Appendix
In this section, the proof of Theorem 1 is stated.
Proof
A Lyapunov function is defined as follows:
$$V = V_{1} + V_{2} + V_{3} + V_{4} ,$$
(46)
where
$$V_{1} = x^{T} (t + \left. s \right|t)P_{1} (t)x(t + \left. s \right|t),$$
$$V_{2} = e_{x}^{T} (t + \left. s \right|t)Q_{1} (t)e_{x} (t + \left. s \right|t),$$
$$V_{3} = e_{f}^{T} (t + \left. s \right|t)G_{1} (t)e_{f} (t + \left. s \right|t),$$
$$V_{4} = \sum\limits_{m = 0}^{h} {x^{T} (t + s - \left. m \right|t)\overline{S}(t)x(t + s - \left. m \right|t) + } \sum\limits_{m = 0}^{h} {e_{x}^{T} } (t + s - \left. m \right|t)\overline{N}(t)e_{x} (t + s - \left. m \right|t).$$
where \(P_{1} (t),\) \(Q_{1} (t),\) \(\overline{S}(t),\) \(G_{1} (t)\) and \(\overline{N}(t)\) are positive definite matrices. Then, \(\Delta V\) is written as:
$$\Delta V = \Delta V_{1} + \Delta V_{2} + \Delta V_{3} + \Delta V_{4} .$$
(47)
\(\Delta V_{1}\) includes system states, which are calculated by substituting the closed-loop dynamics (32) as:
$$\begin{aligned} \Delta V_{1} (x(t + \left. s \right|t)) & = E\{ V_{1} (x(t + \left. {s + 1} \right|t))\} - V_{1} (x(t + \left. s \right|t) \\ & = \sum\limits_{i = 1}^{p} {\sum\limits_{j = 1}^{p} {\omega_{i} } } m_{j} E\{ [A_{ui} x(t + \left. s \right|t) + (A_{dij} + \tilde{v}(t)B_{ij} )x(t + \left. {s - h} \right|t) \\ & \quad - (\tilde{v}(t) + \overline{v})B_{ij} e(t + \left. {s - h} \right|t) - (\tilde{v}(t) + \overline{v})B_{ij} e_{x} (t + \left. {s - h} \right|t) \\ & \quad + B_{i} e_{f} (t + \left. s \right|t) + B_{wi} w(t + \left. s \right|t)]^{T} P_{1} (t)[A_{ui} x(t + \left. s \right|t) \\ & \quad + (A_{dij} + \tilde{v}(t)B_{ij} )x(t + \left. {s - h} \right|t) - (\tilde{v}(t) + \overline{v})B_{ij} e(t + \left. {s - h} \right|t) \\ & \quad - (\tilde{v}(t) + \overline{v})B_{ij} e_{x} (t + \left. {s - h} \right|t) + B_{i} e_{f} (t + \left. s \right|t) + B_{ij} w(t + \left. s \right|t)]\} \\ & \quad - x^{T} (t + \left. s \right|t)P_{1} (t)x(t + \left. s \right|t) \le \xi_{{}}^{T} (t + \left. s \right|t)E\{ \Xi_{1ij} \} \xi (t + \left. s \right|t), \\ \end{aligned}$$
(48)
where
$$\xi (t + \left. s \right|t) = \left[ \begin{gathered} x(t + \left. s \right|t) \\ x(t + \left. {s - h} \right|t) \\ e_{x} (t + \left. s \right|t) \\ e_{x} (t + \left. {s - h} \right|t) \\ e(t + \left. s \right|t) \\ e(t + \left. {s - h} \right|t) \\ e_{f} (t + \left. s \right|t) \\ w(t + \left. s \right|t) \\ \end{gathered} \right],\quad \Pi_{1ij}^{T} = \left[ \begin{gathered} A_{ui}^{T} \\ A_{dij}^{T} + \tilde{\nu }(t)B_{ij}^{T} \\ 0 \\ - (\tilde{\nu }(t) + \overline{\nu })B_{ij}^{T} \\ 0 \\ - (\tilde{\nu }(t) + \overline{\nu })B_{ij}^{T} \\ B_{i}^{T} \\ B_{wi}^{T} \\ \end{gathered} \right],$$
$$\Xi_{1ij} = \Pi_{1ij}^{T} P_{1} (t)\Pi_{1ij} + diag\{ - P_{1} (t),\;0,\;0,\;0,\;0,\;0,\;0,\;0\} .$$
The second part \(\Delta V_{2}\) includes the states estimation error of the system. Therefore, by substituting (35) the following equation is derived:
$$\begin{aligned} \Delta V_{2} (e_{x} (t\left. { + s} \right|t)) & = E\{ V_{2} (e_{x} (t\left. { + s + 1} \right|t))\} - V_{2} (e_{x} (t\left. { + s} \right|t)) \\ & = \sum\limits_{i = 1}^{p} {\omega_{i} } E\{ [(A_{ui} - L_{i} C)e_{x} (t\left. { + s} \right|t) + A_{di} e_{x} (t\left. { + s - h} \right|t) + B_{i} e_{f} (t\left. { + s} \right|t) \\ & \quad + L_{i} C(\tilde{\iota }(t) + (\overline{\iota } + 1))x(t\left. { + s} \right|t)]^{T} Q_{1} (t)[(A_{ui} - L_{i} C)e_{x} (t\left. { + s} \right|t) \\ & \quad + A_{di} e_{x} (t\left. { + s - h} \right|t) + B_{i} e_{f} (t\left. { + s} \right|t) + L_{i} C(\tilde{\iota }(t) + (\overline{\iota } + 1))x(t\left. { + s} \right|t)]\} \\ & \quad - e_{x}^{T} (t\left. { + s} \right|t)Q_{1} (t)e_{x} (t\left. { + s} \right|t) \le \xi_{{}}^{T} (t\left. { + s} \right|t)E\{ \Xi_{2i} \} \xi (t\left. { + s} \right|t), \\ \end{aligned}$$
(49)
where
$$\Pi_{2i}^{T} = \left[ \begin{gathered} C^{T} L_{i}^{T} (\tilde{\iota }(t) + (\overline{\iota } + 1)) \\ 0 \\ A_{ui}^{T} - C^{T} L_{i}^{T} \\ A_{di} \\ 0 \\ 0 \\ B_{i}^{T} \\ B_{wi}^{T} \\ \end{gathered} \right],\quad \Xi_{2i} = \Pi_{2i}^{T} Q_{1} (t)\Pi_{2i} + diag\{ 0,\;0,\;Q_{1} (t),\;0,\;0,\;0,\;0,\;0\} .$$
The third part \(\Delta V_{3}\) includes the fault estimation error of the actuators (36), which yields the following result by substituting the fault dynamics (34) and using Lemma 1 and Assumption 1:
