Appendix 1
Proof of Theorem 1
Consider the Lyapunov–Krasovskii functional candidate for the system (10) as
$$\begin{aligned} V(t,e_{t},i)=\sum _{n=1}^{7} V_{n}(t,e_{t},i), \end{aligned}$$
(26)
where
$$\begin{aligned} V_{1}(t,e_{t},i)&= e_{m}^{T}(t)P_{1i}e_{m}(t)+e_{p}^{T}(t)P_{2i}e_{p}(t),\\ V_{2}(t,e_{t},i)&= \int \limits _{t-\rho _{i}}^{t}e_{m}^{T}(s)Q_{1}e_{m}(s)\mathrm{d}s+\int \limits _{t-h_{i}}^{t}e_{p}^{T}(s)Q_{2}e_{p}(s)\mathrm{d}s\\&+ \, \bar{\pi }\int \limits _{-\rho }^{0}\int \limits _{t+\theta }^{t}e_{m}^{T}(s)Q_{1}e_{m}(s)\mathrm{d}s\mathrm{d}\theta +\bar{\pi }\int \limits _{-h}^{0}\int \limits _{t+\theta }^{t}e_{p}^{T}(s)Q_{2}e_{p}(s)\mathrm{d}s\mathrm{d}\theta , \\ V_{3}(t,e_{t},i)&= \int \limits _{-\rho _{i}}^{0}\int \limits _{t+\theta }^{t}e_{m}^{T}(s)Q_{3}e_{m}(s)\mathrm{d}s\mathrm{d}\theta +\int \limits _{-h_{i}}^{0}\int \limits _{t+\theta }^{t}e_{p}^{T}(s)Q_{4}e_{p}(s)\mathrm{d}s\mathrm{d}\theta \\&+ \, \bar{\pi }\int \limits _{-\rho }^{0}\int \limits _{\theta }^{0}\int \limits _{t+\uplambda }^{t}e_{m}^{T}(s)Q_{3}e_{m}(s)\mathrm{d}s\mathrm{d}\uplambda \mathrm{d}\theta \\&+ \, \bar{\pi }\int \limits _{-h}^{0}\int \limits _{\theta }^{0}\int \limits _{t+\uplambda }^{t}e_{p}^{T}(s)Q_{4}e_{p}(s)\mathrm{d}s\mathrm{d}\uplambda \mathrm{d}\theta ,\\ V_{4}(t,e_{t},i)&= \int \limits _{t-\alpha \tau _{i}}^{t}e_{m}^{T}(s)K^{T}R_{1}Ke_{m}(s)\mathrm{d}s +\int \limits _{t-\tau _{i}}^{t-\alpha \tau _{i}}e_{m}^{T}(s)K^{T}R_{2}Ke_{m}(s)\mathrm{d}s\\&+ \, \int \limits _{t-\beta \sigma _{i}}^{t}e_{p}^{T}(s)K^{T}R_{3}Ke_{p}(s)\mathrm{d}s +\int \limits _{t-\sigma _{i}}^{t-\beta \sigma _{i}}e_{p}^{T}(s)K^{T}R_{4}Ke_{p}(s)\mathrm{d}s\\&+ \, \int \limits _{t-\tau _{i}(t)}^{t}e_{m}^{T}(s)K^{T}R_{5}Ke_{m}(s)\mathrm{d}s +\int \limits _{t-\sigma _{i}(t)}^{t}e_{p}^{T}(s)K^{T}R_{6}Ke_{p}(s)\mathrm{d}s,\\ V_{5}(t,e_{t},i)&= \bar{\pi }\int \limits _{-\alpha \tau }^{0}\int \limits _{t+\theta }^{t}e_{m}^{T}(s)K^{T}R_{1}Ke_{m}(s)\mathrm{d}s\mathrm{d}\theta +\bar{\pi }\int \limits _{-\tau }^{0}\int \limits _{t+\theta }^{t}e_{m}^{T}(s)K^{T}R_{2}Ke_{m}(s)\mathrm{d}s\mathrm{d}\theta \\&+ \, \bar{\pi }\int \limits _{-\beta \sigma }^{0}\int \limits _{t+\theta }^{t}e_{p}^{T}(s)K^{T}R_{3}Ke_{p}(s)\mathrm{d}s\mathrm{d}\theta +\bar{\pi }\int \limits _{-\sigma }^{0}\int \limits _{t+\theta }^{t}e_{p}^{T}(s)K^{T}R_{4}Ke_{p}(s)\mathrm{d}s\mathrm{d}\theta \\&+ \, \bar{\pi }\int \limits _{-\tau }^{0}\int \limits _{t+\theta }^{t}e_{m}^{T}(s)K^{T}R_{5}Ke_{m}(s)\mathrm{d}s\mathrm{d}\theta +\bar{\pi }\int \limits _{-\sigma }^{0}\int \limits _{t+\theta }^{t}e_{p}^{T}(s)K^{T}R_{6}Ke_{p}(s)\mathrm{d}s\mathrm{d}\theta , \\ V_{6}(t,e_{t},i)&= \tau \int \limits _{-\tau }^{0}\int \limits _{t+\theta }^{t}\dot{e}_{m}^{T}(s)K^{T}S_{1}K\dot{e}_{m}(s)\mathrm{d}s\mathrm{d}\theta +\sigma \int \limits _{-\sigma }^{0}\int \limits _{t+\theta }^{t}\dot{e}_{p}^{T}(s)K^{T}S_{2}K\dot{e}_{p}(s)\mathrm{d}s\mathrm{d}\theta ,\\ V_{7}(t,e_{t},i)&= \int \limits _{t-\sigma _{i}(t)}^{t}g^{T}(Ke_{p}(s))T_{1}g(Ke_{p}(s))\mathrm{d}s +\int \limits _{t-\sigma _{i}}^{t}g^{T}(Ke_{p}(s))T_{2}g(Ke_{p}(s))\mathrm{d}s\\&+ \, \bar{\pi }\int \limits _{-\sigma }^{0}\int \limits _{t+\theta }^{t}g^{T}(Ke_{p}(s))(T_{1}+T_{2})g(Ke_{p}(s))\mathrm{d}s\mathrm{d}\theta . \end{aligned}$$
Differentiating (26) stochastically using It\(\hat{o}\)’s formula, one can get
$$\begin{aligned} \mathrm{d}V(t,e_{t},i)&= \sum _{n=1}^{7}\mathcal {L}V_{n}(t,e_{t},i)\mathrm{d}t+2e_{m}^{T}(t)P_{1i}\bar{E}_{i}Ke_{m}(t)\mathrm{d}\omega _{1}(t) \nonumber \\&+ \, 2e_{p}^{T}(t)P_{2i}\bar{F}_{i}Ke_{p}(t)\mathrm{d}\omega _{2}(t), \end{aligned}$$
(27)
where
$$\begin{aligned} \mathcal {L}V_{1}(t,e_{t},i)&= 2e_{m}^{T}(t)P_{1i}[-\bar{A}_{i}e_{m}(t-\rho _{i})+\bar{B}_{i}g(Ke_{p}(t-\sigma _{i}(t)))] \nonumber \\&+ \, e_{m}^{T}(t)\Big [\sum _{j=1}^{N}\pi _{ij}P_{1j}\Big ] e_{m}(t)+e_{m}^{T}(t)K^{T}\bar{E}_{i}^{T}P_{1i}\bar{E}_{i}Ke_{m}(t)+2e_{p}^{T}(t)P_{2i}\nonumber \\&\times [-\bar{C}_{i}e_{p}(t-h_{i}) +\bar{D}_{i}Ke_{m}(t-\tau _{i}(t))]+e_{p}^{T}(t)\Big [\sum _{j=1}^{N}\pi _{ij}P_{2j}\Big ]e_{p}(t)\nonumber \\&+ \, e_{p}^{T}(t)K^{T}\bar{F}_{i}^{T}P_{2i}\bar{F}_{i}Ke_{p}(t), \end{aligned}$$
(28)
$$\begin{aligned} \mathcal {L}V_{2}(t,e_{t},i)&= e_{m}^{T}(t)Q_{1}e_{m}(t)-e_{m}^{T}(t-\rho _{i})Q_{1}e_{m}(t-\rho _{i}) +e_{p}^{T}(t)Q_{2}e_{p}(t)-e_{p}^{T}(t-h_{i})\nonumber \\&\times Q_{2}e_{p}(t-h_{i}) +\sum _{j=1}^{N}\pi _{ij}\int \limits _{t-\rho _{j}}^{t}e_{m}^{T}(s)Q_{1}e_{m}(s)\mathrm{d}s\nonumber \\&+ \, \sum _{j=1}^{N}\pi _{ij}\int \limits _{t-h_{j}}^{t}e_{p}^{T}(s)Q_{2}e_{p}(s)\mathrm{d}s +\bar{\pi }\rho e_{m}^{T}(t)Q_{1}e_{m}(t)\nonumber \\&- \, \bar{\pi }\int \limits _{t-\rho }^{t}e_{m}^{T}(s)Q_{1}e_{m}(s)\mathrm{d}s +\bar{\pi }h e_{p}^{T}(t)Q_{2}e_{p}(t) -\bar{\pi }\int \limits _{t-h}^{t}e_{p}^{T}(s)Q_{2}e_{p}(s)\mathrm{d}s\nonumber \\&\le e_{m}^{T}(t)Q_{1}e_{m}(t)-e_{m}^{T}(t-\rho _{i})Q_{1}e_{m}(t-\rho _{i}) +e_{p}^{T}(t)Q_{2}e_{p}(t)\nonumber \\&- \, e_{p}^{T}(t-h_{i})Q_{2}e_{p}(t-h_{i})+\bar{\pi }\rho e_{m}^{T}(t)Q_{1}e_{m}(t)+\bar{\pi }h e_{p}^{T}(t)Q_{2}e_{p}(t),\nonumber \\ \end{aligned}$$
