Appendix A: Proof of Lemma 4
Firstly, the regularity and causality of filtering error dynamics (5) are considered. Since the matrix \( \tilde{E} \) is singular, there must exist two non-singular matrices M and N such that
$$ \tilde{E} = M\left[ {\begin{array}{*{20}l} {I_{n + r} } & 0 \\ 0 & 0 \\ \end{array} } \right]N, $$
and write
$$ M^{\text{T}} P_{i} M = \left[ {\begin{array}{*{20}l} {\hat{P}_{i1} } & {\hat{P}_{i2} } \\ {\hat{P}_{i1}^{\text{T}} } & {\hat{P}_{i3} } \\ \end{array} } \right],\quad M^{ - 1} \tilde{A}N^{ - 1} = \left[ {\begin{array}{*{20}l} {A_{1i} } & {A_{2i} } \\ {A_{3i} } & {A_{4i} } \\ \end{array} } \right]. $$
(16)
It can obtain from (8) that
$$ \tilde{A}^{\text{T}} \mathop \sum \limits_{j = 1}^{S} \pi_{ij} P_{j} \tilde{A} - \tilde{E}^{\text{T}} P_{i} \tilde{E} < 0. $$
(17)
Pre- and post-multiplying (17) by \( N^{{ - {\text{T}}}} \) and \( N^{ - 1} \), it can get that
$$ N^{{ - {\text{T}}}} \tilde{A}^{\text{T}} \mathop \sum \limits_{j = 1}^{S} \pi_{ij} P_{j} \tilde{A}N^{ - 1} - N^{{ - {\text{T}}}} \tilde{E}^{\text{T}} P_{i} \tilde{E}N^{ - 1} < 0. $$
(18)
Subscribe (16) to (18), then
$$ N^{{ - {\text{T}}}} N^{\text{T}} \left[ {\begin{array}{*{20}l} {A_{1i} } & {A_{2i} } \\ {A_{3i} } & {A_{4i} } \\ \end{array} } \right]^{\text{T}} M^{\text{T}} \mathop \sum \limits_{j = 1}^{S} \pi_{ij} P_{j} M\left[ {\begin{array}{*{20}l} {A_{1i} } & {A_{2i} } \\ {A_{3i} } & {A_{4i} } \\ \end{array} } \right]NN^{ - 1} - N^{{ - {\text{T}}}} N^{\text{T}} \left[ {\begin{array}{*{20}l} {I_{n + r} } & 0 \\ 0 & 0 \\ \end{array} } \right]^{\text{T}} M^{\text{T}} P_{i} M\left[ {\begin{array}{*{20}l} {I_{n + r} } & 0 \\ 0 & 0 \\ \end{array} } \right]NN^{ - 1} < 0, $$
which implies
$$ \left[ {\begin{array}{*{20}l} {Q_{1i} } & {Q_{2i} } \\ {Q_{2i}^{T} } & {Q_{3i} } \\ \end{array} } \right] < 0, $$
(19)
where
$$ Q_{3i} = A_{2i}^{\text{T}} \mathop \sum \limits_{j = 1}^{S} \pi_{ij} \hat{P}_{j1} A_{2i} + A_{2i}^{\text{T}} \mathop \sum \limits_{j = 1}^{S} \pi_{ij} \hat{P}_{j2} A_{4i} + A_{4i}^{\text{T}} \mathop \sum \limits_{j = 1}^{S} \pi_{ij} \hat{P}_{j2}^{\text{T}} A_{2i} + A_{4i}^{\text{T}} \mathop \sum \limits_{j = 1}^{S} \pi_{ij} \hat{P}_{j3} A_{4i} . $$
It follows from (19) that \( Q_{3i} < 0 \), note that \( \hat{P}_{j1} > 0\left( {j \in S} \right) \), which implies that \( A_{4i} \left( {i \in S} \right) \) is non-singular. Thus, by Definition 1, the inequality (8) guarantees that the filtering error system (5) is regular and causal.
Now, if the inequality (8) holds, define the following Lyapunov functional
$$ V\left( k \right) = \tilde{x}^{\text{T}} \left( k \right)\tilde{E}^{\text{T}} P_{i} \tilde{E}\tilde{x}\left( k \right). $$
Then, when \( \omega \left( k \right) = 0 \), one can get that
$$ \begin{aligned} \varepsilon \left[ {\Delta V\left( k \right)} \right] &= \varepsilon \left\{ {V(} \right.\tilde{x}\left( {k + 1} \right), r_{k + 1} )|\tilde{x}\left( k \right)\left. {r_{k} = i} \right\} - \varepsilon \left\{ {V(\tilde{x}\left( k \right)} \right.,\left. {r_{k} = i)} \right\} \\ & = \tilde{x}^{\text{T}} \left( k \right)\left\{ {\tilde{A}^{\text{T}} \mathop \sum \limits_{j = 1}^{s} \pi_{ij} P_{j} \tilde{A} - \tilde{E}^{\text{T}} P_{i} \tilde{E}} \right\} \tilde{x}\left( k \right). \\ \end{aligned} $$
It follows from (17) that
$$ \begin{aligned} \varepsilon \left[ {\Delta V\left( k \right)} \right] &= \tilde{x}^{\text{T}} \left( k \right)\left\{ {\tilde{A}^{\text{T}} \mathop \sum \limits_{j = 1}^{s} \pi_{ij} P_{j} \tilde{A} - \tilde{E}^{\text{T}} P_{i} \tilde{E}} \right\}\tilde{x}\left( k \right) \\ & < 0. \\ \end{aligned} $$
Thus, from Definition 1, the filtering error system (5) is stochastically stable.
