Appendix A
Proof
Define the following Lyapunov–Krasovskii functional
$$\begin{aligned} V(k) = \sum \limits _{i = 1}^{3} {{V_i}(k)} \end{aligned}$$
(38)
where
$$\begin{aligned} {V_1}(k)&= {\varvec{x}^T}(k){\varvec{P}_1}(k)\varvec{x}(k),\\ {V_2}(k)&= \sum _{l = k - {d_1}}^{k - 1} {{\varvec{x}^T}(l){\varvec{R}_1}\varvec{x}(l)} + \sum \limits _{l = k - {d_2}}^{k - {d_1} - 1} {{\varvec{x}^T}(l){\varvec{R}_2}\varvec{x}(l)} \\&\quad + \sum _{l = k - {\tau _1}}^{k - 1} {{\varvec{x}^T}(l){\varvec{R}_3}\varvec{x}(l)} + \sum \limits _{l = k - {\tau _2}}^{k - {\tau _1} - 1} {{\varvec{x}^T}(l){\varvec{R}_4}\varvec{x}(l)},\\ {V_{3}}(k)&= {d_1}\sum _{s = - {d_1}}^{ - 1} {\sum _{l = k + s}^{k - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_1}\Delta \varvec{x}(l)} } \\&\quad + {d_{12}}\sum _{s = - {d_2}}^{ - {d_1} - 1} {\sum _{l = k + s}^{k - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_2}\Delta \varvec{x}(l)} }\\&\quad + {\tau _1}\sum _{s = - {\tau _1}}^{ - 1} {\sum _{l = k + s}^{k - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_{3}}\Delta \varvec{x}(l)} } \\&\quad + {\tau _{12}}\sum _{s = - {\tau _2}}^{ - {\tau _1} - 1} {\sum _{l = k + s}^{k - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_{4}}\Delta \varvec{x}(l)} } \end{aligned}$$
with \({\varvec{P}_1}(k) = \sum _{i = 1}^p {{\omega _i}(\varvec{{\tilde{\theta }}} (k))} {\varvec{P}_{1i}}\) and \(\Delta \varvec{x}(k) = \varvec{x}(k + 1) - \varvec{x}(k)\).
For the convenience of representation, the following auxiliary variable is defined:
$$\begin{aligned}&{\varvec{\xi } ^T}(k) = \bigg [\begin{array}{ll} {\begin{array}{ll} {{\varvec{x}^T}(k)}&{{\varvec{x}^T}(k - {\tau _1})} \end{array}}&{\begin{array}{ll} {{\varvec{x}^T}(k - {\tau _{2}})}&\ldots \end{array}} \end{array}\nonumber \\&\begin{array}{ll} {\begin{array}{ll} {{\varvec{x}^T}(k - {d_{1}})}&{{\varvec{x}^T}(k - {d_2})} \end{array}}&{\begin{array}{ll} {{\varvec{g}^T}(k - \tau (k))}&\ldots \end{array}} \end{array}\nonumber \\&\begin{array}{ll} {\begin{array}{ll} {{\varvec{f}^T}(k - d(k))}&{{\varvec{x}^T}(k - \tau (k))} \end{array}}&{\begin{array}{ll} {{\varvec{x}^T}(k - d(k))}&\ldots \end{array}} \end{array}\nonumber \\&\begin{array}{lll} {\frac{1}{{{\tau _1} + 1}}\sum \limits _{j = k - {\tau _1}}^k {{\varvec{x}^T}(j)} }&{\frac{1}{{\tau (k) - {\tau _1} + 1}}\sum \limits _{j = k - \tau (k)}^{k - {\tau _1}} {{\varvec{x}^T}(j)} }&\ldots \end{array}\nonumber \\&\begin{array}{lll} {\frac{1}{{{\tau _{2}} - \tau (k) + 1}}\sum \limits _{j = k - {\tau _{2}}}^{k - \tau (k)} {{\varvec{x}^T}(j)} }&{\frac{1}{{{d_1} + 1}}\sum \limits _{j = k - {d_1}}^k {{\varvec{x}^T}(j)} }&\ldots \end{array}\nonumber \\&\begin{array}{lll} {\frac{1}{{d(k) - {d_1} + 1}}\sum \limits _{j = k - d(k)}^{k - {d_1}} {{\varvec{x}^T}(j)} }&{\frac{1}{{{d_{2}} - d(k) + 1}}\sum \limits _{j = k - {d_{2}}}^{k - d(k)} {{\varvec{x}^T}(j)} } \end{array}\bigg ] \end{aligned}$$
(39)
From (38), we have
$$\begin{aligned} \Delta {V_1}(k)&= \varvec{E}\left\{ {{V_1}(k + 1) - {V_1}(k)} \right\} \nonumber \\&= \varvec{E}\left\{ {{\varvec{x}^T}(k + 1){\varvec{P}_1}(k + 1)\varvec{x}(k + 1)} \right. \nonumber \\&\quad \left. { - {\varvec{x}^T}(k){\varvec{P}_1}(k)\varvec{x}(k)} \right\} \nonumber \\&= {\varvec{x}^T}(k)\left[ {{{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k)){\varvec{P}_1}(k + 1){\mathbf {A}}(\varvec{{\tilde{\theta }}} (k)) - {\varvec{P}_1}(k)} \right. \nonumber \\&\quad + \left( {1 - \bar{\alpha } } \right) \left( {1 - {\bar{\beta }} } \right) {\varvec{K}^T}(\varvec{{\hat{\theta }}} (k)){{\mathbf {B}}^T}(\varvec{{\tilde{\theta }}} (k)){\varvec{P}_1}(k + 1)\nonumber \\&\quad \times {\mathbf {B}}(\varvec{{\tilde{\theta }}} (k))\varvec{K}(\varvec{{\hat{\theta }}} (k))\nonumber \\&\quad + {\textit{sym}}{\left\{ {\left( {1 - \bar{\alpha } } \right) \left( {1 - {\bar{\beta }} } \right) {{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k)){\varvec{P}_1}(k + 1)} \right. }\nonumber \\&\quad \times \left. {\left. {{\mathbf {{B}}}(\varvec{{\tilde{\theta }}} (k))\varvec{K}(\varvec{{\hat{\theta }}} (k))} \right\} } \right] x(k)\nonumber \\&\quad + \textit{sym}\left\{ {{\varvec{x}^T}(k)\left[ {\left. {{\bar{\beta }} {{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k)){\varvec{P}_1}(k + 1){\mathbf {B}}(\varvec{{\tilde{\theta }}} (k))} \right] } \right. } \right. \nonumber \\&\quad \times \left. {\varvec{g}(k - \tau (k))} \right\} \nonumber \\&\quad + \textit{sym}\left\{ {{\varvec{x}^T}(k)\left[ {\bar{\alpha } \left( {1 - {\bar{\beta }} } \right) {{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k)){\varvec{P}_1}(k + 1)} \right. } \right. \nonumber \\&\quad \left. {\left. { \times {\mathbf {B}}(\varvec{{\tilde{\theta }}} (k))\varvec{K}(\varvec{{\hat{\theta }}} (k))} \right] \varvec{f}(k - d(k))} \right\} \nonumber \\&\quad + {\varvec{g}^T}(k - \tau (k))\left[ {\left. {{\bar{\beta }} {{\mathbf {B}}^T}(\varvec{{\tilde{\theta }}} (k)){\varvec{P}_1}(k + 1){\mathbf {B}}(\varvec{{\tilde{\theta }}} (k))} \right] } \right. \nonumber \\&\quad \times \varvec{g}(k - \tau (k))\nonumber \\&\quad + {\varvec{f}^T}(k - d(k)){\left[ {\bar{\alpha } \left( {1 - {\bar{\beta }} } \right) {\varvec{K}^T}(\varvec{{\hat{\theta }}} (k)){{\mathbf {B}}^T}(\varvec{{\tilde{\theta }}} (k))} \right. }\nonumber \\&\quad \times {\left. {{\varvec{P}_1}(k + 1){\mathbf {B}}(\varvec{{\tilde{\theta }}} (k))\varvec{K}(\varvec{{\hat{\theta }}} (k))} \right] }\varvec{f}(k - d(k)) \end{aligned}$$