(29)
$$\begin{aligned} \mathcal {L}V_{3}(t,e_{t},i)&= \rho _{i}e_{m}^{T}(t)Q_{3}e_{m}(t) -\int \limits _{t-\rho _{i}}^{t}e_{m}^{T}(s)Q_{3}e_{m}(s)\mathrm{d}s+h_{i}e_{p}^{T}(t)Q_{4}e_{p}(t)\nonumber \\&- \, \int \limits _{t-h_{i}}^{t}e_{p}^{T}(s)Q_{4}e_{p}(s)\mathrm{d}s +\bar{\pi }\frac{\rho ^{2}}{2}e_{m}^{T}(t)Q_{3}e_{m}(t)\nonumber \\&- \, \bar{\pi }\int \limits _{-\rho }^{0}\int \limits _{t+\theta }^{t}e_{m}^{T}(s)Q_{3}e_{m}(s)\mathrm{d}s\mathrm{d}\theta +\bar{\pi }\frac{h^{2}}{2}e_{p}^{T}(t)Q_{4}e_{p}(t)\nonumber \\&- \, \bar{\pi }\int \limits _{-h}^{0}\int \limits _{t+\theta }^{t}e_{p}^{T}(s)Q_{4}e_{p}(s)\mathrm{d}s\mathrm{d}\theta +\sum _{j=1}^{N}\pi _{ij}\int \limits _{-\rho _{j}}^{0}\int \limits _{t+\theta }^{t}e_{m}^{T}(s)Q_{3}e_{m}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&+ \, \sum _{j=1}^{N}\pi _{ij}\int \limits _{-h_{j}}^{0}\int \limits _{t+\theta }^{t}e_{p}^{T}(s)Q_{4}e_{p}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\le e_{m}^{T}(t)(\rho _{i}+\bar{\pi }\frac{\rho ^{2}}{2})Q_{3}e_{m}(t) -\frac{1}{\rho _{i}}\Big [\int \limits _{t-\rho _{i}}^{t}e_{m}^{T}(s)\mathrm{d}s\Big ]Q_{3}\Big [\int \limits _{t-\rho _{i}}^{t}e_{m}(s)\mathrm{d}s\Big ] \nonumber \\&+ \, e_{p}^{T}(t)(h_{i}+\bar{\pi }\frac{h^{2}}{2})Q_{4}e_{p}(t) -\frac{1}{h_{i}}\Big [\int \limits _{t-h_{i}}^{t}e_{p}^{T}(s)\mathrm{d}s\Big ]Q_{4}\Big [\int \limits _{t-h_{i}}^{t}e_{p}(s)\mathrm{d}s\Big ], \nonumber \\ \end{aligned}$$
(30)
$$\begin{aligned} \mathcal {L}V_{4}(t,e_{t},i)&\le e_{m}^{T}(t)K^{T}R_{1}Ke_{m}(t)-e_{m}^{T}(t-\alpha \tau _{i})K^{T}R_{1}Ke_{m}(t-\alpha \tau _{i}) \nonumber \\&+ \, \sum _{j=1}^{N}\pi _{ij}\int \limits _{t-\alpha \tau _{j}}^{t}e_{m}^{T}(s)K^{T}R_{1}Ke_{m}(s)\mathrm{d}s +e_{m}^{T}(t-\alpha \tau _{i})K^{T}R_{2}Ke_{m}(t-\alpha \tau _{i}) \nonumber \\&- \, e_{m}^{T}(t-\tau _{i})K^{T}R_{2}Ke_{m}(t-\tau _{i}) +\sum _{j=1}^{N}\pi _{ij}\int \limits _{t-\tau _{j}}^{t-\alpha \tau _{j}}e_{m}^{T}(s)K^{T}R_{2}Ke_{m}(s)\mathrm{d}s \nonumber \\&+ \, e_{p}^{T}(t)K^{T}R_{3}Ke_{p}(t)-e_{p}^{T}(t-\beta \sigma _{i})K^{T}R_{3}Ke_{p}(t-\beta \sigma _{i}) \nonumber \\&+ \, \sum _{j=1}^{N}\pi _{ij}\int \limits _{t-\beta \sigma _{j}}^{t}e_{p}^{T}(s)K^{T}R_{3}Ke_{p}(s)\mathrm{d}s +e_{p}^{T}(t-\beta \sigma _{i})K^{T}R_{4}Ke_{p}(t-\beta \sigma _{i})\nonumber \\&- \, e_{p}^{T}(t-\sigma _{i})K^{T}R_{4}Ke_{p}(t-\sigma _{i}) +\sum _{j=1}^{N}\pi _{ij}\int \limits _{t-\sigma _{j}}^{t-\beta \sigma _{j}}e_{p}^{T}(s)K^{T}R_{4}Ke_{p}(s)\mathrm{d}s \nonumber \\&+ \, e_{m}^{T}(t)K^{T}R_{5}Ke_{m}(t)-(1-\mu _{1i})e_{m}^{T}(t-\tau _{i}(t))K^{T}R_{5}Ke_{m}(t-\tau _{i}(t))\nonumber \\&+ \, \sum _{j=1}^{N}\pi _{ij}\int \limits _{t-\tau _{j}(t)}^{t}e_{m}^{T}(s)K^{T}R_{5}Ke_{m}(s)\mathrm{d}s+e_{p}^{T}(t)K^{T}R_{6}Ke_{p}(t) -(1-\mu _{2i})\nonumber \\&\,\, \times e_{p}^{T}(t-\sigma _{i}(t))K^{T}R_{6}Ke_{p}(t-\sigma _{i}(t))\nonumber \\&+ \, \sum _{j=1}^{N}\pi _{ij}\int \limits _{t-\sigma _{j}(t)}^{t}e_{p}^{T}(s)K^{T}R_{6}Ke_{p}(s)\mathrm{d}s, \end{aligned}$$
(31)
$$\begin{aligned} \mathcal {L}V_{5}(t,e_{t},i)&= \bar{\pi }\alpha \tau e_{m}^{T}(t)K^{T}R_{1}Ke_{m}(t)-\bar{\pi }\int \limits _{t-\alpha \tau }^{t} e_{m}^{T}(s)K^{T}R_{1}Ke_{m}(s)\mathrm{d}s +\bar{\pi }\tau e_{m}^{T}(t)K^{T}\nonumber \\&\times R_{2}Ke_{m}(t)-\bar{\pi }\int \limits _{t-\tau }^{t}e_{m}^{T}(s)K^{T}R_{2}Ke_{m}(s)\mathrm{d}s +\bar{\pi }\beta \sigma e_{p}^{T}(t)K^{T}R_{3}Ke_{p}(t)\nonumber \\&- \, \bar{\pi }\int \limits _{t-\beta \sigma }^{t}e_{p}^{T}(s)K^{T}R_{3}Ke_{p}(s)\mathrm{d}s+\bar{\pi }\sigma e_{p}^{T}(t)K^{T}R_{4}Ke_{p}(t) \nonumber \\&- \, \bar{\pi }\int \limits _{t-\sigma }^{t}e_{p}^{T}(s)K^{T}R_{4}Ke_{p}(s)\mathrm{d}s+\bar{\pi }\tau e_{m}^{T}(t)K^{T}R_{5}Ke_{m}(t) \nonumber \\&- \, \bar{\pi }\int \limits _{t-\tau }^{t}e_{m}^{T}(s)K^{T}R_{5}Ke_{m}(s)\mathrm{d}s +\bar{\pi }\sigma e_{p}^{T}(t)K^{T}R_{6}Ke_{p}(t)\nonumber \\&- \, \bar{\pi }\int \limits _{t-\sigma }^{t}e_{p}^{T}(s)K^{T}R_{6}Ke_{p}(s)\mathrm{d}s, \end{aligned}$$
(32)
$$\begin{aligned} \mathcal {L}V_{6}(t,e_{t},i)&= \tau ^{2}\dot{e}_{m}^{T}(t)K^{T}S_{1}K\dot{e}_{m}(t) {-}\tau \int \limits _{t{-}\tau }^{t}\dot{e}_{m}^{T}(s)K^{T}S_{1}K\dot{e}_{m}(s)\mathrm{d}s\nonumber \\&{+} \, \sigma ^{2}\dot{e}_{p}^{T}(t)K^{T}S_{2}K\dot{e}_{p}(t) {-}\sigma \int \limits _{t{-}\sigma }^{t}\dot{e}_{p}^{T}(s)K^{T}S_{2}K\dot{e}_{p}(s)\mathrm{d}s\nonumber \\&\le \tau ^{2}\dot{e}_{m}^{T}(t)K^{T}S_{1}K\dot{e}_{m}(t) {-}\tau _{i}\int \limits _{t{-}\tau _{i}}^{t{-}\tau _{i}(t)}\dot{e}_{m}^{T}(s)K^{T}S_{1}K\dot{e}_{m}(s)\mathrm{d}s\nonumber \\&{-} \, \tau _{i}\int \limits _{t{-}\tau _{i}(t)}^{t}\dot{e}_{m}^{T}(s)K^{T} S_{1}K\dot{e}_{m}(s)\mathrm{d}s{+}\sigma ^{2}\dot{e}_{p}^{T}(t)K^{T}S_{2}K\dot{e}_{p}(t)\nonumber \\&{-} \, \sigma _{i}\int \limits _{t{-}\sigma _{i}}^{t{-}\sigma _{i}(t)}\dot{e}_{p}^{T}(s)K^{T}S_{2}K\dot{e}_{p}(s)\mathrm{d}s {-}\sigma _{i}\int \limits _{t{-}\sigma _{i}(t)}^{t}\dot{e}_{p}^{T}(s)K^{T}S_{2}K\dot{e}_{p}(s)\mathrm{d}s\nonumber \\&\le \tau ^{2}\dot{e}_{m}^{T}(t)K^{T}S_{1}K\dot{e}_{m}(t) -\frac{\tau _{i}}{\tau _{i}-\tau _{i}(t)}\left[ e_{m}^{T}(t-\tau _{i}(t))K^{T}-e_{m}^{T}(t-\tau _{i})K^{T}\right] S_{1}\nonumber \\&\times [Ke_{m}(t-\tau _{i}(t))-Ke_{m}(t-\tau _{i})]-\frac{\tau _{i}}{\tau _{i}(t)}\left[ e_{m}^{T}(t)K^{T} -e_{m}^{T}(t-\tau _{i}(t))K^{T}\right] \nonumber \\&\times S_{1}[Ke_{m}(t)-Ke_{m}(t-\tau _{i}(t))] +\sigma ^{2}\dot{e}_{p}^{T}(t)K^{T}S_{2}K\dot{e}_{p}(t)-\frac{\sigma _{i}}{\sigma _{i}-\sigma _{i}(t)}\nonumber \\&\times \left[ e_{p}^{T}(t-\sigma _{i}(t))K^{T}-e_{p}^{T}(t-\sigma _{i})K^{T}\right] S_{2}[Ke_{p}(t-\sigma _{i}(t))-Ke_{p}(t-\sigma _{i})]\nonumber \\&- \, \frac{\sigma _{i}}{\sigma _{i}(t)}\left[ e_{p}^{T}(t)K^{T}-e_{p}^{T}(t-\sigma _{i}(t))K^{T}\right] S_{2}[Ke_{p}(t)-Ke_{p}(t-\sigma _{i}(t))]\end{aligned}$$
(33)
$$\begin{aligned}&\le \tau ^{2}\dot{e}_{m}^{T}(t)K^{T}S_{1}K\dot{e}_{m}(t) -\left[ \begin{array}{c} Ke_{m}(t-\tau _{i}(t))-Ke_{m}(t-\tau _{i})\\ Ke_{m}(t)-Ke_{m}(t-\tau _{i}(t)) \end{array}\right] ^{T}\ \nonumber \\&\times \left[ \begin{array}{cc} V_{1}&{}W_{1}\\ W_{1}^{T}&{}V_{1} \end{array}\right] \ \left[ \begin{array}{c} Ke_{m}(t-\tau _{i}(t))-Ke_{m}(t-\tau _{i})\\ Ke_{m}(t)-Ke_{m}(t-\tau _{i}(t)) \end{array}\right] \nonumber \\&+ \, \sigma ^{2}\dot{e}_{p}^{T}(t)K^{T}S_{2}K\dot{e}_{p}(t) -\left[ \begin{array}{c} Ke_{p}(t-\sigma _{i}(t))-Ke_{p}(t-\sigma _{i})\\ Ke_{p}(t)-Ke_{p}(t-\sigma _{i}(t)) \end{array}\right] ^{T}\ \nonumber \\&\times \left[ \begin{array}{cc} V_{2}&{}W_{2}\\ W_{2}^{T}&{}V_{2} \end{array}\right] \ \left[ \begin{array}{c} Ke_{p}(t-\sigma _{i}(t))-Ke_{p}(t-\sigma _{i})\\ Ke_{p}(t)-Ke_{p}(t-\sigma _{i}(t)) \end{array}\right] ,\ \end{aligned}$$
(34)
where the inequalities in (33) derived from Lemma 2 and that of (34) derived from Lemma 1 are described by
$$\begin{aligned}&- \left[ \begin{array}{c} \sqrt{\frac{\beta _{1}}{\alpha _{1}}}[Ke_{m}(t-\tau _{i}(t))-Ke_{m}(t-\tau _{i})]\\ -\sqrt{\frac{\alpha _{1}}{\beta _{1}}}[Ke_{m}(t)-Ke_{m}(t-\tau _{i}(t))] \end{array}\right] ^{T}\ \left[ \begin{array}{cc} V_{1}&{}W_{1}\\ W_{1}^{T}&{}V_{1} \end{array}\right] \\&\quad \quad \times \left[ \begin{array}{c} \sqrt{\frac{\beta _{1}}{\alpha _{1}}}[Ke_{m}(t-\tau _{i}(t))-Ke_{m}(t-\tau _{i})]\\ -\sqrt{\frac{\alpha _{1}}{\beta _{1}}}[Ke_{m}(t)-Ke_{m}(t-\tau _{i}(t))] \end{array}\right] \le 0, \end{aligned}$$
where \(\alpha _{1}=\frac{\tau _{i}-\tau _{i}(t)}{\tau _{i}}\) and \(\beta _{1}=\frac{\tau _{i}(t)}{\tau _{i}}\). Note that when \(\tau _{i}(t)=\tau _{i}\) or \(\tau _{i}(t)=0\), one can obtain
\(\left[ e_{m}^{T}(t-\tau _{i}(t))K^{T}-e_{m}^{T}(t-\tau _{i})K^{T}\right] =0\) or \(\left[ e_{m}^{T}(t)K^{T}-e_{m}^{T}(t-\tau _{i}(t))K^{T}\right] =0\),
and
$$\begin{aligned}&- \left[ \begin{array}{c} \sqrt{\frac{\beta _{2}}{\alpha _{2}}}[Ke_{p}(t-\sigma _{i}(t))-Ke_{p}(t-\sigma _{i})]\\ -\sqrt{\frac{\alpha _{2}}{\beta _{2}}}[Ke_{p}(t)-Ke_{p}(t-\sigma _{i}(t))] \end{array}\right] ^{T}\ \left[ \begin{array}{c@{\quad }c} V_{2}&{}W_{2}\\ W_{2}^{T}&{}V_{2} \end{array}\right] \\&\quad \quad \times \left[ \begin{array}{c} \sqrt{\frac{\beta _{2}}{\alpha _{2}}}[Ke_{p}(t-\sigma _{i}(t))-Ke_{p}(t-\sigma _{i})]\\ -\sqrt{\frac{\alpha _{2}}{\beta _{2}}}[Ke_{p}(t)-Ke_{p}(t-\sigma _{i}(t))] \end{array}\right] \le 0, \end{aligned}$$
where \(\alpha _{2}=\frac{\sigma _{i}-\sigma _{i}(t)}{\sigma _{i}}\) and \(\beta _{2}=\frac{\sigma _{i}(t)}{\sigma _{i}}\). Note that when \(\sigma _{i}(t)=\sigma _{i}\) or \(\sigma _{i}(t)=0\), one can obtain
\(\left[ e_{p}^{T}(t-\sigma _{i}(t))K^{T}-e_{p}^{T}(t-\sigma _{i})K^{T}\right] =0\) or \(\left[ e_{p}^{T}(t)K^{T}-e_{p}^{T}(t-\sigma _{i}(t))K^{T}\right] =0\).
$$\begin{aligned} \mathcal {L}V_{7}(t,e_{t},i)&\le g^{T}(Ke_{p}(t))T_{1}g(Ke_{p}(t)) -(1-\mu _{1i})g^{T}(Ke_{p}(t-\sigma _{i}(t)))T_{1}\nonumber \\&\times g(Ke_{p}(t-\sigma _{i}(t)))+g^{T}(Ke_{p}(t))T_{2}g(Ke_{p}(t)) -g^{T}(Ke_{p}(t-\sigma _{i}))T_{2}\nonumber \\&\times g(Ke_{p}(t-\sigma _{i}))+\bar{\pi }\sigma g^{T}(Ke_{p}(t))(T_{1}+T_{2})g(Ke_{p}(t))\nonumber \\&- \, \bar{\pi }\int \limits _{t-\sigma }^{t}g^{T}(Ke_{p}(s))(T_{1}+T_{2})g(Ke_{p}(s))\mathrm{d}s\nonumber \\&+ \, \sum _{j=1}^{N}\pi _{ij}\int \limits _{t-\sigma _{j}(t)}^{t}g^{T}(Ke_{p}(s))T_{1}g(Ke_{p}(s))\mathrm{d}s\nonumber \\&+ \, \sum _{j=1}^{N}\pi _{ij}\int \limits _{t-\sigma _{j}}^{t}g^{T}(Ke_{p}(s))T_{2}g(Ke_{p}(s))\mathrm{d}s \nonumber \\&\le e_{p}^{T}(t)K^{T}L^{T}T_{1}LKe_{p}(t) -(1-\mu _{1i})g^{T}(Ke_{p}(t-\sigma _{i}(t)))T_{1}g(Ke_{p}(t\nonumber \\&- \, \sigma _{i}(t))) +e_{p}^{T}(t)K^{T}L^{T}T_{2}LKe_{p}(t)) -g^{T}(Ke_{p}(t-\sigma _{i}))T_{2}g(Ke_{p}(t-\sigma _{i}))\nonumber \\&+ \, \bar{\pi }\sigma e_{p}^{T}(t)K^{T}L^{T}(T_{1}+T_{2})LKe_{p}(t). \end{aligned}$$
(35)
By using upper and lower bound of time-varying delays and \(\pi _{ii}\,\le \,0\), the following relationship holds.
