Mean square exponential and robust stability of stochastic discrete-time genetic regulatory networks with uncertainties

Abstract

This paper aims to analyze global robust exponential stability in the mean square sense of stochastic discrete-time genetic regulatory networks with stochastic delays and parameter uncertainties. Comparing to the previous research works, time-varying delays are assumed to be stochastic whose variation ranges and probability distributions of the time-varying delays are explored. Based on the stochastic analysis approach and some analysis techniques, several sufficient criteria for the global robust exponential stability in the mean square sense of the networks are derived. Moreover, two numerical examples are presented to show the effectiveness of the obtained results.

Appendices

Appendix A: Proof of Theorem 1

Consider the following Lyapunov-Krasovskii functional candidate:

$$ V_k=V_{k,1}+V_{k,2}+V_{k,3}+V_{k,4}+V_{k,5} $$

(16)

where

$$ \begin{aligned} V_{k,1}=&x_k^{{\rm T}}P_1x_k+y_k^{{\rm T}}P_2y_k,\\ V_{k,2}=&\sum^{k-1}_{i=k-\tau_{1}^{k}}x^{{\rm T}}_iQ_{1}x_i +\sum^{-\tau_m}_{j=-\tau_0+1}\sum^{k-1}_{i=k+j}x^{{\rm T}}_iQ_{1}x_i\\ &+\sum^{k-1}_{i=k-\tau_{2}^{k}}x^{{\rm T}}_iQ_{2}x_i +\sum^{-\tau_0-1}_{j=-\tau_M+1}\sum^{k-1}_{i=k+j}x^{{\rm T}}_iQ_{2}x_i,\\ V_{k,3}=&\sum^{k-1}_{i=k-d_{1}^{k}}g^{{\rm T}}(y_i)Q_{3}g(y_i) +\sum^{-d_m}_{j=-d_0+1}\sum^{k-1}_{i=k+j}g^{{\rm T}}(y_i)Q_{3}g(y_i)\\ &+\sum^{k-1}_{i=k-d_{2}^{k}}g^{{\rm T}}(y_i)Q_{4}g(y_i) +\sum^{-d_0-1}_{j=-d_M+1}\sum^{k-1}_{i=k+j}g^{{\rm T}}(y_i)Q_{4}g(y_i),\\ V_{k,4}=&\sum_{j=1}^{\tau_0}\sum_{i=k-j}^{k-1}\eta_i^{{\rm T}}R_1\eta_i +\sum_{j=1}^{\tau_M}\sum_{i=k-j}^{k-1}\eta_i^{{\rm T}}R_2\eta_i,\;\;\eta_k=x_{k+1}-x_k,\\ V_{k,5}=&\sum_{j=1}^{d_0}\sum_{i=k-j}^{k-1}\delta_i^{{\rm T}}R_3\delta_i +\sum_{j=1}^{d_M}\sum_{i=k-j}^{k-1}\delta_i^{{\rm T}}R_4\delta_i,\;\;\delta_k=y_{k+1}-y_k. \\ \end{aligned} $$

Calculating the difference of V _k along the solution of (9) and taking its mathematical expectation yield

$$ \begin{aligned} {\mathbb{E}}\{\Updelta V_{k,1}\} =&{\mathbb{E}}\{{\mathbb{E}}\{V_{k+1,1}\}-V_{k,1}\}\\ =&{\mathbb{E}}\{x_k^{{\rm T}}(A^{{\rm T}}P_1A-P_1)x_k+2\alpha_0x_k^{{\rm T}}A^{{\rm T}}P_1Bg(y_{d,1}) +2\bar{\alpha}_0x_k^{{\rm T}}A^{{\rm T}}P_1Bg(y_{d,2})\\ &+\alpha_0g^{{\rm T}}(y_{d,1})B^{{\rm T}}P_1Bg(y_{d,1})+\bar{\alpha}_0g^{{\rm T}}(y_{d,2})B^{{\rm T}}P_1Bg(y_{d,2}) +\sigma^{{\rm T}}(k,x_{k},y_{k})P_1\sigma(k,x_{k},y_{k})\\ &+y_k^{{\rm T}}(C^{{\rm T}}P_2C-P_2)y_k+2\beta_0y_k^{{\rm T}}C^{{\rm T}}P_2Dx_{\tau,1}+2\bar{\beta}_0y_k^{{\rm T}}C^{{\rm T}}P_2Dx_{\tau,2}\\ &+\beta_0x_{\tau,1}^{{\rm T}}D^{{\rm T}}P_2Dx_{\tau,1}+\bar{\beta}_0x_{\tau,2}^{{\rm T}}D^{{\rm T}}P_2Dx_{\tau,2}\},\\ \end{aligned} $$

(17)

$$ \begin{aligned} {\mathbb{E}}\{\Updelta V_{k,2}\} &={\mathbb{E}}\{{\mathbb{E}}\{V_{k+1,2}\}-V_{k,2}\}\cr &\leq{\mathbb{E}}\{(\tau_0-\tau_m+1)x_k^{{\rm T}}Q_1x_k-x_{\tau,1}^{{\rm T}}Q_1x_{\tau,1} +(\tau_M-\tau_0)x_k^{{\rm T}}Q_2x_k-x_{\tau,2}^{{\rm T}}Q_2x_{\tau,2}\}, \end{aligned} $$