(40)
$$\begin{aligned} \Delta {V_2}(k)&= \varvec{E}\left\{ {{V_2}(k + 1) - {V_2}(k)} \right\} \nonumber \\&= {\varvec{x}^T}(k){\varvec{R}_1}\varvec{x}(k) - {\varvec{x}^T}(k - {d_1})\nonumber \\&\quad \left( {{\varvec{R}_1} - {\varvec{R}_2}} \right) \varvec{x}(k - {d_1})\nonumber \\&\quad - {\varvec{x}^T}(k - {d_2}){\varvec{R}_2}\varvec{x}(k - {d_2})\nonumber \\&\quad + {\varvec{x}^T}(k){\varvec{R}_3}\varvec{x}(k) - {\varvec{x}^T}(k - {\tau _1})\nonumber \\&\quad \left( {{\varvec{R}_3} - {\varvec{R}_4}} \right) \varvec{x}(k - {\tau _1})\nonumber \\&\quad - {\varvec{x}^T}(k - {\tau _2}){\varvec{R}_{4}}\varvec{x}(k - {\tau _2}) \end{aligned}$$
(41)
$$\begin{aligned}&\Delta {V_3}(k) = \varvec{E}\left\{ {{V_3}(k + 1) - {V_3}(k)} \right\} \nonumber \\&\quad = \varvec{E}\left\{ {{d_1}^2\Delta {\varvec{x}^T}(k){\varvec{S}_1}\Delta \varvec{x}(k)} \right\} \nonumber \\&\qquad - {d_1}\sum \limits _{l = k - {d_1}}^{k - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_1}\Delta \varvec{x}(l)}\nonumber \\&\qquad + \varvec{E}\left\{ {{d_{12}}^2\Delta {\varvec{x}^T}(k){\varvec{S}_2}\Delta \varvec{x}(k)} \right\} \nonumber \\&\qquad - {d_{12}}\sum \limits _{l = k - {d_2}}^{k - {d_1} - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_2}\Delta \varvec{x}(l)}\nonumber \\&\qquad + \varvec{E}\left\{ {{\tau _1}^2\Delta {\varvec{x}^T}(k){\varvec{S}_3}\Delta \varvec{x}(k)} \right\} \nonumber \\&\qquad - {\tau _1}\sum \limits _{l = k - {\tau _1}}^{k - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_3}\Delta \varvec{x}(l)} \nonumber \\&\qquad + \varvec{E}\left\{ {{\tau _{12}}^2\Delta {\varvec{x}^T}(k){\varvec{S}_4}\Delta \varvec{x}(k)} \right\} \nonumber \\&\qquad - {\tau _{12}}\sum \limits _{l = k - {\tau _2}}^{k - {\tau _1} - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_4}\Delta \varvec{x}(l)} \nonumber \\&\quad = {\varvec{x}^T}(k)\left[ {{{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k))\varvec{\bar{S}}{\mathbf {A}}(\varvec{{\tilde{\theta }}} (k)) + \varvec{\bar{S}}} \right. \nonumber \\&\qquad + \left( {1 - \bar{\alpha } } \right) \left( {1 - {\bar{\beta }} } \right) {\varvec{K}^T}(\varvec{{\hat{\theta }}} (k)){{\mathbf {B}}^T}(\varvec{{\tilde{\theta }}} (k))\varvec{\bar{S}}{\mathbf {B}}(\varvec{{\tilde{\theta }}} (k))\varvec{K}(\varvec{{\hat{\theta }}} (k))\nonumber \\&\qquad + \textit{sym}\left\{ {\left( {1 - \bar{\alpha } } \right) \left( {1 - {\bar{\beta }} } \right) {{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k))\varvec{\bar{S}}{\mathbf {B}}(\varvec{{\tilde{\theta }}} (k))\varvec{K}(\varvec{{\hat{\theta }}} (k))} \right\} \nonumber \\&\qquad - \textit{sym}\left\{ {\left( {1 - \bar{\alpha } } \right) \left( {1 - {\bar{\beta }} } \right) {\varvec{K}^T}(\varvec{{\hat{\theta }}} (k)){{\mathbf {B}}^T}(\varvec{{\tilde{\theta }}} (k))\varvec{\bar{S}}} \right\} \nonumber \\&\qquad \left. { - \textit{sym}\left\{ {{{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k))\varvec{\bar{S}}} \right\} } \right] \varvec{x}(k)\nonumber \\&\qquad + \textit{sym}\left\{ {{\varvec{x}^T}(k)\left[ {\left. {{\bar{\beta }} \left( {{{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k)) - \varvec{I}} \right) \varvec{\bar{S}}{\mathbf {B}}(\varvec{{\tilde{\theta }}} (k))} \right] } \right. \varvec{g}(k - \tau (k))} \right\} \nonumber \\&\qquad + \textit{sym}\left\{ {{\varvec{x}^T}(k)\left[ {\bar{\alpha } \left( {1 - {\bar{\beta }} } \right) \left( {{{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k)) - \varvec{I}} \right) \varvec{\bar{S}}{\mathbf {B}}(\varvec{{\tilde{\theta }}} (k)) } \right. } \right. \nonumber \\&\qquad \times \left. {\left. {\varvec{K}(\varvec{{\hat{\theta }}} (k))} \right] \varvec{f}(k - d(k))} \right\} \nonumber \\&\qquad + {\varvec{f}^T}(k - d(k))\left[ {\bar{\alpha } \left( {1 - {\bar{\beta }} } \right) {\varvec{K}^T}(\varvec{{\hat{\theta }}} (k)){{\mathbf {B}}^T}(\varvec{{\tilde{\theta }}} (k))\varvec{\bar{S}} } \right. \nonumber \\&\qquad \times \left. {{\mathbf {B}}(\varvec{{\tilde{\theta }}} (k))\varvec{K}(\varvec{{\hat{\theta }}} (k))} \right] \varvec{f}(k - d(k))\nonumber \\&\qquad - {d_1}\sum \limits _{l = k - {d_1}}^{k - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_1}\Delta \varvec{x}(l)}\nonumber \\&\qquad - {d_{12}}\sum \limits _{l = k - {d_2}}^{k - {d_1} - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_2}\Delta \varvec{x}(l)} \nonumber \\&\qquad - {\tau _1}\sum \limits _{l = k - {\tau _1}}^{k - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_3}\Delta \varvec{x}(l)}\nonumber \\&\qquad - {\tau _{12}}\sum \limits _{l = k - {\tau _2}}^{k - {\tau _1} - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_4}\Delta \varvec{x}(l)} \end{aligned}$$
(42)
where \(\varvec{\bar{S}} = {d_1}^2{\varvec{S}_1} + {d_{12}}^2{\varvec{S}_2} + {\tau _1}^2{\varvec{S}_3} + {\tau _{12}}^2{\varvec{S}_4}\).