Next, consider the following performance
$$ {\mathcal{J}} \triangleq \mathop \sum \limits_{k = 0}^{\infty } [\varepsilon (\tilde{z}\left( k \right)^{\text{T}} \tilde{z}\left( k \right)) - \gamma^{2} \omega^{\text{T}} \left( k \right)\omega \left( k \right)]. $$
Under zero-initial condition, it is easy to see
$$ \begin{aligned} {\mathcal{J}} & \le \mathop \sum \limits_{k = 0}^{\infty } [\varepsilon (\tilde{z}\left( k \right)^{\text{T}} \tilde{z}\left( k \right)) - \gamma^{2} \omega^{\text{T}} \left( k \right)\omega \left( k \right) + \varepsilon \Delta V\left( k \right)] \\ & = \mathop \sum \limits_{k = 0}^{\infty } \sigma^{\text{T}} \left( k \right)\varTheta \sigma \left( k \right), \\ \end{aligned} $$
where \( \sigma \left( k \right) = \left[ {\tilde{x}^{\text{T}} \left( {k + 1} \right)\tilde{E} ^{\text{T}} \tilde{x}^{\text{T}} \left( k \right) \omega^{\text{T}} \left( k \right)} \right]^{\text{T}} , \)
$$ \varTheta = \left[ {\begin{array}{*{20}l} 0 & 0 & 0 \\ 0 & {\tilde{A}^{\text{T}} \mathop \sum \limits_{j = 1}^{S} \pi_{ij} P_{j} \tilde{A} - \tilde{E}^{\text{T}} P_{i} \tilde{E} + \tilde{L}^{\text{T}} \tilde{L}} & 0 \\ 0 & 0 & { - \gamma^{2} I} \\ \end{array} } \right]. $$
In addition, the system (5) implies
$$ \left[ {\begin{array}{*{20}l} { - I} & {\tilde{A}} & {\tilde{B}} \\ \end{array} } \right]\sigma \left( k \right) = 0. $$
Set
$$ B = \left[ {\begin{array}{*{20}l} { - I} & {\tilde{A}} & {\tilde{B}} \\ \end{array} } \right]\quad \quad {\mathcal{X}} = \left[ {\begin{array}{*{20}l} G \\ 0 \\ F \\ \end{array} } \right], $$
Then
$$ \varTheta + {\mathcal{X}}B + B^{\text{T}} {\mathcal{X}}^{\text{T}} \, = \,\left[ {\begin{array}{*{20}l} { - G - G^{\text{T}} } & {G\tilde{A}} & { - F^{\text{T}} + G\tilde{B}} \\ * & {\tilde{A}^{\text{T}} \mathop \sum \limits_{j = 1}^{s} \pi_{ij} P_{j} \tilde{A} - \tilde{E}^{\text{T}} P_{i} \tilde{E} + \tilde{L}^{\text{T}} \tilde{L}} & {\tilde{A}^{\text{T}} F^{\text{T}} } \\ * & * & {\tilde{B}^{\text{T}} F^{\text{T}} + F\tilde{B} - \gamma^{2} I} \\ \end{array} } \right]. $$
Since
$$ \left[ {\begin{array}{*{20}l} 0 & 0 & 0 \\ 0 & {\tilde{L}^{\text{T}} \tilde{L}} & 0 \\ 0 & 0 & 0 \\ \end{array} } \right] = - \left[ {\begin{array}{*{20}l} 0 \\ {\tilde{L}^{\text{T}} } \\ 0 \\ \end{array} } \right]\left( { - I} \right)\left[ {\begin{array}{*{20}l} 0 & {\tilde{L}} & 0 \\ \end{array} } \right], $$
From the Schur complement formula, the inequality (8) is equivalent to \( \varTheta + {\mathcal{X}}B + B^{\text{T}} {\mathcal{X}}^{\text{T}} < 0 \).
By Lemma 1, \( \varTheta + {\mathcal{X}}B + B^{\text{T}} {\mathcal{X}}^{\text{T}} < 0 \) is equivalent to
$$ \sigma^{\text{T}} \left( k \right)\varTheta \sigma \left( k \right) < 0. $$
Hence, \( {\mathcal{J}} \) < 0, that is to say, \( \varepsilon \left\{ {z^{\text{T}} \left( k \right)z\left( k \right)} \right\} \le \gamma^{2} \omega^{\text{T}} \left( k \right)\omega \left( k \right) \), the \( H_{\infty } \) performance is satisfied. To sum up, the condition (8) can guarantee the filtering error system (5) is stochastically stable with a prescribed \( H_{\infty } \) performance \( \gamma \). This completes the proof of Lemma 4.
Appendix B: Proof of Theorem 1
Case I If \( i \notin U_{k}^{i} \), \( U_{k}^{i} = \left\{ {k_{1}^{i} } \right. \), \( k_{2}^{i} \),…,\( \left. {k_{m}^{i} } \right\} \). In this case, \( P_{j} - P_{i} \le 0 \) (\( j \in U_{uk}^{i} \), \( j \ne i \)) and \( \sum\nolimits_{{j \in U_{uk}^{i} ,j \ne i}} {\pi_{ij} } + \sum\nolimits_{{j \in U_{k}^{i} }} {\pi_{ij} + \pi_{ii} } = 1 \), then one has
$$ \begin{aligned} \mathop \sum \limits_{j = 1}^{S} \pi_{ij} P_{j} &= \mathop \sum \limits_{{j \in U_{k}^{i} }} \pi_{ij} P_{j} + \pi_{ii} P_{i} + \mathop \sum \limits_{{j \in U_{uk}^{i} ,j \ne i}} \pi_{ij} P_{j} = \mathop \sum \limits_{{j \in U_{k}^{i} }} \pi_{ij} P_{j} + \pi_{ii} P_{i} + \left( {1 - \pi_{ii} - \mathop \sum \limits_{{j \in U_{k}^{i} }} \pi_{ij} } \right)P_{j} \\ & \le \mathop \sum \limits_{{j \in U_{k}^{i} }} \pi_{ij} P_{j} + \pi_{ii} P_{i} + \left( {1 - \mathop \sum \limits_{{j \in U_{k}^{i} }} \pi_{ij} - \pi_{ii} } \right)P_{i} = \mathop \sum \limits_{{j \in U_{k}^{i} }} \pi_{ij} \left( {P_{j} - P_{i} } \right) + P_{i} \\ & = \mathop \sum \limits_{{j \in U_{k}^{i} }} (\hat{\pi }_{ij} + \Delta_{ij} )\left( {P_{j} - P_{i} } \right) + P_{i} = \mathop \sum \limits_{{j \in U_{k}^{i} }} \hat{\pi }_{ij} \left( {P_{j} - P_{i} } \right) + \mathop \sum \limits_{{j \in U_{k}^{i} }} \Delta_{ij} \left( {P_{j} - P_{i} } \right) + P_{i} . \\ \end{aligned} $$
