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Finite-Time \(L_2\)\(L_\infty \) Control for Stochastic Asynchronously Switched Systems with Time-Varying Delay and Nonlinearity

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Abstract

This paper deals with the problems of finite-time \(L_2\)\(L_\infty \) control for stochastic switched systems under asynchronous switching with time-varying delay and nonlinearity. Because of the presence of delay in the switching signal of controllers, the switching of the controllers is asynchronous with the switching of the subsystems. Firstly, based on average dwell time approach, merging switching signal technique and multiple Lyapunov function method, state feedback controllers are designed to guarantee the finite-time boundedness of stochastic switched time-delay systems under asynchronous switching by linearization techniques. Then the finite-time \(L_2\)\(L_\infty \) performance is analyzed. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grants (61573088), (61573087) and (61433004).

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Correspondence to Xianwen Gao.

Appendix

Appendix

Because of the switching delay, after the jth subsystem has been switched to the ith subsystem, the controller \(K_j\) is still active. Thus, we rewrite the closed-loop system as

$$\begin{aligned} \mathrm{d}x(t)=&\,\left[ \bar{A}_{\sigma '(t)}x(t)+\bar{A}_{\mathrm{d}\sigma '(t)}x(t-\mathrm{d}(t))+F_{\sigma '(t)}\nu (t)\right] \mathrm{d}t\nonumber \\&+f_{\sigma '(t)}(t,x(t),x(t-\mathrm{d}(t)))\mathrm{d}\omega (t),\nonumber \\ x(t)=&\,\phi (t), t \in [-\tau _d,0], \end{aligned}$$
(47)

where

$$\begin{aligned} \bar{A}_{ii}= & {} A_{ii}+B_{ii}K_{ii}=A_{i}+B_{i}K_{i},\\ \bar{A}_{ij}= & {} A_{ij}+B_{ij}K_{ij}=A_{i}+B_{i}K_{j},\\ \bar{A}_{dii}= & {} {A}_{dij}={A}_{di},F_{ii}=F_{ij}=F_{i},\\ f_{ii}(t,x(t),x(t-\mathrm{d}(t)))= & {} f_{ij}(t,x(t),x(t-\mathrm{d}(t)))=f_{i}(t,x(t),x(t-\mathrm{d}(t))). \end{aligned}$$

For system (47), the piecewise Lyapunov–Krasovskii function candidate is chosen as follows:

$$\begin{aligned} V(t)=&\,V_{\sigma '(t)}(t)\nonumber \\ =&\,x^T(t){P}_{\sigma '(t)}x(t)\nonumber \\&\quad +\int _{t-\mathrm{d}(t)}^{t}{x^T(s)e^{\lambda _{\sigma '(t)}(s-t)}{Q}_{\sigma '(t)}x(s)}\mathrm{d}s\nonumber \\&\quad +\int _{-\tau _d}^{0}\int _{t+\theta }^{t}{y^T(s)e^{\lambda _{\sigma '(t)}(s-t)}{Z}_{\sigma '(t)}y(s)}\mathrm{d}s\mathrm{d}\theta , \end{aligned}$$
(48)

where \({P}_{\sigma '(t)}, {Q}_{\sigma '(t)}, {Z}_{\sigma '(t)}\) are positive definite matrices to be determined and

$$\begin{aligned} \lambda _{ii}= & {} \lambda _a,\lambda _{ij}=-\lambda _b,\\ y(t)\mathrm{d}t= & {} \mathrm{d}x(t). \end{aligned}$$

