} P_{8} \left[ {\begin{array}{*{20}c} {\dot{w}_{t} } \\ {\dot{v}_{4t} } \\ \end{array} } \right] \hfill \\ \end{gathered} $$
$$ \begin{gathered} = \dot{\mu }(t)\left[ {( * )^{{ \top }} P_{5} \eta_{6t} - ( * )^{{ \top }} P_{6} \eta_{7t} } \right] + \dot{\kappa }(t)\left[ {( * )^{{ \top }} P_{7} \eta_{8t} - ( * )^{{ \top }} P_{8} \eta_{9t} } \right] \hfill \\ \quad + 2\eta_{6t}^{{ \top }} P_{5} \left[ {\begin{array}{*{20}c} {\mu (t)\dot{u}_{t} } \\ {u_{t} - [1 - \dot{\mu }(t)]u_{\mu } - \dot{\mu }(t)v_{1t} } \\ \end{array} } \right] + 2\eta_{7t}^{{ \top }} P_{6} \left[ {\begin{array}{*{20}c} {[\overline{\mu } - \mu (t)]\dot{u}_{t} } \\ {[1 - \dot{\mu }(t)]u_{\mu } - u_{{\overline{\mu }}} + \dot{\mu }(t)v_{2t} } \\ \end{array} } \right] \hfill \\ \quad + 2\eta_{8t}^{{ \top }} P_{7} \left[ {\begin{array}{*{20}c} {\kappa (t)\dot{w}_{t} } \\ {w_{t} - [1 - \dot{\kappa }(t)]w_{\kappa } - \dot{\kappa }(t)v_{3t} } \\ \end{array} } \right] + 2\eta_{9t}^{{ \top }} P_{8} \left[ {\begin{array}{*{20}c} {[\overline{\kappa } - \kappa (t)]\dot{w}_{t} } \\ {[1 - \dot{\kappa }(t)]w_{\kappa } - w_{{\overline{\kappa }}} + \dot{\kappa }(t)v_{4t} } \\ \end{array} } \right], \hfill \\ \end{gathered} $$
(28)
where
$$\begin{gathered} V_{a} (t) = - \int_{{t - \mu (t)}}^{t} {\dot{u}_{\upsilon }^{ { \top }} Q_{3} \dot{u}_{\upsilon } {\text{d}}\upsilon } - \int_{{t - \bar{\mu }}}^{{t - \mu (t)}} {\dot{u}_{\upsilon }^{ { \top }} } Q_{3} \dot{u}_{\upsilon } {\text{d}}\upsilon , \hfill \\ V_{b} (t) = [\bar{\mu } - \mu (t)]\int_{{t - \mu (t)}}^{t} {\dot{u}_{\upsilon }^{ { \top }} Q_{4} \dot{u}_{\upsilon } {\text{d}}\upsilon } + \mu (t)\int_{{t - \bar{\mu }}}^{{t - \mu (t)}} {\dot{u}_{\upsilon }^{ { \top }} } Q_{5} \dot{u}_{\upsilon } {\text{d}}\upsilon , \hfill \\ V_{c} (t) = \int_{{t - \mu (t)}}^{t} {\dot{u}_{\upsilon }^{ { \top }} [(\upsilon - t + \mu (t))Q_{4} + (t - \upsilon )Q_{5} ]\dot{u}_{\upsilon } {\text{d}}\upsilon } \hfill \\ \quad \quad \quad + \int_{{t - \bar{\mu }}}^{{t - \tau (t)}} {\dot{u}_{\upsilon }^{ { \top }} } [(\upsilon - t + \bar{\mu })Q_{4} + (t - \mu (t) - \upsilon )Q_{5} ]\dot{u}_{\upsilon } {\text{d}}\upsilon , \hfill \\ V_{d} (t) = - \int_{{t - \kappa (t)}}^{t} {\dot{w}_{\upsilon }^{ { \top }} } Q_{6} \dot{w}_{\upsilon } {\text{d}}\upsilon - \int_{{t - \bar{\kappa }}}^{{t - \kappa (t)}} {\dot{w}_{\upsilon }^{ { \top }} } Q_{6} \dot{w}_{\upsilon } {\text{d}}\upsilon , \hfill \\ V_{e} (t) = [\bar{\kappa } - \kappa (t)]\int_{{t - \kappa (t)}}^{t} {\dot{w}_{\upsilon }^{ { \top }} } Q_{7} \dot{w}_{\upsilon } {\text{d}}\upsilon + \kappa (t)\int_{{t - \bar{\kappa }}}^{{t - \kappa (t)}} {\dot{w}_{\upsilon }^{ { \top }} } Q_{8} \dot{w}_{\upsilon } {\text{d}}\upsilon , \hfill \\ V_{f} (t) = \int_{{t - \kappa (t)}}^{t} {\dot{w}_{\upsilon }^{ { \top }} } [(\upsilon - t + \kappa (t))Q_{7} + (t - \upsilon )Q_{8} ]\dot{w}_{\upsilon } {\text{d}}\upsilon \hfill \\ \quad \quad \quad + \int_{{t - \bar{\kappa }}}^{{t - \kappa (t)}} {\dot{w}_{\upsilon }^{ { \top }} } [(\upsilon - t + \bar{\kappa })Q_{7} + (t - \kappa (t) - \upsilon )Q_{8} ]\dot{w}_{\upsilon } {\text{d}}\upsilon . \hfill \\ \end{gathered}$$
Integration by part derives the following free-weight-matrix-based zero-value terms (see [39])
$$ \begin{gathered} 0 = u_{t}^{{ \top }} R_{1} u_{t} - u_{\mu }^{{ \top }} R_{1} u_{\mu } - 2\int_{t - \mu (t)}^{t} {u_{\upsilon }^{{ \top }} R_{1} \dot{u}_{\upsilon } } {\text{d}}\upsilon + u_{\mu }^{{ \top }} R_{2} u_{\mu } - u_{{\overline{\mu }}}^{{ \top }} R_{2} u_{{\overline{\mu }}} - 2\int_{{t - \overline{\mu }}}^{t - \mu (t)} {u_{\upsilon }^{{ \top }} R_{2} \dot{u}_{\upsilon } } {\text{d}}\upsilon , \hfill \\ 0 = w_{t}^{{ \top }} R_{3} w_{t} - w_{\kappa }^{{ \top }} R_{3} w_{\kappa } - 2\int_{t - \kappa (t)}^{t} {w_{\upsilon }^{{ \top }} R_{3} \dot{w}_{\upsilon } } {\text{d}}\upsilon + w_{\kappa }^{{ \top }} R_{4} w_{\kappa } - w_{{\overline{\kappa }}}^{{ \top }} R_{4} w_{{\overline{\kappa }}} - 2\int_{{t - \overline{\kappa }}}^{t - \kappa (t)} {w_{\upsilon }^{{ \top }} R_{4} \dot{w}_{\upsilon } } {\text{d}}\upsilon . \hfill \\ \end{gathered} $$
Adding them into \(\dot{V}_{5} (t)\) yields