It follows from Lemma 2 that
$$ \begin{aligned} & \mathop \sum \limits_{{j \in U_{k}^{i} }} \Delta_{ij} \left( {P_{j} - P_{i} } \right) = \mathop \sum \limits_{{j \in U_{k}^{i} }} \left[ {\frac{1}{2}\Delta_{ij} \left( {P_{j} - P_{i} } \right) + \frac{1}{2}\Delta_{ij} \left( {P_{j} - P_{i} } \right)} \right] \\ & \quad \le \mathop \sum \limits_{{j \in U_{k}^{i} }} \left[ {\frac{{\delta_{ij}^{2} }}{4}T_{ij} + \left( {P_{j} - P_{i} } \right)^{\text{T}} T_{ij}^{ - 1} \left( {P_{j} - P_{i} } \right)} \right]. \\ \end{aligned} $$
Thus
$$ \begin{aligned} & \mathop \sum \limits_{j = 1}^{S} \pi_{ij} P_{j} \le \mathop \sum \limits_{{j \in U_{k}^{i} }} \hat{\pi }_{ij} \left( {P_{j} - P_{i} } \right) + \mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}T_{ij} + \mathop \sum \limits_{{j \in U_{k}^{i} }} \left( {P_{j} - P_{i} } \right)^{\text{T}} T_{ij}^{ - 1} \left( {P_{j} - P_{i} } \right) + P_{i} , \\ & \tilde{A}^{\text{T}} \mathop \sum \limits_{j = 1}^{S} \pi_{ij} P_{j} \tilde{A} \le \tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \hat{\pi }_{ij} \left( {P_{j} - P_{i} } \right)\tilde{A} + \tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}T_{ij} \tilde{A} + \tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \left( {P_{j} - P_{i} } \right)^{\text{T}} T_{ij}^{ - 1} \left( {P_{j} - P_{i} } \right)\tilde{A} + \tilde{A}^{\text{T}} P_{i} \tilde{A}. \\ \end{aligned} $$
(20)
By Lemma 4, the following is immediate
$$ \begin{aligned} & \theta \le \beta_{1} = \left[ {\begin{array}{*{20}l} { - G - G^{\text{T}} } & {G\tilde{A}} & { - F^{\text{T}} + G\tilde{B}} & 0 \\ * & {\theta_{1} + \mathop \sum \limits_{{j \in U_{k}^{i} }} \tilde{A}^{\text{T}} \left( {P_{j} - P_{i} } \right)^{\text{T}} T_{ij}^{ - 1} \left( {P_{j} - P_{i} } \right)\tilde{A}} & {\tilde{A}^{\text{T}} F^{\text{T}} } & {\tilde{L}^{\text{T}} } \\ * & * & {\tilde{B}^{\text{T}} F^{\text{T}} + F\tilde{B} - \gamma^{2} I} & 0 \\ * & * & * & { - I} \\ \end{array} } \right], \\ & \theta_{1} = - \tilde{E}^{\text{T}} P_{i} \tilde{E} + \tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \hat{\pi }_{ij} \left( {P_{j} - P_{i} } \right)\tilde{A} + \tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}T_{ij} \tilde{A} + \tilde{A}^{\text{T}} P_{i} \tilde{A}, \\ & \beta_{1} = \left[ {\begin{array}{*{20}l} { - G - G^{\text{T}} } & {G\tilde{A}} & { - F^{\text{T}} + G\tilde{B}} & 0 \\ * & {\theta_{1} } & {\tilde{A}^{\text{T}} F^{\text{T}} } & {\tilde{L}^{\text{T}} } \\ * & * & {\tilde{B}^{\text{T}} F^{\text{T}} + F\tilde{B} - \gamma^{2} I} & 0 \\ * & * & * & { - I} \\ \end{array} } \right] \\ & \quad - \left[ {\begin{array}{*{20}l} 0 & \cdots & 0 \\ {\tilde{A}^{\text{T}} \left( {P_{{k_{1}^{i} }} - P_{i} } \right)^{\text{T}} } & \cdots & {\tilde{A}^{\text{T}} \left( {P_{{k_{m}^{i} }} - P_{i} } \right)^{\text{T}} } \\ 0 & \cdots & 0 \\ 0 & \cdots & 0 \\ \end{array} } \right] \,\, \left[ {\begin{array}{*{20}l} { - T_{{ik_{1}^{i} }}^{ - 1} } & 0 & \cdots & 0 \\ 0 & { - T_{{ik_{2}^{i} }}^{ - 1} } & \cdots & 0 \\ \vdots & \cdots & \ddots & \vdots \\ 0 & 0 & \cdots & { - T_{{ik_{m}^{i} }}^{ - 1} } \\ \end{array} } \right]\,\,\,\left[ {\begin{array}{*{20}l} 0 & \cdots & 0 \\ {\tilde{A}^{\text{T}} \left( {P_{{k_{1}^{i} }} - P_{i} } \right)^{\text{T}} } & \cdots & {\tilde{A}^{\text{T}} \left( {P_{{k_{m}^{i} }} - P_{i} } \right)^{\text{T}} } \\ 0 & \cdots & 0 \\ 0 & \cdots & 0 \\ \end{array} } \right]^{\text{T}} . \\ \end{aligned} $$
(21)
Using Schur complement formula, the condition (9) is equal to \( \beta_{1} < 0 \). Then, it follows from (21) that \( \theta \le \beta_{1} \) < 0. Thus, according to Lemma 4, the filtering error system (5) with GUTRs is stochastically admissible with a prescribed \( H_{\infty } \) performance index \( \gamma \). Thus, the proof of Case I is completed.