$$\begin{aligned} \Delta V_{3} (e_{f} (t\left. { + s} \right|t)) & = E\{ V_{3} (e_{f} (t\left. { + s + 1} \right|t))\} - V_{3} (e_{f} (t\left. { + s} \right|t)) \\ & = E\{ [f(t\left. { + s + 1} \right|t) - \hat{f}(t\left. { + s} \right|t) + \sum\limits_{i + 1}^{p} {\omega_{i} \Gamma C((\tilde{\iota }(t) + (\overline{\iota } + 1))} x(t\left. { + s} \right|t) \\ & \quad + e_{x} (t\left. { + s} \right|t))]^{T} G_{1} (t)[f(t\left. { + s + 1} \right|t) - \hat{f}(t\left. { + s} \right|t) \\ & \quad + \sum\limits_{i = 1}^{p} {\omega_{i} \Gamma C((\tilde{\iota }(t) + (\overline{\iota } + 1))} x(t\left. { + s} \right|t) + e_{x} (t\left. { + s} \right|t))]\} \\ & \quad - e_{f}^{T} (t\left. { + s} \right|t)G_{1} (t)e_{f} (t\left. { + s} \right|t) = E\{ [f_{r} + e_{f} (t\left. { + s} \right|t) \\ & \quad + \sum\limits_{i = 1}^{p} {\omega_{i} \Gamma C((\tilde{\iota }(t) + (\overline{\iota } + 1))} x(t\left. { + s} \right|t) + e_{x} (t\left. { + s} \right|t))]^{T} G_{1} (t)[f_{r} \\ & \quad + e_{f} (t\left. { + s} \right|t) + \sum\limits_{i = 1}^{p} {\omega_{i} \Gamma C((\tilde{\iota }(t) + (\overline{\iota } + 1))} x(t\left. { + s} \right|t) + e_{x} (t\left. { + s} \right|t))]\} \\ & \quad - e_{f}^{T} (t\left. { + s} \right|t)G_{1} (t)e_{f} (t\left. { + s} \right|t)^{{}} \le \sum\limits_{i = 1}^{p} {\omega_{i} } E\{ [(\tilde{\iota }(t) + (\overline{\iota } + 1))(x(t\left. { + s} \right|t) \\ & \quad + e_{x} (t\left. { + s} \right|t))]\} + e_{f}^{T} (t\left. { + s} \right|t)2G_{1} (t)e_{f} (t\left. { + s} \right|t) \\ & \quad + f_{r}^{T} 3G_{1} (t)f_{r} \le \xi^{T} (t\left. { + s} \right|t)E\{ \Xi_{3i} \} \xi^{T} (t\left. { + s} \right|t) + f_{r}^{T} 3G_{1} (t)f_{r} \\ & \le \xi_{{}}^{T} (t\left. { + s} \right|t)E\{ \Xi_{3i} \} \xi (t\left. { + s} \right|t) + \delta , \\ \end{aligned}$$
(50)
where
$$\begin{aligned} & \Xi_{3i} = diag\{ g^{2} C^{T} \Gamma^{T} 3G_{1} (t)\Gamma C,\;0,\;g^{2} C^{T} \Gamma^{T} 3G_{1} (t)\Gamma C,\;0,\;0,\;0,\;2G_{1} (t),\;0\} , \\ & \delta = f_{r\max }^{2} \lambda_{\max } (3G_{1} ) > 0. \\ \end{aligned}$$
The fourth part of the Lyapunov function \(\Delta V_{4} ,\) includes the delay term, which is expressed as follows:
$$\begin{aligned} \Delta V_{4} & = E\{ x^{T} (t + \left. s \right|t)\overline{S}(t)x(t + \left. s \right|t) - x^{T} (t + \left. {s - h} \right|t)\overline{S}(t)x(t + \left. {s - h} \right|t)\} \\ & \quad + E\{ e_{x}^{T} (t + \left. s \right|t)\overline{N}(t)e_{x} (t + \left. s \right|t) - e_{x}^{T} (t + \left. {s - h} \right|t)\overline{N}(t)e_{x} (t + \left. {s - h} \right|t)\} \\ & \quad \le \xi_{{}}^{T} (t + \left. s \right|t)E\{ \Xi_{4} \} \xi (t + \left. s \right|t), \\ \end{aligned}$$
(51)
where
$$\Xi_{4} = diag\{ \overline{S}(t),\; - \overline{S}(t),\;\overline{N}(t),\; - \overline{N}(t),\;0,\;0,\;0,\;0\} .$$
Now, the performance indices criterion is established for the closed-loop system. Using (48)–(51) and adding the \({\mathcal{H}}_{\infty }\) performance index given by (15), the function \(\Delta \overline{V}\) is defined as:
$$\begin{aligned} \Delta \overline{V} & \triangleq E\{ x(\left. {t + s + 1} \right|t)^{T} P_{1} (t)x(\left. {t + s + 1} \right|t)\} - x^{T} (\left. {t + s} \right|t)P_{1} (t)x(\left. {t + s} \right|t) \\ & \quad + E\{ e_{x} (\left. {t + s + 1} \right|t)^{T} Q_{1} (t)e_{x} (\left. {t + s + 1} \right|t)\} - e_{x}^{T} (\left. {t + s} \right|t)Q_{1} (t)e_{x} (\left. {t + s} \right|t) \\ & \quad + E\{ e_{f} (\left. {t + s + 1} \right|t)^{T} G_{1} (t)e_{f} (\left. {t + s + 1} \right|t)\} - e_{f}^{T} (\left. {t + s} \right|t)G_{1} (t)e_{f} (\left. {t + s} \right|t) \\ & \quad + E\{ x^{T} (\left. {t + s} \right|t)\overline{S}(t)x(\left. {t + s} \right|t) - x^{T} (\left. {t + s - h} \right|t)\overline{S}(t)x(\left. {t + s - h} \right|t)\} \\ & \quad + E\{ e_{x}^{T} (\left. {t + s} \right|t)\overline{N}(t)e_{x} (\left. {t + s} \right|t) - e_{x}^{T} (\left. {t + s - h} \right|t)\overline{N}(t)e_{x} (\left. {t + s - h} \right|t)\} \\ & \quad + E\{ y^{T} (\left. {t + s} \right|t)y(\left. {t + s} \right|t)\} - \gamma^{T} w^{T} (\left. {t + s} \right|t)w(\left. {t + s} \right|t). \\ \end{aligned}$$
(52)
Adding the event trigger condition (11) to \(\Delta \overline{V}\) yields:
$$\begin{aligned} & \Delta \overline{V} + \sigma [(\hat{x}(\left. {t + s} \right|t) - e(\left. {t + s} \right|t))^{T} \theta (\hat{x}(\left. {t + s} \right|t) - e(\left. {t + s} \right|t))] - e^{T} (\left. {t + s} \right|t)\theta e(\left. {t + s} \right|t) \\ & \quad \le \sum\limits_{i = 1}^{p} {\sum\limits_{j = 1}^{p} {\omega_{i} m_{j} } } \xi^{T} (\left. {t + s} \right|t)E\{ \overline{\Xi }_{ij} \} \xi (\left. {t + s} \right|t) + \delta , \\ \end{aligned}$$
(53)
where
$$\overline{\Xi }_{ij} = \Xi_{1ij} + \Xi_{2i} + \Xi_{3i} + \Xi_{4} = \Pi_{1ij}^{T} P_{1} (t)\Pi_{1ij} + \Pi_{2i}^{T} Q_{1} (t)\Pi_{2i} + \Psi ,$$
$$\Psi = \left[ {\begin{array}{*{20}c} { - P_{1} (t) + \sigma \theta + \overline{S}(t) + g^{2} \Gamma^{T} C^{T} 3G_{1} (t)C\Gamma } & * & * & * & * & * & * & * \\ 0 & { - \overline{S}(t) - Q_{1} (t)} & * & * & * & * & * & * \\ { - \sigma \theta } & 0 & {\overline{N}(t) + g^{2} \Gamma^{T} C^{T} 3G_{1} (t)C\Gamma - \sigma \theta } & * & * & * & * & * \\ 0 & 0 & 0 & { - \overline{N}(t)} & * & * & * & * \\ { - \sigma \theta } & 0 & { - \sigma \theta } & 0 & {(\sigma - 1)\theta } & * & * & * \\ 0 & 0 & 0 & 0 & 0 & 0 & * & * \\ 0 & 0 & 0 & 0 & 0 & 0 & {2G_{1} (t)} & * \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - \gamma^{2} } \\ \end{array} } \right].$$
It is first needed to show that \(\sum\nolimits_{i = 1}^{p} {\sum\nolimits_{j = 1}^{p} {\omega_{i} } m_{j} } \overline{\Xi }_{ij} < 0\) is established in Eq. (53). To this aim, since \(\sum\nolimits_{i = 1}^{p} {\omega_{i} } = 1\) and \(\sum\nolimits_{j = 1}^{p} {m_{j} = 1} ,\) then we have \(\sum\nolimits_{i = 1}^{p} {\sum\nolimits_{j = 1}^{p} {\omega_{i} } } (\omega_{i} - m_{j} )\Lambda_{ij} = 0,\) where \(\Lambda_{ij}\) are arbitrary symmetric matrices with proper dimensions, defined as follows:
$$\Lambda_{ij} = \left[ {\begin{array}{*{20}c} {T_{ij}^{11} } & * & * & * & * & * & * & * \\ {T_{ij}^{21} } & {T_{ij}^{22} } & * & * & * & * & * & * \\ {T_{ij}^{31} } & {T_{ij}^{32} } & {T_{ij}^{33} } & * & * & * & * & * \\ {T_{ij}^{41} } & {T_{ij}^{42} } & {T_{ij}^{43} } & {T_{ij}^{44} } & * & * & * & * \\ {T_{ij}^{51} } & {T_{ij}^{52} } & {T_{ij}^{53} } & {T_{ij}^{54} } & {T_{ij}^{55} } & * & * & * \\ {T_{ij}^{61} } & {T_{ij}^{62} } & {T_{ij}^{63} } & {T_{ij}^{64} } & {T_{ij}^{65} } & {T_{ij}^{66} } & * & * \\ {T_{ij}^{71} } & {T_{ij}^{72} } & {T_{ij}^{73} } & {T_{ij}^{74} } & {T_{ij}^{75} } & {T_{ij}^{76} } & {T_{ij}^{77} } & * \\ {T_{ij}^{81} } & {T_{ij}^{82} } & {T_{ij}^{83} } & {T_{ij}^{84} } & {T_{ij}^{85} } & {T_{ij}^{86} } & {T_{ij}^{87} } & {T_{ij}^{88} } \\ \end{array} } \right].$$
(54)
Therefore, \(\sum\nolimits_{i = 1}^{p} {\sum\nolimits_{j = 1}^{p} {\omega_{i} } } m_{j} \overline{\Xi }_{ij}\) can be expanded as:
$$\begin{array}{*{20}l} {\sum\limits_{{i = 1}}^{p} {\sum\limits_{{j = 1}}^{p} {\omega _{i} m_{j} \bar{\Xi }_{{ij}} } = } \sum\limits_{{i = 1}}^{p} {\sum\limits_{{j = 1}}^{p} {\omega _{i} m_{j} \bar{\Xi }_{{ij}} } + \sum\limits_{{i = 1}}^{p} {\sum\limits_{{j = 1}}^{p} {\omega _{i} (\omega _{j} - m_{j} + \lambda _{j} \omega _{j} - \lambda _{j} \omega _{j} )\Lambda _{{ij}} } } } } \hfill \\ { = \sum\limits_{{i = 1}}^{p} {\omega _{i}^{2} (\lambda _{i} \bar{\Xi }_{{ii}} - \lambda _{i} \Lambda _{{ii}} + \Lambda _{{ii}} )} + \sum\limits_{{i = 1}}^{{p - 1}} {\sum\limits_{{j = i + 1}}^{p} {\omega _{i} \omega _{j} (\lambda _{j} \bar{\Xi }_{{ij}} - \lambda _{j} \Lambda _{{ij}} + \Lambda _{{ij}} + \lambda _{i} \bar{\Xi }_{{ji}} - \lambda _{i} \Lambda _{{ji}} + \Lambda _{{ji}} )} } } \hfill \\ {\quad + \sum\limits_{{i = 1}}^{p} {\sum\limits_{{j = 1}}^{p} {(m_{j} - \lambda _{j} \omega _{j} )(\bar{\Xi }_{{ij}} - \Lambda _{{ij}} ).} } } \hfill \\ \end{array}$$
(55)
Considering the following equations:
$$\begin{array}{l} Q = \rho P_1^{ - 1}(t),\quad G = \rho P_1^{ - 1}{\Gamma ^T}{C^T}3{G_1}(t)P_1^{ - 1}C\Gamma ,\quad P = \rho Q_1^{ - 1},\quad {K_i}Q(t) = {Y_i},\quad P(t){L_i} = {M_i}(t),\\ S = \rho {\bar{S}}P_1^{ - 1}(t),\quad N = \rho {\bar{N}}Q_1^{ - 1}(t),\quad \rho P_1^{ - 1}\theta P_1^{ - 1} = {\bar{\theta}} ,\quad \rho P_1^{ - 1}T_{ij}^{11}P_1^{ - 1} = {\bar{T}}_{ij}^{11},\quad \rho P_1^{ - 1}T_{ij}^{21}P_1^{ - 1} = {\bar{T}}_{ij}^{21},\\ \rho P_1^{ - 1}T_{ij}^{22}P_1^{ - 1} = {\bar{T}}_{ij}^{22},\quad T_{ij}^{31}P_1^{ - 1} = {\bar{T}}_{ij}^{31},\quad T_{ij}^{32}P_1^{ - 1} = {\bar{T}}_{ij}^{32},\quad \rho P_1^{ - 1}T_{ij}^{33}P_1^{ - 1} = {\bar{T}}_{ij}^{33},\quad \rho Q_1^{ - 1}T_{ij}^{41}Q_1^{ - 1} = {\bar{T}}_{ij}^{41},\\ \rho Q_1^{ - 1}T_{ij}^{42}Q_1^{ - 1} = {\bar{T}}_{ij}^{42},\quad \rho Q_1^{ - 1}T_{ij}^{43}Q_1^{ - 1} = {\bar{T}}_{ij}^{43},\quad \rho Q_1^{ - 1}T_{ij}^{44}Q_1^{ - 1} = {\bar{T}}_{ij}^{44},\quad \rho Q_1^{ - 1}T_{ij}^{51}P_1^{ - 1} = {\bar{T}}_{ij}^{51},\quad \rho Q_1^{ - 1}T_{ij}^{52}P_1^{ - 1} = {\bar{T}}_{ij}^{52},\\ \rho Q_1^{ - 1}T_{ij}^{53}P_1^{ - 1} = {\bar{T}}_{ij}^{53},\quad \rho Q_1^{ - 1}T_{ij}^{54}P_1^{ - 1} = {\bar{T}}_{ij}^{54},\quad \rho Q_1^{ - 1}T_{ij}^{55}P_1^{ - 1} = {\bar{T}}_{ij}^{55},\quad \rho Q_1^{ - 1}T_{ij}^{61}P_1^{ - 1} = {\bar{T}}_{ij}^{61},\quad \rho Q_1^{ - 1}T_{ij}^{62}P_1^{ - 1} = {\bar{T}}_{ij}^{62},\\ \rho Q_1^{ - 1}T_{ij}^{63}P_1^{ - 1} = {\bar{T}}_{ij}^{63},\quad \rho Q_1^{ - 1}T_{ij}^{64}P_1^{ - 1} = {\bar{T}}_{ij}^{64},\quad \rho Q_1^{ - 1}T_{ij}^{65}P_1^{ - 1} = {\bar{T}}_{ij}^{65},\quad \rho Q_1^{ - 1}T_{ij}^{66}P_1^{ - 1} = {\bar{T}}_{ij}^{66},\quad \rho Q_1^{ - 1}T_{ij}^{71}P_1^{ - 1} = {\bar{T}}_{ij}^{71},\\ \rho Q_1^{ - 1}T_{ij}^{72}P_1^{ - 1} = {\bar{T}}_{ij}^{72},\quad \rho Q_1^{ - 1}T_{ij}^{73}P_1^{ - 1} = {\bar{T}}_{ij}^{73},\quad \rho Q_1^{ - 1}T_{ij}^{74}P_1^{ - 1} = {\bar{T}}_{ij}^{74},\quad \rho Q_1^{ - 1}T_{ij}^{75}P_1^{ - 1} = {\bar{T}}_{ij}^{75},\quad \rho Q_1^{ - 1}T_{ij}^{76}P_1^{ - 1} = {\bar{T}}_{ij}^{76},\\ \rho Q_1^{ - 1}T_{ij}^{77}P_1^{ - 1} = {\bar{T}}_{ij}^{77},\quad \rho Q_1^{ - 1}T_{ij}^{81}P_1^{ - 1} = {\bar{T}}_{ij}^{81},\quad \rho Q_1^{ - 1}T_{ij}^{82}P_1^{ - 1} = {\bar{T}}_{ij}^{82},\quad \rho Q_1^{ - 1}T_{ij}^{83}P_1^{ - 1} = {\bar{T}}_{ij}^{83},\quad \rho Q_1^{ - 1}T_{ij}^{84}P_1^{ - 1} = {\bar{T}}_{ij}^{84},\\ \rho Q_1^{ - 1}T_{ij}^{85}P_1^{ - 1} = {\bar{T}}_{ij}^{85},\quad \rho Q_1^{ - 1}T_{ij}^{86}P_1^{ - 1} = {\bar{T}}_{ij}^{86},\quad \rho Q_1^{ - 1}T_{ij}^{87}P_1^{ - 1} = {\bar{T}}_{ij}^{87},\quad \rho Q_1^{ - 1}T_{ij}^{88}P_1^{ - 1} = {\bar{T}}_{ij}^{88},\quad {\bar{G}} = \rho 2{G_1}(t), \end{array}$$
and multiplying both sides of the matrix inequalities (39), (40), and (41) in \(diag\left\{ {\rho^{{\frac{ - 1}{2}}} P_{1} (t),\;\rho^{{\frac{ - 1}{2}}} P_{1} (t),\;Q_{1} (t)\rho^{{\frac{ - 1}{2}}} ,\;Q_{1} (t)\rho^{{\frac{ - 1}{2}}} , \ldots ,\rho^{{\frac{ - 1}{2}}} I} \right\}\) and its transposes, we will have:
$$\left[ {\begin{array}{*{20}c} {\bar{\Upsilon }_{{11}} } & * \\ {\bar{\Upsilon }_{{21}} } & {\bar{\Upsilon }_{{22}} } \\ \end{array} } \right] < 0,$$
(56)
$$\left[ {\begin{array}{*{20}c} {\overline{F}_{11} } & * & * \\ {\overline{F}_{21} } & {\overline{F}_{22} } & * \\ {\overline{F}_{31} } & 0 & {\overline{F}_{33} } \\ \end{array} } \right] < 0,$$
(57)
$$\left[ {\begin{array}{*{20}c} {\overline{\Delta }_{11} } & * \\ {\overline{\Delta }_{21} } & {\overline{\Delta }_{22} } \\ \end{array} } \right] < 0,$$
(58)
where:
$$\overline{\Upsilon }_{11} = \left[ {\begin{array}{*{20}c} {\overline{\Upsilon }_{11}^{1} } & {(\overline{\Upsilon }_{11}^{2} )^{T} } \\ {\overline{\Upsilon }_{11}^{2} } & {\overline{\Upsilon }_{11}^{3} } \\ \end{array} } \right],$$
$$\begin{aligned} & \overline{\Upsilon }_{11}^{1} = \left[ {\begin{array}{*{20}c} {\overline{\Upsilon }_{11}^{11} } & * & * & * & * & * \\ {\overline{\lambda }_{i} T_{ii}^{21} } & {\overline{\lambda }_{i} {T}_{ii}^{22} + \lambda_{i} ( - \overline{S} - Q_{1} )} & * & * & * & * \\ {\overline{\lambda }_{i} T_{ii}^{31} - \lambda_{i} \sigma \theta } & {\overline{\lambda }_{i} T_{ii}^{32} } & {\overline{\Upsilon }_{11}^{13} } & * & * & * \\ {\overline{\lambda }_{i} T_{ii}^{41} } & {\overline{\lambda }_{i} T_{ii}^{42} } & {\overline{\lambda }_{i} T_{ii}^{43} } & {\overline{\lambda }_{i} T_{ii}^{44} - \overline{\lambda }_{i} \overline{N}} & * & * \\ {\overline{\lambda }_{i} T_{ii}^{51} - \sigma \lambda_{i} \theta } & {\overline{\lambda }_{i} T_{ii}^{52} } & {\overline{\lambda }_{i} T_{ii}^{53} - \lambda_{i} \sigma \theta } & {\overline{\lambda }_{i} T_{ii}^{54} } & {\overline{\lambda }_{i} T_{ii}^{55} + (\sigma - 1)\lambda_{i} \theta } & * \\ {\overline{\lambda }_{i} T_{ii}^{61} } & {\overline{\lambda }_{i} T_{ii}^{62} } & {\overline{\lambda }_{i} T_{ii}^{63} } & {\overline{\lambda }_{i} T_{ii}^{64} } & {\overline{\lambda }_{i} T_{ii}^{65} } & {\overline{\lambda }_{i} T_{ii}^{66} } \\ \end{array} } \right], \\ & \overline{\Upsilon }_{11}^{11} = \lambda_{i} ( - P_{1} + \overline{S} + \sigma \theta + g^{2} \Gamma^{T} C^{T} 3G_{1} C\Gamma ) + \overline{\lambda }_{i} T_{ii}^{11} ,\quad \overline{\Upsilon }_{11}^{13} = \overline{\lambda }_{i} T_{ii}^{33} + \lambda_{i} (\overline{N} + g^{2} \Gamma^{T} C^{T} 3G_{1} C\Gamma - \sigma \theta ), \\ \end{aligned}$$