Based on the assumptions of network attacks in (18), we have
$$\begin{aligned}&\bar{\alpha } {\varvec{x}^T}(k - d(k)){\varvec{U}_1}^T{\varvec{U}_1}\varvec{x}(k - d(k))\nonumber \\&\quad - \bar{\alpha } {\varvec{f}^T}(\varvec{x}(k - d(k)))\varvec{f}(\varvec{x}(k - d(k))) \ge 0 \end{aligned}$$
(43)
$$\begin{aligned}&{\bar{\beta }} {\varvec{x}^T}(k - \tau (k)){\varvec{U}_2}^T{\varvec{U}_2}\varvec{x}(k - \tau (k))\nonumber \\&\quad - {\bar{\beta }} {\varvec{g}^T}(\varvec{x}(k - \tau (k)))\varvec{g}(\varvec{x}(k - \tau (k))) \ge 0 \end{aligned}$$
(44)
Based on (40), (41), (42), (43) and (44), we have
$$\begin{aligned}&\Delta V(k) = \sum \limits _{i = 1}^3 {\Delta {V_i}(k)}\nonumber \\&\quad \le {\varvec{\xi } ^T}(k)\left[ {{\varvec{\eta } ^T}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))\varvec{\bar{P}}(k + 1)\varvec{\eta } (\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))} \right. \nonumber \\&\qquad + {\varvec{\eta } ^T}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))\varvec{\bar{\bar{S}}}\varvec{\eta } (\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))\nonumber \\&\qquad - {\varvec{e}_1}^T{\varvec{P}_1}(k){\varvec{e}_1}- d(k){\varvec{e}_1}^T{\varvec{P}_{2}}(k){\varvec{e}_1}^T\nonumber \\&\qquad - \tau (k){\varvec{e}_1}^T{\varvec{P}_{3}}(k){\varvec{e}_1}^T\nonumber \\&\qquad + {\varvec{e}_1}^T\varvec{\bar{S}}{\varvec{e}_1}\nonumber \\&\qquad - \textit{sym}\bigg \{ ({1 - \bar{\alpha }})( {1 - {\bar{\beta }}}){\varvec{e}_1}^T{\varvec{K}^T}(\varvec{{\hat{\theta }}} (k))\nonumber \\&\qquad \qquad \quad \qquad {{\mathbf {B}}^T}(\varvec{{\tilde{\theta }}} (k))\varvec{\bar{S}}{\varvec{e}_1} \bigg \}\nonumber \\&\qquad - \textit{sym}\left\{ {{\varvec{e}_1}^T{{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k))\varvec{\bar{S}}{\varvec{e}_1}} \right\} \nonumber \\&\qquad - \textit{sym}\left\{ {\left. {{\bar{\beta }} {\varvec{e}_1}^T\varvec{\bar{S}}{\mathbf {B}}(\varvec{{\tilde{\theta }}} (k)){\varvec{e}_6}} \right] } \right\} \nonumber \\&\qquad - \textit{sym}\left\{ {{\varvec{e}_1}^T\bar{\alpha } \left( {1 - {\bar{\beta }} } \right) \varvec{\bar{S}}{\mathbf {B}}(\varvec{{\tilde{\theta }}} (k))\varvec{K}(\varvec{{\hat{\theta }}} (k)){\varvec{e}_7}} \right\} \nonumber \\&\qquad + {\varvec{e}_1}^T\left( {{\varvec{R}_1} + {\varvec{R}_3}} \right) {\varvec{e}_1} - {\varvec{e}_4}^T\left( {{\varvec{R}_1} - {\varvec{R}_2}} \right) {\varvec{e}_4}\nonumber \\&\qquad - {\varvec{e}_{5}}^T{\varvec{R}_2}{\varvec{e}_{5}}\nonumber \\&\qquad - {\varvec{e}_{2}}^T\left( {{\varvec{R}_3} - {\varvec{R}_4}} \right) {\varvec{e}_{2}}+ \bar{\alpha } {\varvec{e}_9}^T{\varvec{U}_1}^T{\varvec{U}_1}{\varvec{e}_9}- \bar{\alpha } {\varvec{e}_7}^T{\varvec{e}_7}\nonumber \\&\qquad + {\bar{\beta }} {\varvec{e}_8}^T{\varvec{U}_2}^T{\varvec{U}_2}{\varvec{e}_8}- {\bar{\beta }} {\varvec{e}_6}^T{\varvec{e}_6}\left. { - {\varvec{e}_{3}}^T{\varvec{R}_{4}}{\varvec{e}_{3}}} \right] \varvec{\xi } (k)\nonumber \\&\qquad - {d_1}\sum \limits _{l = k - {d_1}}^{k - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_1}\Delta \varvec{x}(l)} \nonumber \\&\qquad - {d_{12}}\sum \limits _{l = k - {d_2}}^{k - {d_1} - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_2}\Delta \varvec{x}(l)} \nonumber \\&\qquad - {\tau _1}\sum \limits _{l = k - {\tau _1}}^{k - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_3}\Delta \varvec{x}(l)}\nonumber \\&\qquad - {\tau _{12}}\sum \limits _{l = k - {\tau _2}}^{k - {\tau _1} - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_4}\Delta \varvec{x}(l)} \end{aligned}$$
(45)
where
$$\begin{aligned}&\varvec{\bar{P}}(k + 1) = \textit{diag}\left\{ {{\bar{\beta }} {\varvec{P}_1}(k + 1),(1 - {\bar{\beta }} )\bar{\alpha } {\varvec{P}_1}(k + 1),} \right. \\&\qquad \qquad \qquad \qquad \qquad \left. {(1 - {\bar{\beta }} )(1 - \bar{\alpha } ){\varvec{P}_1}(k + 1)} \right\} ,\\&\varvec{\bar{\bar{S}}} = \textit{diag}\left\{ {{\bar{\beta }} \varvec{\bar{S}},(1 - {\bar{\beta }} )\bar{\alpha } \varvec{\bar{S}},(1 - {\bar{\beta }} )(1 - \bar{\alpha } )\varvec{\bar{S}}} \right\} ,\\&\varvec{\eta } (\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k)) = \left[ {\begin{array}{ll} {{\varvec{e}_1}^T{{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k)) + {\varvec{e}_6}^T{{\mathbf {B}}^T}(\varvec{{\tilde{\theta }}} (k))}&\ldots \end{array}} \right. \\&\qquad \qquad {\begin{array}{ll} {{\varvec{e}_1}^T{{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k)) + {\varvec{e}_7}^T{\varvec{K}^T}(\varvec{{\hat{\theta }}} (k)){{\mathbf {B}}^T}(\varvec{{\tilde{\theta }}} (k))}&\ldots \end{array}}\\&\qquad \qquad {\left. {{\varvec{e}_1}^T{{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k)) + {\varvec{e}_1}^T{\varvec{K}^T}(\varvec{{\hat{\theta }}} (k)){{\mathbf {B}}^T}(\varvec{{\tilde{\theta }}} (k))} \right] ^T}. \end{aligned}$$
Using Lemmas 1 and 2 for the summation terms in (45), we have
$$\begin{aligned}&- {d_1}\sum \limits _{l = k - {d_1}}^{k - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_1}\Delta \varvec{x}(l)}\nonumber \\&\le - {\varvec{\xi } ^T}(k)\left[ {{\varvec{\prod } _{11}}^T{\varvec{S}_1}{\varvec{\prod } _{11}} + 3{\varvec{\prod } _{12}}^T{\varvec{S}_1}{\varvec{\prod } _{12}}} \right] \varvec{\xi } (k) \end{aligned}$$
(46)