Introducing the free-weighting matrix \(T_{ii}\) and \(T_{ij}\) yields

$$\begin{aligned} \varLambda _{ii}(t)=&\, 2y^T(t)T^T_{ii}\{[\bar{A}_{ii}x(t)+\bar{A}_{dii}x(t-\mathrm{d}(t))+F_{i}\nu (t)-y(t)]\mathrm{d}{t}\nonumber \\&\quad +f_{i}(t,x(t),x(t-\mathrm{d}(t)))\mathrm{d}\omega (t)\}=0, \end{aligned}$$
(49)
$$\begin{aligned} \varLambda _{ij}(t)=&\,2y^T(t)T^T_{ij}\{[\bar{A}_{ij}x(t)+\bar{A}_{dij}x(t-\mathrm{d}(t))+F_{i}\nu (t)-y(t)]\mathrm{d}{t}\nonumber \\&\quad +f_{i}(t,x(t),x(t-\mathrm{d}(t)))\mathrm{d}\omega (t)\}=0. \end{aligned}$$
(50)

By It\(\hat{o}\)’s formula [23], we have

$$\begin{aligned} \mathrm{d}V_{ii}(t)=&\,\mathrm{LV}_{ii}(t)\mathrm{d}t+2x^T(t){P}_{ii}f_{i}(t,x(t),x(t-\mathrm{d}(t)))x(t)\mathrm{d}\omega (t), \end{aligned}$$
(51)
$$\begin{aligned} \mathrm{d}V_{ij}(t)=&\,\mathrm{LV}_{ij}(t)\mathrm{d}t+2x^T(t){P}_{ij}f_{i}(t,x(t),x(t-\mathrm{d}(t)))x(t)\mathrm{d}\omega (t). \end{aligned}$$
(52)

where

$$\begin{aligned} \mathrm{LV}_{ii}(t)=&\,\frac{1}{2}tr\left( x^T(t)f^T_i(t,x(t),x(t-\mathrm{d}(t))\right) \frac{\partial ^2{V_{ii}}}{\partial {x^2}}f_i(t,x(t),x(t-\mathrm{d}(t)))x(t))\\&+\frac{\partial {V_{ii}}}{{\partial }t}(t)+\left( \frac{\partial {V_{ii}}}{\partial {x}}(t)\right) ^T\times [(A_{i}+B_iK_i)x(t)+A_{di}x(t-\mathrm{d}(t)]\\ \le&\,2x^T(t){P}_{ii}[\bar{A}_{ii}x(t)+\bar{A}_{dii}x(t-\mathrm{d}(t))+F_{i}\nu (t)]\\&-(1-\dot{{d}}(t))e^{-\lambda _a\tau }x^T(t-\mathrm{d}(t)){Q}_{ii}x(t-\mathrm{d}(t))+x^T(t){Q}_{ii}x(t)\\&+\mathrm{trace}[f^T_{i}(t,x(t),x(t-\mathrm{d}(t)))P_{ii}f_{i}(t,x(t),x(t-\mathrm{d}(t)))]\\&-\lambda _a\int _{-\tau _d}^{0}\int _{t+\theta }^{t}{y^T(s)e^{\lambda _{a}(s-t)}Z_{ii}y(s)}\mathrm{d}s\mathrm{d}\theta \\&-\lambda _a\int _{t-\mathrm{d}(t)}^{t}x^T(\theta )e^{\lambda _a(s-t)}Q_{ii}x(\theta )\mathrm{d}\theta \\&-e^{-\lambda _a\tau _d}\int _{t-\mathrm{d}(t)}^{t}{y^T(\theta )Z_{ii}y(\theta )}\mathrm{d}\theta \\&+{y}^T(t)(\tau _d{Z}_{ii}-T^T_{ii}-T_{ii})y(t),\\ \mathrm{LV}_{ij}(t)=&\,\frac{1}{2}tr(x^T(t)f^T_i(t,x(t),x(t-\mathrm{d}(t)))\frac{\partial ^2{V_{ij}}}{\partial {x^2}}f_i(t,x(t),x(t-\mathrm{d}(t)))x(t))\\&+\frac{\partial {V_{ij}}}{{\partial }t}(t)+\left( \frac{\partial {V_{ij}}}{\partial {x}}(t)\right) ^T\times [(A_{i}+B_iK_j)x(t)+A_{di}x(t-\mathrm{d}(t)]\\ \le&\, 2x^T(t){P}_{ij}[\bar{A}_{ij}x(t)+\bar{A}_{dij}x(t-\mathrm{d}(t))+F_{i}\nu (t)]\\&-(1-\dot{{d}}(t))e^{\lambda _b\tau }x^T(t-\mathrm{d}(t)){Q}_{ij}x(t-\mathrm{d}(t))+x^T(t){Q}_{ij}x(t)\\&+\mathrm{trace}[f^T_{i}(t,x(t),x(t-\mathrm{d}(t)))P_{ij}f_{i}(t,x(t),x(t-\mathrm{d}(t)))]\\&+\lambda _b\int _{-\tau _d}^{0}\int _{t+\theta }^{t}{y^T(s)e^{-\lambda _{b}(s-t)}Z_{ij}y(s)}\mathrm{d}s\mathrm{d}\theta \\&+\lambda _b\int _{t-\mathrm{d}(t)}^{t}x^T(\theta )e^{-\lambda _b(s-t)}Q_{ij}x(\theta )\mathrm{d}\theta \\&-e^{\lambda _b\tau _d}\int _{t-\mathrm{d}(t)}^{t}{y^T(\theta )Z_{ij}y(\theta )}\mathrm{d}\theta \\&+{y}^T(t)(\tau _d{Z}_{ij}-T^T_{ij}-T_{ij})y(t). \end{aligned}$$