$$\begin{gathered} - \bar{\mu }\int_{{t - \bar{\mu }}}^{t} {\eta _{{4\upsilon }}^{ { \top }} } S\eta _{{4\upsilon }} {\text{d}}\upsilon - \bar{\kappa }\int_{{t - \bar{\kappa }}}^{t} {\eta _{{5\upsilon }}^{ { \top }} } U\eta _{{5\upsilon }} {\text{d}}\upsilon \hfill \\ \quad = u_{t}^{ { \top }} R_{1} u_{t} + u_{\mu }^{ { \top }} (R_{2} - R_{1} )u_{\mu } - u_{{\bar{\mu }}}^{ { \top }} R_{2} u_{{\bar{\mu }}} - \int_{{t - \mu (t)}}^{t} {\eta _{{4\upsilon }}^{ { \top }} \Phi _{1} \eta _{{4\upsilon }} } {\text{d}}\upsilon - \int_{{t - \bar{\mu }}}^{{t - \mu (t)}} {\eta _{{4\upsilon }}^{ { \top }} \Phi _{2} \eta _{{4\upsilon }} } {\text{d}}\upsilon \hfill \\ \quad \;\;\; + w_{t}^{ { \top }} R_{3} w_{t} + w_{\kappa }^{ { \top }} (R_{4} - R_{3} )w_{\kappa } - w_{{\bar{\kappa }}}^{ { \top }} R_{4} w_{{\bar{\kappa }}} - \int_{{t - \kappa (t)}}^{t} {\eta _{{5\upsilon }}^{ { \top }} \Phi _{3} \eta _{{5\upsilon }} } {\text{d}}\upsilon - \int_{{t - \bar{\kappa }}}^{{t - \kappa (t)}} {\eta _{{5\upsilon }}^{ { \top }} \Phi _{4} \eta _{{5\upsilon }} } {\text{d}}\upsilon . \hfill \\ \end{gathered}$$
Utilizing Lemma 1 derives
$$\begin{gathered} \int_{{t - \mu (t)}}^{t} {\eta _{{4\upsilon }}^{ { \top }} \Phi _{1} \eta _{{4\upsilon }} } {\text{d}}\upsilon + \int_{{t - \bar{\mu }}}^{{t - \mu (t)}} {\eta _{{4\upsilon }}^{ { \top }} \Phi _{2} \eta _{{4\upsilon }} } {\text{d}}\upsilon \hfill \\ \quad \ge \frac{1}{{\mu (t)}}( * )^{ { \top }} \Phi _{1} \left[ {\begin{array}{*{20}c} {\mu (t)v_{{1t}} } \\ {u_{t} - u_{\mu } } \\ \end{array} } \right] + \frac{3}{{\mu (t)}}( * )^{ { \top }} \Phi _{1} \left[ {\begin{array}{*{20}c} {\mu (t)(v_{{1t}} - v_{{5t}} )} \\ {u_{t} + u_{\mu } - 2\mu (t)v_{{1t}} } \\ \end{array} } \right] \hfill \\ \quad \;\; + \frac{1}{{\bar{\mu } - \mu (t)}}( * )^{ { \top }} \Phi _{2} \left[ {\begin{array}{*{20}c} {[\bar{\mu } - \mu (t)]v_{{2t}} } \\ {u_{\mu } - u_{{\bar{\mu }}} } \\ \end{array} } \right] + \frac{3}{{\bar{\mu } - \mu (t)}}( * )^{ { \top }} \Phi _{2} \left[ {\begin{array}{*{20}c} {[\bar{\mu } - \mu (t)](v_{{2t}} - v_{{6t}} )} \\ {u_{\mu } + u_{{\bar{\mu }}} - 2[\bar{\mu } - \mu (t)]v_{{2t}} } \\ \end{array} } \right] \hfill \\ \quad = \mu (t)v_{{1t}}^{ { \top }} (\bar{\mu }S_{1} )v_{{1t}} + 2v_{{1t}}^{ { \top }} (\bar{\mu }S_{2} + R_{1} )(u_{t} - u_{\mu } ) + \frac{{\bar{\mu }}}{{\mu (t)}}( * )^{ { \top }} S_{3} (u_{t} - u_{\mu } ) \hfill \\ \quad \;\; + 3\mu (t)( * )^{ { \top }} (\bar{\mu }S_{1} )(v_{{1t}} - v_{{5t}} ) + 6(v_{{1t}} - v_{{5t}} )^{ { \top }} (\bar{\mu }S_{2} + R_{1} )[u_{t} + u_{\mu } - 2\mu (t)v_{{1t}} ] \hfill \\ \quad \;\; + \frac{{3\bar{\mu }}}{{\mu (t)}}( * )^{ { \top }} S_{3} (u_{t} + u_{\mu } ) - 12(u_{t} + u_{\mu } )^{ { \top }} (\bar{\mu }S_{3} )v_{{1t}} + 12\mu (t)v_{{1t}}^{ { \top }} (\bar{\mu }S_{3} )v_{{1t}} \hfill \\ \quad \;\; + [\bar{\mu } - \mu (t)]v_{{2t}}^{ { \top }} (\bar{\mu }S_{1} )v_{{2t}} + 2v_{{2t}}^{ { \top }} (\bar{\mu }S_{2} + R_{2} )(u_{\mu } - u_{{\bar{\mu }}} ) + \frac{{\bar{\mu }}}{{\bar{\mu } - \mu (t)}}( * )^{ { \top }} S_{3} (u_{\mu } - u_{{\bar{\mu }}} ) \hfill \\ \quad \;\; + 3[\bar{\mu } - \mu (t)]( * )^{ { \top }} (\bar{\mu }S_{1} )(v_{{2t}} - v_{{6t}} ) + 6(v_{{2t}} - v_{{6t}} )^{ { \top }} (\bar{\mu }S_{2} + R_{2} )\{ u_{\mu } + u_{{\bar{\mu }}} - 2[\bar{\mu } - \mu (t)]v_{{2t}} \} \hfill \\ \quad \;\; + \frac{{3\bar{\mu }}}{{\bar{\mu } - \mu (t)}}( * )^{ { \top }} S_{3} (u_{\mu } + u_{{\bar{\mu }}} ) - 12(u_{\mu } + u_{{\bar{\mu }}} )^{ { \top }} (\bar{\mu }S_{3} )v_{{2t}} + 12[\bar{\mu } - \mu (t)]v_{{2t}}^{ { \top }} (\bar{\mu }S_{3} )v_{{2t}} , \hfill \\ \end{gathered}$$
(29)