Case II If \( i \in U_{k}^{i} \), \( U_{uk}^{i} \ne \emptyset \) and \( U_{k}^{i} = \left\{ {k_{1}^{i} } \right. \), \( k_{2}^{i} \),…,\( \left. {k_{m}^{i} } \right\} \). There must be an \( l \in U_{uk}^{i} \) for \( \forall j \in U_{uk}^{i} \), \( P_{l} \ge P_{j} \). Let
$$ \mathop \sum \limits_{j = 1}^{S} \pi_{ij} P_{j} = \mathop \sum \limits_{{j \in U_{k}^{i} }} \pi_{ij} P_{j} + \mathop \sum \limits_{{j \in U_{uk}^{i} }} \pi_{ij} P_{j} . $$
Since \( \sum\nolimits_{j = 1}^{s} {\pi_{ij} } = 1 \), then \( \sum\nolimits_{{j \in U_{uk}^{i} }} {\pi_{ij} } = 1 - \sum\nolimits_{{j \in U_{k}^{i} }} {\pi_{ij} } \). Similar to Case I, one can derive that
$$ \begin{aligned} \mathop \sum \limits_{j = 1}^{S} \pi_{ij} P_{j} & \le \mathop \sum \limits_{{j \in U_{k}^{i} }} \pi_{ij} P_{j} + \left( {1 - \mathop \sum \limits_{{j \in U_{uk}^{i} }} \pi_{ij} } \right)P_{l} = \mathop \sum \limits_{{j \in U_{k}^{i} }} \pi_{ij} P_{j} - \mathop \sum \limits_{{j \in U_{k}^{i} }} \pi_{ij} P_{l} + P_{l} \\ & = \mathop \sum \limits_{{j \in U_{k}^{i} }} \pi_{ij} \left( {P_{j} - P_{l} } \right) + P_{l} = \mathop \sum \limits_{{j \in U_{k}^{i} }} (\hat{\pi }_{ij} + \Delta_{ij} )\left( {P_{j} - P_{l} } \right) + P_{l} \\ & = \mathop \sum \limits_{{j \in U_{k}^{i} }} \hat{\pi }_{ij} \left( {P_{j} - P_{l} } \right) + \mathop \sum \limits_{{j \in U_{k}^{i} }} \Delta_{ij} \left( {P_{j} - P_{l} } \right) + P_{l} \\ & \le \mathop \sum \limits_{{j \in U_{k}^{i} }} \hat{\pi }_{ij} \left( {P_{j} - P_{l} } \right) + \mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}V_{ijl} + \mathop \sum \limits_{{j \in U_{k}^{i} }} \left( {P_{j} - P_{l} } \right)^{\text{T}} V_{ijl}^{ - 1} \left( {P_{j} - P_{l} } \right) + P_{l} . \\ \end{aligned} $$
Thus
$$ \tilde{A}^{\text{T}} \mathop \sum \limits_{j = 1}^{S} \pi_{ij} P_{j} \tilde{A} \le \tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \hat{\pi }_{ij} \left( {P_{j} - P_{l} } \right)\tilde{A} + \tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}V_{ijl} \tilde{A} + \tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \left( {P_{j} - P_{l} } \right)^{\text{T}} V_{ijl}^{ - 1} \left( {P_{j} - P_{l} } \right)\tilde{A} + \tilde{A}^{\text{T}} P_{l} \tilde{A}. $$
(22)
It follows from Lemma 4 that
$$ \begin{aligned} & \theta \le \beta_{2} = \left[ {\begin{array}{*{20}l} { - G - G^{\text{T}} } & {G\tilde{A}} & { - F^{\text{T}} + G\tilde{B}} & 0 \\ * & {\theta_{2} + \mathop \sum \limits_{{j \in U_{k}^{i} }} \tilde{A}^{\text{T}} \left( {P_{j} - P_{i} } \right)^{\text{T}} V_{ijl}^{ - 1} \left( {P_{j} - P_{i} } \right)\tilde{A}} & {\tilde{A}^{\text{T}} F^{\text{T}} } & {\tilde{L}^{\text{T}} } \\ * & * & {\tilde{B}^{\text{T}} F^{\text{T}} + F\tilde{B} - \gamma^{2} I} & 0 \\ * & * & * & { - I} \\ \end{array} } \right], \\ & \theta_{2} = - \tilde{E}^{\text{T}} P_{i} \tilde{E} + \tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \hat{\pi }_{ij} \left( {P_{j} - P_{l} } \right)\tilde{A} + \tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}V_{ijl} \tilde{A} + \tilde{A}^{\text{T}} P_{l} \tilde{A}. \\ \end{aligned} $$
(23)
Similar to Case I, the condition (10) is equal to \( \beta_{2} < 0 \), and \( \theta \le \beta_{2} \) < 0 holds. Hence, the condition (8) in Lemma 4 is satisfied. Thus, the proof of Case II is completed.
