Theorem 1
Under Assumption 1, the given filter (3) with scalars \(\nu _{\hat{\ell }}\), fulfilling \(0<\nu _{\hat{\ell }}<1, \hat{\ell }=0,1,2\) and \(\nu _{0}+\nu _{1}+\nu _{2}=1\), is stochastically stable for any time-varying delays \(\hbar _{kp}(t)\) satisfying (5), if there exist matrices \({\mathbb{G}}>0\), \({\mathbf{S}}_{k}>0\), \({\mathbf{W}}_{k}>0\), \(\mathbf{P}_{ip}>0\), \({\mathbf{Q}}_{ki}>0\), \({\mathbf{R}}_{ki}>0\), \({\mathbf{Z}}_{ki}>0\), \({\mathbf{M}}_{ki}\) such that , \(k=1,2,3\), and the following inequalities hold:
$$\begin{aligned} &{\mathbb{G}}-\mathbf{P}_{ip}< 0, \end{aligned}$$
(6)
$$\begin{aligned} &\nabla _{a}:=\sum^{N_{r}}_{j=1}\pi _{ij} [{\mathbf{Q}}_{kj}+{ \mathbf{R}}_{kj} ]-{ \mathbf{S}}_{k}< 0, \end{aligned}$$
(7)
$$\begin{aligned} &\nabla _{b}:=\sum^{N_{r}}_{j=1}\pi _{ij}{\mathbf{R}}_{kj}-{ \mathbf{S}}_{k}< 0, \end{aligned}$$
(8)
$$\begin{aligned} &\nabla _{c}:=\sum^{N_{r}}_{j=1}\pi _{ij}{\mathbf{Z}}_{kj}-{\hbar }_{k}^{-1}{ \mathbf{W}}_{k}< 0, \end{aligned}$$
(9)
$$\begin{aligned} & \begin{bmatrix} -\nu _{0}\mathbb{G}&0&0& \bar{E} ^{T}\tilde{\exists }_{0}^{T} \\ \spadesuit & -\nu _{1}\mathbb{G}&0& \bar{E}_{\hbar _{1}}^{T} \tilde{\exists }_{0}^{T} \\ \spadesuit &\spadesuit &-\nu _{2}\mathbb{G}\mathbbm{ } &\bar{E}_{\hbar _{2}}^{T} \tilde{\exists }_{0}^{T} \\ \spadesuit &\spadesuit &\spadesuit & -I \end{bmatrix}< 0, \end{aligned}$$
(10)
$$\begin{aligned} & \begin{bmatrix} {\chi }_{ij} & \chi _{ij}^{a^{T}}& \mathbb{A} ^{T}\mathbf{P}_{ip} & \mathbb{E} ^{T}\tilde{\exists }_{1}^{T} \\ \spadesuit & -\exists _{3}-\gamma ^{2}I & \bar{B}^{T} \mathbf{P}_{ip} &0 \\ \spadesuit &\spadesuit & \varrho -2\mathbf{P}_{ip} & 0 \\ \spadesuit &\spadesuit &\spadesuit & -I \end{bmatrix}< 0, \end{aligned}$$
(11)
where \(\varrho =\sum_{k=1}^{2} (\bar{\hbar }_{k}^{2}\mathbf{Z}_{ki}+ \frac{1}{2}\bar{\hbar }_{k}^{2}\mathbf{W}_{k} )\) and
$$\begin{aligned} &{\chi }_{ij}= \begin{bmatrix} {\chi }_{ij}^{1} & {\chi }_{21} & {\mathbf{M}}_{1i} & {\chi }_{22} & { \mathbf{M}}_{2i} & \mathbf{P}_{ip}\bar{T}_{0}-a\hat{\mathcal{U}}_{2} & \mathbf{P}_{ip}\bar{T}_{1}-b\hat{\mathcal{U}}_{2} \\ \spadesuit & \chi _{31} & \mathbf{Z}_{1i} - \mathbf{M}_{1i} & 0 & 0 & 0 & 0 \\ \spadesuit & \spadesuit & -\mathbf{Z}_{1i} -\mathbf{R}_{1i} & 0 & 0 & 0 & 0 \\ \spadesuit & \spadesuit & \spadesuit & \chi _{32}& \mathbf{Z}_{2i} - \mathbf{M}_{2i} & 0 & 0 \\ \spadesuit & \spadesuit & \spadesuit & \spadesuit & -\mathbf{Z}_{2i} - \mathbf{R}_{2i} & 