## Appendix A Proof of Theorem 1  
For \(t\in[t_{k},t_{k+1}]\) and \(t=t_{k}+\tau\), it is known that \(x(t)=\zeta(\tau)x(t_{k})+\zeta(\tau)\), where,  
\[\zeta(\tau)=I+\int_{0}^{\tau}e^{As}ds(A-BK),\zeta(\tau)=\int_{0}^{\tau}e^{As} EW(s)ds\]  
Therefore for the Lyapunov function  
\[V(x(t))=x(t)^{T}Px(t)=[\zeta(\tau)x(t_{k})+\zeta(\tau)]^{T}P[\zeta(\tau)x(t_{k })+\zeta(\tau)]\leq x(x_{k})^{T}\zeta(\tau)^{T}P\zeta(\tau)x(x_{k})+\zeta^{T}P\xi.\]  
We denote:  
\[V^{{}^{\prime}}(x(t_{k}),\tau)=x(x_{k})^{T}\zeta(\tau)^{T}P\zeta(\tau)x(x_{k} )+\xi^{T}P\xi.\]  
We define \(\varpi\) as the mean of EW(s) in the paper. For the given \(\beta>0\) and \(\gamma>0\), if,  
\[\dot{V}^{{}^{\prime}}\leq-2\beta\dot{V}^{{}^{\prime}}+\gamma/2||W||^{2}+\gamma /2||\varpi||^{2}, \tag{10}\]  
where \(\xi=\int_{0}^{\tau}e^{As}ds\varpi\), then,  
\[V^{{}^{\prime}}(x(t_{k}),\tau)\leq e^{-\beta\tau}V^{{}^{\prime}}(x(t_{k}),0)+ \int_{0}^{\tau}(\gamma/2e^{-\beta s}||W(s)||^{2}+\gamma/2||\varpi||^{2})ds,\]so that,  
\[V(x(t))\leq V^{{}^{\prime}}(x(t_{k}),\tau)\leq e^{-\beta\tau}V(x(t_{k}))+\int_{0} ^{\tau}\gamma/2e^{-\beta\beta}||W(s)||^{2}ds+\int_{0}^{\tau}\gamma/2e^{-\beta \beta}||\omega(s)||^{2}ds.\]  
which means the dynamic system is of \(\beta\)-exponential stability. When \(||\alpha||^{2}\leq n\eta\), we have \(||x||^{2}\leq n\gamma\eta/(\varepsilon\beta)\) with initial state \(x(t_{k})=0\). The inequality (A1) is:  
\[d[x(t_{k})^{T}\xi(\tau)^{T}P\xi(\tau)x(t_{k})]/dt+2\beta x(t_{k})^{T}\xi(\tau)^ {T}P\xi(\tau)x(t_{k})\leq-d(\xi^{T}P\xi)/dt-2\beta(\xi^{T}P\xi)+\gamma/2||W||^{ 2}+\gamma/2||\omega||^{2},\] (A2)  
and the inequality is:  
\[-d(\xi^{T}P\xi)/dt-2\beta(\xi^{T}P\xi)+\gamma/2||W||^{2}\geq\xi^{ T}P\xi-(2\beta+1)(\xi^{T}P\xi)+\gamma/2||W||^{2}\geq\] \[(\gamma/2-re\varepsilon^{\pi\tau})||W||^{2}+[\gamma/2-(2\beta+1) (e^{\pi\tau}-1)r]e||\omega||^{2}.\]  
It should be noted that the left-hand side of inequality (A2) is equivalent to the left-hand side of inequality (9). If:  
\[d[x(t_{k})^{T}\xi(\tau)^{T}P\xi(\tau)x(t_{k})]/dt+2\beta x(t_{k})^ {T}\xi(\tau)^{T}P\xi(\tau)x(t_{k})\leq\] \[(\gamma/2-re\varepsilon^{\pi\tau})||W||^{2}+[\gamma/2-(2\beta+1) (e^{\pi\tau}-1)r]e||\omega||^{2},\]  
then the inequality (A1) satisfies. When \(\gamma/2-re\varepsilon^{\pi\tau}>0\), and \(\gamma/2-(2\beta+1)(e^{\pi\tau}-1)r>0\), the above inequality holds if:  
\[d[x(t_{k})^{T}\xi(\tau)^{T}P\xi(\tau)x(t_{k})]/dt+2\beta x(t_{k})^{T}\xi(\tau) ^{T}P\xi(\tau)x(t_{k})\leq 0.\]  
When \(\gamma/2-re\varepsilon^{\pi\tau}>0\), and \(\gamma/2-(2\beta+1)(e^{\pi\tau}-1)r\leq 0\), the above inequality holds if:  
\[d[x(t_{k})^{T}\xi(\tau)^{T}P\xi(\tau)x(t_{k})]/dt+2\beta x(t_{k})^{T}\xi(\tau) ^{T}P\xi(\tau)x(t_{k})\leq[\gamma/2-(2\beta+1)(e^{\pi\tau}-1)re]n\eta.\]  
When \(\gamma/2-re\varepsilon^{\pi\tau}\leq 0\), and \(\gamma/2-(2\beta+1)(e^{\pi\tau}-1)r>0\), the above inequality holds if:  
\[d[x(t_{k})^{T}\xi(\tau)^{T}P\xi(\tau)x(t_{k})]/dt+2\beta x(t_{k})^{T}\xi(\tau )^{T}P\xi(\tau)x(t_{k})\leq(\gamma/2-re\varepsilon^{\pi\tau})n\eta.\]  
When \(\gamma/2-re\varepsilon^{\pi\tau}\leq 0\), and \(\gamma/2-(2\beta+1)(e^{\pi\tau}-1)r\leq 0\), the above inequality holds if:  
\[d[x(t_{k})^{T}\xi(\tau)^{T}P\xi(\tau)x(t_{k})]/dt+2\beta x(t_{k}) ^{T}\xi(\tau)^{T}P\xi(\tau)x(t_{k})\leq\] \[(\gamma/2-re\varepsilon^{\pi\tau})n\eta+[\gamma/2-(2\beta+1)(e^{ \pi\tau}-1)re]n\eta.\]