$$\overline{\Upsilon }_{11}^{2} = \left[ {\begin{array}{*{20}c} {\overline{\lambda }_{i} T_{ii}^{61} } & {\overline{\lambda }_{i} T_{ii}^{62} } & {\overline{\lambda }_{i} T_{ii}^{63} } & {\overline{\lambda }_{i} T_{ii}^{64} } & {\overline{\lambda }_{i} T_{ii}^{65} } & {\overline{\lambda }_{i} T_{ii}^{66} } \\ {\overline{\lambda }_{i} T_{ii}^{71} } & {\overline{\lambda }_{i} T_{ii}^{72} } & {\overline{\lambda }_{i} T_{ii}^{73} } & {\overline{\lambda }_{i} T_{ii}^{74} } & {\overline{\lambda }_{i} T_{ii}^{75} } & {\overline{\lambda }_{i} T_{ii}^{76} } \\ \end{array} } \right],$$
$$\overline{\Upsilon }_{11}^{3} = \left[ {\begin{array}{*{20}c} {\overline{\lambda }_{i} T_{ii}^{67} + \overline{\lambda }_{i} 2G_{1} } & * \\ {\overline{\lambda }_{i} T_{ii}^{76} } & { - \overline{\lambda }_{i} \gamma^{2} - \overline{\lambda }_{i} T_{ii}^{77} } \\ \end{array} } \right],$$
$$\overline{\Upsilon }_{21} = \left[ {\begin{array}{*{20}c} {\sqrt {\lambda_{i} } A_{ui} } & {\sqrt {\lambda_{i} } (A_{di} + \overline{\nu }B_{i} K_{i} )} & 0 & { - \sqrt {\lambda_{i} } \overline{\nu }B_{i} K_{i} } & 0 & { - \sqrt {\lambda_{i} } \overline{\nu }B_{i} K_{i} } & {\sqrt {\lambda_{i} } B_{i} } & {\sqrt {\lambda_{i} } B_{wi} } \\ 0 & {\sqrt {\lambda_{i} } gB_{i} K_{i} } & 0 & { - \sqrt {\lambda_{i} } gB_{i} K_{i} } & 0 & { - \sqrt {\lambda_{i} } gB_{i} K_{i} } & 0 & 0 \\ {(\overline{\iota } + 1)\sqrt {\lambda_{i} } L_{i} C} & 0 & {\sqrt {\lambda_{i} } (A_{ui} - L_{i} C)} & {\sqrt {\lambda_{i} } A_{di} } & 0 & 0 & {\sqrt {\lambda_{i} } B_{i} } & {\sqrt {\lambda_{i} } B_{wi} } \\ {\sqrt {\lambda_{i} } lL_{i} C} & 0 & {\sqrt {\lambda_{i} } (A_{ui} - L_{i} C)} & {\sqrt {\lambda_{i} } A_{di} } & 0 & 0 & {\sqrt {\lambda_{i} } B_{i} } & {\sqrt {\lambda_{i} } B_{wi} } \\ \end{array} } \right],$$
$$\overline{\Upsilon }_{22} = diag\{ - P_{1}^{ - 1} ,\; - P_{1}^{ - 1} ,\; - I,\; - I\} ,$$
$$\overline{F}_{11} = \left[ {\begin{array}{*{20}c} {\overline{F}_{11}^{11} } & * & * & * & * & * & * & * \\ {\overline{F}_{11}^{21} } & {\overline{F}_{11}^{22} } & * & * & * & * & * & * \\ {\overline{F}_{11}^{31} } & {\overline{F}_{11}^{32} } & {\overline{F}_{11}^{33} } & * & * & * & * & * \\ {\overline{F}_{11}^{41} } & {\overline{F}_{11}^{42} } & {\overline{F}_{11}^{43} } & {\overline{F}_{11}^{44} } & * & * & * & * \\ {\overline{F}_{11}^{51} } & {\overline{F}_{11}^{52} } & {\overline{F}_{11}^{53} } & {\overline{F}_{11}^{54} } & {\overline{F}_{11}^{55} } & * & * & * \\ {\overline{F}_{11}^{61} } & {\overline{F}_{11}^{62} } & {\overline{F}_{11}^{63} } & {\overline{F}_{11}^{64} } & {\overline{F}_{11}^{65} } & {\overline{F}_{11}^{66} } & * & * \\ {\overline{F}_{11}^{71} } & {\overline{F}_{11}^{72} } & {\overline{F}_{11}^{73} } & {\overline{F}_{11}^{74} } & {\overline{F}_{11}^{75} } & {\overline{F}_{11}^{76} } & {\overline{F}_{11}^{77} } & * \\ {\overline{F}_{11}^{81} } & {\overline{F}_{11}^{82} } & {\overline{F}_{11}^{83} } & {\overline{F}_{11}^{84} } & {\overline{F}_{11}^{85} } & {\overline{F}_{11}^{86} } & {\overline{F}_{11}^{87} } & {\overline{F}_{11}^{88} } \\ \end{array} } \right],$$
$$\begin{aligned} & \overline{F}_{11}^{11} = \lambda_{ij} ( - P_{1} + \sigma \theta + \overline{S} + g^{2} \Gamma^{T} C^{T} 3G_{1} C\Gamma ) + \overline{\lambda }_{j} T_{ij}^{11} + \overline{\lambda }_{i} T_{ji}^{11} ,\quad \overline{F}_{11}^{21} = - \lambda_{ij} \sigma \theta + \overline{\lambda }_{j} T_{ij}^{21} + \overline{\lambda }_{i} T_{ji}^{21} , \\ & \overline{F}_{11}^{22} = \lambda_{ij} ( - Q_{1} - \overline{S}) + \overline{\lambda }_{j} T_{ij}^{22} + \overline{\lambda }_{i} T_{ji}^{22} ,\quad \overline{F}_{11}^{31} = - \lambda_{ij} \sigma \theta +