0 & 0 \\ \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & -a I & 0 \\ \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & -b I \end{bmatrix}, \\ &\chi _{ij}^{a} = \begin{bmatrix} -\exists _{2}^{T} \bar{E}+\bar{B}^{T} \mathbf{P}_{ip}& -\exists _{2}^{T} \bar{E}_{\hbar _{1}}& 0 & -\exists _{2}^{T} \bar{E}_{\hbar _{2}}& 0 & 0 & 0 \end{bmatrix}, \\ &\mathbb{A} = \begin{bmatrix} \bar{A}& \bar{A}_{\hbar _{1}}& 0 & \bar{A}_{\hbar _{2}}& 0 & 0& 0 \end{bmatrix},\qquad \mathbb{E}= \begin{bmatrix} \bar{E}& \bar{E}_{\hbar _{1}}& 0 & \bar{E}_{\hbar _{2}}& 0 & 0 & 0 \end{bmatrix}, \\ &\chi _{ij}^{1}= \Biggl(\sum^{N_{r}}_{j=1} \pi _{ij}\mathbf{P}_{jp} + \sum^{N_{\sigma }}_{q=1} \lambda _{pq}\mathbf{P}_{iq}+\sum _{k=1}^{2} \{\mathbf{Q}_{ki}+ \mathbf{R}_{ki}-\mathbf{Z}_{ki}+ \bar{\hbar }_{k} \mathbf{S}_{k}\} \Biggr)\\ &\phantom{\chi _{ij}^{1}=}{}+ \mathbf{P}_{ip}\bar{A} + \bar{A} ^{T} \mathbf{P}_{ip}-(a+b)\hat{\mathcal{U}}_{1}, \\ &\chi _{2k}= \mathbf{P}_{ip}\bar{A}_{\hbar _{k}} + \mathbf{Z}_{ki} - \mathbf{M}_{ki},\quad k=1,2, \\ &\chi _{3k}= -(1-\mu _{k})\mathbf{Q}_{ki}-2 \mathbf{Z}_{ki} + \mathbf{M}_{ki}+ \mathbf{M}^{T}_{ki},\quad k=1,2. \end{aligned}$$
Proof
We construct the following Lyapunov–Krasovskii functional candidate for system (3):
$$\begin{aligned} \mathcal{G}\bigl(t,\hat{m}(t),\sigma (t),r(t)\bigr) = \hat{m}(t)^{T}\mathbf{P}_{(r(t), \sigma (t))}\hat{m}(t)+\sum ^{5}_{l = 1}\mathcal{G}_{l}(t), \end{aligned}$$
(12)
where
$$\begin{aligned} &\mathcal{G}_{1}(t) = \sum_{k=1}^{2} \biggl( \int ^{t}_{t-\hbar _{k \sigma (t)}(t)} \hat{m}(\alpha )^{T} \mathbf{Q}_{kr_{t}}\hat{m}( \alpha ) \biggr)\,d\alpha , \\ &\mathcal{G}_{2}(t) = \sum_{k=1}^{2} \biggl( \int ^{t}_{t-\hbar _{k}} \hat{m}(\alpha )^{T} \mathbf{R}_{kr_{t}}\hat{m}(\alpha ) \biggr)\,d \alpha , \\ &\mathcal{G}_{3}(t) = \sum_{k=1}^{2} \biggl(\hbar _{k} \int ^{0}_{- \hbar _{k}} \int ^{t}_{t+\beta } \dot{\hat{m}}(\alpha )^{T} \mathbf{Z}_{kr_{t}} \dot{\hat{m}}(\alpha ) \biggr)\,d\alpha\, d\beta , \\ &\mathcal{G}_{4}(t) = \sum_{k=1}^{2} \biggl( \int ^{0}_{-\hbar _{k}} \int ^{t}_{t+\beta }\hat{m}(\alpha )^{T} \mathbf{S}_{k} \hat{m}(\alpha ) \biggr)\,d\alpha\, d\beta , \\ &\mathcal{G}_{5}(t) = \sum_{k=1}^{2} \biggl( \int ^{0}_{-\hbar _{k}} \int ^{0}_{\theta } \int ^{t}_{t+\beta }\dot{\hat{m}}(\alpha )^{T} \mathbf{W}_{k}\dot{\hat{m}}(\alpha ) \biggr)\,d\alpha\, d\beta \,d \theta , \end{aligned}$$