Combining Assumption 4 and (15), there is \(\varepsilon _{ii}>0\) for

$$\begin{aligned}&f^T_{i}(t,x(t),x(t-\mathrm{d}(t)))P_{ii}f_{i}(t,x(t),x(t-\mathrm{d}(t)))\nonumber \\&\le {\varepsilon _{ii}[x^T(t)G^T_{1i}G_{1i}x(t)+x^T(t-\mathrm{d}(t))G^T_{2i}G_{2i}x(t-\mathrm{d}(t))]}. \end{aligned}$$
(53)

By Lemma 2, we can get

$$\begin{aligned}&-\int _{t-\mathrm{d}(t)}^{t}{y^T(\theta )Z_{ii}y(\theta )}\mathrm{d}\theta \nonumber \\&\le -\tau _d^{-1}{\int _{t-\mathrm{d}(t)}^{t}{y^T(\theta )}\mathrm{d}\theta }Z_{ii}{\int _{t-\mathrm{d}(t)}^{t}{y(\theta )}\mathrm{d}\theta }\nonumber \\&=\tau _d^{-1}\left[ \begin{matrix} x(t)\\ x(t-\mathrm{d}(t)) \end{matrix} \right] ^T\left[ \begin{matrix} -Z_{ii} &{}Z_{ii}\\ Z_{ii} &{}-Z_{ii} \end{matrix} \right] \left[ \begin{matrix} x(t)\\ x(t-\mathrm{d}(t)) \end{matrix} \right] . \end{aligned}$$
(54)

Construct function \(\varGamma _{ii}\), we have

$$\begin{aligned} \varGamma _{ii}(t)=&\,\mathrm{LV}_{ii}(t)+\lambda _aV_{ii}(t)+\beta \nu ^T(t)P_{ii}\nu (t)+\varLambda _{ii}(t)-2y^T(t)T^T_{ii}f_{i}(t,x(t),x(t-\mathrm{d}(t)))\mathrm{d}\omega (t)\nonumber \\&\quad \le 2[x^T(t)P_{ii}+y^T(t)T^T_{ii}][\bar{A}_{ii}x(t)+\bar{A}_{dii}x(t-\mathrm{d}(t))+F_{i}v(t)]\nonumber \\&\qquad -(1-h)e^{-\lambda _a\tau _d}x^T(t-\mathrm{d}(t))Q_{ii}x(t-\mathrm{d}(t))+x^T(t)(Q_{ii}+\lambda _aP_{ii})x(t)\nonumber \\&\qquad +{\varepsilon _{ii}[x^T(t)G^T_{1i}G_{1i}x(t)+x^T(t-\mathrm{d}(t))G^T_{2i}G_{2i}x(t-\mathrm{d}(t))]}\nonumber \\&\qquad +\tau _d^{-1}e^{-\lambda _a\tau _d}\left[ \begin{matrix} x(t)\\ x(t-\mathrm{d}(t)) \end{matrix} \right] ^T\left[ \begin{matrix} -Z_{ii} &{}Z_{ii}\\ Z_{ii} &{}-Z_{ii} \end{matrix} \right] \left[ \begin{matrix} x(t)\\ x(t-\mathrm{d}(t)) \end{matrix} \right] \nonumber \\&\qquad +{y}^T(t)({\tau _d}Z_{ii}-T^T_{ii}-T_{ii})y(t)\nonumber \\&\qquad +\beta \nu ^T(t)P_{ii}\nu (t)\nonumber \\ =&\,\eta ^T(t)\varPhi _{ii}\eta (t), \end{aligned}$$
(55)