$$\begin{aligned} \sum _{j=1}^{N}\pi _{ij}\int \limits _{t-\rho _{j}}^{t}e_{m}^{T}(s)Q_{1}e_{m}(s)\mathrm{d}s&= \sum _{j\ne i}\pi _{ij}\int \limits _{t-\rho _{j}}^{t}e_{m}^{T}(s)Q_{1}e_{m}(s)\mathrm{d}s\nonumber \\&+ \, \pi _{ii}\int \limits _{t-\rho _{i}}^{t}e_{m}^{T}(s)Q_{1}e_{m}(s)\mathrm{d}s \nonumber \\&\le \bar{\pi }\int \limits _{t-\rho }^{t}e_{m}^{T}(s)Q_{1}e_{m}(s)\mathrm{d}s, \end{aligned}$$
(36)
$$\begin{aligned} \sum _{j=1}^{N}\pi _{ij}\int \limits _{-\rho _{j}}^{0}\int \limits _{t+\theta }^{t}e_{m}^{T}(s)Q_{3}e_{m}(s)\mathrm{d}s\mathrm{d}\theta&= \sum _{j\ne i}\pi _{ij}\int \limits _{-\rho _{j}}^{0}\int \limits _{t+\theta }^{t}e_{m}^{T}(s)Q_{3}e_{m}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&+ \, \pi _{ii}\int \limits _{-\rho _{j}}^{0}\int \limits _{t+\theta }^{t}e_{m}^{T}(s)Q_{3}e_{m}(s)\mathrm{d}s\mathrm{d}\theta \nonumber \\&\le \bar{\pi }\int \limits _{-\rho }^{0}\int \limits _{t+\theta }^{t}e_{m}^{T}(s)Q_{3}e_{m}(s)\mathrm{d}s\mathrm{d}\theta , \end{aligned}$$
(37)
$$\begin{aligned} \sum _{j=1}^{N}\pi _{ij}\int \limits _{t-\tau _{j}}^{t-\alpha \tau _{j}}e_{m}^{T}(s)K^{T}R_{2}Ke_{m}(s)\mathrm{d}s&= \sum _{j\ne i}\pi _{ij}\int \limits _{t-\tau _{j}}^{t-\alpha \tau _{j}}e_{m}^{T}(s)K^{T}R_{2}Ke_{m}(s)\mathrm{d}s\nonumber \\&+ \, \pi _{ii}\int \limits _{t-\tau _{j}}^{t-\alpha \tau _{j}}e_{m}^{T}(s)K^{T}R_{2}Ke_{m}(s)\mathrm{d}s\nonumber \\&\le \bar{\pi }\int \limits _{t-\tau }^{t}e_{m}^{T}(s)K^{T}R_{2}Ke_{m}(s)\mathrm{d}s\nonumber \\&+ \, \pi _{ii}\int \limits _{t-\tau _{j}}^{t-\alpha \tau _{j}}e_{m}^{T}(s)K^{T}R_{2}Ke_{m}(s)\mathrm{d}s\nonumber \\&\le \bar{\pi }\int \limits _{t-\tau }^{t}e_{m}^{T}(s)K^{T}R_{2}Ke_{m}(s)\mathrm{d}s. \end{aligned}$$
(38)
Hence (29), (30), and (35) are obtained from the relationship (36)–(38). Therefore, from (27)–(32), (34), and (35), one can obtain
$$\begin{aligned} E\{\mathcal {L}V(t,e_{t},i)\}&\le \zeta ^{T}(t)\Sigma ^{i}\zeta (t), \end{aligned}$$
(39)
where
$$\begin{aligned} \zeta ^{T}(t)&= \left[ e_{m}^{T}(t)\ \ \ e_{p}^{T}(t)\ \ \ e_{m}^{T}(t-\rho _{i})\ \ \ e_{p}^{T}(t-h_{i})\ \ \ e_{m}^{T}(t-\alpha \tau _{i})K^{T}\ \ \ e_{m}^{T}(t-\tau _{i})K^{T}\right. \\&e_{p}^{T}(t-\beta \sigma _{i})K^{T}\ \ \ e_{p}^{T}(t-\sigma _{i})K^{T}\ \ \ e_{m}^{T}(t-\tau _{i}(t))K^{T}\ \ \ e_{p}^{T}(t-\sigma _{i}(t))K^{T}\\&\left. \int \limits _{t-\rho _{i}}^{t}e_{m}^{T}(s)\mathrm{d}s\ \ \ \int \limits _{t-h_{i}}^{t}e_{p}^{T}(s)\mathrm{d}s\ \ \ g^{T}(Ke_{p}(t-\sigma _{i}(t)))\ \ \ g^{T}(Ke_{p}(t-\sigma _{i}))\right] , \end{aligned}$$
and \(\Sigma ^{i}\) is defined in (12).
Now, we show that the filtering error system (10) with \(\nu _{1}(t)\,=\,\nu _{2}(t)=0\) is globally stochastically stable in the mean-square sense.
If \(\nu _{1}(t)\,=\,\nu _{2}(t)=0\), we can conclude from (12) that \(\Sigma ^{i}\,<\,0\), which implies that
$$\begin{aligned} E\{\mathcal {L}V(t,e_{t},i)\}\le \zeta ^{T}(t)\Sigma ^{i}\zeta (t)<0. \end{aligned}$$
(40)
Thus, from (40), we have
$$\begin{aligned} E\{\mathcal {L}V(t,e_{t},i)\}\le -\alpha _{1}E\{\parallel e_{m}(t) \parallel ^{2}+\parallel e_{p}(t) \parallel ^{2}\}, \end{aligned}$$
(41)
where \(\alpha _{1}\ =\min _{i\in S}\{\uplambda _{min}(-\Sigma ^{i})\}>0\). Integrating both sides of (41) from 0 to \(t\), we have by Dynkin’s formula,
$$\begin{aligned} E\{V(t,e_{t},i)\}\le -\alpha _{1}\int \limits _{0}^{t}E\{\parallel e_{m}(s)\parallel ^{2}+\parallel e_{p}(s)\parallel ^{2}\}\mathrm{d}s +E\{V(0,e_{0},i)\}. \end{aligned}$$
(42)
From the definition of (26), we have
$$\begin{aligned} E\{V(t,e_{t},i)\}\ge \alpha _{2}E\{\parallel e_{m}(t) \parallel ^{2}+\parallel e_{p}(t) \parallel ^{2}\}, \end{aligned}$$
(43)
where \(\alpha _{2}\ =\min _{i\in S}\{\uplambda _{\mathrm{{min}}}(P_{1i}+P_{2i})\}>0\). Inequalities (42) and (43) imply that
$$\begin{aligned} \lim _{t\rightarrow \infty }\int \limits _{0}^{t}E\{\parallel e_{m}(t) \parallel ^{2}+\parallel e_{p}(t) \parallel ^{2}\}\mathrm{d}s&\le k_{2}k_{1}^{-1}E\{V(0,e_{0},i)\}<\infty , \end{aligned}$$
(44)
where \(k_{1}=\alpha _{1}\alpha _{2}^{-1}\), \(k_{2}=\alpha _{2}^{-1}\), and
$$\begin{aligned} E\{V(0,e_{0},i)\}&= E\Big \{e_{m}^{T}(0)P_{1i}e_{m}(0)+e_{p}^{T}(0)P_{2i}e_{p}(0)+\int \limits _{-\rho _{i}}^{0}e_{m}^{T}(s)Q_{1}e_{m}(s)\mathrm{d}s\\&+ \, \int \limits _{-h_{i}}^{0}e_{p}^{T}(s)Q_{2}e_{p}(s)\mathrm{d}s +\bar{\pi }\int \limits _{-\rho }^{0}\int \limits _{\theta }^{0}e_{m}^{T}(s)Q_{1}e_{m}(s)\mathrm{d}s\mathrm{d}\theta \\&+ \, \bar{\pi }\int \limits _{-h}^{0}\int \limits _{\theta }^{0}e_{p}^{T}(s)Q_{2}e_{p}(s)\mathrm{d}s\mathrm{d}\theta +\int \limits _{-\rho _{i}}^{0}\int \limits _{\theta }^{0}e_{m}^{T}(s)Q_{3}e_{m}(s)\mathrm{d}s\mathrm{d}\theta \\&+ \, \int \limits _{-h_{i}}^{0}\int \limits _{\theta }^{0}e_{p}^{T}(s)Q_{4}e_{p}(s)\mathrm{d}s\mathrm{d}\theta +\bar{\pi }\int \limits _{-\rho }^{0}\int \limits _{\theta }^{0}\int \limits _{\uplambda }^{0}e_{m}^{T}(s)Q_{3}e_{m}(s)\mathrm{d}s\mathrm{d}\uplambda \mathrm{d}\theta \\&+ \, \bar{\pi }\int \limits _{-h}^{0}\int \limits _{\theta }^{0}\int \limits _{\uplambda }^{0}e_{p}^{T}(s)Q_{4}e_{p}(s)\mathrm{d}s\mathrm{d}\uplambda \mathrm{d}\theta +\int \limits _{-\alpha \tau _{i}}^{0}e_{m}^{T}(s)K^{T}R_{1}Ke_{m}(s)\mathrm{d}s\\&+ \, \int \limits _{-\tau _{i}}^{-\alpha \tau _{i}}e_{m}^{T}(s)K^{T}R_{2}Ke_{m}(s)\mathrm{d}s +\int \limits _{-\beta \sigma _{i}}^{0}e_{p}^{T}(s)K^{T}R_{3}Ke_{p}(s)\mathrm{d}s \\&+ \, \int \limits _{-\sigma _{i}}^{-\beta \sigma _{i}}e_{p}^{T}(s)K^{T}R_{4}Ke_{p}(s)\mathrm{d}s +\int \limits _{-\tau _{i}(0)}^{0}e_{m}^{T}(s)K^{T}R_{5}Ke_{m}(s)\mathrm{d}s \\&+ \, \int \limits _{-\sigma _{i}(0)}^{0}e_{p}^{T}(s)K^{T}R_{6}Ke_{p}(s)\mathrm{d}s +\bar{\pi }\int \limits _{-\alpha \tau }^{0}\int \limits _{\theta }^{0}e_{m}^{T}(s)K^{T}R_{1}Ke_{m}(s)\mathrm{d}s\mathrm{d}\theta \\&+ \, \bar{\pi }\int \limits _{-\tau }^{0}\int \limits _{\theta }^{0}e_{m}^{T}(s)K^{T}R_{2}Ke_{m}(s)\mathrm{d}s\mathrm{d}\theta +\bar{\pi }\int \limits _{-\beta \sigma }^{0}\int \limits _{\theta }^{0}e_{p}^{T}(s)K^{T}R_{3}Ke_{p}(s)\mathrm{d}s\mathrm{d}\theta \\&+ \, \bar{\pi }\int \limits _{-\sigma }^{0}\int \limits _{\theta }^{0}e_{p}^{T}(s)K^{T}R_{4}Ke_{p}(s)\mathrm{d}s\mathrm{d}\theta +\bar{\pi }\int \limits _{-\tau }^{0}\int \limits _{\theta }^{0}e_{m}^{T}(s)K^{T}R_{5}Ke_{m}(s)\mathrm{d}s\mathrm{d}\theta \\&+ \, \bar{\pi }\int \limits _{-\sigma }^{0}\int \limits _{\theta }^{0}e_{p}^{T}(s)K^{T}R_{6}Ke_{p}(s)\mathrm{d}s\mathrm{d}\theta +\tau \int \limits _{-\tau }^{0}\int \limits _{\theta }^{0}\dot{e}_{m}^{T}(s)K^{T}S_{1}K\dot{e}_{m}(s)\mathrm{d}s\mathrm{d}\theta \\&+ \, \sigma \int \limits _{-\sigma }^{0}\int \limits _{\theta }^{0}\dot{e}_{p}^{T}(s)K^{T}S_{2}K\dot{e}_{p}(s)\mathrm{d}s\mathrm{d}\theta +\int \limits _{-\sigma _{i}(0)}^{0}g^{T}(Ke_{p}(s))T_{1}g(Ke_{p}(s))\mathrm{d}s\\&+ \, \int \limits _{-\sigma _{i}}^{0}g^{T}(Ke_{p}(s))T_{2}g(Ke_{p}(s))\mathrm{d}s +\bar{\pi }\int \limits _{-\sigma }^{0}\int \limits _{\theta }^{0}g^{T}(Ke_{p}(s))(T_{1}+T_{2})g(Ke_{p}(s))\mathrm{d}s\mathrm{d}\theta \Big \}\\&\le \{\uplambda _{\mathrm{{max}}}(P_{1i})+(\rho _{i}+\bar{\pi }\rho ^{2})\uplambda _{\mathrm{{max}}}(Q_{1})+(\rho _{i}^{2} +\bar{\pi }\rho ^{3})\uplambda _{\mathrm{{max}}}(Q_{3}) +(\alpha \tau _{i}+\bar{\pi }\alpha ^{2}\tau ^{2})\\&\times \uplambda _{\mathrm{{max}}}(R_{1})+((\tau _{i}-\alpha \tau _{i})+\bar{\pi }\tau ^{2})\uplambda _{\mathrm{{max}}}(R_{2}) +(\tau _{i}+\bar{\pi }\tau ^{2})\uplambda _{\mathrm{{max}}}(R_{5})\\&+ \, \tau ^{3}\uplambda _{\mathrm{{max}}}(S_{1})\} E\{\parallel \chi _{m}(t)\parallel _{\tau ^{*}}^{2}\}+ \{\uplambda _{\mathrm{{max}}}(P_{2i})+(h_{i}+\bar{\pi }h^{2})\uplambda _{\mathrm{{max}}}(Q_{2})+(h_{i}^{2} +\bar{\pi }h^{3})\\&\times \uplambda _{\mathrm{{max}}}(Q_{4}) +(\beta \sigma _{i}+\bar{\pi }\beta ^{2}\sigma ^{2})\uplambda _{\mathrm{{max}}}(R_{3}) +((\sigma _{i}-\beta \sigma _{i})+\bar{\pi }\sigma ^{2})\uplambda _{\mathrm{{max}}}(R_{4})\\&+ \, (\sigma _{i}+\bar{\pi }\sigma ^{2})\uplambda _{\mathrm{{max}}}(R_{6})+\sigma ^{3}\uplambda _{\mathrm{{max}}}(S_{2}) +\sigma _{i}\mathrm{{max}}_{j\in \Lambda } L_{j}^{2}\uplambda _{\mathrm{{max}}}(T_{1}+T_{2})\\&+ \, \bar{\pi }\sigma ^{2}\mathrm{{max}}_{j\in \Lambda } L_{j}^{2}\uplambda _{\mathrm{{max}}}(T_{1}+T_{2})\} E\{\parallel \chi _{p}(t)\parallel _{\sigma ^{*}}^{2}\}<\infty . \end{aligned}$$
This implies that the trivial solution of (10) is locally stable.
Considering the continuity of the function \(g(\cdot )\), the solutions \(e_{m}(t)\) and \(e_{p}(t)\) of system (10) are bounded on \([0,\infty )\). The uniform boundedness of solutions of (10) implies that the derivatives of the solutions of (10) are bounded on \([0,\infty )\), which lead to the uniform continuity of solutions \(e_{m}(t)\) and \(e_{p}(t)\) on \([0,\infty )\). By Barbalat’s lemma, we have
$$\begin{aligned} \lim _{t\rightarrow \infty }E\{\parallel e_{m}(t) \parallel ^{2}+\parallel e_{p}(t) \parallel ^{2}\}=0. \end{aligned}$$
Thus, the filtering error system (10) with \(\nu _{1}(t)=\nu _{2}(t)=0\) is globally stochastically stable in the mean-square sense by Definition 1. This completes the proof. \(\square \)
1.1 Appendix 2
Proof of Theorem 2
Choose the same Lyapunov–Krasovskii functional candidate as in (26) for the filtering error system (10) and employing the similar approach as in Theorem 1 (with \(\nu _{1}(t)\ne 0,\,\nu _{2}(t)\ne 0\)), by using the weak infinitesimal operator \(\mathcal {L}\), the stochastic differential of \(V(t,e_{t},i)\) is given by
$$\begin{aligned} \mathcal {L}V(t,e_{t},i)&= \zeta ^{T}(t)\Sigma ^{i}\zeta (t)+2e_{m}^{T}(t)P_{1i}\bar{M}_{i}\nu _{1}(t)+ 2e_{p}^{T}(t)P_{2i}\bar{N}_{i}\nu _{2}(t)+2e_{m}^{T}(t-\rho _{i})\nonumber \\&(-\tau ^{2}\bar{A}_{i}^{T}K^{T}S_{1}K\bar{M}_{i})\nu _{1}(t) +2g^{T}(Ke_{p}(t-\sigma _{i}(t))(\tau ^{2}\bar{B}_{i}^{T}K^{T}S_{1}K\bar{M}_{i})\nu _{1}(t)\nonumber \\&+ \, \nu _{1}^{T}(t)(\tau ^{2}\bar{M}_{i}^{T}K^{T}S_{1}K\bar{M}_{i})\nu _{1}(t) +2e_{p}^{T}(t-h_{i})(-\sigma ^{2}\bar{C}_{i}^{T}K^{T}S_{2}K\bar{N}_{i})\nu _{2}(t)\nonumber \\&+ \, 2e_{m}^{T}(t-\tau _{i}(t))K^{T}(\sigma ^{2}\bar{D}_{i}^{T}K^{T}S_{2}K\bar{N}_{i})\nu _{2}(t)\nonumber \\&+ \, \nu _{2}^{T}(t)(\sigma ^{2}\bar{N}_{i}^{T}K^{T}S_{2}K\bar{N}_{i})\nu _{2}(t). \end{aligned}$$
(45)
Next, we shall deal with the \(H_{\infty }\) performance for the filtering error system (10). Let us define