in which \(\mathbf{P}_{(r(t),\sigma (t))}\), \(\mathbf{Q}_{kr_{t}}\), \(\mathbf{R}_{kr_{t}}\), \(\mathbf{Z}_{kr_{t}}\), \(\mathbf{S}_{k}\), and \(\mathbf{Z}_{k}\) are to be determined, when \(r(t)=i\) and \(\sigma (t)=p\). Let \(\mathcal{A}\) be the weak infinitesimal generator of the random process \(\{\hat{m}(t),\sigma (t),r(t)\}\) (see, e.g., [23]). Then, by using similar techniques as those in [23, 38], we have
$$\begin{aligned} \begin{aligned} &\mathcal{A}\bigl\{ \mathcal{G}(t, \hat{m}(t),\sigma (t),r(t)\bigr\} \\ &\quad= \hat{m}(t)^{T} \Biggl(\sum ^{N_{r}}_{j=1}\pi _{ij}\mathbf{P}_{jp} + \sum^{N_{\sigma }}_{q=1}\lambda _{pq} \mathbf{P}_{iq}+\mathbf{Q}_{ki}+ \mathbf{R}_{ki}+ \bar{\hbar }_{k}\mathbf{S}_{k} \Biggr)\hat{m}(t) \\ &\qquad{} +2\hat{m}(t)^{T}\mathbf{P}_{ip}\dot{\hat{m}}(t)- \bigl(1- \bar{\hbar }_{k}(t)\bigr)\hat{m}\bigl(t-\hbar _{kp}(t) \bigr)^{T}\mathbf{Q}_{ki} \hat{m}\bigl(t-\hbar _{k}(t)\bigr) \\ &\qquad{} -\sum_{k=1}^{2}\hat{m}(t-\bar{\hbar }_{k})^{T}\mathbf{R}_{ki} \hat{m}(t-\bar{\hbar }_{k}) +\sum_{k=1}^{2}\dot{ \hat{m}}(t)^{T} \biggl( \bar{\hbar }_{k}^{2} \mathbf{Z}_{ki}+\frac{1}{2}\bar{\hbar }_{k}^{2} \mathbf{W}_{k} \biggr)\dot{\hat{m}}(t) \\ &\qquad{} -\sum_{k=1}^{2}\bar{\hbar }_{k} \int ^{t}_{t-\bar{\hbar }_{k}} \dot{\hat{m}}(\alpha )^{T} \mathbf{Z}_{ki}\dot{\hat{m}}(\alpha )\,d \alpha +\sum _{k=1}^{2} \int ^{t}_{t-\hbar _{kp}(t)}\hat{m}(\alpha )^{T} \nabla _{a}\hat{m}(\alpha )\,d\alpha \\ &\qquad{} +\sum_{k=1}^{2} \int ^{t-\hbar _{kp}(t)}_{t-\bar{\hbar }_{k}} \hat{m}(\alpha )^{T}\nabla _{b}\hat{m}(\alpha )\,d\alpha +\sum_{k=1}^{2} \bar{\hbar }_{k} \int ^{0}_{-\bar{\hbar }_{k}} \int ^{t}_{t+\beta } \dot{\hat{m}}(\alpha )^{T} \nabla _{c}\dot{\hat{m}}(\alpha )\,d\alpha \,d \beta . \end{aligned} \end{aligned}$$
(13)
Recalling (14) and applying Lemma 1 for each \(k=1,2\), we have
$$\begin{aligned} -\bar{\hbar }_{k} \int ^{t}_{t-\bar{\hbar }_{k}}\dot{m}(\alpha )^{T} \mathbf{Z}_{ki}\dot{m}(\alpha )\,d\alpha \leq \aleph (t)^{T} \mathcal{M}_{k}\aleph (t), \end{aligned}$$
(14)
where
$$\begin{aligned} &\aleph (t) = \begin{bmatrix} m(t)^{T} & m(t-{\hbar }_{kp})^{T} & m(t-\bar{\hbar }_{k})^{T} & \omega (t)^{T} \end{bmatrix}^{T}, \\ &\mathcal{M}_{k} = \begin{bmatrix} -\mathbf{Z}_{ki} & \mathbf{Z}_{ki} - \mathbf{M}_{ki} & \mathbf{M}_{ki} & 0 \\ \spadesuit & -2\mathbf{Z}_{ki} + \mathbf{M}_{ki}+ \mathbf{M}^{T}_{ki} & \mathbf{Z}_{ki}-\mathbf{M}_{ki} & 0 \\ \spadesuit & \spadesuit & -\mathbf{Z}_{ki} & 0 \\ \spadesuit & \spadesuit & \spadesuit & 0 \end{bmatrix}. \end{aligned}$$