Case III If \( i \in U_{k}^{i} \), \( U_{uk}^{i} = \emptyset \), similar to Case I and Case II, one can get that
$$ \begin{aligned} \mathop \sum \limits_{j = 1}^{S} \pi_{ij} P_{j} &= \mathop \sum \limits_{j = 1,j \ne i}^{S} \pi_{ij} P_{j} + \pi_{ii} P_{i} = \mathop \sum \limits_{j = 1,j \ne i}^{S} \pi_{ij} (P_{j} - P_{i} ) + P_{i} = \mathop \sum \limits_{j = 1,j \ne i}^{S} (\hat{\pi }_{ij} + \Delta_{ij} )(P_{j} - P_{i} ) + P_{i} \\ & \le \mathop \sum \limits_{j = 1,j \ne i}^{S} \hat{\pi }_{ij} (P_{j} - P_{i} ) + \mathop \sum \limits_{j = 1,j \ne i}^{S} \frac{{\delta_{ij}^{2} }}{4}R_{ij} + \mathop \sum \limits_{j = 1,j \ne i}^{S} \left( {P_{j} - P_{i} } \right)^{\text{T}} R_{ij}^{ - 1} \left( {P_{j} - P_{i} } \right) + P_{i} , \\ & \tilde{A}^{\text{T}} \mathop \sum \limits_{j = 1}^{S} \pi_{ij} P_{j} \tilde{A} \le \tilde{A}^{\text{T}} \mathop \sum \limits_{j = 1,j \ne i}^{S} \hat{\pi }_{ij} (P_{j} - P_{i} )\tilde{A} + \tilde{A}^{\text{T}} \mathop \sum \limits_{j = 1,j \ne i}^{S} \frac{{\delta_{ij}^{2} }}{4}R_{ij} \tilde{A} + \tilde{A}^{\text{T}} \mathop \sum \limits_{j = 1,j \ne i}^{S} \left( {P_{j} - P_{i} } \right)^{\text{T}} R_{ij}^{ - 1} \left( {P_{j} - P_{i} } \right)\tilde{A} + \tilde{A}^{\text{T}} P_{i} \tilde{A}. \\ \end{aligned} $$
(24)
From Lemma 4
$$ \begin{aligned} & \theta \le \beta_{3} = \left[ {\begin{array}{*{20}l} { - G - G^{\text{T}} } & {G\tilde{A}} & { - F^{T} + G\tilde{B}} & 0 \\ * & {\theta_{3} + \mathop \sum \limits_{j = 1,j \ne i}^{s} \tilde{A}^{\text{T}} \left( {P_{j} - P_{i} } \right)^{\text{T}} R_{ij}^{ - 1} \left( {P_{j} - P_{i} } \right)\tilde{A}} & {\tilde{A}^{\text{T}} F^{\text{T}} } & {\tilde{L}^{\text{T}} } \\ * & * & {\tilde{B}^{\text{T}} F^{\text{T}} + F\tilde{B} - \gamma^{2} I} & 0 \\ * & * & * & { - I} \\ \end{array} } \right], \\ & \theta_{3} = - \tilde{E}^{\text{T}} P_{i} \tilde{E} + \tilde{A}^{\text{T}} \mathop \sum \limits_{j = 1,j \ne i}^{s} \hat{\pi }_{ij} \left( {P_{j} - P_{i} } \right)\tilde{A} + \tilde{A}^{\text{T}} \mathop \sum \limits_{j = 1,j \ne i}^{s} \frac{{\delta_{ij}^{2} }}{4}R_{ij} \tilde{A} + \tilde{A}^{\text{T}} P_{i} \tilde{A}. \\ \end{aligned} $$
(25)
Similar to Case I and Case II, the condition (11) is equal to \( \beta_{3} < 0 \), and then \( \theta \le \beta_{3} < 0 \) holds. Hence, the condition (8) in Lemma 4 is satisfied. Thus, the proof of Case III is completed.
In conclusion, it completes the proof of Theorem 1.
Appendix C: Proof of Theorem 2
Choose the structure of \( G \), \( F \) and \( P_{i} \) in (8) as follows:
$$ G = \left[ {\begin{array}{*{20}l} {G_{11} } & {G_{22} } \\ {G_{21} } & {G_{22} } \\ \end{array} } \right],\quad F = \left[ {\begin{array}{*{20}l} {F_{11} } & {G_{22} } \\ {F_{21} } & {G_{22} } \\ \end{array} } \right],\quad P_{i} = \left[ {\begin{array}{*{20}l} {P_{i1} } & 0 \\ 0 & {P_{i2} } \\ \end{array} } \right], $$
(26)
where \( P_{i2} = \alpha_{i} G_{22} \left( {i \in U_{k}^{i} } \right) \).
Case I From the formulas (6), (15) and (26), one has
$$ \begin{aligned} & - G - G^{\text{T}} = \left[ {\begin{array}{*{20}l} { - G_{11} - G_{11}^{\text{T}} } & { - G_{22} - G_{21}^{\text{T}} } \\ * & { - G_{22} - G_{22}^{\text{T}} } \\ \end{array} } \right],\quad - \tilde{F} + G\tilde{B} = \left[ {\begin{array}{*{20}l} { - F_{11}^{\text{T}} + G_{11} B_{i} + b_{\mathrm{fi}} D_{i} } & { - F_{21}^{\text{T}} } \\ { - G_{22}^{\text{T}} + G_{21} B_{i} + b_{\mathrm{fi}} D_{i} } & { - G_{22}^{\text{T}} } \\ \end{array} } \right], \\ & G\tilde{A} = \left[ {\begin{array}{*{20}l} {G_{11} A_{i} + b_{\mathrm{fi}} C_{i} } & {a_{\mathrm{fi}} } \\ {G_{21} A_{i} + b_{\mathrm{fi}} C_{i} } & {a_{\mathrm{fi}} } \\ \end{array} } \right],\quad \tilde{L}^{\text{T}} = \left[ {\begin{array}{*{20}l} {L_{i}^{\text{T}} } \\ { - l_{\mathrm{fi}}^{\text{T}} } \\ \end{array} } \right],\quad \tilde{A}^{\text{T}} F^{\text{T}} = \left[ {\begin{array}{*{20}l} {A_{i}^{\text{T}} F_{11}^{\text{T}} + C_{i}^{\text{T}} b_{\mathrm{fi}}^{\text{T}} } & {A_{i}^{\text{T}} F_{21}^{\text{T}} + C_{i}^{\text{T}} b_{\mathrm{fi}}^{\text{T}} } \\ {a_{\mathrm{fi}}^{\text{T}} } & {a_{\mathrm{fi}}^{\text{T}} } \\ \end{array} } \right], \\ & \tilde{A}^{\text{T}} G^{\text{T}} = \left[ {\begin{array}{*{20}l} {A_{i}^{\text{T}} G_{11}^{\text{T}} + C_{i}^{\text{T}} b_{\mathrm{fi}}^{\text{T}} } & {A_{i}^{\text{T}} G_{21}^{\text{T}} + C_{i}^{\text{T}} b_{\mathrm{fi}}^{\text{T}} } \\ {a_{\mathrm{fi}}^{\text{T}} } & {a_{\mathrm{fi}}^{\text{T}} } \\ \end{array} } \right],\quad T_{ij} = \left[ {\begin{array}{*{20}l} {T_{ij1} } & {T_{ij2} } \\ * & {T_{ij3} } \\ \end{array} } \right], \\ & \tilde{A}^{\text{T}} \left( {P_{k} - P_{i} } \right)^{\text{T}} = \left[ {\begin{array}{*{20}l} {A_{i}^{\text{T}} \left( {P_{k1} - P_{i1} } \right)^{\text{T}} } & {\left( {\alpha_{k} - \alpha_{i} } \right)C_{i}^{\text{T}} b_{\mathrm{fi}}^{\text{T}} } \\ 0 & {\left( {\alpha_{k} - \alpha_{i} } \right)a_{\mathrm{fi}}^{\text{T}} } \\ \end{array} } \right],(k \in U_{k}^{i} )\quad \tilde{A}^{\text{T}} P_{i}^{\text{T}} = \left[ {\begin{array}{*{20}l} {A_{i}^{\text{T}} P_{i1}^{\text{T}} } & {\alpha_{i} C_{i}^{\text{T}} b_{\mathrm{fi}}^{\text{T}} } \\ 0 & {\alpha_{i} a_{\mathrm{fi}}^{\text{T}} } \\ \end{array} } \right], \\ & \tilde{B}^{\text{T}} F^{\text{T}} + F\tilde{B} - \gamma^{2} I = \left[ {\begin{array}{*{20}l} {B_{i}^{\text{T}} F_{11}^{\text{T}} + D_{i}^{\text{T}} b_{\mathrm{fi}}^{\text{T}} + F_{11} B_{i} + b_{\mathrm{fi}} D_{i} - \gamma^{2} I} \hfill & {B_{i}^{\text{T}} F_{21}^{\text{T}} + D_{i}^{\text{T}} b_{\mathrm{fi}}^{\text{T}} } \hfill \\ * \hfill & { - \gamma^{2} I} \hfill \\ \end{array} } \right], \\ & \mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}T_{ij} - G - G^{\text{T}} = \left[ {\begin{array}{*{20}l} {\mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}T_{ij1} - G_{11} - G_{11}^{\text{T}} } \hfill & {\mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}T_{ij2} - G_{22} - G_{21}^{\text{T}} } \hfill \\ * \hfill & {\mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}T_{ij3} - G_{22} - G_{22}^{\text{T}} } \hfill \\ \end{array} } \right]. \\ \end{aligned} $$
So, the inequality (12) can be rewritten as
$$ \left[ {\begin{array}{*{20}l} { - G - G^{\text{T}} } \hfill & {G\tilde{A}} \hfill & { - \tilde{F} + G\tilde{B}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & \ldots \hfill & 0 \hfill \\ * \hfill & {\varOmega_{22} } \hfill & {\tilde{A}^{\text{T}} F^{\text{T}} } \hfill & {\tilde{L}^{\text{T}} } \hfill & {\tilde{A}^{\text{T}} G^{\text{T}} } \hfill & {\tilde{A}^{\text{T}} P^{\text{T}} } \hfill & {\tilde{A}^{\text{T}} \left( {P_{{k_{1}^{i} }} - P_{i} } \right)^{\text{T}} } \hfill & \ldots \hfill & {\tilde{A}^{\text{T}} \left( {P_{{k_{m}^{i} }} - P_{i} } \right)^{\text{T}} } \hfill \\ * \hfill & * \hfill & {\tilde{B}^{\text{T}} F^{\text{T}} + F\tilde{B} - \gamma^{2} I} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & \ldots \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & { - I} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & \ldots \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & {\theta^{\prime } } \hfill & 0 \hfill & 0 \hfill & \ldots \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & { - P_{i} } \hfill & 0 \hfill & \ldots \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & { - T_{{ik_{1}^{i} }} } \hfill & \ldots \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & \ddots \hfill & \vdots \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & { - T_{{ik_{1}^{i} }} } \hfill \\ \end{array} } \right] < 0, $$
(27)
where \( \theta^{\prime } = \sum\nolimits_{{j \in U_{k}^{i} }} {\frac{{\delta_{ij}^{2} }}{4}T_{ij} - G - G^{\text{T}} } \).
Substituting (6) and (26) into \( \tilde{A}^{\text{T}} \left( {P_{j} - P_{i} } \right)\tilde{A} \) yields
$$ \begin{aligned} & \tilde{A}^{\text{T}} \left( {P_{j} - P_{i} } \right)\tilde{A} = \left[ {\begin{array}{*{20}l} {A_{i} } & 0 \\ {B_{\mathrm{fi}} C_{i} } & {A_{\mathrm{fi}} } \\ \end{array} } \right]^{\text{T}} \left[ {\begin{array}{*{20}l} {P_{j1} - P_{i1} } & 0 \\ 0 & {P_{j2} - P_{i2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {A_{i} } & 0 \\ {B_{\mathrm{fi}} C_{i} } & {A_{\mathrm{fi}} } \\ \end{array} } \right] \\ & = \left[ {\begin{array}{*{20}l} {A_{i}^{\text{T}} (P_{j1} - P_{i1} )A_{i} + C_{i}^{\text{T}} B_{\mathrm{fi}}^{\text{T}} (P_{j2} - P_{i2} )B_{\mathrm{fi}} C_{i} } \hfill & {C_{i}^{\text{T}} B_{\mathrm{fi}}^{\text{T}} (P_{j2} - P_{i2} )A_{\mathrm{fi}} } \hfill \\ {A_{\mathrm{fi}}^{\text{T}} (P_{j2} - P_{i2} )B_{\mathrm{fi}} C_{i} } \hfill & {A_{\mathrm{fi}}^{\text{T}} (P_{j2} - P_{i2} )A_{\mathrm{fi}} } \hfill \\ \end{array} } \right] \\ & = \left[ {\begin{array}{*{20}l} {A_{i}^{\text{T}} (P_{j1} - P_{i1} )A_{i} } & 0 \\ 0 & 0 \\ \end{array} } \right] + \left[ {\begin{array}{*{20}l} {C_{i}^{\text{T}} B_{\mathrm{fi}}^{\text{T}} (P_{j2} - P_{i2} )B_{\mathrm{fi}} C_{i} } & {C_{i}^{\text{T}} B_{\mathrm{fi}}^{\text{T}} (P_{j2} - P_{i2} )A_{\mathrm{fi}} } \\ {A_{\mathrm{fi}}^{\text{T}} (P_{j2} - P_{i2} )B_{\mathrm{fi}} C_{i} } & {A_{\mathrm{fi}}^{\text{T}} (P_{j2} - P_{i2} )A_{\mathrm{fi}} } \\ \end{array} } \right] \\ & = \left[ {\begin{array}{*{20}l} {A_{i}^{\text{T}} (P_{j1} - P_{i1} )A_{i} } & 0 \\ 0 & 0 \\ \end{array} } \right] + \left[ {\begin{array}{*{20}l} {C_{i}^{\text{T}} B_{\mathrm{fi}}^{\text{T}} } \\ {A_{\mathrm{fi}}^{\text{T}} } \\ \end{array} } \right](P_{j2} - P_{i2} )\left[ {\begin{array}{*{20}l} {B_{\mathrm{fi}} C_{i} } & {A_{\mathrm{fi}} } \\ \end{array} } \right]. \\ \end{aligned} $$
Since \( P_{j} - P_{i} \le 0 \), then \( (P_{j2} - P_{i2} ) \le 0 \), which implies \( \left[ {\begin{array}{*{20}l} {C_{i}^{\text{T}} B_{\mathrm{fi}}^{\text{T}} } \\ {A_{\mathrm{fi}}^{\text{T}} } \\ \end{array} } \right](P_{j2} - P_{i2} )\left[ {\begin{array}{*{20}l} {B_{\mathrm{fi}} C_{i} } & {A_{\mathrm{fi}} } \\ \end{array} } \right] \le 0 \), so
$$ \tilde{A}^{\text{T}} \left( {P_{j} - P_{i} } \right)\tilde{A} \le \left[ {\begin{array}{*{20}l} {A_{i}^{\text{T}} (P_{j1} - P_{i1} )A_{i} } & 0 \\ 0 & 0 \\ \end{array} } \right]. $$
(28)
In addition,
$$ - \tilde{E}^{\text{T}} P_{i} \tilde{E} = - \left[ {\begin{array}{*{20}l} {E^{\text{T}} } & 0 \\ 0 & I \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {P_{i1} } & 0 \\ 0 & {P_{i2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} E & 0 \\ 0 & I \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} { - E^{\text{T}} P_{i1} E} & 0 \\ 0 & { - P_{i2} } \\ \end{array} } \right]. $$
(29)
It can be obtained from (28) and (29) that
$$ - \tilde{E}^{\text{T}} P_{i} \tilde{E} + \tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \hat{\pi }_{ij} \left( {P_{j} - P_{i} } \right)\tilde{A} \le \left[ {\begin{array}{*{20}l} { - E^{\text{T}} P_{i1} E + A_{i}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \hat{\pi }_{ij} \left( {P_{j1} - P_{i1} } \right)A_{i} } & 0 \\ 0 & { - P_{i2} } \\ \end{array} } \right] = \varOmega_{22} . $$
(30)
From Lemma 3, one can get that
$$ - G\left( {\mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}T_{ij} } \right)^{ - 1} G^{\text{T}} \le \mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}T_{ij} - G - G^{\text{T}} . $$
(31)
It follows from (30), (31) that
$$ \left[ {\begin{array}{*{20}l} { - G - G^{\text{T}} } \hfill & {G\tilde{A}} \hfill & { - \tilde{F} + G\tilde{B}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & \ldots \hfill & 0 \hfill \\ * \hfill & {\theta^{\prime \prime } } \hfill & {\tilde{A}^{\text{T}} F^{\text{T}} } \hfill & {\tilde{L}^{\text{T}} } \hfill & {\tilde{A}^{\text{T}} G^{\text{T}} } \hfill & {\tilde{A}^{\text{T}} P^{\text{T}} } \hfill & {\tilde{A}^{\text{T}} \left( {P_{{k_{1}^{i} }} - P_{i} } \right)^{\text{T}} } \hfill & \ldots \hfill & {\tilde{A}^{\text{T}} \left( {P_{{k_{m}^{i} }} - P_{i} } \right)^{\text{T}} } \hfill \\ * \hfill & * \hfill & {\tilde{B}^{\text{T}} F^{\text{T}} + F\tilde{B} - \gamma^{2} I} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & \ldots \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & { - I} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & \ldots \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & {\theta^{\prime \prime \prime } } \hfill & 0 \hfill & 0 \hfill & \ldots \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & { - P_{i} } \hfill & 0 \hfill & \ldots \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & { - T_{{ik_{1}^{i} }} } \hfill & \ldots \hfill & 0 \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & \ddots \hfill & \vdots \hfill \\ * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & * \hfill & { - T_{{ik_{1}^{i} }} } \hfill \\ \end{array} } \right] < 0, $$
(32)
where \( \theta^{\prime \prime } = - \tilde{E}^{\text{T}} P_{i} \tilde{E} + \tilde{A}^{\text{T}} \sum\nolimits_{{j \in U_{k}^{i} }} {\hat{\pi }_{ij} \left( {P_{j} - P_{i} } \right)\tilde{A}} \), \( \theta^{\prime \prime \prime } = - G\left( {\sum\nolimits_{{j \in U_{k}^{i} }} {\frac{{\delta_{ij}^{2} }}{4}T_{ij} } } \right)^{ - 1} G^{\text{T}} \).