$$\begin{aligned}&\quad - {d_{12}}\sum \limits _{l = k - {d_2}}^{k - {d_1} - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_2}\Delta \varvec{x}(l)} \nonumber \\&= - {d_{12}}\sum \limits _{l = k - {d_2}}^{k - d(k) - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_2}\Delta \varvec{x}(l)} \nonumber \\&\quad - {d_{12}}\sum \limits _{l = k - d(k)}^{k - {d_1} - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_2}\Delta \varvec{x}(l)}\nonumber \\&\le - \frac{{{d_{12}}}}{{{d_2} - d(k)}}{\varvec{\xi } ^T}(k)\left[ {{\varvec{\prod } _{{21}}}^T{\varvec{S}_{2}}{\varvec{\prod } _{{21}}} + 3{\varvec{\prod } _{{22}}}^T{\varvec{S}_{2}}{\varvec{\prod } _{{22}}}} \right] \varvec{\xi } (k)\nonumber \\&\quad - \frac{{{d_{12}}}}{{d(k) - {d_{1}}}}{\varvec{\xi } ^T}(k)\left[ {{\varvec{\prod } _{{23}}}^T{\varvec{S}_{2}}{\varvec{\prod } _{{23}}} + 3{\varvec{\prod } _{{24}}}^T{\varvec{S}_{2}}{\varvec{\prod } _{{24}}}} \right] \varvec{\xi } (k)\nonumber \\&\le - {\varvec{\xi } ^T}(k)\left[ {{\varvec{\prod } _{{21}}}^T\left( {{\varvec{S}_{2}}+\frac{{d(k) - {d_{1}}}}{{{d_{12}}}}\left( {{\varvec{S}_{2}} - {\varvec{Y}_{1}}{\varvec{S}_{2}}^{ - 1}{\varvec{Y}_{1}}^T} \right) } \right) {\varvec{\prod } _{{21}}}} \right. \nonumber \\&\quad +{\varvec{\prod } _{{23}}}^T\left( {{\varvec{S}_{2}}+\frac{{{d_2} - d(k)}}{{{d_{12}}}}\left( {{\varvec{S}_{2}} - {\varvec{Y}_{2}}{\varvec{S}_{2}}^{ - 1}{\varvec{Y}_{2}}^T} \right) } \right) {\varvec{\prod } _{{23}}}\nonumber \\&\quad +\textit{sym}\left\{ {{\varvec{\prod } _{{21}}}^T\left( {\frac{{{d_2} - d(k)}}{{{d_{12}}}}{\varvec{Y}_{1}} + \frac{{d(k) - {d_1}}}{{{d_{12}}}}{\varvec{Y}_{2}}} \right) {\varvec{\prod } _{{23}}}} \right\} \nonumber \\&\quad +{\varvec{\prod } _{{22}}}^T\left( {3{\varvec{S}_{2}}+\frac{{d(k) - {d_{1}}}}{{{d_{12}}}}\left( {3{\varvec{S}_{2}} - {\varvec{Y}_{3}}\frac{{{\varvec{S}_{2}}^{ - 1}}}{3}{\varvec{Y}_{3}}^T} \right) } \right) {\varvec{\prod } _{{22}}}\nonumber \\&\quad +{\varvec{\prod } _{{24}}}^T\left( {{3}{\varvec{S}_{2}}+\frac{{{d_2} - d(k)}}{{{d_{12}}}}\left( {{3}{\varvec{S}_{2}} - {\varvec{Y}_{4}}\frac{{{\varvec{S}_{2}}^{ - 1}}}{3}{\varvec{Y}_{4}}^T} \right) } \right) {\varvec{\prod } _{{24}}}\nonumber \\&\quad + \textit{sym}\left\{ {{\varvec{\prod } _{{22}}}^T\left( {\frac{{{d_2} - d(k)}}{{{d_{12}}}}{\varvec{Y}_{3}}} \right. } \right. \nonumber \\&\quad \qquad \qquad {\left. {\left. {\left. { + \frac{{d(k) - {d_1}}}{{{d_{12}}}}{\varvec{Y}_{4}}} \right) {\varvec{\prod } _{{24}}}} \right\} } \right] \varvec{\xi } (k)} \end{aligned}$$
(47)
$$\begin{aligned}&- {\tau _1}\sum \limits _{l = k - {\tau _1}}^{k - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_3}\Delta \varvec{x}(l)}\nonumber \\&\quad \le - {\varvec{\xi } ^T}(k)\left[ {{\varvec{\prod } _{31}}^T{\varvec{S}_3}{\varvec{\prod } _{31}} + 3{\varvec{\prod } _{32}}^T{\varvec{S}_3}{\varvec{\prod } _{32}}} \right] \varvec{\xi } (k) \end{aligned}$$
(48)
$$\begin{aligned}&\quad - {\tau _{12}}\sum \limits _{l = k - {\tau _2}}^{k - {\tau _1} - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_4}\Delta \varvec{x}(l)} \nonumber \\&= - {\tau _{12}}\sum \limits _{l = k - {\tau _2}}^{k - \tau (k) - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_2}\Delta \varvec{x}(l)} \nonumber \\&\quad - {\tau _{12}}\sum \limits _{l = k - \tau (k)}^{k - {\tau _1} - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_2}\Delta \varvec{x}(l)}\nonumber \\&\le - \frac{{{\tau _{12}}}}{{{\tau _2} - \tau (k)}}{\xi ^T}(k)\left[ {{\varvec{\prod } _{{41}}}^T{\varvec{S}_{4}}{\varvec{\prod } _{{41}}} + 3{\varvec{\prod } _{{42}}}^T{\varvec{S}_{4}}{\varvec{\prod } _{{42}}}} \right] \varvec{\xi } (k)\nonumber \\&\quad - \frac{{{\tau _{12}}}}{{\tau (k) - {\tau _{1}}}}{\varvec{\xi } ^T}(k)\left[ {{\varvec{\prod } _{{43}}}^T{\varvec{S}_{4}}{\varvec{\prod } _{{43}}} + 3{\varvec{\prod } _{{44}}}^T{\varvec{S}_{4}}{\varvec{\prod } _{{44}}}} \right] \varvec{\xi } (k)\nonumber \\&\le - {\varvec{\xi } ^T}(k)\left[ {{\varvec{\prod } _{{41}}}^T\left( {{\varvec{S}_{4}}+\frac{{\tau (k) - {\tau _{1}}}}{{{\tau _{12}}}}\left( {{\varvec{S}_{4}}} \right. } \right. } \right. \nonumber \\&\qquad \qquad \qquad \left. {\left. { - {\varvec{Y}_{5}}{\varvec{S}_{4}}^{ - 1}{\varvec{Y}_{5}}^T} \right) } \right) {\varvec{\prod } _{{41}}}\nonumber \\&\quad +{\varvec{\prod } _{{43}}}^T\left( {{\varvec{S}_{4}}+\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}\left( {{\varvec{S}_{4}} - {\varvec{Y}_{6}}{\varvec{S}_{4}}^{ - 1}{\varvec{Y}_{6}}^T} \right) } \right) {\varvec{\prod } _{{43}}}\nonumber \\&\quad + \textit{sym}\left\{ {{\varvec{\prod } _{{41}}}^T\left( {\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}{\varvec{Y}_{5}} + \frac{{\tau (k) - {\tau _1}}}{{{\tau _{12}}}}{\varvec{Y}_{6}}} \right) {\varvec{\prod } _{{43}}}} \right\} \nonumber \\&\quad + {\varvec{\prod } _{{42}}}^T\left( {3{\varvec{S}_{4}}+\frac{{\tau (k) - {\tau _{1}}}}{{{\tau _{12}}}}\left( {3{\varvec{S}_{4}} - {\varvec{Y}_{7}}\frac{{{\varvec{S}_{4}}^{ - 1}}}{3}{\varvec{Y}_{7}}^T} \right) } \right) {\varvec{\prod } _{{42}}}\nonumber \\&\quad +{\varvec{\prod } _{{44}}}^T\left( {{3}{\varvec{S}_{4}}+\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}\left( {{3}{\varvec{S}_{4}} - {\varvec{Y}_{8}}\frac{{{\varvec{S}_{4}}^{ - 1}}}{3}{\varvec{Y}_{8}}^T} \right) } \right) {\varvec{\prod } _{{44}}}\nonumber \\&\quad +\textit{sym}\left\{ {{\varvec{\prod } _{{42}}}^T\left( {\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}{\varvec{Y}_{7}}} \right. } \right. \nonumber \\&\qquad \qquad \qquad {\left. {\left. {\left. { + \frac{{\tau (k) - {\tau _1}}}{{{\tau _{12}}}}{\varvec{Y}_{8}}} \right) {\varvec{\prod } _{{44}}}} \right\} } \right] \varvec{\xi } (k)} \end{aligned}$$
(49)
Based on (45), (46), (47), (48) and (49), we have
$$\begin{aligned} \Delta V(k)\le {\varvec{\xi } ^T}(k)\varvec{{\hat{\vartheta }}} (\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k),\varvec{{\tilde{\theta }}} (k + 1),d(k),\tau (k))\varvec{\xi } (k) \end{aligned}$$
(50)
where