\overline{\lambda }_{j} T_{ij}^{31} + \overline{\lambda }_{i} T_{ji}^{31} , \\ & \overline{F}_{11}^{32} = \overline{\lambda }_{j} T_{ij}^{32} + \overline{\lambda }_{i} T_{ji}^{32} ,\quad \overline{F}_{11}^{33} = \overline{\lambda }_{j} T_{ij}^{33} + \overline{\lambda }_{i} T_{ji}^{33} + \lambda_{ij} (\overline{N} + g^{2} \Gamma^{T} C^{T} 3G_{1} C\Gamma - \sigma \theta ), \\ & \overline{F}_{11}^{41} = \overline{\lambda }_{j} T_{ij}^{41} + \overline{\lambda }_{i} T_{ji}^{41} ,\quad \overline{F}_{11}^{42} = \overline{\lambda }_{j} T_{ij}^{42} + \overline{\lambda }_{i} T_{ji}^{42} ,\quad \overline{F}_{11}^{43} = \overline{\lambda }_{j} T_{ij}^{43} + \overline{\lambda }_{i} T_{ji}^{43} ,\quad \overline{F}_{11}^{44} = \overline{\lambda }_{j} T_{ij}^{44} + \overline{\lambda }_{i} T_{ji}^{44} - \lambda_{ij} \overline{N}, \\ & \overline{F}_{11}^{51} = \overline{\lambda }_{j} \overline{T}_{ij}^{51} + \overline{\lambda }_{i} \overline{T}_{ji}^{51} - (\sigma \theta )\lambda_{ij} ,\quad \overline{F}_{11}^{52} = \overline{\lambda }_{j} T_{ij}^{52} + \overline{\lambda }_{i} T_{ji}^{52} ,\quad \overline{F}_{11}^{53} = \overline{\lambda }_{j} T_{ij}^{53} + \overline{\lambda }_{i} T_{ji}^{53} - (\sigma \theta )\lambda_{ij} , \\ & \overline{F}_{11}^{54} = \overline{\lambda }_{j} T_{ij}^{54} + \overline{\lambda }_{i} T_{ji}^{54} ,\quad \overline{F}_{11}^{55} = \overline{\lambda }_{j} T_{ij}^{55} + \overline{\lambda }_{i} T_{ji}^{55} + \lambda_{ij} (\sigma - 1)\theta ,\quad \overline{F}_{11}^{61} = \overline{\lambda }_{j} T_{ij}^{61} + \overline{\lambda }_{i} T_{ji}^{61} ,\quad \overline{F}_{11}^{62} = \overline{\lambda }_{j} T_{ij}^{62} + \overline{\lambda }_{i} T_{ji}^{62} , \\ \end{aligned}$$
$$\begin{aligned} & \overline{F}_{11}^{63} = \overline{\lambda }_{j} T_{ij}^{63} + \overline{\lambda }_{i} T_{ji}^{63} ,\quad \overline{F}_{11}^{64} = \overline{\lambda }_{j} T_{ij}^{64} + \overline{\lambda }_{i} T_{ji}^{64} ,\quad \overline{F}_{11}^{65} = \overline{\lambda }_{j} T_{ij}^{65} + \overline{\lambda }_{i} T_{ji}^{65} ,\quad \overline{F}_{11}^{66} = \overline{\lambda }_{j} T_{ij}^{66} + \overline{\lambda }_{i} T_{ji}^{66} \\ & \overline{F}_{11}^{71} = \overline{\lambda }_{j} T_{ij}^{71} + \overline{\lambda }_{i} \overline{T}_{ji}^{71} ,\quad \overline{F}_{11}^{72} = \overline{\lambda }_{j} T_{ij}^{72} + \overline{\lambda }_{i} T_{ji}^{72} ,\quad \overline{F}_{11}^{73} = \overline{\lambda }_{j} T_{ij}^{73} + \overline{\lambda }_{i} T_{ji}^{73} ,\quad \overline{F}_{11}^{74} = \overline{\lambda }_{j} T_{ij}^{74} + \overline{\lambda }_{i} T_{ji}^{74} , \\ & \overline{F}_{11}^{75} = \overline{\lambda }_{j} T_{ij}^{75} + \overline{\lambda }_{i} T_{ji}^{75} ,\quad \overline{F}_{11}^{76} = \overline{\lambda }_{j} T_{ij}^{76} + \overline{\lambda }_{i} T_{ji}^{76} ,\quad \overline{F}_{11}^{77} = \overline{\lambda }_{j} T_{ij}^{77} + \overline{\lambda }_{i} T_{ji}^{77} + \lambda_{ij} 2G_{1} , \\ & \overline{F}_{11}^{81} = \overline{\lambda }_{j} T_{ij}^{81} + \overline{\lambda }_{i} \overline{T}_{ji}^{81} ,\quad \overline{F}_{11}^{82} = \overline{\lambda }_{j} T_{ij}^{82} + \overline{\lambda }_{i} T_{ji}^{82} ,\quad \overline{F}_{11}^{83} = \overline{\lambda }_{j} T_{ij}^{83} + \overline{\lambda }_{i} T_{ji}^{83} ,\quad \overline{F}_{11}^{84} = \overline{\lambda }_{j} T_{ij}^{84} + \overline{\lambda }_{i} T_{ji}^{84} , \\ & \overline{F}_{11}^{85} = \overline{\lambda }_{j} T_{ij}^{85} + \overline{\lambda }_{i} T_{ji}^{85} ,\quad \overline{F}_{11}^{86} = \overline{\lambda }_{j} T_{ij}^{86} + \overline{\lambda }_{i} T_{ji}^{86} ,\quad \overline{F}_{11}^{87} = \overline{\lambda }_{j} T_{ij}^{87} + \overline{\lambda }_{i} T_{ji}^{87} ,\quad \overline{F}_{11}^{88} = \overline{\lambda }_{j} T_{ij}^{88} + \overline{\lambda }_{i} T_{ji}^{88} - \lambda_{ij} \gamma^{2} , \\ \end{aligned}$$
$$\overline{F}_{21} = \left[ {\begin{array}{*{20}c} {\sqrt {\lambda_{j} } A_{ui} } & {\sqrt {\lambda_{j} } (A_{di} + \overline{\nu }B_{i} K_{j} )} & 0 & { - \sqrt {\lambda_{j} } \overline{\nu }B_{i} K_{j} } & 0 & { - \sqrt {\lambda_{j} } \overline{\nu }B_{i} K_{j} } & {\sqrt {\lambda_{j} } B_{i} } & {\sqrt {\lambda_{j} } B_{wi} } \\ 0 & {\sqrt {\lambda_{j} } gB_{i} K_{j} } & 0 & { - \sqrt {\lambda_{j} } gB_{i} K_{j} } & 0 & { - \sqrt {\lambda_{j} } gB_{i} K_{j} } & 0 & 0 \\ {(\overline{\iota } + 1)\sqrt {\lambda_{j} } L_{j} C} & 0 & {\sqrt {\lambda_{j} } (A_{ui} - L_{j} C)} & {\sqrt {\lambda_{j} } A_{di} } & 0 & 0 & {\sqrt {\lambda_{j} } B_{i} } & {\sqrt {\lambda_{j} } B_{wi} } \\ {\sqrt {\lambda_{j} } lL_{j} C} & 0 & {\sqrt {\lambda_{j} } (A_{ui} - L_{j} C)} & {\sqrt {\lambda_{j} } A_{di} } & 0 & 0 & {\sqrt {\lambda_{j} } B_{i} } & {\sqrt {\lambda_{j} } B_{wi} } \\ \end{array} } \right],$$
$$\overline{F}_{22} = diag\{ - P_{1}^{ - 1} ,\; - P_{1}^{ - 1} ,\; - I,\; - I\} ,$$