$$ \begin{gathered} \int_{t - \kappa (t)}^{t} {\eta_{5\upsilon }^{{ \top }} \Phi_{3} \eta_{5\upsilon } } {\text{d}}\upsilon + \int_{{t - \overline{\kappa }}}^{t - \kappa (t)} {\eta_{5\upsilon }^{{ \top }} \Phi_{4} \eta_{5\upsilon } } {\text{d}}\upsilon \hfill \\ \quad \ge \frac{1}{\kappa (t)}( * )^{{ \top }} \Phi_{3} \left[ {\begin{array}{*{20}c} {\kappa (t)v_{3t} } \\ {w_{t} - w_{\kappa } } \\ \end{array} } \right] + \frac{3}{\kappa (t)}( * )^{{ \top }} \Phi_{3} \left[ {\begin{array}{*{20}c} {\kappa (t)(v_{3t} - v_{7t} )} \\ {w_{t} + w_{\kappa } - 2\kappa (t)v_{3t} } \\ \end{array} } \right] \hfill \\ \quad \;\; + \frac{1}{{\overline{\kappa } - \kappa (t)}}( * )^{{ \top }} \Phi_{4} \left[ {\begin{array}{*{20}c} {[\overline{\kappa } - \kappa (t)]v_{4t} } \\ {w_{\kappa } - w_{{\overline{\kappa }}} } \\ \end{array} } \right] + \frac{3}{{\overline{\kappa } - \kappa (t)}}( * )^{{ \top }} \Phi_{4} \left[ {\begin{array}{*{20}c} {[\overline{\kappa } - \kappa (t)](v_{4t} - v_{8t} )} \\ {w_{\kappa } + w_{{\overline{\kappa }}} - 2[\overline{\kappa } - \kappa (t)]v_{4t} } \\ \end{array} } \right] \hfill \\ \quad = \kappa (t)v_{3t}^{{ \top }} (\overline{\kappa }U_{1} )v_{3t} + 2v_{3t}^{{ \top }} (\overline{\kappa }U_{2} + R_{3} )(w_{t} - w_{\kappa } ) + \frac{{\overline{\kappa }}}{\kappa (t)}( * )^{{ \top }} U_{3} (w_{t} - w_{\kappa } ) \hfill \\ \quad \;\; + 3\kappa (t)( * )^{{ \top }} (\overline{\kappa }U_{1} )(v_{3t} - v_{7t} ) + 6(v_{3t} - v_{7t} )^{{ \top }} (\overline{\kappa }U_{2} + R_{3} )[w_{t} + w_{\kappa } - 2\kappa (t)v_{3t} ] \hfill \\ \quad \;\; + \frac{{3\overline{\kappa }}}{\kappa (t)}( * )^{{ \top }} U_{3} (w_{t} + w_{\kappa } ) - 12(w_{t} + w_{\kappa } )^{{ \top }} (\overline{\kappa }U_{3} )v_{3t} + 12\kappa (t)v_{3t}^{{ \top }} (\overline{\kappa }U_{3} )v_{3t} \hfill \\ \quad \;\; + [\overline{\kappa } - \kappa (t)]v_{4t}^{{ \top }} (\overline{\kappa }U_{1} )v_{4t} + 2v_{4t}^{{ \top }} (\overline{\kappa }U_{2} + R_{4} )(w_{\kappa } - w_{{\overline{\kappa }}} ) + \frac{{\overline{\kappa }}}{{\overline{\kappa } - \kappa (t)}}( * )^{{ \top }} U_{3} (w_{\kappa } - w_{{\overline{\kappa }}} ) \hfill \\ \quad \;\; + 3[\overline{\kappa } - \kappa (t)]( * )^{{ \top }} (\overline{\kappa }U_{1} )(v_{4t} - v_{8t} ) + 6(v_{4t} - v_{8t} )^{{ \top }} (\overline{\kappa }U_{2} + R_{4} )\{ w_{\kappa } + w_{{\overline{\kappa }}} - 2[\overline{\kappa } - \kappa (t)]v_{4t} \} \hfill \\ \quad \;\; + \frac{{3\overline{\kappa }}}{{\overline{\kappa } - \kappa (t)}}( * )^{{ \top }} U_{3} (w_{\kappa } + w_{{\overline{\kappa }}} ) - 12(w_{\kappa } + w_{{\overline{\kappa }}} )^{{ \top }} (\overline{\kappa }U_{3} )v_{4t} + 12[\overline{\kappa } - \kappa (t)]v_{4t}^{{ \top }} (\overline{\kappa }U_{3} )v_{4t} . \hfill \\ \end{gathered} $$
(30)
Using the RCI yields
$$ \frac{{\overline{\mu }}}{\mu (t)}( * )^{{ \top }} S_{3} (u_{t} - u_{\mu } ) + \frac{{\overline{\mu }}}{{\overline{\mu } - \mu (t)}}( * )^{{ \top }} S_{3} (u_{\mu } - u_{{\overline{\mu }}} ) \ge ( * )^{{ \top }} \left[ {\begin{array}{*{20}c} {S_{3} } & {X_{3} } \\ * & {S_{3} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u_{t} - u_{\mu } } \\ {u_{\mu } - u_{{\overline{\mu }}} } \\ \end{array} } \right], $$
(31)
$$ \frac{{3\overline{\mu }}}{\mu (t)}( * )^{{ \top }} S_{3} (u_{t} + u_{\mu } ) + \frac{{3\overline{\mu }}}{{\overline{\mu } - \mu (t)}}( * )^{{ \top }} S_{3} (u_{\mu } + u_{{\overline{\mu }}} ) \ge 3( * )^{{ \top }} \left[ {\begin{array}{*{20}c} {S_{3} } & {X_{3} } \\ * & {S_{3} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u_{t} + u_{\mu } } \\ {u_{\mu } + u_{{\overline{\mu }}} } \\ \end{array} } \right], $$
(32)
$$\,\frac{{\bar{\kappa }}}{{\kappa (t)}}( * )^{ { \top }} U_{3} (w_{t} - w_{\kappa } ) + \frac{{\bar{\kappa }}}{{\bar{\kappa } - \kappa (t)}}( * )^{ { \top }} U_{3} (w_{\kappa } - w_{{\bar{\kappa }}} ) \ge ( * )^{ { \top }} \left[ {\begin{array}{*{20}c} {U_{3} } & {X_{4} } \\ * & {U_{3} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {w_{t} - w_{\kappa } } \\ {w_{\kappa } - w_{{\bar{\kappa }}} } \\ \end{array} } \right],\,$$
(33)
$$ \frac{{3\overline{\kappa }}}{\kappa (t)}( * )^{{ \top }} U_{3} (w_{t} + w_{\kappa } ) + \frac{{3\overline{\kappa }}}{{\overline{\kappa } - \kappa (t)}}( * )^{{ \top }} U_{3} (w_{\kappa } + w_{{\overline{\kappa }}} ) \ge 3( * )^{{ \top }} \left[ {\begin{array}{*{20}c} {U_{3} } & {X_{4} } \\ * & {U_{3} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {w_{t} + w_{\kappa } } \\ {w_{\kappa } + w_{{\overline{\kappa }}} } \\ \end{array} } \right]\,. $$
(34)
Applying Lemma 1 to \(V_{a} (t)\) and \(V_{d} (t)\) yields