From Assumption 1, we get
$$\begin{aligned} \begin{bmatrix} \hat{m}(t) \\ \hat{p}(\mathcal{H}(\hat{m}(t))) \end{bmatrix}^{T} \begin{bmatrix} \hat{\mathcal{U}}_{1} & \hat{\mathcal{U}}_{2} \\ \hat{\mathcal{U}}_{2} & I \end{bmatrix} \begin{bmatrix} \hat{m}(t) \\ \hat{p}(\mathcal{H}(\hat{m}(t))) \end{bmatrix} \leq 0, \end{aligned}$$
(15)
where \((\hat{\mathcal{U}}_{1},\hat{\mathcal{U}}_{2})=(\mathcal{H}^{T} \hat{\mathcal{U}}_{1}\mathcal{H},-\mathcal{H}^{T}\hat{\mathcal{U}}_{2})\). Furthermore, \((\hat{\mathcal{U}}_{1},\hat{\mathcal{U}}_{2})=( \frac{\mathcal{U}_{1}^{T}\mathcal{U}_{2}+\mathcal{U}_{2}^{T}\mathcal{U}_{1}}{2}, \frac{\mathcal{U}_{1}^{T}+\mathcal{U}_{2}^{T}}{2})\). So for the parameters \(a>0\) and \(b>0\), this yields
$$\begin{aligned} & {-}a \begin{bmatrix} \hat{m}(t) \\ \hat{p}(\mathcal{H}(\hat{m}(t))) \end{bmatrix}^{T} \begin{bmatrix} \hat{\mathcal{U}}_{1} & \hat{\mathcal{U}}_{2} \\ \hat{\mathcal{U}}_{2} & I \end{bmatrix} \begin{bmatrix} \hat{m}(t) \\ \hat{p}(\mathcal{H}(\hat{m}(t))) \end{bmatrix} \geq 0, \end{aligned}$$
(16)
$$\begin{aligned} &{-}b \begin{bmatrix} \hat{m}(t) \\ \hat{p}(\mathcal{H}(\hat{m}(t-\hbar _{1p}(t)))) \end{bmatrix}^{T} \begin{bmatrix} \hat{\mathcal{U}}_{1} & \hat{\mathcal{U}}_{2} \\ \hat{\mathcal{U}}_{2} & I \end{bmatrix} \begin{bmatrix} \hat{m}(t) \\ \hat{p}(\mathcal{H}(\hat{m}(t-\hbar _{1p}(t)))) \end{bmatrix} \geq 0. \end{aligned}$$
(17)
To simplify the notation, define
It follows from (12)–(17) that then
$$\begin{aligned} \mathcal{A}\bigl\{ \mathcal{G}(t,\hat{m}(t),\sigma (t),r(t)\bigr\} - \mathbb{J}(t) \leq \zeta (t)^{T} \tilde{\digamma }_{ij} \zeta (t), \end{aligned}$$
(18)
where
$$\begin{aligned} \tilde{\digamma }_{ij}= \begin{bmatrix} \chi _{ij} & \chi _{ij}^{a^{T}} \\ \spadesuit & -\exists _{3}-\gamma ^{2}I \end{bmatrix}+ \begin{bmatrix} \mathbb{A}^{T} \\ \mathbb{B}^{T} \end{bmatrix} \varrho \begin{bmatrix} \mathbb{A}^{T} \\ \mathbb{B}^{T} \end{bmatrix}^{T}+ \begin{bmatrix} \mathbb{E}^{T} \\ 0 \end{bmatrix}\tilde{\exists }_{1}^{T} \tilde{\exists }_{1} \begin{bmatrix} \mathbb{E}^{T} \\ 0 \end{bmatrix}^{T}. \end{aligned}$$
Note that
$$\begin{aligned} \varrho =\mathbf{P}_{ip} \bigl[ \mathbf{P}_{ip}\varrho ^{-1} \mathbf{P}_{ip} \bigr]^{-1} \mathbf{P}_{ip}\leq \mathbf{P}_{ip}[2 \mathbf{P}_{ip}- \varrho ]^{-1}\mathbf{P}_{ip}. \end{aligned}$$
Applying the Schur complement equivalence to (11) yields \(\tilde{\digamma }_{ij}<0\).