$$\overline{F}_{31} = \left[ {\begin{array}{*{20}c} {\sqrt {\lambda_{i} } A_{uj} } & {\sqrt {\lambda_{i} } (A_{dj} + \overline{\nu }B_{j} K_{i} )} & 0 & { - \sqrt {\lambda_{i} } \overline{\nu }B_{j} K_{i} } & 0 & { - \sqrt {\lambda_{i} } \overline{\nu }B_{j} K_{i} } & {\sqrt {\lambda_{i} } B_{j} } & {\sqrt {\lambda_{i} } B_{wj} } \\ 0 & {\sqrt {\lambda_{i} } gB_{j} K_{i} } & 0 & { - \sqrt {\lambda_{i} } gB_{j} K_{i} } & 0 & { - \sqrt {\lambda_{i} } gB_{j} K_{i} } & 0 & 0 \\ {(\overline{\iota } + 1)\sqrt {\lambda_{i} } L_{i} C} & 0 & {\sqrt {\lambda_{i} } (A_{uj} - L_{i} C)} & {\sqrt {\lambda_{i} } A_{dj} } & 0 & 0 & {\sqrt {\lambda_{i} } B_{j} } & {\sqrt {\lambda_{i} } B_{wj} } \\ {\sqrt {\lambda_{i} } lL_{i} C} & 0 & {\sqrt {\lambda_{i} } (A_{uj} - L_{i} C)} & {\sqrt {\lambda_{i} } A_{dj} } & 0 & 0 & {\sqrt {\lambda_{i} } B_{j} } & {\sqrt {\lambda_{i} } B_{wj} } \\ \end{array} } \right],$$
$$\overline{F}_{33} = diag\{ - P_{1}^{ - 1} ,\; - P_{1}^{ - 1} ,\; - I,\; - I\} ,$$
$$\begin{aligned} & \overline{\Delta }_{11} = \left[ {\begin{array}{*{20}c} {\overline{\Delta }_{11}^{11} } & * & * & * & * & * & * & * \\ { - T_{ij}^{21} } & {(Q_{1} - \overline{S}) - T_{ij}^{22} } & * & * & * & * & * & * \\ { - T_{ij}^{31} - \sigma \theta } & { - T_{ij}^{32} } & {\overline{\Delta }_{11}^{33} } & * & * & * & * & * \\ { - T_{ij}^{41} } & { - T_{ij}^{42} } & { - T_{ij}^{43} } & { - T_{ij}^{44} - \overline{N}} & * & * & * & * \\ {\begin{array}{*{20}c} { - T_{ij}^{51} - \sigma \theta } \\ { - T_{ij}^{61} } \\ \end{array} } & {\begin{array}{*{20}c} { - T_{ij}^{52} } \\ { - T_{ij}^{62} } \\ \end{array} } & {\begin{array}{*{20}c} { - T_{ij}^{53} - \sigma \theta } \\ { - T_{ij}^{63} } \\ \end{array} } & {\begin{array}{*{20}c} { - T_{ij}^{54} } \\ { - T_{ij}^{64} } \\ \end{array} } & {\begin{array}{*{20}c} { - T_{ij}^{55} + (\sigma - 1)\theta } \\ { - T_{ij}^{65} } \\ \end{array} } & {\begin{array}{*{20}c} * \\ { - T_{ij}^{66} } \\ \end{array} } & {\begin{array}{*{20}c} * \\ * \\ \end{array} } & {\begin{array}{*{20}c} * \\ * \\ \end{array} } \\ { - T_{ij}^{71} } & { - T_{ij}^{72} } & { - T_{ij}^{73} } & { - T_{ij}^{74} } & { - T_{ij}^{75} } & { - T_{ij}^{76} } & { - T_{ij}^{77} + 2G_{1} } & * \\ { - T_{ij}^{81} } & { - T_{ij}^{82} } & { - T_{ij}^{83} } & { - T_{ij}^{84} } & { - T_{ij}^{85} } & { - T_{ij}^{86} } & { - T_{ij}^{87} } & { - \gamma^{2} - T_{ij}^{88} } \\ \end{array} } \right], \\ & \overline{\Delta }_{11}^{33} = - T_{ij}^{33} + \overline{N} + g^{2} \Gamma^{T} C^{T} 3G_{1} C\Gamma - \sigma \theta ,\quad \overline{\Delta }_{11}^{11} = ( - P_{1} + \sigma \theta + \overline{S} + g^{2} \Gamma^{T} C^{T} 3G_{1} C\Gamma ) - T_{ij}^{11} , \\ \end{aligned}$$
$$\overline{\Delta }_{21} = \left[ {\begin{array}{*{20}c} {A_{ui} } & {(A_{di} + \overline{\nu }B_{i} K_{j} )} & 0 & { - \overline{\nu }B_{i} K_{j} } & 0 & { - \overline{\nu }B_{i} K_{j} } & {B_{i} } & {B_{wi} } \\ 0 & {gB_{i} K_{j} } & 0 & { - gB_{i} K_{j} } & 0 & { - gB_{i} K_{j} } & 0 & 0 \\ {(\overline{\iota } + 1)L_{j} C} & 0 & {(A_{ui} - L_{j} C)} & {A_{di} } & 0 & 0 & {B_{i} } & {B_{wi} } \\ {lL_{j} C} & 0 & {(A_{ui} - L_{j} C)} & {A_{di} } & 0 & 0 & {B_{i} } & {B_{wi} } \\ \end{array} } \right],$$
$$\overline{\Delta }_{22} = diag\{ - P_{1}^{ - 1} ,\; - P_{1}^{ - 1} ,\; - I,\; - I\} .$$
Using the Schur complement theory and Eqs. (56)–(58), the following equations are extracted:
$$\lambda_{i} \overline{\Xi }_{ii} - \lambda_{i} \Lambda_{ii} + \Lambda_{ii} < 0;\quad i = 1, \ldots ,p,$$
(59)
$$- \lambda_{j} \Lambda_{ij} + \Lambda_{ij} + \lambda_{i} \overline{\Xi }_{ji} - \lambda_{i} \Lambda_{ji} + \Lambda_{ji} + \lambda_{j} \overline{\Xi }_{ij} < 0;\quad 1 \le i < j < p,$$
(60)
$$- \Lambda_{ij} + \overline{\Xi }_{ij} < 0;\quad i,\;j = 1, \ldots ,p.$$
(61)
Therefore, under the condition \(m_{j} - \lambda_{j} \omega_{i} \ge 0,\) from Remark 1, the inequality \(\sum\nolimits_{i = 1}^{p} {\sum\nolimits_{j = 1}^{p} {\omega_{i} m_{j} } } \Xi_{ij} < 0\) is satisfied. Hence, if \(\left\| {\xi (\left. {t + s} \right|t)} \right\|^{2} \ge (\lambda_{\min } (\overline{\Xi }_{ij} ))^{ - 1} \delta\) the following inequality is established:
$$\begin{aligned} \Delta \overline{V} & = \Delta V + E\{ y^{T} (\left. {t + s} \right|t)y(\left. {t + s} \right|t)\} - \gamma^{T} w^{T} (\left. {t + s} \right|t)w(\left. {t + s} \right|t) \\ & \le \left\{ {\sum\limits_{i = 1}^{p} {\sum\limits_{j = 1}^{p} {\xi^{T} (\left. {t + s} \right|t)\omega_{i} m_{j} E\{ \overline{\Xi }_{ij} \} \xi (\left. {t + s} \right|t)} } } \right\} + \delta \\ & \le - \lambda_{\min } ( - \overline{\Xi }_{ij} )\left\| {\xi (\left. {t + s} \right|t)} \right\|^{2} + \delta \le 0. \\ \end{aligned}$$