where \(\eta ^T(t)=\left[ \begin{matrix} x^T(t)&x^T(t-\mathrm{d}(t))&\nu ^T(t)&y^T(t) \end{matrix} \right] \) and

$$\begin{aligned} \varPhi _{ii}=&\,\left[ \begin{matrix} \varPi _{1ii} &{}\varPi _{2ii} &{}P_{ii}F_{i} &{}\bar{A}^T_{ii}T_{ii}\\ * &{}\varPi _{3ii} &{}0 &{}\bar{A}^T_{dii}T_{ii}\\ * &{}* &{}\beta {P_{ii}} &{}F^T_{i}T_{ii}\\ * &{}* &{}* &{}{\tau _d}Z_{ii}-T^T_{ii}-T_{ii} \end{matrix} \right] ,\\ \varPi _{1ii}=&\,P_{ii}\bar{A}_{ii}+\bar{A}^T_{ii}P_{ii}+\varepsilon _{ii}G^T_{1i}G_{1i}+Q_{ii}+\lambda _aP_{ii}-\tau _d^{-1}Z_{ii}e^{-\lambda _a\tau _d},\\ \varPi _{2ii}=&\,P_{ii}\bar{A}_{ii}+\tau _d^{-1}Z_{ii}e^{-\lambda _a\tau _d},\\ \varPi _{3ii}=&\,\varepsilon _{ii}G^T_{2i}G_{2i}-(1-h)e^{-\lambda _a\tau _d}Q_{ii}-\tau _d^{-1}Z_{ii}e^{-\lambda _a\tau _d}. \end{aligned}$$

Multiplying both sides of \(\varPhi _{ii}\) by \(\varDelta =\mathrm{diag}\{\bar{P}_{ii},\bar{P}_{ii},\bar{P}_{ii},\bar{P}_{ii}\}\) and letting \(M_{ii}= K_{ii}\bar{P}_{ii},T_{ii}={\alpha }P_{ii}\), we derive

$$\begin{aligned} \varDelta ^T\varPhi _{ii}\varDelta =&\,\left[ \begin{array}{cccc} \bar{\varPi }_{1ii} &{}\bar{\varPi }_{2ii} &{}F_{i}\bar{P}_{ii} &{}\bar{P}_{ii}A^T_{ii}+{\alpha }M^T_{ii}B_{ii}\\ * &{}\bar{\varPi }_{3ii} &{}0 &{}\alpha \bar{P}_{ii}A^T_{dii}\\ * &{}* &{}\beta {\bar{P}_{ii}} &{}\alpha \bar{P}_{ii}F^T_{i}\\ * &{}* &{}* &{}{\tau _d}\bar{P}_{ii}Z_{ii}\bar{P}_{ii}-2\alpha \bar{P}_{ii} \end{array} \right] , \end{aligned}$$
(56)