$$\begin{aligned}&\varvec{{\hat{\vartheta }}} (\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k),\varvec{{\tilde{\theta }}} (k + 1),d(k),\tau (k))\\&\quad = {\varvec{\eta } ^T}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))\varvec{\bar{P}}(k + 1)\varvec{\eta } (\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))\\&\qquad + {\varvec{\eta } ^T}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))\varvec{\bar{\bar{S}}}\varvec{\eta } (\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))\\&\qquad + {\varvec{\bar{T}}_{1}}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k),d(k),\tau (k)) + {\varvec{\bar{T}}_{2}}(d(k),\tau (k))\\&{\varvec{\bar{T}}_{1}}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k),d(k),\tau (k))\\&\quad = - {\varvec{e}_1}^T{\varvec{P}_1}(k){\varvec{e}_1} + {\varvec{e}_1}^T\varvec{\bar{S}}{\varvec{e}_1} \nonumber \\&\qquad - \textit{sym}\left\{ {{\varvec{e}_1}^T{{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k))\varvec{\bar{S}}{\varvec{e}_1}} \right\} \\&\qquad - \textit{sym}\left\{ {{\bar{\beta }} {\varvec{e}_1}^T\varvec{\bar{S}}{\mathbf {B}}(\varvec{{\tilde{\theta }}} (k)){\varvec{e}_6}} \right\} \\&\qquad - \textit{sym}\left\{ {\left( {1 - \bar{\alpha } } \right) \left( {1 - {\bar{\beta }} } \right) {\varvec{e}_1}^T{\varvec{K}^T}(\varvec{{\hat{\theta }}} (k)){{\mathbf {B}}^T}(\varvec{{\tilde{\theta }}} (k))\varvec{\bar{S}}{\varvec{e}_1}} \right\} \\&\qquad - \textit{sym}\left\{ {{\varvec{e}_1}^T\bar{\alpha } \left( {1 - {\bar{\beta }} } \right) \varvec{\bar{S}}{\mathbf {B}}(\varvec{{\tilde{\theta }}} (k))\varvec{K}(\varvec{{\hat{\theta }}} (k)){\varvec{e}_7}} \right\} \\&\qquad - {\varvec{e}_{3}}^T{\varvec{R}_{4}}{\varvec{e}_{3}} + {\varvec{e}_1}^T\left( {{\varvec{R}_1} + {\varvec{R}_3}} \right) {\varvec{e}_1} - {\varvec{e}_4}^T\left( {{\varvec{R}_1} - {\varvec{R}_2}} \right) {\varvec{e}_4}\\&\qquad - {\varvec{e}_{5}}^T{\varvec{R}_2}{\varvec{e}_{5}}- {\varvec{e}_{2}}^T\left( {{\varvec{R}_3} - {\varvec{R}_4}} \right) {\varvec{e}_{2}}+ \bar{\alpha } {\varvec{e}_9}^T{\varvec{U}_1}^T{\varvec{U}_1}{\varvec{e}_9}\\&\qquad - \bar{\alpha } {\varvec{e}_7}^T{\varvec{e}_7}+ {\bar{\beta }} {\varvec{e}_8}^T{\varvec{U}_2}^T{\varvec{U}_2}{\varvec{e}_8}- {\bar{\beta }} {\varvec{e}_6}^T{\varvec{e}_6} - {\varvec{\prod } _{11}}^T{\varvec{S}_1}{\varvec{\prod } _{11}}\\&\qquad - 3{\varvec{\prod } _{12}}^T{\varvec{S}_1}{\varvec{\prod } _{12}} - {\varvec{\prod } _{{21}}}^T\left( {{\varvec{S}_{2}}+\frac{{d(k) - {d_{1}}}}{{{d_{12}}}}{\varvec{S}_{2}}} \right) {\varvec{\prod } _{{21}}}\\&\qquad - {\varvec{\prod } _{{23}}}^T\left( {{\varvec{S}_{2}}+\frac{{{d_2} - d(k)}}{{{d_{12}}}}{\varvec{S}_{2}}} \right) {\varvec{\prod } _{{23}}}\\&\qquad - \textit{sym}\left\{ {{\varvec{\prod } _{{21}}}^T\left( {\frac{{{d_2} - d(k)}}{{{d_{12}}}}{\varvec{Y}_{1}} + \frac{{d(k) - {d_1}}}{{{d_{12}}}}{\varvec{Y}_{2}}} \right) {\varvec{\prod } _{{23}}}} \right\} \\&\qquad - {\varvec{\prod } _{{22}}}^T\left( {3{\varvec{S}_{2}}\mathrm{{ + 3}}\frac{{d(k) - {d_{1}}}}{{{d_{12}}}}{\varvec{S}_{2}}} \right) {\varvec{\prod } _{{22}}}\\&\qquad - {\varvec{\prod } _{{24}}}^T\left( {{3}{\varvec{S}_{2}}\mathrm{{ + 3}}\frac{{{d_2} - d(k)}}{{{d_{12}}}}{\varvec{S}_{2}}} \right) {\varvec{\prod } _{{24}}}\\&\qquad - \textit{sym}\left\{ {{\varvec{\prod } _{{22}}}^T\left( {\frac{{{d_2} - d(k)}}{{{d_{12}}}}{\varvec{Y}_{3}} + \frac{{d(k) - {d_1}}}{{{d_{12}}}}{\varvec{Y}_{4}}} \right) {\varvec{\prod } _{{24}}}} \right\} \\&\qquad - {\varvec{\prod } _{31}}^T{\varvec{S}_3}{\varvec{\prod } _{31}} - 3{\varvec{\prod } _{32}}^T{\varvec{S}_3}{\varvec{\prod } _{32}}\\&\qquad - {\varvec{\prod } _{{41}}}^T\left( {{\varvec{S}_{4}}+\frac{{\tau (k) - {\tau _{1}}}}{{{\tau _{12}}}}{\varvec{S}_{4}}} \right) {\varvec{\prod } _{{41}}}\\&\qquad - {\varvec{\prod } _{{43}}}^T\left( {{\varvec{S}_{4}}+\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}{\varvec{S}_{4}}} \right) {\varvec{\prod } _{{43}}}\\&\qquad - \textit{sym}\left\{ {{\varvec{\prod } _{{41}}}^T\left( {\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}{\varvec{Y}_{5}} + \frac{{\tau (k) - {\tau _1}}}{{{\tau _{12}}}}{\varvec{Y}_{6}}} \right) {\varvec{\prod } _{{43}}}} \right\} \\&\qquad - {\varvec{\prod } _{{42}}}^T\left( {3{\varvec{S}_{4}}\mathrm{{ + 3}}\frac{{\tau (k) - {\tau _{1}}}}{{{\tau _{12}}}}{\varvec{S}_{4}}} \right) {\varvec{\prod } _{{42}}}\\&\qquad - {\varvec{\prod } _{{44}}}^T\left( {{3}{\varvec{S}_{4}}\mathrm{{ + 3}}\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}{\varvec{S}_{4}}} \right) {\varvec{\prod } _{{44}}}\\&\qquad - \textit{sym}\left\{ {{\varvec{\prod } _{{42}}}^T\left( {\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}{\varvec{Y}_{7}} + \frac{{\tau (k) - {\tau _1}}}{{{\tau _{12}}}}{\varvec{Y}_{8}}} \right) {\varvec{\prod } _{{44}}}} \right\} ,\\&{\varvec{\bar{T}}_{2}}(d(k),\tau (k))\\&\quad ={\varvec{\prod } _{{21}}}^T\left( {\frac{{d(k) - {d_{1}}}}{{{d_{12}}}}{\varvec{Y}_{1}}{\varvec{S}_{2}}^{ - 1}{\varvec{Y}_{1}}^T} \right) {\varvec{\prod } _{{21}}}\\&\qquad + {\varvec{\prod } _{{23}}}^T\left( {\frac{{{d_2} - d(k)}}{{{d_{12}}}}{\varvec{Y}_{2}}{\varvec{S}_{2}}^{ - 1}{\varvec{Y}_{2}}^T} \right) {\varvec{\prod } _{{23}}}\\&\qquad +{\varvec{\prod } _{{22}}}^T\left( {\frac{{d(k) - {d_{1}}}}{{{d_{12}}}}{\varvec{Y}_{3}}\frac{{{\varvec{S}_{2}}^{ - 1}}}{3}{\varvec{Y}_{3}}^T} \right) {\varvec{\prod } _{{22}}}\\&\qquad +{\varvec{\prod } _{{24}}}^T\left( {\frac{{{d_2} - d(k)}}{{{d_{12}}}}{\varvec{Y}_{4}}\frac{{{\varvec{S}_{2}}^{ - 1}}}{3}{\varvec{Y}_{4}}^T} \right) {\varvec{\prod } _{{24}}}\\&\qquad + {\varvec{\prod } _{{41}}}^T\left( {\frac{{\tau (k) - {\tau _{1}}}}{{{\tau _{12}}}}{\varvec{Y}_{5}}{\varvec{S}_{4}}^{ - 1}{\varvec{Y}_{5}}^T} \right) {\varvec{\prod } _{{41}}}\\&\qquad +{\varvec{\prod } _{{43}}}^T\left( {\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}{\varvec{Y}_{6}}{\varvec{S}_{4}}^{ - 1}{\varvec{Y}_{6}}^T} \right) {\varvec{\prod } _{{43}}}\\&\qquad + {\varvec{\prod } _{{42}}}^T\left( {\frac{{\tau (k) - {\tau _{1}}}}{{{\tau _{12}}}}{\varvec{Y}_{7}}\frac{{{\varvec{S}_{4}}^{ - 1}}}{3}{\varvec{Y}_{7}}^T} \right) {\varvec{\prod } _{{42}}}\\&\qquad + {\varvec{\prod } _{{44}}}^T\left( {\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}{\varvec{Y}_{8}}\frac{{{\varvec{S}_{4}}^{ - 1}}}{3}{\varvec{Y}_{8}}^T} \right) {\varvec{\prod } _{{44}}}. \end{aligned}$$