$$\begin{aligned} J(T)=E\Big \{\int \limits _{0}^{T}[\tilde{z}_{m}^{T}(t)\tilde{z}_{m}(t)-\gamma _{1}^{2}\nu _{1}^{T}(t)\nu _{1}(t) +\tilde{z}_{p}^{T}(t)\tilde{z}_{p}(t)-\gamma _{2}^{2}\nu _{2}^{T}(t)\nu _{2}(t)]\mathrm{d}t\Big \}\nonumber \\ \end{aligned}$$
(46)
for any nonzero \(\nu _{1}(t),\,\nu _{2}(t)\in L_{2}[0,\,\infty )\). From the zero initial conditions, we have
$$\begin{aligned} J(T)&= E\Big \{\int \limits _{0}^{T}[\tilde{z}_{m}^{T}(t)\tilde{z}_{m}(t)-\gamma _{1}^{2}\nu _{1}^{T}(t)\nu _{1}(t) +\tilde{z}_{p}^{T}(t)\tilde{z}_{p}(t)-\gamma _{2}^{2}\nu _{2}^{T}(t)\nu _{2}(t)\nonumber \\&+ \, \mathcal {L}V(t,e_{t},i)]\mathrm{d}t\Big \}-V(T,e_{T},i) \nonumber \\&\le E\Big \{\int \limits _{0}^{T}[\tilde{z}_{m}^{T}(t)\tilde{z}_{m}(t)-\gamma _{1}^{2}\nu _{1}^{T}(t)\nu _{1}(t) +\tilde{z}_{p}^{T}(t)\tilde{z}_{p}(t)-\gamma _{2}^{2}\nu _{2}^{T}(t)\nu _{2}(t)\nonumber \\&+ \, \mathcal {L}V(t,e_{t},i)]\mathrm{d}t\Big \}\nonumber \\&= E\Big \{\eta ^{T}(t)\varOmega ^{i}\eta (t)\mathrm{d}t\Big \}, \end{aligned}$$
(47)
where
$$\begin{aligned}&\eta ^{T}(t)=[e_{m}^{T}(t) \quad e_{p}^{T}(t) \quad e_{m}^{T}(t-\rho _{i})\quad e_{p}^{T}(t-h_{i}) \quad e_{m}^{T}(t-\alpha \tau _{i})K^{T} \quad e_{m}^{T}(t-\tau _{i})K^{T}\\&e_{p}^{T}(t-\beta \sigma _{i})K^{T} \quad e_{p}^{T}(t-\sigma _{i})K^{T}\quad e_{m}^{T}(t-\tau _{i}(t))K^{T}\quad e_{p}^{T}(t-\sigma _{i}(t))K^{T} \quad \int \limits _{t-\rho _{i}}^{t}e_{m}^{T}(s)\mathrm{d}s\\&\qquad \int \limits _{t-h_{i}}^{t}e_{p}^{T}(s)\mathrm{d}s \quad g^{T}(Ke_{p}(t-\sigma _{i}(t)))\quad g^{T}(Ke_{p}(t-\sigma _{i}))\quad \nu ^{T}_{1}(t)\quad \nu ^{T}_{2}(t)], \end{aligned}$$
$$\begin{aligned}&\!\!\!\varOmega ^{i}=\nonumber \\&\left[ \begin{array}{llllllllllllllll} \varOmega _{11}^{i}&{}0&{}\Sigma _{13}^{i}&{}0&{}0&{}\Sigma _{16}^{i}&{}0&{}0&{}\Sigma _{19}^{i}&{}0&{}0&{}0&{}\Sigma _{1,13}^{i}&{}0&{}\varOmega _{1,15}^{i}&{}0\\ *&{}\varOmega _{22}^{i}&{}0&{}\Sigma _{24}^{i}&{}0&{}0&{}0&{}\Sigma _{28}^{i}&{}\Sigma _{29}^{i}&{}\Sigma _{2,10}^{i}&{}0&{}0&{}0&{}0&{}0&{}\varOmega _{2,16}^{i}\\ *&{}*&{}\Sigma _{33}^{i}&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}\Sigma _{3,13}^{i}&{}0&{}\varOmega _{3,15}^{i}&{}0\\ *&{}*&{}*&{}\Sigma _{44}^{i}&{}0&{}0&{}0&{}0&{}\Sigma _{49}^{i}&{}0&{}0&{}0&{}0&{}0&{}0&{}\varOmega _{4,16}^{i}\\ *&{}*&{}*&{}*&{}\Sigma _{55}^{i}&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}\Sigma _{66}^{i}&{}0&{}0&{}\Sigma _{69}^{i}&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}\Sigma _{77}^{i}&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}\Sigma _{88}^{i}&{}0&{}\Sigma _{8,10}^{i}&{}0&{}0&{}0&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}\Sigma _{99}^{i}&{}0&{}0&{}0&{}0&{}0&{}0&{}\varOmega _{9,16}^{i}\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}\Sigma _{10,10}^{i}&{}0&{}0&{}0&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}\Sigma _{11,11}^{i}&{}0&{}0&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}\Sigma _{12,12}^{i}&{}0&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}\Sigma _{13,13}^{i}&{}0&{}\varOmega _{13,15}^{i}&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}\Sigma _{14,14}^{i}&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}\varOmega _{15,15}^{i}&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}\varOmega _{16,16}^{i}\\ \end{array}\right] \!,\nonumber \\ \end{aligned}$$
(48)
with
$$\begin{aligned} \varOmega _{11}^{i}&= K^{T}\bar{E}_{i}^{T}P_{1i}\bar{E}_{i}K+\sum _{j=1}^{N}\pi _{ij}P_{1j}+(1+\bar{\pi }\rho )Q_{1} +(\rho _{i}+\bar{\pi }\frac{\rho ^{2}}{2})Q_{3}+K^{T}(R_{1}+R_{5})K \\&+ \, \bar{\pi }\tau K^{T}(\alpha R_{1}+R_{2}+R_{5})K-K^{T}V_{1}K+\bar{H}_{mi}^{T}\bar{H}_{mi},\, \varOmega _{1,15}^{i}=P_{1i}\bar{M}_{i}, \\ \varOmega _{22}^{i}&= K^{T}\bar{F}_{i}^{T}P_{2i}\bar{F}_{i}K+\sum _{j=1}^{N}\pi _{ij}P_{2j}+(1+\bar{\pi }h)Q_{2} +(h_{i}+\bar{\pi }\frac{h^{2}}{2})Q_{4}+K^{T}(R_{3}+R_{6})K \\&+ \, \bar{\pi }\sigma K^{T}(\beta R_{3}+R_{4}+R_{6})K-K^{T}V_{2}K+(1+\bar{\pi }\sigma )K^{T}L^{T}(T_{1}+T_{2})L K+\bar{H}_{pi}^{T}\bar{H}_{pi},\\ \varOmega _{2,16}^{i}&= P_{2i}\bar{N}_{i},\,\varOmega _{3,15}^{i}=-\tau ^{2}\bar{A}_{i}^{T}K^{T}S_{1}K\bar{M}_{i},\, \varOmega _{4,16}^{i}=-\sigma ^{2}\bar{C}_{i}^{T}K^{T}S_{2}K\bar{N}_{i},\\ \varOmega _{9,16}^{i}&= \sigma ^{2}\bar{D}_{i}^{T}K^{T}S_{2}K\bar{N}_{i},\, \varOmega _{13,15}^{i}=\tau ^{2}\bar{B}_{i}^{T}K^{T}S_{1}K\bar{M}_{i},\, \varOmega _{15,15}^{i}=\tau ^{2}\bar{M}_{i}^{T}K^{T}S_{1}K\bar{M}_{i}-\gamma _{1}^{2}I,\\ \varOmega _{16,16}^{i}&= \sigma ^{2}\bar{N}_{i}^{T}K^{T}S_{2}K\bar{N}_{i}-\gamma _{2}^{2}I, \end{aligned}$$
and the other parameters are defined as in Theorem 1.
It is obvious that if \(\varOmega ^{i}<0\), then one can conclude that \(J(t)<0\). Note that when disturbances are zero, that is \(\nu _{1}(t)=0\), and \(\nu _{2}(t)=0\), then it is easy to deduce from Theorem 1 that the filtering error system (10) is globally stochastically stable in the mean-square sense.
However, the inequality \(\varOmega ^{i}<0\) is not an LMI, and it is very difficult to get solutions. Hence we will find an equivalent LMI representation for \(\varOmega ^{i}<0\).