where

$$\begin{aligned} \bar{\varPi }_{1ii}=&\,{A}_{ii}\bar{P}_{ii}+\bar{P}_{ii}{A}^T_{ii}+B_{ii}M_{ii}+M^T_{ii}B^T_{ii}+\varepsilon _{ii}\bar{P}_{ii}G^T_{1i}G_{1i}\bar{P}_{ii}+\bar{P}_{ii}Q_{ii}\bar{P}_{ii}\\&+\lambda _a\bar{P}_{ii}-\tau _d^{-1}\bar{P}_{ii}Z_{ii}\bar{P}_{ii}e^{-\lambda _a\tau _d},\\ \bar{\varPi }_{2ii}=&\,{A}_{ii}\bar{P}_{ii}+B_{ii}M_{ii}+\tau _d^{-1}\bar{P}_{ii}Z_{ii}\bar{P}_{ii}e^{-\lambda _a\tau _d},\\ \bar{\varPi }_{3ii}=&\,\varepsilon _{ii}\bar{P}_{ii}G^T_{2i}G_{2i}\bar{P}_{ii}-(1-h)e^{-\lambda _a\tau _d}\bar{P}_{ii}Q_{ii}\bar{P}_{ii}-\tau _d^{-1}\bar{P}_{ii}Z_{ii}\bar{P}_{ii}e^{-\lambda _a\tau _d}. \end{aligned}$$

By using the Schur complement Lemma, letting \(\bar{Q}_{ii}= \bar{P}_{ii}Q_{ii}\bar{P}_{ii},\bar{Z}_{ii}= \bar{P}_{ii}Z_{ii}\bar{P}_{ii},\bar{\varepsilon }_{ii}=\varepsilon ^{-1}_{ii}\), (56) is equivalent to

$$\begin{aligned} \varDelta ^T\varPhi _{ii}\varDelta =&\,\left[ \begin{array}{cccccc} \tilde{\varPi }_{1ii} &{}\tilde{\varPi }_{2ii} &{}F_{i}\bar{P}_{ii} &{}\bar{P}_{ii}A^T_{ii}+{\alpha }M^T_{ii}B_{ii} &{}\bar{P}_{ii}G^T_{1i} &{}0\\ * &{}\tilde{\varPi }_{3ii} &{}0 &{}\alpha \bar{P}_{ii}A^T_{dii} &{}0 &{}\bar{P}_{ii}G^T_{2i}\\ * &{}* &{}\beta {\bar{P}_{ii}} &{}\alpha \bar{P}_{ii}F^T_{i} &{}0 &{}0\\ * &{}* &{}* &{}{\tau _d}\bar{Z}_{ii}-2\alpha \bar{P}_{ii} &{}0 &{}0\\ * &{}* &{}* &{}* &{}-\bar{\varepsilon }_{ii}I &{}0\\ * &{}* &{}* &{}* &{}* &{}-\bar{\varepsilon }_{ii}I \end{array} \right] . \end{aligned}$$
(57)

where

$$\begin{aligned} \tilde{\varPi }_{1ii}=&\,{A}_{ii}\bar{P}_{ii}+\bar{P}_{ii}{A}^T_{ii}+B_{ii}M_{ii}+M^T_{ii}B^T_{ii}+\lambda _a\bar{P}_{ii}+\bar{Q}_{ii}-\tau _d^{-1}e^{-\lambda _a\tau _d}\bar{Z}_{ii},\\ \tilde{\varPi }_{2ii}=&\,{A}_{ii}\bar{P}_{ii}+B_{ii}M_{ii}+\tau _d^{-1}e^{-\lambda _a\tau _d}\bar{Z}_{ii},\\ \tilde{\varPi }_{3ii}=&\,-(1-h)e^{-\lambda _a\tau _d}\bar{Q}_{ii}-\tau _d^{-1}e^{-\lambda _a\tau _d}\bar{Z}_{ii}. \end{aligned}$$

From (51), (57) and Theorem 1, we can obtain

$$\begin{aligned} \mathrm{LV}_{ii}(t)+\lambda _aV_{ii}(t)+\beta \nu ^T(t){P}_{ii}\nu (t)-2y^T(t)T^T_{ii}f_{i}(t,x(t),x(t-\mathrm{d}(t)))\mathrm{d}\omega (t)<0 \end{aligned}$$
(58)