Define \(\varvec{\hat{G}} = {\varvec{\bar{S}}^{ - 1}}\); \({\varvec{\hat{P}}_1}(k) = \varvec{\hat{G}}{\varvec{P}_1}(k){\varvec{\hat{G}}^T}\); \({\varvec{\hat{R}}_i} = \varvec{\hat{G}}{\varvec{R}_i}{\varvec{\hat{G}}^T}\), \(i = 1,2,3,4\); \({\varvec{\hat{S}}_i} = \varvec{\varvec{\varvec{\hat{G}}}}{\varvec{S}_i}{\varvec{\hat{G}}^T}\), \(i = 1,2,3,4\); \({\varvec{\hat{Y}}_i} \in {\mathbb {R}^{n \times n}}\), \(i = 1,\ldots ,8\); \(\varvec{\bar{G}} = \left[ {\begin{array}{lllllll} {{\varvec{e}_1}^T\varvec{\hat{G}}}&\cdots&{{\varvec{e}_5}^T\varvec{\hat{G}}}&I&{{\varvec{e}_7}^T\varvec{\hat{G}}}&\cdots&{{\varvec{e}_{15}}^T\varvec{\hat{G}}} \end{array}} \right] \). Obviously, we have \(\varvec{\hat{G}} = {d_1}^2{\varvec{\hat{S}}_1} + {d_{12}}^2{\varvec{\hat{S}}_2} + {\tau _1}^2{\varvec{\hat{S}}_3} + {\tau _{12}}^2{\varvec{\hat{S}}_4}\). From the facts of \( - \varvec{\hat{G}}{\varvec{\hat{P}}_1}^{ - 1}(k + 1)\varvec{\hat{G}} \le - \varvec{\hat{G}} - \varvec{\hat{G}} + {\varvec{\hat{P}}_1}(k + 1)\) and using Schur complement to (27)–(30), we have
$$\begin{aligned} \left[ {\begin{array}{lll} {{{\varvec{\hat{\bar{T}}}}_{1}} + {{\varvec{\hat{\bar{T}}}}_{2}}}&{}{}&{} * \\ {\varvec{\hat{\Xi }} (\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))}&{}{ - \varvec{\Theta } (\varvec{{\tilde{\theta }}} (k + 1))}&{}{}\\ {\varvec{\hat{\Xi }} (\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))}&{}\varvec{0}&{}{ - \varvec{\hat{\bar{S}}}} \end{array}} \right] < \varvec{0} \end{aligned}$$
(51)
where
$$\begin{aligned}&{{{\varvec{\hat{\bar{T}}}}_{1}} + {{\varvec{\hat{\bar{T}}}}_{2}}}\\&\quad ={{{\varvec{\hat{\bar{T}}}}_{1}}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k),d(k),\tau (k)) + {{\varvec{\hat{\bar{T}}}}_{2}}(d(k),\tau (k))}\\&\varvec{\Theta } (\varvec{{\tilde{\theta }}} (k + 1))\\&\quad = \textit{diag}\left\{ {\frac{1}{{{\bar{\beta }} }}\varvec{\hat{G}}{{\varvec{\hat{P}}}_1}^{ - 1}(k + 1)\varvec{\hat{G}},\frac{1}{{(1 - {\bar{\beta }} )\bar{\alpha } }}\varvec{\hat{G}}{{\varvec{\hat{P}}}_1}^{ - 1}(k + 1)\varvec{\hat{G}},} \right. \\&\qquad \qquad \qquad \left. {\frac{1}{{(1 - {\bar{\beta }} )(1 - \bar{\alpha } )}}\varvec{\hat{G}}{{\varvec{\hat{P}}}_1}^{ - 1}(k + 1)\varvec{\hat{G}}} \right\} \\&{\varvec{\hat{\bar{T}}}_{1}}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k),d(k),\tau (k))\\&\quad = - {\varvec{e}_1}^T{\varvec{\hat{P}}_1}(k){\varvec{e}_1} + {\varvec{e}_1}^T\varvec{\hat{G}}{\varvec{e}_1}\\&\qquad - \textit{sym}\left\{ {{\varvec{e}_1}^T\varvec{\hat{G}}{{\mathbf {A}}^T}(\varvec{{\tilde{\theta }}} (k)){\varvec{e}_1}} \right\} - \textit{sym}\left\{ {{\bar{\beta }} {\varvec{e}_1}^T{\mathbf {B}}(\varvec{{\tilde{\theta }}} (k)){\varvec{e}_6}} \right\} \\&\qquad - \textit{sym}\left\{ {\left( {1 - \bar{\alpha } } \right) \left( {1 - {\bar{\beta }} } \right) {\varvec{e}_1}^T{{\varvec{\hat{K}}}^T}(\varvec{{\hat{\theta }}} (k)){{\mathbf {B}}^T}(\varvec{{\tilde{\theta }}} (k)){\varvec{e}_1}} \right\} \\&\qquad - \textit{sym}\left\{ {{\varvec{e}_1}^T\bar{\alpha } \left( {1 - {\bar{\beta }} } \right) {\mathbf {B}}(\varvec{{\tilde{\theta }}} (k))\varvec{\hat{K}}(\varvec{{\hat{\theta }}} (k)){\varvec{e}_7}} \right\} \\&\qquad - {\varvec{e}_{3}}^T{\varvec{\hat{R}}_{4}}{\varvec{e}_{3}} + {\varvec{e}_1}^T\left( {{{\varvec{\hat{R}}}_1} + {{\varvec{\hat{R}}}_3}} \right) {\varvec{e}_1} - {\varvec{e}_4}^T\left( {{{\varvec{\hat{R}}}_1} - {{\varvec{\hat{R}}}_2}} \right) {\varvec{e}_4}\\&\qquad - {\varvec{e}_{5}}^T{\varvec{\hat{R}}_2}{\varvec{e}_{5}} - {\varvec{e}_{2}}^T\left( {{{\varvec{\hat{R}}}_3} - {{\varvec{\hat{R}}}_4}} \right) {\varvec{e}_{2}} + \bar{\alpha } {\varvec{e}_9}^T\varvec{\hat{G}}{\varvec{U}_1}^T{\varvec{U}_1}\varvec{\hat{G}}{\varvec{e}_9}\\&\qquad - \bar{\alpha } {\varvec{e}_7}^T\varvec{\hat{G}}\varvec{\hat{G}}{\varvec{e}_7} + {\bar{\beta }} {\varvec{e}_8}^T\varvec{\hat{G}}{\varvec{U}_2}^T{\varvec{U}_2}\varvec{\hat{G}}{\varvec{e}_8} - {\bar{\beta }} {\varvec{e}_6}^T{\varvec{e}_6}\\&\qquad - {\varvec{\prod } _{11}}^T{\varvec{\hat{S}}_1}{\varvec{\prod } _{11}} - 3{\varvec{\prod } _{12}}^T{\varvec{\hat{S}}_1}{\varvec{\prod } _{12}}\\&\qquad - {\varvec{\prod } _{{21}}}^T\left( {{{\varvec{\hat{S}}}_{2}}+\frac{{d(k) - {d_{1}}}}{{{d_{12}}}}{{\varvec{\hat{S}}}_{2}}} \right) {\varvec{\prod } _{{21}}}\\&\qquad - {\varvec{\prod } _{{23}}}^T\left( {{{\varvec{\hat{S}}}_{2}}+\frac{{{d_2} - d(k)}}{{{d_{12}}}}{{\varvec{\hat{S}}}_{2}}} \right) {\varvec{\prod } _{{23}}}\\&\qquad - \textit{sym}\left\{ {{\varvec{\prod } _{{21}}}^T\left( {\frac{{{d_2} - d(k)}}{{{d_{12}}}}{{\varvec{\hat{Y}}}_{1}} + \frac{{d(k) - {d_1}}}{{{d_{12}}}}{{\varvec{\hat{Y}}}_{2}}} \right) {\varvec{\prod } _{{23}}}} \right\} \\&\qquad - {\varvec{\prod } _{{22}}}^T\left( {3{{\varvec{\hat{S}}}_{2}}\mathrm{{ + 3}}\frac{{d(k) - {d_{1}}}}{{{d_{12}}}}{{\varvec{\hat{S}}}_{2}}} \right) {\varvec{\prod } _{{22}}}\\ \end{aligned}$$