$$ V_{a} (t) \le \xi_{t}^{{ \top }} \left( {{\text{sym}}\{ F_{1}^{{ \top }} e_{a} + F_{2}^{{ \top }} e_{b} \} + \mu (t)F_{1}^{{ \top }} {\mathbf{\mathbb{Q}}}_{3}^{ - 1} F_{1} + [\overline{\mu } - \mu (t)]F_{2}^{{ \top }} {\mathbf{\mathbb{Q}}}_{3}^{ - 1} F_{2} } \right)\xi_{t} , $$
(35)
$$ V_{d} (t) \le \xi_{t}^{{ \top }} \left( {{\text{sym}}\{ F_{3}^{{ \top }} e_{c} + F_{4}^{{ \top }} e_{d} \} + \kappa (t)F_{3}^{{ \top }} {\mathbf{\mathbb{Q}}}_{6}^{ - 1} F_{3} + [\overline{\kappa } - \kappa (t)]F_{4}^{{ \top }} {\mathbf{\mathbb{Q}}}_{6}^{ - 1} F_{4} } \right)\xi_{t} . $$
(36)
Denoting \(\vartheta = \frac{\mu (t)}{{\overline{\mu }}},\) when \(0 < \mu (t) < \overline{\mu },\) applying Lemma 1 to \(V_{b} (t)\) derives
$$ \begin{gathered} V_{b} (t) \ge \xi_{t}^{\rm T} \left\{ {\frac{{\overline{\mu } - \mu (t)}}{\mu (t)}e_{a}^{{ \top }} {\mathbf{\mathbb{Q}}}_{4} e_{a} + \frac{\mu (t)}{{\overline{\mu } - \mu (t)}}e_{b}^{{ \top }} {\mathbf{\mathbb{Q}}}_{5} e_{b} } \right\}\xi_{t} \hfill \\ \quad \quad \, = \xi_{t}^{\rm T} \left\{ {\left( {\frac{1}{\vartheta } - 1} \right)e_{a}^{{ \top }} {\mathbf{\mathbb{Q}}}_{4} e_{a} + \left( {\frac{1}{1 - \vartheta } - 1} \right)e_{b}^{{ \top }} {\mathbf{\mathbb{Q}}}_{5} e_{b} } \right\}\xi_{t} . \hfill \\ \end{gathered} $$
(37)
Based on the first inequality of (9), applying the RCI derives
$$ \frac{1}{\vartheta }e_{a}^{{ \top }} {\mathbf{\mathbb{Q}}}_{4} e_{a} + \frac{1}{1 - \vartheta }e_{b}^{{ \top }} {\mathbf{\mathbb{Q}}}_{5} e_{b} \ge ( * )^{{ \top }} \left[ {\begin{array}{*{20}c} {{\mathbf{\mathbb{Q}}}_{4} } & {X_{1} } \\ * & {{\mathbf{\mathbb{Q}}}_{5} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {e_{a} } \\ {e_{b} } \\ \end{array} } \right]. $$
(38)
Similarly, denoting \(\varpi = \frac{\kappa (t)}{{\overline{\kappa }}},\) when \(0 < \kappa (t) < \overline{\kappa },\) applying Lemma 1 to \(V_{e} (t)\) yields
$$\begin{gathered} V_{e} (t) \ge \xi _{t} ^{ { \top }} \left\{ {\frac{{\bar{\kappa } - \kappa (t)}}{{\kappa (t)}}e_{c} ^{ { \top }} {\mathbf{\mathbb{Q}}}_{7} e_{c} + \frac{{\kappa (t)}}{{\bar{\kappa } - \kappa (t)}}e_{d} ^{ { \top }} {\mathbf{\mathbb{Q}}}_{8} e_{d} } \right\}\xi _{t} \hfill \\ \quad \quad \, = \xi _{t} ^{ { \top }} \left\{ {\left( {\frac{1}{\varpi } - 1} \right)e_{c} ^{ { \top }} {\mathbf{\mathbb{Q}}}_{7} e_{c} + \left( {\frac{1}{{1 - \varpi }} - 1} \right)e_{d} ^{ { \top }} {\mathbf{\mathbb{Q}}}_{8} e_{d} } \right\}\xi _{t} . \hfill \\ \end{gathered}$$
(39)
Based on the second inequality of (9), applying the RCI derives
$$ \frac{1}{\varpi }e_{c}^{{ \top }} {\mathbf{\mathbb{Q}}}_{7} e_{c} + \frac{1}{1 - \varpi }e_{d}^{{ \top }} {\mathbf{\mathbb{Q}}}_{8} e_{d} \ge ( * )^{{ \top }} \left[ {\begin{array}{*{20}c} {{\mathbf{\mathbb{Q}}}_{7} } & {X_{2} } \\ * & {{\mathbf{\mathbb{Q}}}_{8} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {e_{c} } \\ {e_{d} } \\ \end{array} } \right]. $$
(40)
By the use of Lemma 1, one gets
$$ \begin{gathered} V_{c} (t) \ge 2( * )^{{ \top }} \left[ {\begin{array}{*{20}c} {Q_{4} } & {\mathbf{0}} \\ {\mathbf{0}} & {2Q_{4} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u_{t} - v_{1t} } \\ {u_{t} + v_{1t} - 2v_{5t} } \\ \end{array} } \right] + 2( * )^{{ \top }} \left[ {\begin{array}{*{20}c} {Q_{5} } & {\mathbf{0}} \\ {\mathbf{0}} & {2Q_{5} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u_{\mu } - v_{1t} } \\ {u_{\mu } - 4v_{1t} + 3v_{5t} } \\ \end{array} } \right] \hfill \\ \quad \quad \quad + 2( * )^{{ \top }} \left[ {\begin{array}{*{20}c} {Q_{4} } & {\mathbf{0}} \\ {\mathbf{0}} & {2Q_{4} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u_{\mu } - v_{2t} } \\ {u_{\mu } + 2v_{2t} - 3v_{6t} } \\ \end{array} } \right] + 2( * )^{{ \top }} \left[ {\begin{array}{*{20}c} {Q_{5} } & {\mathbf{0}} \\ {\mathbf{0}} & {2Q_{5} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u_{{\overline{\mu }}} - v_{2t} } \\ {u_{{\overline{\mu }}} - 4v_{2t} + 3v_{6t} } \\ \end{array} } \right]. \hfill \\ \end{gathered} $$
(41)