Similarly, we can get

$$\begin{aligned} \mathrm{LV}_{ij}(t)-\lambda _bV_{ij}(t)+\beta \nu ^T(t){P}_{ij}\nu (t)-2y^T(t)T^T_{ij}f_{i}(t,x(t),x(t-\mathrm{d}(t)))\mathrm{d}\omega (t)<0 \end{aligned}$$
(59)

Considering formula (23) and formula (32), we can get

$$\begin{aligned} \int _{t_k+\tau _s}^{t}{}\mathrm{d}[e^{\lambda _a\theta }V_{ii}(\theta )]=&\,\int _{t_k+\tau _s}^{t}{\lambda _ae^{\lambda _a\theta }V_{ii}(\theta )}\mathrm{d}\theta +\int _{t_k+\tau _s}^{t}{e^{\lambda _a\theta }}\mathrm{d}V_{ii}(\theta )\nonumber \\&\quad <-\int _{t_k+\tau _s}^{t}{2e^{\lambda _a\theta }[x^T(\theta )P_{ii}}-y^T(t)T^T_{ii}]f_{i}(\theta ,x(\theta ),x(\theta -\mathrm{d}(\theta )))\mathrm{d}\omega (\theta )\nonumber \\&\qquad -\int _{t_k+\tau _s}^{t}{{\beta }e^{\lambda _a\theta }\nu ^T(\theta ){P}_{ii}\nu (\theta )}\mathrm{d}\theta . \end{aligned}$$
(60)

Taking expectations, we can get

$$\begin{aligned} E\left\{ \int _{t_k+\tau _s}^{t}{}\mathrm{d}[e^{\lambda _a\theta }V_{ii}(\theta )]\right\}&= e^{\lambda _at}E\{V_{ii}(t)\}-e^{\lambda _a(t_k+\tau _s)}E\{V_{ii}(t_k+\tau _s)\}\nonumber \\&<E\left\{ \int _{t_k+\tau _s}^{t}{{-\beta }e^{\lambda _a\theta }\nu ^T(\theta ){P}_{ii}\nu (\theta )}\mathrm{d}\theta \right\} . \end{aligned}$$
(61)

Namely

$$\begin{aligned} E\{V_{ii}(t)\}<e^{\lambda _a(t_k+\tau _s-t)}E\{V_{ii}(t_k+\tau _s)\}-{\beta }E\left\{ \int _{t_k+\tau _s}^{t}{\nu ^T(\theta ){P}_{ii}\nu (\theta )}\mathrm{d}\theta \right\} . \end{aligned}$$
(62)

Similarly, we can get

$$\begin{aligned} E\{V_{ij}(t)\}<e^{-\lambda _b(t_k-t)}E\{V_{ij}(t_k)\}-{\beta }e^{\lambda _b\tau _s}E\left\{ \int _{t_k}^{t}{\nu ^T(\theta ){P}_{ij}\nu (\theta )}\mathrm{d}\theta \right\} . \end{aligned}$$
(63)

Considering (48) with Theorem 1, we have

$$\begin{aligned} E\{V_{ii}(t_k+\tau _s)\}\le&{\kappa }E\{V_{ij}(t_k+\tau _s)\},\nonumber \\ E\{V_{ij}(t_k)\}\le&{\kappa }e^{(\lambda _a+\lambda _b)\tau _d}E\{V_{jj}(t_k)\}. \end{aligned}$$
(64)

Combining (62), (63), (64) and Assumption 3, by some mathematical manipulation, we can get that when \(t \in [t_k+\tau _s, t_{k+1})\),