(62)
Summing up both sides of the inequality (62) from \(t = 0\) to \(t = \infty\) under zero initial condition and considering \(E\{ x_{\infty }^{T} (t)Q_{1} (t)x_{\infty } (t)\} \ge 0\), \(E\{ (e_{x} (t))_{\infty }^{T} Q_{1} (t)(e_{x} (t))_{\infty } \} \ge 0\), and \(E\{ (e_{f} (t))_{\infty }^{T} Q_{1} (t)(e_{f} (t))_{\infty } \} \ge 0\), we will have:
$$E\left\{ {\sum\limits_{t = 0}^{\infty } {y(t + \left. s \right|t)^{T} (t + \left. s \right|t)} } \right\} \le \sum\limits_{t = 0}^{\infty } {\gamma^{2} w^{T} (t + \left. s \right|t)w(t + \left. s \right|t)} .$$
This expression shows satisfying the \({\mathcal{H}}_{\infty }\) performance index defined in (15). Moreover, considering inequality (42), and employing the Schur complement theory, and assuming that disturbance is norm-bounded (Assumption 1), the following inequality will be extracted:
$$E\left\{ {\sum\limits_{t = 0}^{\infty } {y(t + \left. s \right|t)^{T} (t + \left. s \right|t)} } \right\} \le x^{T} (t)P(t)x(t) + \gamma^{2} \overline{w } \le \rho (t).$$
Therefore, the \({\mathcal{H}}_{2}\) performance index described in (16) is established.
By pre- and post-multiplying the inequality (43) in \(diag\left\{ {\rho^{{ - \frac{1}{2}}} P(t),\;\rho^{{ - \frac{1}{2}}} P(t)} \right\}\) and its transpose, respectively, we will have:
$$\left[ {\begin{array}{*{20}c} { - Q^{ - 1} } & * \\ {\sqrt {\overline{\nu }} K_{i} } & { - u_{\max }^{2} I} \\ \end{array} } \right] < 0,$$
(63)
using the Schur complement theory, the inequality (63) is written as:
$$- \rho^{ - 1} P(t) + \frac{{\overline{\nu }}}{{u_{\max }^{2} }}K^{T} (t)K(t) \le 0.$$
(64)
By multiplying the right side in \(x(\left. {t + s} \right|t)\) and multiplying the left side in its transpose, the following inequality will yield:
$$- x^{T} (\left. {t + s} \right|t)\rho^{ - 1} P(t)x(\left. {t + s} \right|t) + \frac{{\overline{\nu }}}{{u_{\max }^{2} }}((K(t)x(\left. {t + s} \right|t))^{T} (K(t)x(\left. {t + s} \right|t)) \le 0.$$
(65)
Using the control law (31), we will have:
$$\frac{{\overline{\nu }}}{{u_{\max }^{2} }}(u^{T} (\left. {t + s} \right|t)u(\left. {t + s} \right|t)) \le x^{T} (\left. {t + s} \right|t)\rho^{ - 1} P(t)x(\left. {t + s} \right|t) \le 1,$$
(66)
that yields the following expression:
$$\overline{\nu }(u^{T} (\left. {t + s} \right|t)u(\left. {t + s} \right|t)) \le u_{\max }^{2} ,$$
(67)
where the input constraint (17) is satisfied.
Now, to prove the stochastic stability, we use \(\Delta \overline{V} \le 0\) obtained in (61) and consider \(w(\left. {t + s} \right|t) = 0.\) Therefore, we have:
$$E\{ \Delta V\} = E\{ \xi^{T} (\left. {t + s} \right|t)\overline{\Xi }_{ij} \xi (\left. {t + s} \right|t)\} + \delta \le 0,$$
(68)
which is rewritten as:
$$E\{ \Delta V\} \le - \lambda_{\min } ( - \overline{\Xi }_{ij} )\xi^{T} (\left. {t + s} \right|t)\xi (\left. {t + s} \right|t) + \delta .$$
(69)
Therefore, based on the definition of the vector \(\xi (\left. {t + s} \right|t),\) we have:
$$E\{ \Delta V\} \le - \lambda_{\min } ( - \overline{\Xi }_{ij} )x^{T} (\left. {t + s} \right|t)x(\left. {t + s} \right|t).$$
(70)
Calculating the mathematical expectation of both sides of the above inequality and summing up both sides between \(t = 0\) and \(t = d\) for each \(d \ge 0,\) we will have:
$$E\{ V(x(\left. {t + s + d} \right|t))\} - V(x(0)) \le - \lambda_{\min } ( - \overline{\Xi }_{ij} )E\left\{ {\sum\limits_{t = 0}^{d} {x^{T} (\left. {t + s} \right|t)x(\left. {t + s} \right|t)} } \right\},$$
(71)
which is equivalent to:
$$E\left\{ {\sum\limits_{t = 0}^{d} {\left\| {x(\left. {t + s} \right|t)} \right\|}^{2} } \right\} \le (\lambda_{\min } ( - \overline{\Xi }_{ij} ))^{ - 1} (V(x(0)) - E\{ V(x(\left. {t + s + d} \right|t))\} ).$$
(72)
The above equation, \(x(0)\) indicates the initial condition. Considering \(E\{ x_{\infty }^{T} (t)P(t)x_{\infty } (t)\} \ge 0\) we have:
$$E\left\{ {\sum\limits_{t = 0}^{d} {\left\| {x(\left. {t + s} \right|t)} \right\|}^{2} \left| {x(0)} \right.} \right\} \le x^{T} (0)Yx(0).$$
(73)
Hence, based on Definition 1, for the bounded matrix \(Y = (\lambda_{\min } ( - \overline{\Xi }_{ij} ))^{ - 1} P(t) > 0,\) the condition (73) is satisfied that leads to the stochastic stability of the closed loop system ■