$$ \begin{gathered} V_{f} (t) \ge 2( * )^{{ \top }} \left[ {\begin{array}{*{20}c} {Q_{7} } & {\mathbf{0}} \\ {\mathbf{0}} & {2Q_{7} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {w_{t} - v_{3t} } \\ {w_{t} + v_{3t} - 2v_{7t} } \\ \end{array} } \right] + 2( * )^{{ \top }} \left[ {\begin{array}{*{20}c} {Q_{8} } & {\mathbf{0}} \\ {\mathbf{0}} & {2Q_{8} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {w_{\kappa } - v_{3t} } \\ {w_{\kappa } - 4v_{3t} + 3v_{7t} } \\ \end{array} } \right] \hfill \\ \quad \quad \quad + 2( * )^{{ \top }} \left[ {\begin{array}{*{20}c} {Q_{7} } & {\mathbf{0}} \\ {\mathbf{0}} & {2Q_{7} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {w_{\kappa } - v_{4t} } \\ {w_{\kappa } + 2v_{4t} - 3v_{8t} } \\ \end{array} } \right] + 2( * )^{{ \top }} \left[ {\begin{array}{*{20}c} {Q_{8} } & {\mathbf{0}} \\ {\mathbf{0}} & {2Q_{8} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {w_{{\overline{\kappa }}} - v_{4t} } \\ {w_{{\overline{\kappa }}} - 4v_{4t} + 3v_{8t} } \\ \end{array} } \right]. \hfill \\ \end{gathered} $$
(42)
Based on the right inequalities of (10), the \(\dot{\mu }(t) -\) and \(\dot{\kappa }(t) -\) dependent terms in \(\dot{V}(t,u_{t} ,w_{t} )\) can be combined and estimated as follows:
$$ \dot{\mu }(t)\xi_{t}^{{ \top }} \Xi_{\mu } \xi_{t} \le \mu_{d} \xi_{t}^{{ \top }} \Xi_{\mu } \xi_{t} , $$
(43)
$$ \dot{\kappa }(t)\xi_{t}^{{ \top }} \Xi_{\kappa } \xi_{t} \le \kappa_{d} \xi_{t}^{{ \top }} \Xi_{\kappa } \xi_{t} . $$
(44)
From assumption condition (4), the following inequalities hold for diagonal matrices \(Y_{k} > {\mathbf{0}}\) \((k = 1,...,6)\) with appropriate dimensions
$$ \begin{gathered} \alpha_{i} (s) = - 2[\rho (w_{s} ) - \Theta_{1} w_{s} ]^{{ \top }} Y_{i} [\rho (w_{s} ) - \Theta_{2} w_{s} ] \ge 0, \hfill \\ \beta_{j} (s,r) = - 2\{ [\rho (w_{s} ) - \rho (w_{r} )] - \Theta_{1} (w_{s} - w_{r} )\}^{{ \top }} Y_{3 + j} \{ [\rho (w_{s} ) - \rho (w_{r} )] - \Theta_{2} (w_{s} - w_{r} )\} \ge 0, \hfill \\ \end{gathered} $$
where \(i,j = 1,2,3.\)
Hence, the next inequalities are true:
$$ \alpha_{1} (t) + \alpha_{2} (t - \kappa (t)) + \alpha_{3} (t - \overline{\kappa }) \ge 0, $$
(45)
$$ \beta_{1} (t,t - \kappa (t)) + \beta_{2} (t - \kappa (t),t - \overline{\kappa }) + \beta_{3} (t,t - \overline{\kappa }) \ge 0. $$
(46)
Using (21–46) yields
$$ \dot{V}(t) \le \xi_{t}^{{ \top }} \Xi (\mu (t),\kappa (t))\xi_{t} , $$
(47)
where
$$ \Xi (\mu (t),\kappa (t)) = \Xi_{0} + \Xi^{\prime}_{1} + \mu (t)\Xi^{\prime}_{2} + [\overline{\mu } - \mu (t)]\Xi^{\prime}_{3} + \kappa (t)\Xi^{\prime}_{4} + [\overline{\kappa } - \kappa (t)]\Xi^{\prime}_{5} $$
with
$$ \begin{gathered} \Xi^{\prime}_{2} = \Xi_{2} + \Xi_{3} + F_{1}^{{ \top }} {\mathbb{Q}}_{3}^{ - 1} F_{1} ,\;\;\Xi^{\prime}_{3} = \Xi_{4} + \Xi_{5} + F_{2}^{{ \top }} {\mathbb{Q}}_{3}^{ - 1} F_{2} , \hfill \\ \Xi^{\prime}_{4} = \Xi_{6} + \Xi_{7} + F_{3}^{{ \top }} {\mathbb{Q}}_{6}^{ - 1} F_{3} ,\;\;\Xi^{\prime}_{5} = \Xi_{8} + \Xi_{9} + F_{4}^{{ \top }} {\mathbb{Q}}_{6}^{ - 1} F_{4} . \hfill \\ \end{gathered} $$
On account of the assumptions prior to Theorem 1, it is straightforward that inequality (47) is still true for \(\mu (t) = 0\) or \(\mu (t) = \overline{\mu }\) or \(\kappa (t) = 0\) or \(\kappa (t) = \overline{\kappa }.\)
Note that \(\Xi (\mu (t),\kappa (t))\) is linear about \(\mu (t)\) and \(\kappa (t)\), respectively, inequality \(\Xi (\mu (t),\kappa (t)) < {\mathbf{0}}\) can be handled non-conservatively by the following four corresponding boundary conditions:
$$ \begin{gathered} \Xi (0,0) = \Xi_{0} + \Xi^{\prime}_{1} + \overline{\mu }\Xi^{\prime}_{3} + \overline{\kappa }\Xi^{\prime}_{5} < {\mathbf{0}},\;\;\Xi (0,\overline{\kappa }) = \Xi_{0} + \Xi^{\prime}_{1} + \overline{\mu }\Xi^{\prime}_{3} + \overline{\kappa }\Xi^{\prime}_{4} < {\mathbf{0}}, \hfill \\ \Xi (\overline{\mu },0) = \Xi_{0} + \Xi^{\prime}_{1} + \overline{\mu }\Xi^{\prime}_{2} + \overline{\kappa }\Xi^{\prime}_{5} < {\mathbf{0}},\;\;\Xi (\overline{\mu },\overline{\kappa }) = \Xi_{0} + \Xi^{\prime}_{1} + \overline{\mu }\Xi^{\prime}_{2} + \overline{\kappa }\Xi^{\prime}_{4} < {\mathbf{0}}. \hfill \\ \end{gathered} $$
That is