$$\begin{aligned} \nonumber E\{V(t)\}&<e^{-\lambda _a(t-t_k-\tau _s)}E\{V_{ii}(t_k+\tau _s)\}-{\beta }E\left\{ \int _{t_k+\tau _s}^{t}{\nu ^T(\theta ){P}_{ii}\nu (\theta )}\mathrm{d}\theta \right\} \\ \nonumber&<{\kappa }e^{-\lambda _a(t-t_k-\tau _s)}E\{V_{ij}(t_k+\tau _s)\}-{\beta }E\left\{ \int _{t_k+\tau _s}^{t}{\nu ^T(\theta ){P}_{ii}\nu (\theta )}\mathrm{d}\theta \right\} \\ \nonumber&<{\kappa }e^{-\lambda _a(t-t_k-\tau _s)+\lambda _b\tau _s}E\{V_{ij}(t_k)\}\\ \nonumber&\quad -{\beta }{\kappa }e^{\lambda _b\tau _s}E\left\{ \int _{t_k}^{t_k+\tau _s}{\nu ^T(\theta ){P}_{ij}\nu (\theta )}\mathrm{d}\theta \right\} -{\beta }E\left\{ \int _{t_k+\tau _s}^{t}{\nu ^T(\theta ){P}_{ii}\nu (\theta )}\mathrm{d}\theta \right\} \\ \nonumber&<{\kappa }^2e^{-\lambda _a(t-t_k-\tau _s)+\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}E\{V_{jj}(t_k)\}\\ \nonumber&\quad -{\beta }{\kappa }e^{\lambda _b\tau _s}E\left\{ \int _{t_k}^{t_k+\tau _s}{\nu ^T(\theta ){P}_{ij}\nu (\theta )}\mathrm{d}\theta \right\} -{\beta }E\left\{ \int _{t_k+\tau _s}^{t}{\nu ^T(\theta ){P}_{ii}\nu (\theta )}\mathrm{d}\theta \right\} \\ \nonumber&<\cdots \\ \nonumber&<\kappa ^{(2k+1)}e^{-\lambda _a[t-t_0+(k+1)\tau _s]+k\lambda _b\tau _s+(k+1)(\lambda _a+\lambda _b)\tau _d}E\{V(t_0)\}\\ \nonumber&\quad -\beta \lambda _2[1+\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}+\cdots +\kappa ^{2k}e^{k\lambda _b\tau _s+k(\lambda _a+\lambda _b)\tau _d}]d_\nu \\ \nonumber&\quad -\beta \lambda _2{\kappa }e^{\lambda _b\tau _s}[1+\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}+\cdots \\ \nonumber&\quad +\kappa ^{2(k-1)}e^{(k-1)\lambda _b\tau _s+(k-1)(\lambda _a+\lambda _b)\tau _d}]d_\nu \\ \nonumber&<-\beta \lambda _2{d_\nu }\frac{1-[\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}]^{\left( N_0+1+\frac{t-t_0}{\tau _a}\right) }}{1-\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}}\\ \nonumber&\quad -\beta \lambda _2{\kappa }e^{\lambda _b\tau _s}d_\nu \frac{1-[\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}]^{\left( N_0+\frac{t-t_0}{\tau _a}\right) }}{1-\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}}\\ \nonumber&\quad +{\kappa }e^{(\lambda _a+\lambda _b)\tau _s}e^{[2\ln \kappa +(\lambda _a+\lambda _b)(\tau _s+\tau _d)]N_0}\\&\quad e^{\left[ \frac{2\ln \kappa }{\tau _a}+[(\lambda _a+\lambda _b)(\tau _s+\tau _d)\right] \frac{1}{\tau _a}-\lambda _a](t-t_0)}E\{V(t_0)\}. \end{aligned}$$
(65)

Similarly, we can get that when \(t \in [t_k, t_k+\tau _s)\),

$$\begin{aligned} \nonumber E\{V(t)\} <&-\beta \lambda _2{d_\nu }\frac{1-[\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}]^{\left( N_0+\frac{t-t_0}{\tau _a}\right) }}{1-\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}}\\ \nonumber&-\beta \lambda _2{\kappa }e^{\lambda _b\tau _s}d_\nu \frac{1-[\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}]^{\left( N_0+\frac{t-t_0}{\tau _a}\right) }}{1-\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}}\\ \nonumber&+e^{(\lambda _a+\lambda _b)\tau _s}e^{[2\ln \kappa +(\lambda _a+\lambda _b)(\tau _s+\tau _d)]N_0}\\&e^{\left[ \frac{2\ln \kappa }{\tau _a}+\left[ (\lambda _a+\lambda _b)(\tau _s+\tau _d)\right] \frac{1}{\tau _a}-\lambda _a\right] (t-t_0)}E\{V(t_0)\}. \end{aligned}$$
(66)