For each \( j \in U_{k}^{i} \), it is obvious that
$$ \begin{aligned} & \left[ {\begin{array}{*{20}l} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {\tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \left( {P_{j} - P_{i} } \right)^{\text{T}} T_{ij}^{ - 1} \left( {P_{j} - P_{i} } \right)\tilde{A}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ & = - \left[ {\begin{array}{*{20}l} 0 & \cdots & 0 \\ {\tilde{A}^{\text{T}} \left( {P_{{k_{1}^{i} }} - P_{i} } \right)^{\text{T}} } & \cdots & {\tilde{A}^{\text{T}} \left( {P_{{k_{1}^{i} }} - P_{i} } \right)^{\text{T}} } \\ 0 & \cdots & 0 \\ 0 & \cdots & 0 \\ 0 & \cdots & 0 \\ 0 & \cdots & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} { - T_{{ik_{1}^{i} }}^{ - 1} } & \cdots & 0 \\ 0 & \ddots & 0 \\ 0 & \cdots & { - T_{{ik_{m}^{i} }}^{ - 1} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} 0 & \cdots & 0 \\ {\tilde{A}^{\text{T}} \left( {P_{{k_{1}^{i} }} - P_{i} } \right)^{\text{T}} } & \cdots & {\tilde{A}^{\text{T}} \left( {P_{{k_{1}^{i} }} - P_{i} } \right)^{\text{T}} } \\ 0 & \cdots & 0 \\ 0 & \cdots & 0 \\ 0 & \cdots & 0 \\ 0 & \cdots & 0 \\ \end{array} } \right]^{\text{T}} . \\ \end{aligned} $$
(33)
Then, according to Schur complement formula, it follows from (32) and (33) that
$$ \left[ {\begin{array}{*{20}l} { - G - G^{\text{T}} } & {G\tilde{A}} & { - F^{\text{T}} + G\tilde{B}} & 0 & 0 & 0 \\ * & {\theta^{\prime \prime } + \tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \left( {P_{j} - P_{i} } \right)^{\text{T}} T_{ij}^{ - 1} \left( {P_{j} - P_{i} } \right)\tilde{A}} & {\tilde{A}^{\text{T}} F^{\text{T}} } & {\tilde{L}^{\text{T}} } & {\tilde{A}^{\text{T}} G^{\text{T}} } & {\tilde{A}^{\text{T}} P^{\text{T}} } \\ * & * & {\tilde{B}^{\text{T}} F^{\text{T}} + F\tilde{B} - \gamma^{2} I} & 0 & 0 & 0 \\ * & * & * & { - I} & 0 & 0 \\ * & * & * & * & {\theta^{\prime \prime \prime } } & 0 \\ * & * & * & * & * & { - P_{i} } \\ \end{array} } \right] < 0. $$
(34)
Since \( \tilde{A}^{\text{T}} \left( {\sum\nolimits_{{j \in U_{k}^{i} }} {\frac{{\delta_{ij}^{2} }}{4}T_{ij} } } \right)\tilde{A} = \tilde{A}^{\text{T}} G^{\text{T}} G^{{ - {\text{T}}}} \left( {\sum\nolimits_{{j \in U_{k}^{i} }} {\frac{{\delta_{ij}^{2} }}{4}T_{ij} } } \right)G^{ - 1} G\tilde{A} \) and \( \tilde{A}^{\text{T}} P_{i} \tilde{A} = \tilde{A}^{\text{T}} P_{i}^{\text{T}} P_{i}^{ - 1} P_{i} \tilde{A} \) one has
$$ \begin{aligned} & \left[ {\begin{array}{*{20}l} 0 & 0 & 0 & 0 \\ 0 & {\tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}T_{ij} \tilde{A}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] = - \left[ {\begin{array}{*{20}l} 0 \\ {\tilde{A}^{\text{T}} G^{\text{T}} } \\ 0 \\ 0 \\ \end{array} } \right]G^{{ - {\text{T}}}} \left( {\mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}T_{ij} } \right)G^{ - 1} \left[ {\begin{array}{*{20}l} 0 & {G\tilde{A}} & 0 & 0 \\ \end{array} } \right], \\ & \left[ {\begin{array}{*{20}l} 0 & 0 & 0 & 0 & 0 \\ 0 & {\tilde{A}^{\text{T}} P_{i} \tilde{A}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] = - \left[ {\begin{array}{*{20}l} 0 \\ {\tilde{A}^{\text{T}} P_{i}^{\text{T}} } \\ 0 \\ 0 \\ 0 \\ \end{array} } \right]P_{i}^{ - 1} \left[ {\begin{array}{*{20}l} 0 & {P_{i} \tilde{A}} & 0 & 0 & 0 \\ \end{array} } \right]. \\ \end{aligned} $$
(35)
Using Schur complement formula, it follows from (34) and (35) that
$$ \left[ {\begin{array}{*{20}l} { - G - G^{\text{T}} } & {G\tilde{A}} & { - F^{\text{T}} + G\tilde{B}} & 0 \\ * & {\theta^{\prime \prime \prime \prime } } & {\tilde{A}^{\text{T}} F^{\text{T}} } & {\tilde{L}^{\text{T}} } \\ * & * & {\tilde{B}^{\text{T}} F^{\text{T}} + F\tilde{B} - \gamma^{2} I} & 0 \\ * & * & * & { - I} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}l} 0 & 0 & 0 & 0 \\ 0 & {\tilde{A}^{\text{T}} \mathop \sum \limits_{{j \in U_{k}^{i} }} \frac{{\delta_{ij}^{2} }}{4}T_{ij} \tilde{A} + \tilde{A}^{\text{T}} P_{i} \tilde{A}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] < 0, $$
(36)
where \( \theta^{\prime \prime \prime \prime } = - \tilde{E}^{\text{T}} P_{i} \tilde{E} + \tilde{A}^{\text{T}} \sum\nolimits_{{j \in U_{k}^{i} }} {\hat{\pi }_{ij} \left( {P_{j} - P_{i} } \right)\tilde{A}} + \tilde{A}^{\text{T}} \sum\nolimits_{{j \in U_{k}^{i} }} {\left( {P_{j} - P_{i} } \right)^{\text{T}} T_{ij}^{ - 1} \left( {P_{j} - P_{i} } \right)} \tilde{A} \).
Inequality (24) can be rewritten as \( \tilde{A}^{\text{T}} \sum\nolimits_{j = 1}^{S} {\pi_{ij} P_{j} \tilde{A} \le \theta^{\prime \prime \prime \prime } } + \tilde{E}^{\text{T}} P_{i} \tilde{E} + \tilde{A}^{\text{T}} \sum\nolimits_{{j \in U_{k}^{i} }} {\frac{{\delta_{ij}^{2} }}{4}T_{ij} \tilde{A} + \tilde{A}^{\text{T}} P_{i} \tilde{A}} \), which implies that (36) can guarantee (8). Thus, the filtering error system (5) with GUTRs is stochastically admissible with a prescribed performance \( H_{\infty } \) index \( \gamma \) in Case I.
The proofs of Case II and Case III are similar to that of Case I, so they are omitted here.
Thus, it completes the proof of Theorem 2.