$$ \begin{gathered} \Xi (0,0) = \Xi_{a} + \overline{\mu }F_{2}^{{ \top }} {\mathbb{Q}}_{3}^{ - 1} F_{2} + \overline{\kappa }F_{4}^{{ \top }} {\mathbb{Q}}_{6}^{ - 1} F_{4} < {\mathbf{0}}, \hfill \\ \Xi (0,\overline{\kappa }) = \Xi_{b} + \overline{\mu }F_{2}^{{ \top }} {\mathbb{Q}}_{3}^{ - 1} F_{2} + \overline{\kappa }F_{3}^{{ \top }} {\mathbb{Q}}_{6}^{ - 1} F_{3} < {\mathbf{0}}, \hfill \\ \Xi (\overline{\mu },0) = \Xi_{c} + \overline{\mu }F_{1}^{{ \top }} {\mathbb{Q}}_{3}^{ - 1} F_{1} + \overline{\kappa }F_{4}^{{ \top }} {\mathbb{Q}}_{6}^{ - 1} F_{4} < {\mathbf{0}}, \hfill \\ \Xi (\overline{\mu },\overline{\kappa }) = \Xi_{d} + \overline{\mu }F_{1}^{{ \top }} {\mathbb{Q}}_{3}^{ - 1} F_{1} + \overline{\kappa }F_{3}^{{ \top }} {\mathbb{Q}}_{6}^{ - 1} F_{3} < {\mathbf{0}}. \hfill \\ \end{gathered} $$
From the Schur complement Lemma, \(\Xi (0,0) < {\mathbf{0}}\) is equivalent to
$$\left[ {\begin{array}{*{20}c} {\Xi _{b} + \bar{\kappa }F_{3} ^{ { \top }} {\mathbb{Q}}_{6} ^{{ - 1}} F_{3} } & {\bar{\mu }F_{2} ^{ { \top }} } \\ * & { - \bar{\mu } {\mathbb{Q}}_{3} } \\ \end{array} } \right] < {\mathbf{0}}. $$
That is
$$ \left[ {\begin{array}{*{20}c} {\Xi_{a} } & {\overline{\mu }F_{2}^{{ \top }} } \\ * & { - \overline{\mu }{\mathbb{Q}}_{3} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\overline{\kappa }F_{4}^{{ \top }} } \\ {\mathbf{0}} \\ \end{array} } \right](\overline{\kappa }{\mathbb{Q}}_{6} )^{ - 1} \left[ {\begin{array}{*{20}c} {\overline{\kappa }F_{4}^{{ \top }} } \\ {\mathbf{0}} \\ \end{array} } \right]^{{ \top }} < {\mathbf{0}}. $$
(48)
Again based on the Schur complement lemma, inequality (48) converts to the first inequality of (11). Similarly, \(\Xi (0,\overline{\kappa }) < {\mathbf{0}}\) is equivalent to
$$ \left[ {\begin{array}{*{20}c} {\Xi_{b} + \overline{\kappa }F_{3}^{{ \top }} {\mathbb{Q}}_{6}^{ - 1} F_{3} } & {\overline{\mu }F_{2}^{{ \top }} } \\ * & { - \overline{\mu }{\mathbb{Q}}_{3} } \\ \end{array} } \right] < {\mathbf{0}}. $$
That is
$$\left[ {\begin{array}{*{20}c} {\Xi _{b} } & {\bar{\mu }F_{2} ^{ { \top }} } \\ * & { - \bar{\mu } {\mathbb{Q}}_{3} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\bar{\kappa }F_{3} ^{ { \top }} } \\ {\mathbf{0}} \\ \end{array} } \right](\bar{\kappa } {\mathbb{Q}}_{6} )^{{ - 1}} \left[ {\begin{array}{*{20}c} {\bar{\kappa }F_{3} ^{ { \top }} } \\ {\mathbf{0}} \\ \end{array} } \right]^{ { \top }} < {\mathbf{0}}.$$
(49)
Again in view of the Schur complement lemma, inequality (49) converts to the second inequality of (11). Similarly, \(\Xi (\overline{\mu },0) < {\mathbf{0}}\) is equivalent to
$$ \left[ {\begin{array}{*{20}c} {\Xi_{c} + \overline{\kappa }F_{4}^{{ \top }} {\mathbb{Q}}_{6}^{ - 1} F_{4} } & {\overline{\mu }F_{1}^{{ \top }} } \\ * & { - \overline{\mu }{\mathbb{Q}}_{3} } \\ \end{array} } \right] < {\mathbf{0}}. $$
That is
$$ \left[ {\begin{array}{*{20}c} {\Xi_{c} } & {\overline{\mu }F_{1}^{{ \top }} } \\ * & { - \overline{\mu }{\mathbb{Q}}_{3} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\overline{\kappa }F_{4}^{{ \top }} } \\ {\mathbf{0}} \\ \end{array} } \right](\overline{\kappa }{\mathbb{Q}}_{6} )^{ - 1} \left[ {\begin{array}{*{20}c} {\overline{\kappa }F_{4}^{{ \top }} } \\ {\mathbf{0}} \\ \end{array} } \right]^{{ \top }} < {\mathbf{0}}. $$
(50)
Again on account of the Schur complement lemma, inequality (50) turns into the first inequality of (12). Similarly, \(\Xi (\overline{\mu },\overline{\kappa }) < {\mathbf{0}}\) is equivalent to
$$ \left[ {\begin{array}{*{20}c} {\Xi_{d} + \overline{\kappa }F_{3}^{{ \top }} {\mathbb{Q}}_{6}^{ - 1} F_{3} } & {\overline{\mu }F_{1}^{{ \top }} } \\ * & { - \overline{\mu }{\mathbb{Q}}_{3} } \\ \end{array} } \right] < {\mathbf{0}}. $$
That is
$$ \left[ {\begin{array}{*{20}c} {\Xi_{d} } & {\overline{\mu }F_{1}^{{ \top }} } \\ * & { - \overline{\mu }{\mathbb{Q}}_{3} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\overline{\kappa }F_{3}^{{ \top }} } \\ {\mathbf{0}} \\ \end{array} } \right](\overline{\kappa }{\mathbb{Q}}_{6} )^{ - 1} \left[ {\begin{array}{*{20}c} {\overline{\kappa }F_{3}^{{ \top }} } \\ {\mathbf{0}} \\ \end{array} } \right]^{{ \top }} < {\mathbf{0}}. $$
(51)
Again because of the Schur complement lemma, inequality (51) becomes the second inequality of (12). Thus, inequalities (8–12) mean that network (3) is asymptotic stable. This finishes the proof of Theorem 1.