$$\begin{aligned}&\qquad - {\varvec{\prod } _{{24}}}^T\left( {{3}{{\varvec{\hat{S}}}_{2}}\mathrm{{ + 3}}\frac{{{d_2} - d(k)}}{{{d_{12}}}}{{\varvec{\hat{S}}}_{2}}} \right) {\varvec{\prod } _{{24}}}\\&\qquad - \textit{sym}\left\{ {{\varvec{\prod } _{{22}}}^T\left( {\frac{{{d_2} - d(k)}}{{{d_{12}}}}{{\varvec{\hat{Y}}}_{3}} + \frac{{d(k) - {d_1}}}{{{d_{12}}}}{{\varvec{\hat{Y}}}_{4}}} \right) {\varvec{\prod } _{{24}}}} \right\} \\&\qquad - {\varvec{\prod } _{31}}^T{\varvec{\hat{S}}_3}{\varvec{\prod } _{31}} - 3{\varvec{\prod } _{32}}^T{\varvec{\hat{S}}_3}{\varvec{\prod } _{32}}\\&\qquad - {\varvec{\prod } _{{41}}}^T\left( {{{\varvec{\hat{S}}}_{4}}+\frac{{\tau (k) - {\tau _{1}}}}{{{\tau _{12}}}}{{\varvec{\hat{S}}}_{4}}} \right) {\varvec{\prod } _{{41}}}\\&\qquad - {\varvec{\prod } _{{43}}}^T\left( {{{\varvec{\hat{S}}}_{4}}+\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}{{\varvec{\hat{S}}}_{4}}} \right) {\varvec{\prod } _{{43}}}\\&\qquad - \textit{sym}\left\{ {{\varvec{\prod } _{{41}}}^T\left( {\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}{{\varvec{\hat{Y}}}_{5}} + \frac{{\tau (k) - {\tau _1}}}{{{\tau _{12}}}}{{\varvec{\hat{Y}}}_{6}}} \right) {\varvec{\prod } _{{43}}}} \right\} \\&\qquad - {\varvec{\prod } _{{42}}}^T\left( {3{{\varvec{\hat{S}}}_{4}}\mathrm{{ + 3}}\frac{{\tau (k) - {\tau _{1}}}}{{{\tau _{12}}}}{{\varvec{\hat{S}}}_{4}}} \right) {\varvec{\prod } _{{42}}}\\&\qquad - {\varvec{\prod } _{{44}}}^T\left( {{3}{{\varvec{\hat{S}}}_{4}}\mathrm{{ + 3}}\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}{{\varvec{\hat{S}}}_{4}}} \right) {\varvec{\prod } _{{44}}}\\&\qquad - \textit{sym}\left\{ {{\varvec{\prod } _{{42}}}^T\left( {\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}{{\varvec{\hat{Y}}}_{7}} + \frac{{\tau (k) - {\tau _1}}}{{{\tau _{12}}}}{{\varvec{\hat{Y}}}_{8}}} \right) {\varvec{\prod } _{{44}}}} \right\} ,\\&{\varvec{\hat{ \bar{T}}}_{2}}(d(k),\tau (k))\\&\quad = {\varvec{\prod } _{{21}}}^T\left( {\frac{{d(k) - {d_{1}}}}{{{d_{12}}}}{{\varvec{\hat{Y}}}_{1}}{{\varvec{\hat{S}}}_{2}}^{ - 1}{{\varvec{\hat{Y}}}_{1}}^T} \right) {\varvec{\prod } _{{21}}}\\&\qquad +{\varvec{\prod } _{{23}}}^T\left( {\frac{{{d_2} - d(k)}}{{{d_{12}}}}{{\varvec{\hat{Y}}}_{2}}{{\varvec{\hat{S}}}_{2}}^{ - 1}{{\varvec{\hat{Y}}}_{2}}^T} \right) {\varvec{\prod } _{{23}}}\\&\qquad + {\varvec{\prod } _{{22}}}^T\left( {\frac{{d(k) - {d_{1}}}}{{{d_{12}}}}{{\varvec{\hat{Y}}}_{3}}\frac{{{{\varvec{\hat{S}}}_{2}}^{ - 1}}}{3}{{\varvec{\hat{Y}}}_{3}}^T} \right) {\varvec{\prod } _{{22}}}\\&\qquad +{\varvec{\prod } _{{24}}}^T\left( {\frac{{{d_2} - d(k)}}{{{d_{12}}}}{{\varvec{\hat{Y}}}_{4}}\frac{{{{\varvec{\hat{S}}}_{2}}^{ - 1}}}{3}{{\varvec{\hat{Y}}}_{4}}^T} \right) {\varvec{\prod } _{{24}}}\\&\qquad + {\varvec{\prod } _{{41}}}^T\left( {\frac{{\tau (k) - {\tau _{1}}}}{{{\tau _{12}}}}{{\varvec{\hat{Y}}}_{5}}{{\varvec{\hat{S}}}_{4}}^{ - 1}{{\varvec{\hat{Y}}}_{5}}^T} \right) {\varvec{\prod } _{{41}}}\\&\qquad + {\varvec{\prod } _{{43}}}^T\left( {\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}{{\varvec{\hat{Y}}}_{6}}{{\varvec{\hat{S}}}_{4}}^{ - 1}{{\varvec{\hat{Y}}}_{6}}^T} \right) {\varvec{\prod } _{{43}}}\\&\qquad +{\varvec{\prod } _{{42}}}^T\left( {\frac{{\tau (k) - {\tau _{1}}}}{{{\tau _{12}}}}{{\varvec{\hat{Y}}}_{7}}\frac{{{{\varvec{\hat{S}}}_{4}}^{ - 1}}}{3}{{\varvec{\hat{Y}}}_{7}}^T} \right) {\varvec{\prod } _{{42}}}\\&\qquad +{\varvec{\prod } _{{44}}}^T\left( {\frac{{{\tau _2} - \tau (k)}}{{{\tau _{12}}}}{{\varvec{\hat{Y}}}_{8}}\frac{{{{\varvec{\hat{S}}}_{4}}^{ - 1}}}{3}{{\varvec{\hat{Y}}}_{8}}^T} \right) {\varvec{\prod } _{{44}}} \end{aligned}$$
Pre and postmultiplying both sides of (51) with \(\textit{diag}\left\{ {\begin{array}{lll} {{{\varvec{\bar{G}}}^{ - 1}},}&{\varvec{I},}&\varvec{I} \end{array}} \right\} \) and its transposition, we have
$$\begin{aligned} \left[ {\begin{array}{lll} {{{\varvec{\bar{T}}}_{1}} + {{\varvec{\bar{T}}}_{2}}}&{}{}&{} * \\ {\varvec{\eta } (\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))}&{}{ - {{\varvec{\bar{P}}}^{ - 1}}(k + 1)}&{}{}\\ {\varvec{\eta } (\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))}&{}\varvec{0}&{}{ - {{\varvec{\bar{\bar{S}}}}^{ - 1}}} \end{array}} \right] < \varvec{0} \end{aligned}$$
(52)
where
$$\begin{aligned} {{{\varvec{\bar{T}}}_{1}} + {{\varvec{\bar{T}}}_{2}}}={{{\varvec{\bar{T}}}_{1}}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k),d(k),\tau (k)) + {{\varvec{\bar{T}}}_{2}}(d(k),\tau (k))} \end{aligned}$$
Using Schur Complement to (52), we have
$$\begin{aligned} \varvec{{\hat{\vartheta }}} (\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k),\varvec{{\tilde{\theta }}} (k + 1),d(k),\tau (k)) < 0 \end{aligned}$$
(53)
From (53), we can find \(\rho > 0\) such that
$$\begin{aligned}&\varvec{{\hat{\vartheta }}} (\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k),\varvec{{\tilde{\theta }}} (k + 1),d(k),\tau (k)) \nonumber \\&\quad < \textit{diag}\left\{ {\begin{array}{llll} { - \rho {\varvec{I}_n}}&\varvec{0}&\cdots&\varvec{0} \end{array}} \right\} \end{aligned}$$
(54)
Combining (50) and (54), we have
$$\begin{aligned}&\varvec{E}\left\{ {V(k + 1) - V(k)} \right\} \nonumber \\&\quad \le {\varvec{\xi } ^T}(k)\varvec{{\hat{\vartheta }}} (\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k),\varvec{{\tilde{\theta }}} (k + 1),d(k),\nonumber \\&\qquad \Delta d(k),\tau (k),\Delta \tau (k))\varvec{\xi } (k)\nonumber \\&\quad \le {\varvec{\xi } ^T}(k)\textit{diag}\left\{ {\begin{array}{llll} { - \rho {\varvec{I}_n}}&\varvec{0}&\cdots&\varvec{0} \end{array}} \right\} \varvec{\xi } (k)\nonumber \\&\quad \le - \rho ||\varvec{x}(k)|{|^2} \end{aligned}$$
(55)
From (55), we obtain
$$\begin{aligned} \varvec{E}\left\{ {V(k + 1)} \right\} - \varvec{E}\left\{ {V(k)} \right\} \le - \rho \varvec{E}\left\{ {||\varvec{x}(k)|{|^2}} \right\} \end{aligned}$$
(56)
For \(N > 0\), the following inequality can be derived.