For each \(r_{t}=i\), \(i\in S\), define
$$\begin{aligned} P_{1i}&= \left[ \begin{array}{cc} X_{1i}&{} \quad Y_{1i}-X_{1i}\\ Y_{1i}-X_{1i}&{} \quad X_{1i}-Y_{1i} \end{array}\right] >0, \quad P_{2i}= \left[ \begin{array}{cc} X_{2i}&{} \quad Y_{2i}-X_{2i}\\ Y_{2i}-X_{2i}&{} \quad X_{2i}-Y_{2i} \end{array}\right] >0, \end{aligned}$$
$$\begin{aligned} Q_{1}&= \left[ \begin{array}{cc} Q_{11}^{1m}&{} \quad Q_{12}^{1m}\\ {Q_{12}^{1m}}^{T}&{} \quad Q_{22}^{1m} \end{array}\right] >0, \quad Q_{2}= \left[ \begin{array}{cc} Q_{11}^{2p}&{} \quad Q_{12}^{2p}\\ {Q_{12}^{2p}}^{T}&{} \quad Q_{22}^{2p} \end{array}\right] >0, \end{aligned}$$
$$\begin{aligned} Q_{3}&= \left[ \begin{array}{cc} Q_{11}^{3m}&{} \quad Q_{12}^{3m}\\ {Q_{12}^{3m}}^{T}&{} \quad Q_{22}^{3m} \end{array}\right] >0, \quad Q_{4}= \left[ \begin{array}{cc} Q_{11}^{4p}&{} \quad Q_{12}^{4p}\\ {Q_{12}^{4p}}^{T}&{} \quad Q_{22}^{4p} \end{array}\right] >0. \end{aligned}$$
Pre- and post-multiplying (48) by \(\{\Delta _{i}, \mathrm{{diag}}\{I\}_{14\times 14}\}\), where
$$\begin{aligned} \Delta _{i}=\left[ \begin{array}{cccccccc} Y_{1i}^{-1}&{}\quad Y_{1i}^{-1}&{}\quad 0&{}\quad 0&{}\quad Y_{1i}^{-1}&{}\quad Y_{1i}^{-1}&{}\quad 0&{}\quad 0\\ I&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad I&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad Y_{2i}^{-1}&{} \quad Y_{2i}^{-1}&{} \quad 0&{} \quad 0&{} \quad Y_{2i}^{-1}&{} \quad Y_{2i}^{-1}\\ 0&{} \quad 0&{} \quad I&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad I&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad I&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad I&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad I&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad I\\ \end{array}\right] \end{aligned}$$
and its transpose, we can see that the following LMI holds
$$\begin{aligned} \tilde{\varPsi }^{i}&= \left[ \begin{array}{cc} \tilde{\varPsi }_{1}^{i}&{} \quad \tilde{\varPsi }_{2}^{i}\\ *&{} \quad \tilde{\varPsi }_{3}^{i} \end{array}\right] <0, \end{aligned}$$
(49)
where
$$\begin{aligned}&\!\!\!\tilde{\varPsi }_{1}^{i}=\nonumber \\&\left[ \begin{array}{ccccccccccc} \tilde{\varPsi }_{11}^{i}&{}\tilde{\varPsi }_{12}^{i}&{}0&{}0&{} \tilde{\varPsi }_{15}^{i} &{}\tilde{\varPsi }_{16}^{i} &{}0&{}0&{}0&{}Y_{1i}^{-1}W_{1}^{T}&{}0\\ *&{}\tilde{\varPsi }_{22}^{i}&{}0&{}0&{}\tilde{\varPsi }_{25}^{i}&{} \tilde{\varPsi }_{26}^{i} &{}0&{}0&{}0&{}W_{1}^{T}&{}0\\ *&{}*&{}\tilde{\varPsi }_{33}^{i}&{}\tilde{\varPsi }_{34}^{i}&{}0&{}0&{}\tilde{\varPsi }_{37}^{i} &{} \tilde{\varPsi }_{38}^{i} &{}0&{}0&{}0\\ *&{}*&{}*&{}\tilde{\varPsi }_{44}^{i}&{}0&{}0&{}\tilde{\varPsi }_{47}^{i} &{}\tilde{\varPsi }_{48}^{i} &{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}\tilde{\varPsi }_{55}^{i}&{}-Q_{12}^{1m}&{}0&{}0&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}-Q_{22}^{1m}&{}0&{}0&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}\tilde{\varPsi }_{77}^{i}&{}-Q_{12}^{2p}&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}-Q_{22}^{2p}&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}-R_{1}+R_{2}&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}-R_{2}-V_{1}&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}-R_{3}+R_{4}\\ \end{array}\right] \!, \end{aligned}$$
$$\begin{aligned} \tilde{\varPsi }_{2}^{i}=\left[ \begin{array}{ccccccccccc} 0&{} \tilde{\varPsi }_{1,13}^{i} &{}0&{}0&{}0&{}0&{}0&{}\tilde{\varPsi }_{1,19}^{i}&{}0&{}\tilde{\varPsi }_{1,21}^{i}&{}0\\ 0&{}-W_{1}^{T}+V_{1}&{}0&{}0&{}0&{}0&{}0&{}\tilde{\varPsi }_{2,19}^{i}&{}0&{}\tilde{\varPsi }_{2,21}^{i}&{}0\\ Y_{2i}^{-1}W_{2}^{T}&{}\tilde{\varPsi }_{3,13}^{i}&{} \tilde{\varPsi }_{3,14}^{i} &{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}\tilde{\varPsi }_{3,22}^{i}\\ W_{2}^{T}&{}\tilde{\varPsi }_{4,13}^{i}&{}-W_{2}^{T}+V_{2}&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}\tilde{\varPsi }_{4,22}^{i}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}\tilde{\varPsi }_{5,19}^{i} &{}0&{}\tilde{\varPsi }_{5,21}^{i} &{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}-\sigma ^{2}C_{i}^{T}S_{2}D_{i}&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}-\sigma ^{2}C_{i}^{T}S_{2}N_{i}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}V_{1}-W_{1}&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ \end{array}\right] , \end{aligned}$$
$$\begin{aligned}&\!\!\!\tilde{\varPsi }_{3}^{i}=\nonumber \\&\left[ \begin{array}{lllllllllll} -R_{4}-V_{2}&{}0&{}-W_{2}+V_{2}&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ *&{}\tilde{\varPsi }_{13,13}^{i}&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0 &{}\tilde{\varPsi }_{13,22}^{i}\\ *&{}*&{}\tilde{\varPsi }_{14,14}^{i}&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ *&{}*&{}*&{}-\frac{1}{\rho _{i}}Q_{11}^{3m}&{}-\frac{1}{\rho _{i}}Q_{12}^{3m}&{}0&{}0&{}0&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}-\frac{1}{\rho _{i}}Q_{22}^{3m}&{}0&{}0&{}0&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}-\frac{1}{h_{i}}Q_{11}^{4p}&{}-\frac{1}{h_{i}}Q_{12}^{4p}&{}0&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}-\frac{1}{h_{i}}Q_{22}^{4p}&{}0&{}0&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}\tilde{\varPsi }_{19,19}^{i}&{}0&{} \tilde{\varPsi }_{19,21}^{i} &{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}-T_{2}&{}0&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}\tilde{\varPsi }_{21,21}^{i}&{}0\\ *&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}*&{}\tilde{\varPsi }_{22,22}^{i}\\ \end{array}\right] \!, \end{aligned}$$
with