Appendix 4
$$ \begin{gathered} \Xi_{0}^{g} = {\text{sym}}\left\{ {e_{1}^{{ \top }} P_{1g} e_{u} + e_{4}^{{ \top }} P_{2g} e_{v} } \right\} + \sum\limits_{l = 1}^{2} {\sum\limits_{{q \in {\mathfrak{N}}_{k}^{g} }}^ {\tilde{\psi }_{gq} (P_{lq} - Z_{lq} )} } , \hfill \\ \Xi_{a}^{g} = \Xi_{0}^{g} + \Xi^{\prime}_{1} + \overline{\mu }\Xi_{5} + \overline{\kappa }\Xi_{9} ,\quad \Xi_{b}^{g} = \Xi_{0}^{g} + \Xi^{\prime}_{1} + \overline{\mu }\Xi_{5} + \overline{\kappa }\Xi_{7} , \hfill \\ \Xi_{c}^{g} = \Xi_{0}^{g} + \Xi^{\prime}_{1} + \overline{\mu }\Xi_{3} + \overline{\kappa }\Xi_{9} ,\quad \Xi_{d}^{g} = \Xi_{0}^{g} + \Xi^{\prime}_{1} + \overline{\mu }\Xi_{3} + \overline{\kappa }\Xi_{7} . \hfill \\ \end{gathered} $$
Appendix 5: Proof of Theorem 2
Consider the following LKF:
$$ V_{g} (t) = V_{1g} (t) + V_{2g} (t) + \sum\limits_{j = 3}^{7} {V_{j} } (t), $$
where \(V_{1g} (t) = u_{t}^{{ \top }} P_{1g} u_{t} ,\;V_{2g} (t) = w_{t}^{{ \top }} P_{2g} w_{t} ,\) and \(V_{j} (t)(j = 3,4,...,7)\) are defined in Theorem 1.
Let \(\nabla\) designate the weak infinitesimal generator of the SP \(\{ r(t) = g\}_{t \ge 0}\), i.e.,
$$ \nabla V_{g} (t): = \mathop {\lim }\limits_{{\tau \to 0^{ + } }} \frac{1}{\tau }[{\mathfrak{E}}\left\{ {V_{g} (t + \tau )|r(t) = g} \right\} - V_{g} (t)]. $$
Then
$$ \nabla V_{g} (t) = \nabla V_{1g} (t) + \nabla V_{2g} (t) + \sum\limits_{j = 3}^{7} {\nabla V_{j} } (t). $$
(52)
For each \(r(t) = g \in {\mathfrak{N}}\), applying the law of conditional expectation and total probability gives
$$\begin{gathered} \nabla V_{{1g}} (t) = \mathop {\lim }\limits_{{\tau \to 0^{ + } }} \frac{1}{\tau }\left\{ { {\mathfrak{E}}\left[ {\sum\limits_{{q = 1,q \ne g}}^{N} {\Pr \{ r_{{j + 1}} = q,b_{{j + 1}} \le b + \tau |r_{j} = g,b_{{j + 1}} > b\} ( * )^{ { \top }} P_{{1q}} u_{{t + \tau }} } } \right.} \right. \hfill \\ \quad \quad \quad \quad \quad \quad \quad \;\;\left. {\left. { + \Pr \{ r_{{j + 1}} = g,b_{{j + 1}} \le b + \tau |r_{j} = g,b_{{j + 1}} > b\} ( * )^{ { \top }} P_{{1g}} u_{{t + \tau }} } \right] - ( * )^{ { \top }} P_{{1g}} u_{t} } \right\} \hfill \\ = \mathop {\lim }\limits_{{\tau \to 0^{ + } }} \frac{1}{\tau }\left\{ { {\mathfrak{E}}\left[ {\sum\limits_{{q = 1,q \ne g}}^{N} {\frac{{\Pr \{ b < b_{{j + 1}} \le b + \tau |r_{{j + 1}} = q,r_{j} = g\} }}{{\Pr \{ b_{{j + 1}} > b|r_{j} = g\} }}\frac{{\Pr \{ r_{{j + 1}} = q,r_{j} = g\} }}{{\Pr \{ r_{j} = g\} }}( * )^{ { \top }} P_{{1q}} u_{{t + \tau }} } } \right.} \right. \hfill \\ \quad \quad \quad \quad \;\;\left. {\left. {\;\quad + \frac{{\Pr \{ b_{{j + 1}} > b + \tau |r_{j} = g\} }}{{\Pr \{ b_{{j + 1}} > b|r_{j} = g\} }}( * )^{ { \top }} P_{{1g}} u_{{t + \tau }} } \right] - ( * )^{ { \top }} P_{{1g}} u_{t} } \right\} \hfill \\ = \mathop {\lim }\limits_{{\tau \to 0^{ + } }} \frac{1}{\tau }\left\{ { {\mathfrak{E}}\left[ {\sum\limits_{{q = 1,q \ne g}}^{N} {\frac{{\gamma _{{gq}} (D_{g} (b + \tau ) - D_{g} (b))}}{{1 - D_{g} (b)}}( * )^{ { \top }} P_{{1q}} u_{{t + \tau }} } + \frac{{1 - D_{g} (b + \tau )}}{{1 - D_{g} (b)}}( * )^{ { \top }} P_{{1g}} u_{{t + \tau }} } \right] - ( * )^{ { \top }} P_{{1g}} u_{t} } \right\}. \hfill \\ \end{gathered}$$
With a small \(\tau ,\) the first-order approximation of \(u_{t + \tau }\) is \(u_{t + \tau } = u_{t} + \tau \dot{u}_{t} + o(\tau ).\) Thus we come to
$$ \begin{gathered} \nabla V_{1g} (t) = \mathop {\lim }\limits_{{\tau \to 0^{ + } }} \frac{1}{\tau }\left\{ {{\mathfrak{E}}\left[ {\sum\limits_{q = 1,q \ne g}^{N} {\frac{{\gamma_{gq} (D_{g} (b + \tau ) - D_{g} (b))}}{{1 - D_{g} (b)}}( * )^{{ \top }} P_{1q} (u_{t} + \tau \dot{u}_{t} )} } \right.} \right. \hfill \\ \quad \quad \quad \quad \quad \quad \left. {\left. {\quad {\kern 1pt} \quad + \frac{{1 - D_{g} (b + \tau )}}{{1 - D_{g} (b)}}( * )^{{ \top }} P_{1g} (u_{t} + \tau \dot{u}_{t} )} \right] - ( * )^{{ \top }} P_{1g} u_{t} } \right\}. \hfill \\ \end{gathered} $$