Considering (48), we can get

$$\begin{aligned} E\{V(t)\}\ge&{(\lambda _1+\lambda _4\tau _de^{-\lambda _a\tau _d})E\{x^T(t)Rx(t)\}}, \end{aligned}$$
(67)
$$\begin{aligned} E\{V(t_0)\}\le&{\left( \lambda _1+\lambda _3\tau _d+\frac{1}{2}\tau ^2_d\lambda _5\right) c_1}. \end{aligned}$$
(68)

Combining (66), (67) and (68), we can get

$$\begin{aligned} \nonumber E\{x^T(t)Rx(t)\}<&[{\kappa }e^{(\lambda _a+\lambda _b)\tau _s}e^{[2\ln \kappa +(\lambda _a+\lambda _b)(\tau _s+\tau _d)]N_0}\\ \nonumber&e^{\left[ \frac{2\ln \kappa }{\tau _a}+[(\lambda _a+\lambda _b)(\tau _s+\tau _d)]\frac{1}{\tau _a}-\lambda _a\right] (t-t_0)}E\{V(t_0)\}\\ \nonumber&-\beta \lambda _2{d_\nu }\frac{1-[\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}]^{\left( N_0+1+\frac{t-t_0}{\tau _a}\right) }}{1-\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}}\\ \nonumber&-\beta \lambda _2{\kappa }e^{\lambda _b\tau _s}d_\nu \frac{1-[\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}]^{\left( N_0+\frac{t-t_0}{\tau _a}\right) }}{1-\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}}]\\ \nonumber&(\lambda _1+\lambda _4\tau _de^{-\lambda _a\tau _d})^{-1}\\ <&c_2. \end{aligned}$$
(69)

Simplifying formula (69), we get \(\tau _a>\tau ^*_a\) and \(\tau ^*_a\) is the numerical solution of the following equation.

$$\begin{aligned} \xi _1e^{[2\ln \kappa +(\lambda _a+\lambda _b)(\tau _s+\tau _d)]\frac{T}{\tau ^*_a}} +\xi _2e^{[2\ln \kappa +\lambda _b\tau _s+(\lambda _a+\lambda _b)(\tau _s+\tau _d)]\frac{T}{\tau ^*_a}} =\xi _3 \end{aligned}$$
(70)

where

$$\begin{aligned} \xi _1=&\,c_1{\kappa }e^{(\lambda _a+\lambda _b)\tau _s-\lambda _aT}\left( \lambda _1+\lambda _3\tau _d+\frac{1}{2}\tau ^2_d\lambda _5\right) (\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}-1),\\ \xi _2=&\,-\beta \lambda _2d_{\nu }e^{\lambda _b\tau _s}(\kappa +e^{2\ln \kappa +(\lambda _a+\lambda _b)\tau _d}),\\ \xi _3=&\,(\lambda _1+\lambda _4\tau _de^{-\lambda _a\tau _d})(\kappa ^2e^{\lambda _b\tau _s+(\lambda _a+\lambda _b)\tau _d}-1)c_2 -\beta \lambda _2d_\nu (1+{\kappa }e^{\lambda _b\tau _s}). \end{aligned}$$

Therefore, if the ADT satisfies (70), by Definition 1, system (11) is finite-time bounded with respect to \([c_1,c_2,T,d,R,\sigma ]\).

The proof is completed. \(\square \)

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He, H., Gao, X. & Qi, W. Finite-Time \(L_2\)\(L_\infty \) Control for Stochastic Asynchronously Switched Systems with Time-Varying Delay and Nonlinearity. Circuits Syst Signal Process 37, 112–134 (2018). https://doi.org/10.1007/s00034-017-0549-y

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