Thus, by following the same procedure as in [35], we can show that system (3) with is stochastically stable in the sense of Definition 2 [35]. □
Theorem 2
Under Assumption 1, the given filter (3) is stochastically stable in the sense of dissipative property for any time-varying delays \(\hbar _{kp}(t)\) satisfying (5), if there exist matrices \(\mathbf{P}_{ip}= \operatorname{diag}\{ \mathbf{P}_{ip_{1}} , \mathbf{P}_{ip_{2}} \} >0\), \({\mathbb{G}}>0\), \({\mathbf{S}}_{k}>0\), \({\mathbf{W}}_{k}>0\), \({\mathbf{Q}}_{ki}>0\), \({\mathbf{R}}_{ki}>0\), \({\mathbf{Z}}_{ki}>0\), \({\mathbf{M}}_{ki}\), \(\mathbb{A}_{fj}\), \(\mathbb{B}_{fj}\), \(\mathbb{C}_{fj}\), \(\mathbb{D}_{fj}\), \(\mathbb{A}_{\hbar fj}\), and \(\mathbb{C}_{\hbar fj}\) such that , \(k=1,2,3\), and the following inequalities hold:
$$\begin{aligned} & \begin{bmatrix} -\nu _{0}{\mathbb{G}}&0&0& \Xi _{ai} \tilde{\exists }_{0}^{T} \\ \spadesuit & -\nu _{1}{\mathbb{G}}&0& \Xi _{bi} \tilde{\exists }_{0}^{T} \\ \spadesuit &\spadesuit &-\nu _{2}{\mathbb{G}} &\Xi _{ci} \tilde{\exists }_{0}^{T} \\ \spadesuit &\spadesuit &\spadesuit & -I \end{bmatrix}< 0, \end{aligned}$$
(19)
$$\begin{aligned} & \begin{bmatrix} \check{\chi }_{ij}^{1} & \check{\chi }_{21} & {\mathbf{M}}_{1i} & \check{\chi }_{22} & {\mathbf{M}}_{2i} & \check{\varphi }_{1} & \check{\varphi }_{2} & \Xi _{1ij}\exists _{2}+ \Xi _{4ij} & \Xi _{1ij} ^{T} & \Xi _{ai}\tilde{\exists }_{1}^{T} \\ \spadesuit & \check{\chi }_{31} & \check{\phi }_{1} & 0 & 0 & 0 & 0 & - \Xi _{2ij}\exists _{2} & \Xi _{2ij} ^{T} & \Xi _{bi}\tilde{\exists }_{1}^{T} \\ \spadesuit & \spadesuit & \check{\phi }_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \spadesuit & \spadesuit & \spadesuit & \check{\chi }_{32}& \check{\phi }_{3} & 0 & 0 & -\Xi _{3ij}\exists _{2} & \Xi _{3ij} ^{T} & \Xi _{ci}\tilde{\exists }_{1}^{T} \\ \spadesuit & \spadesuit & \spadesuit & \spadesuit & \check{\phi }_{4} & 0 & 0 & 0 & 0 & 0 \\ \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & -a I & 0 & 0 & 0 & 0 \\ \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & -b I& 0 & 0 & 0 \\ \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & -\exists _{3}-\gamma ^{2}I & \Xi _{4ij} ^{T} &0 \\ \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & \varrho -2\mathbf{P}_{ip} & 0 \\ \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & \spadesuit & -I \end{bmatrix} < 0, \\ &\check{\chi }_{ij}^{1}= \Biggl(\sum ^{N_{r}}_{j=1}\pi _{ij} \mathbf{P}_{jp} +\sum^{N_{\sigma }}_{q=1}\lambda _{pq} \mathbf{P}_{iq}+ \sum_{k=1}^{2}\{ \mathbf{Q}_{ki}+\mathbf{R}_{ki}-\mathbf{Z}_{ki}+ \bar{\hbar }_{k}\mathbf{S}_{k}\} \Biggr) + \Xi _{1ij} + \Xi _{1ij} ^{T}-(a+b) \hat{ \mathcal{U}}_{1}, \\ &\chi _{2k}= \Xi _{(k+1)ij} + {\mathbf{Z}}_{ki} - \check{\mathbf{M}}_{ki},\quad k=1,2,\qquad \check{\varphi }_{1}= \Xi _{5ij}-a\hat{\mathcal{U}}_{2}, \qquad\check{\varphi }_{2}= \Xi _{6ij}-b\hat{\mathcal{U}}_{2}, \\ &\check{\phi }_{1} = {\mathbf{Z}}_{1i} - { \mathbf{M}}_{1i} , \qquad\check{\phi }_{2}= -{\mathbf{Z}}_{1i} -{\mathbf{R}}_{1i},\qquad \check{\phi }_{3}= { \mathbf{Z}}_{2i} - {\mathbf{M}}_{2i} ,\qquad \check{\phi }_{4}= -{\mathbf{Z}}_{2i} -{\mathbf{R}}_{2i}, \\ &\check{\chi }_{3k}= -(1-\mu _{k}){\mathbf{Q}}_{ki}-2{ \mathbf{Z}}_{ki} + {\mathbf{M}}_{ki}+ {\mathbf{M}}^{T}_{ki},\quad k=1,2, \\ &\Xi _{1ij} = \begin{bmatrix} -\mathbf{P}_{ip_{1}}A_{i} & 0 \\ \mathbb{B}_{fj}C_{i} & \mathbb{A}_{fj} \end{bmatrix},\qquad \Xi _{2ij} = \begin{bmatrix} \mathbf{P}_{ip_{1}}A_{\hbar _{1i}} & 0 \\ 0 & 0 \end{bmatrix},\qquad \Xi _{3ij} = \begin{bmatrix} 0 & 0 \\ 0 & \mathbb{A}_{\hbar fj} \end{bmatrix} , \\ &\Xi _{4ij} = \begin{bmatrix} \mathbf{P}_{ip_{1}}B_{i} \\ 0 \end{bmatrix}, \qquad\Xi _{5ij} = \begin{bmatrix} \mathbf{P}_{ip_{1}}T_{0} \\ 0 \end{bmatrix}, \qquad\Xi _{6ij} = \begin{bmatrix} \mathbf{P}_{ip_{1}}T_{1} \\ 0 \end{bmatrix} , \\ &\Xi _{aij} = \begin{bmatrix} E_{i}^{T}-C_{i}^{T}\mathbb{D}_{fj} \\ -\mathbb{C}_{fj}^{T} \end{bmatrix},\qquad \Xi _{bij} = \begin{bmatrix} E_{\hbar _{1i}}^{T} \\ 0 \end{bmatrix}, \qquad\Xi _{cij} = \begin{bmatrix} 0 \\ -\mathbb{C}_{\hbar fj}^{T} \end{bmatrix}. \end{aligned}$$
(20)
Some of the parameters are the same as mentioned in the previous theorem.
Proof
$$\begin{aligned} \mathbf{P}_{ip}= \operatorname{diag}\{ \mathbf{P}_{ip_{1}} , \mathbf{P}_{ip_{2}} \}. \end{aligned}$$
Based on the above matrix, we further define the matrix variables mentioned in Theorem 1:
$$\begin{aligned} \textstyle\begin{cases} \mathbb{A}_{fj} =\mathbf{P}_{ip_{2}}A_{fj}, \\ \mathbb{A}_{\hbar fj} =\mathbf{P}_{ip_{2}}A_{\hbar fj} , \\ \mathbb{B}_{fj} =\mathbf{P}_{ip_{2}}B_{fj}, \end{cases}\displaystyle \qquad\textstyle\begin{cases} \mathbb{C}_{fj} =C_{fj}, \\ \mathbb{C}_{\hbar fj} =C_{\hbar fj} , \\ \mathbb{D}_{fj} =D_{fj}, \end{cases}\displaystyle \end{aligned}$$
(21)
From the above, it means that all the conditions in Theorem 1 are satisfied. Therefore, by Theorem 1, the filtering error system (3) is extended dissipative for any time-varying delays \(\hbar _{kp}(t)\) which are mode dependent and satisfying (5). The proof is completed. □