Considering the condition that \(\mathop {\lim }\limits_{{\tau \to 0^{ + } }} [D_{g} (b + \tau ) - D_{g} (b)] = 0,\) it can be easily inferred that
$$ \nabla V_{1g} (t) = 2u_{t}^{{ \top }} {\mathfrak{E}}\left\{ {\mathop {\lim }\limits_{{\tau \to 0^{ + } }} \left[ {\frac{{1 - D_{g} (b + \tau )}}{{1 - D_{g} (b)}}P_{1g} } \right]} \right\}\dot{u}_{t} + ( * )^{{ \top }} {\mathfrak{E}}\left\{ {\mathop {\lim }\limits_{{\tau \to 0^{ + } }} {\mathbf{\mathbb{P}}}_{1g} (\tau )} \right\}u_{t} , $$
where
$$ {\mathbf{\mathbb{P}}}_{1g} (\tau ) = \sum\limits_{q = 1,q \ne g}^{N} {\frac{{\gamma_{gq} [D_{g} (b + \tau ) - D_{g} (b)]}}{{\tau [1 - D_{g} (b)]}}P_{1q} } + \frac{{D_{g} (b) - D_{g} (b + \tau )}}{{\tau [1 - D_{g} (b)]}}P_{1g} . $$
According to the properties of the CDF, one obtains
$$ \mathop {\lim }\limits_{{\tau \to 0^{ + } }} \frac{{1 - D_{g} (b + \tau )}}{{1 - D_{g} (b)}} = 1,\;\;\mathop {\lim }\limits_{{\tau \to 0^{ + } }} \frac{{D_{g} (b + \tau ) - D_{g} (b)}}{{\tau [1 - D_{g} (b)]}} = \psi_{g} (b), $$
where \(\psi_{g} (b)\) expresses the TR of the system jump from mode \(g\), which satisfies (see [39]) \(\psi_{gq} (b) =\) \(\gamma_{gq} \psi_{g} (b),\) \(g \ne q,\) then it follows that
$$ \nabla V_{1g} (t) = 2u_{t}^{{ \top }} P_{1g} \dot{u}_{t} + ( * )^{{ \top }} \sum\limits_{q = 1}^{N} {\tilde{\psi }_{gq} P_{1q} u_{t} ,} $$
(53)
where \(\tilde{\psi }_{gq} : = {\mathfrak{E}}[\psi_{gq} (b)] = \int_{0}^{ + \infty } {\psi_{gq} (b)f_{g} (b)} {\text{d}}b.\)
Similarly, one has
$$ \nabla V_{2g} (t) = 2w_{t}^{{ \top }} P_{2g} \dot{w}_{t} + ( * )^{{ \top }} \sum\limits_{q = 1}^{N} {\tilde{\psi }_{gq} P_{2q} w_{t} } . $$
(54)
If the information of TRs is not completely available, it is easy to find \(\sum\nolimits_{q = 1}^{N} {\tilde{\psi }_{gq} } = 0,\;\tilde{\psi }_{gq} \ge 0,\;\forall q \ne g,\;q \in {\mathfrak{N}}\) from \(\sum\nolimits_{q = 1}^{N} {\psi_{gq} (b)} = 0,\;\psi_{gq} (b) \ge 0,\;\forall q \ne g,\;q \in {\mathfrak{N}}.\) Thus for any matrices \(Z_{lg} = Z_{lg}^{{ \top }}\) satisfying (13) and (14), one has
$$\begin{aligned} \sum\limits_{{q = 1}}^{N} {\tilde{\psi }_{{gq}} P_{{lq}} } & = \sum\limits_{{q = 1}}^{N} {\tilde{\psi }_{{gq}} (P_{{lq}} - Z_{{lq}} )} \hfill \\ & \quad = \sum\limits_{{q \in {\mathfrak{N}}_{k}^{g} }}^{{}} {\tilde{\psi }_{{gq}} (P_{{lq}} - Z_{{lq}} )} + \sum\limits_{{q \in {\mathfrak{N}}_{u}^{g} }}^{{}} {\tilde{\psi }_{{gq}} (P_{{lq}} - Z_{{lq}} )} \hfill \\ & \quad \le \sum\limits_{{q \in {\mathfrak{N}}_{k}^{g} }}^{{}} {\tilde{\psi }_{{gq}} (P_{{lq}} - Z_{{lq}} )} . \hfill \\ \end{aligned}$$
(55)
Combining (24–46), (52–55) and taking mathematical expectation on both sides yields
$$ {\mathfrak{E}\{ }\nabla V_{g} (t)\} \le \xi_{t}^{{ \top }} \Xi_{g} (\mu (t)\kappa (t))\xi_{t} , $$
(56)
where
$$\Xi _{g} (\mu (t),\kappa (t)) = \Xi _{0}^{g} + \Xi ^{\prime}_{1} + \mu (t)\Xi ^{\prime\prime}_{2} + [\bar{\mu } - \mu (t)]\Xi ^{\prime\prime}_{3} + \kappa (t)\Xi ^{\prime\prime}_{4} + [\bar{\kappa } - \kappa (t)]\Xi ^{\prime\prime}_{5} ,$$
with
$$ \Xi^{\prime\prime}_{2} = \Xi_{3} + F_{1}^{{ \top }} {\mathbb{Q}}_{3}^{ - 1} F_{1} ,\;\;\Xi^{\prime\prime}_{3} = \Xi_{5} + F_{2}^{{ \top }} {\mathbb{Q}}_{3}^{ - 1} F_{2} ,\;\;\Xi^{\prime\prime}_{4} = \Xi_{7} + F_{3}^{{ \top }} {\mathbb{Q}}_{6}^{ - 1} F_{3} ,\;\;\Xi^{\prime\prime}_{5} = \Xi_{9} + F_{4}^{{ \top }} {\mathbb{Q}}_{6}^{ - 1} F_{4} . $$
Next, following the same line as Theorem 1, one can conclude Theorem 2.