$$\begin{aligned} \tilde{\varPsi }_{11}^{i}&= -A_{i}Y_{1i}^{-1}-Y_{1i}^{-1}A_{i}^{T}+Y_{1i}^{-1}E_{i}^{T}Y_{1i}E_{i}Y_{1i}^{-1} +\bar{\pi }\rho Y_{1i}^{-1}(Q_{11}^{1m}+Q_{12}^{1m}+{Q_{12}^{1m}}^{T}+Q_{22}^{1m})Y_{1i}^{-1}\\&+ \, (\rho _{i}+\bar{\pi }\frac{\rho ^{2}}{2})Y_{1i}^{-1}(Q_{11}^{3m} +Q_{12}^{3m}+{Q_{12}^{3m}}^{T}+Q_{22}^{3m})Y_{1i}^{-1}+Y_{1i}^{-1}(R_{1}+R_{5})Y_{1i}^{-1}\\&+ \, \bar{\pi }\tau Y_{1i}^{-1}(\alpha R_{1}+R_{2}+R_{5})Y_{1i}^{-1}-Y_{1i}^{-1}V_{1}Y_{1i}^{-1} +Y_{1i}^{-1}\sum _{j=1}^{N}\pi _{ij}Y_{1j}Y_{1i}^{-1} +Y_{1i}^{-1}H_{mi}^{T}H_{mi}Y_{1i}^{-1}\\&+ \, \tau ^{2}Y_{1i}^{-1}A_{i}^{T}S_{1}A_{i}Y_{1i}^{-1}, \\ \tilde{\varPsi }_{12}^{i}&= -A_{i}-Y_{1i}^{-1}A_{i}^{T}Y_{1i}+Y_{1i}^{-1}E_{i}^{T}Y_{1i}E_{i}+\bar{\pi }\rho Y_{1i}^{-1}(Q_{11}^{1m}+{Q_{12}^{1m}}^{T}) +(\rho _{i}+\bar{\pi }\frac{\rho ^{2}}{2})Y_{1i}^{-1}(Q_{11}^{3m}\\&+ \, {Q_{12}^{3m}}^{T}) +Y_{1i}^{-1}(R_{1}+R_{5})+\bar{\pi }\tau Y_{1i}^{-1}(\alpha R_{1}+R_{2}+R_{5})-Y_{1i}^{-1}V_{1}+Y_{1i}^{-1}\sum _{j=1}^{N}\pi _{ij}Y_{1j} \\&-\, Y_{1i}^{-1}C_{mi}^{T}B_{fi}^{T}(Y_{1i}-X_{1i})+\tau ^{2}Y_{1i}^{-1}A_{i}^{T}S_{1}A_{i},\\ \tilde{\varPsi }_{15}^{i}&= -A_{i}-Y_{1i}^{-1}Q_{11}^{1m}+\tau ^{2}Y_{1i}^{-1}A_{i}^{T}S_{1}A_{i}-Y_{1i}^{-1}{Q_{12}^{1m}}^{T},\tilde{\varPsi }_{16}^{i} = -Y_{1i}^{-1}Q_{12}^{1m} -Y_{1i}^{-1}Q_{22}^{1m} ,\\ \tilde{\varPsi }_{22}^{i}&= -Y_{1i}A_{i}-A_{i}^{T}Y_{1i}+E_{i}^{T}Y_{1i}E_{i}+\bar{\pi }\rho Q_{11}^{1m} +(\rho _{i}+\bar{\pi }\frac{\rho ^{2}}{2})Q_{11}^{3m}+R_{1}+R_{5}+\bar{\pi }\tau (\alpha R_{1}\\&+ \, R_{2}+R_{5}) -V_{1}+\sum _{j=1}^{N}\pi _{ij}X_{1j}+A_{fi}^{T}(Y_{1i}-X_{1i})-C_{mi}^{T}B_{fi}^{T}(Y_{1i}-X_{1i})+(Y_{1i}-X_{1i})A_{fi}\\&- \, (Y_{1i}-X_{1i})B_{fi}C_{mi}+\tau ^{2}A_{i}^{T}S_{1}A_{i}, \\ \tilde{\varPsi }_{25}^{i}&= -Y_{1i}A_{i}+(Y_{1i}-X_{1i})A_{fi}-(Y_{1i}-X_{1i})B_{fi}C_{mi}-Q_{11}^{1m}+\tau ^{2}A_{i}^{T}S_{1}A_{i},\tilde{\varPsi }_{26}^{i}= -(Y_{1i}-X_{1i})A_{fi} -Q_{12}^{1m} , \\ \tilde{\varPsi }_{33}^{i}&= -C_{i}Y_{2i}^{-1}-Y_{2i}^{-1}C_{i}^{T}+Y_{2i}^{-1}F_{i}^{T}Y_{2i}F_{i}Y_{2i}^{-1} +\bar{\pi }hY_{2i}^{-1}(Q_{11}^{2p}+Q_{12}^{2p}+{Q_{12}^{2p}}^{T}+Q_{22}^{2p})Y_{2i}^{-1}\\&+ \, (h_{i}+\bar{\pi }\frac{h^{2}}{2})Y_{2i}^{-1}(Q_{11}^{4p} +Q_{12}^{4p}+{Q_{12}^{4p}}^{T}+Q_{22}^{4p})Y_{2i}^{-1}+Y_{2i}^{-1}(R_{3}+R_{6})Y_{2i}^{-1}\\&+ \, \bar{\pi }\sigma Y_{2i}^{-1}(\beta R_{3}+R_{4}+R_{6})Y_{2i}^{-1}-Y_{2i}^{-1}V_{2}Y_{2i}^{-1} +(1+\bar{\pi }\sigma )Y_{2i}^{-1}L^{T}(T_{1}+T_{2})LY_{2i}^{-1}\\&+ \, Y_{2i}^{-1}\sum _{j=1}^{N}\pi _{ij}Y_{2j}Y_{2i}^{-1}+Y_{2i}^{-1}H_{pi}^{T}H_{pi}Y_{2i}^{-1} +\sigma ^{2}Y_{2i}^{-1}C_{i}^{T}S_{2}C_{i}Y_{2i}^{-1}, \end{aligned}$$
$$\begin{aligned} \tilde{\varPsi }_{34}^{i}&= -C_{i}-Y_{2i}^{-1}C_{i}^{T}Y_{2i}+Y_{2i}^{-1}F_{i}^{T}Y_{2i}F_{i}+\bar{\pi }hY_{2i}^{-1}(Q_{11}^{2p} +{Q_{12}^{2p}}^{T})+(h_{i}+\bar{\pi }\frac{h^{2}}{2})Y_{2i}^{-1}\\&\times (Q_{11}^{4p}+{Q_{12}^{4p}}^{T})+Y_{2i}^{-1}(R_{3}+R_{6}) +\bar{\pi }\sigma Y_{2i}^{-1}(\beta R_{3}+R_{4}+R_{6})-Y_{2i}^{-1}V_{2}\\&+ \, (1+\bar{\pi }\sigma )Y_{2i}^{-1}L^{T}(T_{1}+T_{2})L+Y_{2i}^{-1}\sum _{j=1}^{N}\pi _{ij}Y_{2j} -Y_{2i}^{-1}C_{pi}^{T}D_{fi}^{T}(Y_{2i}-X_{2i})\\&+ \, \sigma ^{2}Y_{2i}^{-1}C_{i}^{T}S_{2}C_{i},\\ \tilde{\varPsi }_{37}^{i}&= -C_{i}-Y_{2i}^{-1}Q_{11}^{2p}+\sigma ^{2}Y_{2i}^{-1}C_{i}^{T}S_{2}C_{i} -Y_{2i}^{-1}{Q_{12}^{2p}}^{T},\tilde{\varPsi }_{38}^{i} = -Y_{2i}^{-1}Q_{12}^{2p} -Y_{2i}^{-1}Q_{22}^{2p},\\ \tilde{\varPsi }_{44}^{i}&= -Y_{2i}C_{i}-C_{i}^{T}Y_{2i}+F_{i}^{T}Y_{2i}F_{i}+\bar{\pi }hQ_{11}^{2p} +(h_{i}+\bar{\pi }\frac{h^{2}}{2})Q_{11}^{4p}+R_{3}+R_{6}+\bar{\pi }\sigma (\beta R_{3}\\&+ \, R_{4}+R_{6})-V_{2} +(1+\bar{\pi }\sigma )L^{T}(T_{1}+T_{2})L+\sum _{j=1}^{N}\pi _{ij}X_{2j}+C_{fi}^{T}(Y_{2i}-X_{2i})\\&+ \, (Y_{2i}-X_{2i})C_{fi}-(Y_{2i}-X_{2i})D_{fi}C_{pi}-C_{pi}^{T}D_{fi}^{T}(Y_{2i}-X_{2i})+\sigma ^{2}C_{i}^{T}S_{2}C_{i},\\ \tilde{\varPsi }_{47}^{i}&\!=\! \!-\!Y_{2i}C_{i}\!-\!Q_{11}^{2p}\!+\!\sigma ^{2}C_{i}^{T}S_{2}C_{i}+(Y_{2i}-X_{2i})C_{fi} -(Y_{2i}-X_{2i})D_{fi}C_{pi},\tilde{\varPsi }_{48}^{i} = -(Y_{2i}-X_{2i})C_{fi} -Q_{12}^{2p},\\ \tilde{\varPsi }_{55}^{i}&\!\!= \!\!\! -\!Q_{11}^{1m}\!\!+\!\!\tau ^{2}A_{i}^{T}S_{1}A_{i},\, \tilde{\varPsi }_{77}^{i}\!=\!\!-Q_{11}^{2p}\!+\!\!\sigma ^{2}C_{i}^{T}S_{2}C_{i},\, \tilde{\varPsi }_{1,13}^{i}\!\!=\!\! -Y_{1i}^{-1}W_{1}^{T}\! +\! Y_{1i}^{-1}V_{1},\, \tilde{\varPsi }_{1,19}^{i}\!=\!B_{i}\!\!-\!\!\tau ^{2}Y_{1i}^{-1}A_{i}^{T}S_{1}B_{i},\\ \tilde{\varPsi }_{1,21}^{i}&= M_{i}-\tau ^{2}Y_{1i}^{-1}A_{i}^{T}S_{1}M_{i},\, \tilde{\varPsi }_{2,19}^{i}=Y_{1i}B_{i}-\tau ^{2}A_{i}^{T}S_{1}B_{i},\\ \tilde{\varPsi }_{2,21}^{i}&\!=\!Y_{1i}M_{i}\!-\!(Y_{1i}\!-\!X_{1i})B_{fi}G_{mi}\!-\!\tau ^{2}A_{i}^{T}S_{1}M_{i},\, \tilde{\varPsi }_{3,13}^{i}\!=\!\!D_{i}\!-\!\sigma ^{2}Y_{2i}^{-1}C_{i}^{T}S_{2}D_{i}, \tilde{\varPsi }_{3,14}^{i}= -Y_{2i}^{-1}W_{2}^{T} + Y_{2i}^{-1}V_{2},\\ \tilde{\varPsi }_{3,22}^{i}&= N_{i}-\sigma ^{2}Y_{2i}^{-1}C_{i}^{T}S_{2}N_{i},\, \tilde{\varPsi }_{4,13}^{i}=Y_{2i}D_{i}-\sigma ^{2}C_{i}^{T}S_{2}D_{i},\\ \tilde{\varPsi }_{4,22}^{i}&= Y_{2i}N_{i}-\sigma ^{2}C_{i}^{T}S_{2}N_{i}-(Y_{2i}-X_{2i})D_{fi}G_{pi},\tilde{\varPsi }_{5,19}^{i} = -\tau ^{2}A_{i}^{T}S_{1}B_{i}, \tilde{\varPsi }_{5,21}^{i} = -\tau ^{2}A_{i}^{T}S_{1}M_{i},\\ \tilde{\varPsi }_{13,13}^{i}&= -(1-\mu _{1i})R_{5}-2V_{1}+\sigma ^{2}D_{i}^{T}S_{2}D_{i}+W_{1}+W_{1}^{T},\tilde{\varPsi }_{13,22}^{i} =\sigma ^{2}D_{i}^{T}S_{2}N_{i},\\ \tilde{\varPsi }_{14,14}^{i}&= -(1-\mu _{2i})R_{6}-2V_{2}+W_{2}+W_{2}^{T},\, \tilde{\varPsi }_{19,19}^{i}=-(1-\mu _{1i})T_{1}+\tau ^{2}B_{i}^{T}S_{1}B_{i},\tilde{\varPsi }_{19,21}^{i} =\tau ^{2}B_{i}^{T}S_{1}M_{i},\\ \tilde{\varPsi }_{21,21}^{i}&= \tau ^{2}M_{i}^{T}S_{1}M_{i}-\gamma _{1}^{2}I,\, \tilde{\varPsi }_{22,22}^{i}=\sigma ^{2}N_{i}^{T}S_{2}N_{i}-\gamma _{2}^{2}I. \end{aligned}$$
Define \(\bar{A}_{fi}\,=\,(Y_{1i}-X_{1i})A_{fi},\,\bar{B}_{fi}\,=\,(Y_{1i}-X_{1i})B_{fi},\, \bar{C}_{fi}\,=\,(Y_{2i}-X_{2i})C_{fi},\,\bar{D}_{fi}\,=\,(Y_{2i}-X_{2i})D_{fi}\). Then, pre- and post-multiplying (49) by diag\(\{Y_{1i},I,Y_{2i},\underbrace{I,\cdot \cdot \cdot \cdot ,I}_{19\,terms}\}\), one can obtain the LMI (17) in Theorem 2. Hence, for \(T>0\), \(J(T)<0\). That is,
$$\begin{aligned} {\parallel z_{m}(t)-z_{mf}(t)\parallel }\le \gamma _{1}{\parallel \nu _{1}(t)\parallel },\quad {\parallel z_{p}(t)-z_{pf}(t)\parallel }\le \gamma _{2}{\parallel \nu _{2}(t)\parallel }, \end{aligned}$$
for all nonzero \(\nu _{1}(t)\) and \(\nu _{2}(t)\). Thus, the filtering error system (10) is globally stochastically stable in the mean-square sense with \(H_{\infty }\) performances \(\gamma _{1}\) and \(\gamma _{2}\) by Definition 2. This completes the proof. \(\square \)