$$\begin{aligned} \varvec{E}\left\{ {\sum \limits _{k = 0}^N {||\varvec{x}(k)|{|^2}} } \right\}&\le \frac{1}{\rho }\left( {\varvec{E}\left\{ {V(0)} \right\} - \varvec{E}\left\{ {V(N + 1)} \right\} } \right) \nonumber \\&\le \frac{1}{\rho }\varvec{E}\left\{ {V(0)} \right\} \end{aligned}$$
(57)
Based on the definitions of Lyapunov–Krasovskii functional in (38), we have
$$\begin{aligned} {V_1}(0)&= {\varvec{x}^T}(0){\varvec{P}_1}(0)\varvec{x}(0) = \sum \limits _{i = 1}^p {{\omega _i}(\varvec{{\tilde{\theta }}} (0))} {\varvec{x}^T}(0){\varvec{P}_{1i}}(0)\varvec{x}(0)\nonumber \\&\le \mathop {\max }\limits _{i = 1,2,\ldots ,p} \left\{ {{\lambda _{\max }}({\varvec{P}_{1i}})} \right\} ||\varvec{x}(0)||_a^2\end{aligned}$$
(58)
$$\begin{aligned} {V_2}(0)&= \sum \limits _{l = - {d_1}}^{ - 1} {{\varvec{x}^T}(l){\varvec{R}_1}\varvec{x}(l)} + \sum \limits _{l = - {d_2}}^{ - {d_1} - 1} {{\varvec{x}^T}(l){\varvec{R}_2}\varvec{x}(l)} \nonumber \\&\quad + \sum \limits _{l = - {\tau _1}}^{ - 1} {{\varvec{x}^T}(l){\varvec{R}_3}\varvec{x}(l)} + \sum \limits _{l = - {\tau _2}}^{ - {\tau _1} - 1} {{\varvec{x}^T}(l){\varvec{R}_4}\varvec{x}(l)} \nonumber \\&\le {d_1}{\lambda _{\max }}({\varvec{R}_1})||\varvec{x}(0)||_a^2 + {d_{12}}{\lambda _{\max }}({\varvec{R}_2})||\varvec{x}(0)||_a^2\nonumber \\&\quad + {\tau _1}{\lambda _{\max }}({\varvec{R}_3})||\varvec{x}(0)||_a^2 + {\tau _{12}}{\lambda _{\max }}({\varvec{R}_4})||\varvec{x}(0)||_a^2\end{aligned}$$
(59)
$$\begin{aligned} {V_{3}}(0)&= {d_1}\sum \limits _{s = - {d_1}}^{ - 1} {\sum \limits _{l = s}^{ - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_1}\Delta \varvec{x}(l)} }\nonumber \\&\quad + {d_{12}}\sum \limits _{s = - {d_2}}^{ - {d_1} - 1} {\sum \limits _{l = s}^{ - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_2}\Delta \varvec{x}(l)} } \nonumber \\&\quad + {\tau _1}\sum \limits _{s = - {\tau _1}}^{ - 1} {\sum \limits _{l = s}^{ - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_{3}}\Delta \varvec{x}(l)} } \nonumber \\&\quad + {\tau _{12}}\sum \limits _{s = - {\tau _2}}^{ - {\tau _1} - 1} {\sum \limits _{l = s}^{ - 1} {\Delta {\varvec{x}^T}(l){\varvec{S}_{4}}\Delta \varvec{x}(l)} }\nonumber \\&\le \frac{{{d_1}^2({d_1} + 1)}}{2}{\lambda _{\max }}({\varvec{S}_1})||\varvec{x}(0)||_a^2\nonumber \\&\quad + \frac{{{d_{12}}^2({d_1} + {d_2} + 1)}}{2}{\lambda _{\max }}({\varvec{S}_2})||\varvec{x}(0)||_a^2\nonumber \\&\quad + \frac{{{\tau _1}^2({\tau _1} + 1)}}{2}{\lambda _{\max }}({\varvec{S}_3})||\varvec{x}(0)||_a^2\nonumber \\&\quad + \frac{{{\tau _{12}}^2({\tau _1} + {\tau _2} + 1)}}{2}{\lambda _{\max }}({\varvec{S}_4})||\varvec{x}(0)||_a^2 \end{aligned}$$
(60)
where \({\lambda _{\max }}(\cdot )\) and \({\lambda _{\min }}(\cdot )\) stand for maximum and minimum eigenvalues of a real square matrix, respectively.
Based on (58), (59) and (60), we have
$$\begin{aligned} V(0) \le \kappa ||\varvec{x}(0)||_a^2 \end{aligned}$$
(61)
where
$$\begin{aligned} \kappa&= \max \limits _{i = 1,2,\ldots ,p} \left\{ {{\lambda _{\max }}({\varvec{P}_{1i}})} \right\} + {d_1}{\lambda _{\max }}({\varvec{R}_1}) \\&\quad + {d_{12}}{\lambda _{\max }}({\varvec{R}_2})\\&\quad + {\tau _1}{\lambda _{\max }}({\varvec{R}_3})||\varvec{x}(0)||_a^2 + {\tau _{12}}{\lambda _{\max }}({\varvec{R}_4})||\varvec{x}(0)||_a^2\\&\quad + \frac{{{d_1}^2({d_1} + 1)}}{2}{\lambda _{\max }}({\varvec{S}_1}) \\&\quad + \frac{{{d_{12}}^2({d_1} + {d_2} + 1)}}{2}{\lambda _{\max }}({\varvec{S}_2})\\&\quad + \frac{{{\tau _1}^2({\tau _1} + 1)}}{2}{\lambda _{\max }}({\varvec{S}_3})\\&\quad + \frac{{{\tau _{12}}^2({\tau _1} + {\tau _2} + 1)}}{2}{\lambda _{\max }}({\varvec{S}_4}). \end{aligned}$$
From (57) and (61), we have
$$\begin{aligned} \varvec{E}\left\{ {\sum \limits _{k = 0}^\infty {||\varvec{x}(k)|{|^2}} } \right\} \le \sigma \varvec{E}\left\{ {||\varvec{x}(0)||_a^2} \right\} \end{aligned}$$
(62)
where \(\sigma = \frac{\kappa }{\rho }\). The whole proof is completed. \(\square \)
Appendix B
Proof
Inspired by literature [53], we define the following equations:
$$\begin{aligned} {\omega _i}(\varvec{{\tilde{\theta }}} (k))&= {h_i}(\varvec{{\hat{\theta }}} (k))\left( {\lambda _i^1\beta _i^{(1)}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))} \right. \nonumber \\&\quad \left. { +\, \lambda _i^2\beta _i^{(2)}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))} \right) \end{aligned}$$
(63)
$$\begin{aligned} {\omega _i}(\varvec{{\tilde{\theta }}} (k + 1))&= {h_i}(\varvec{{\hat{\theta }}} (k))\left( {\gamma _i^1\alpha _i^{(1)}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))} \right. \nonumber \\&\quad \left. { +\, \gamma _i^2\alpha _i^{(2)}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))} \right) \end{aligned}$$
(64)
where \(\alpha _i^{(1)}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))\), \(\alpha _i^{(2)}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))\), \(\beta _i^{(1)}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))\) and \(\beta _i^{(2)}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))\) are nonnegative nonlinear functions satisfying \(\sum _{s = 1}^2 {\alpha _i^{(s)}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))} =1\) and \(\sum _{s = 1}^2 {\beta _i^{(s)}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k))} =1\). For the convenience of expression, define
and
The \({\varvec{\Psi } _q}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k),\varvec{{\tilde{\theta }}} (k + 1))\) in Theorem 1 can be rewritten as:
It can be seen from (32)–(34) and (65) that \({\varvec{\Psi } _q}(\varvec{{\tilde{\theta }}} (k),\varvec{{\hat{\theta }}} (k),\varvec{{\tilde{\theta }}} (k + 1)) < 0\). The whole proof is completed.\(\square \)