(18)

$$ \begin{aligned} {\mathbb{E}}\{\Updelta V_{k,3}\} =&{\mathbb{E}}\{{\mathbb{E}}\{V_{k+1,3}\}-V_{k,3}\}\\ \leq&{\mathbb{E}}\{(d_0-d_m+1)g^{{\rm T}}(y_k)Q_{3}g(y_k)-g^{{\rm T}}(y_{d,1})Q_{3}g(y_{d,1})\\ &+(d_M-d_0)g^{{\rm T}}(y_k)Q_{4}g(y_k)-g^{{\rm T}}(y_{d,2})Q_{4}g(y_{d,2})\}, \end{aligned} $$

(19)

$$ \begin{aligned} {\mathbb{E}}\{\Updelta V_{k,4}\} &={\mathbb{E}}\{{\mathbb{E}}\{V_{k+1,4}\}-V_{k,4}\}\\ &={\mathbb{E}}\left\{\eta_k^{{\rm T}}(\tau_0R_1+\tau_MR_2)\eta_k-\sum_{j=k-\tau_0}^{k-1}\eta_i^{{\rm T}}R_1\eta_i -\sum_{j=k-\tau_M}^{k-1}\eta_i^{{\rm T}}R_2\eta_i\right\}, \end{aligned} $$

(20)

$$ \begin{aligned} {\mathbb{E}}\{\Updelta V_{k,5}\} =&{\mathbb{E}}\{{\mathbb{E}}\{V_{k+1,5}\}-V_{k,5}\}\\ =&{\mathbb{E}}\left\{\delta_k^{{\rm T}}(d_0R_3+d_MR_4)\delta_k-\sum_{j=k-d_0}^{k-1}\delta_i^{{\rm T}}R_3\delta_i- \sum_{j=k-d_M}^{k-1}\delta_i^{{\rm T}}R_4\delta_i\right\}. \end{aligned} $$

(21)

Considering Assumption 2 and (10), it can be easily obtained

$$ \begin{aligned} {\mathbb{E}}\{\sigma^{{\rm T}}(k,x_{k},y_{k})P_1\sigma(k,x_{k},y_{k})\} &\leq{\mathbb{E}}\{\lambda_{\max}(P_1)\sigma^{{\rm T}}(k,x_{k},y_{k})\sigma(k,x_{k},y_{k})\}\\ &\leq {\mathbb{E}}\{\mu_{*}x^{{\rm T}}_kH_{1}x_k+\mu_{*}y^{{\rm T}}_kH_{2}y_k\}. \end{aligned} $$

(22)

Making use of Lemma 2, we can derive

$$ 2\alpha_0x_k^{{\rm T}}A^{{\rm T}}P_1Bg(y_{d,1})\leq\alpha_0x_k^{{\rm T}}A^{{\rm T}}P_1Ax_k+\alpha_0g^{{\rm T}}(y_{d,1})B^{{\rm T}}P_1Bg(y_{d,1}), $$

(23)

$$ 2\bar{\alpha}_0x_k^{{\rm T}}A^{{\rm T}}P_1Bg(y_{d,2})\leq\bar{\alpha}_0x_k^{{\rm T}}A^{{\rm T}}P_1Ax_k+\bar{\alpha}_0g^{{\rm T}}(y_{d,2})B^{{\rm T}}P_1Bg(y_{d,2}), $$

(24)

$$ 2\beta_0y_k^{{\rm T}}C^{{\rm T}}P_2Dx_{\tau,1}\leq\beta_0y_k^{{\rm T}}C^{{\rm T}}P_2Cy_k+\beta_0x_{\tau,1}^{{\rm T}}D^{{\rm T}}P_2Dx_{\tau,1}, $$

(25)

$$ 2\bar{\beta}_0y_k^{{\rm T}}C^{{\rm T}}P_2Dx_{\tau,2}\leq\bar{\beta}_0y_k^{{\rm T}}C^{{\rm T}}P_2Cy_k+\bar{\beta}_0x_{\tau,2}^{{\rm T}}D^{{\rm T}}P_2Dx_{\tau,2}. $$

(26)

Obviously, the following zero equations hold.

$$ 2\xi_1^{{\rm T}}(k)M\left[x_k-x_{\tau,1}-\sum_{i=k-\tau_1^k}^{k-1}(x_{i+1}-x_i)\right]=0, $$

(27)

$$ 2\xi_1^{{\rm T}}(k)N\left[x_k-x_{\tau,2}-\sum_{i=k-\tau_2^k}^{k-1}(x_{i+1}-x_i)\right]=0, $$

(28)

$$ 2\xi_2^{{\rm T}}(k)S\left[y_k-y_{d,1}-\sum_{i=k-d_1^k}^{k-1}(y_{i+1}-y_i)\right]=0, $$

(29)

$$ 2\xi_2^{{\rm T}}(k)Z\left[y_k-y_{d,2}-\sum_{i=k-d_2^k}^{k-1}(y_{i+1}-y_i)\right]=0. $$

(30)

where

$$ \begin{aligned} \xi_1^{{\rm T}}(k)&=\left[\begin{array}{cccc} x_k^{{\rm T}} & \eta_k^{{\rm T}} & x_{\tau,1}^{{\rm T}} & x_{\tau,2}^{{\rm T}}\end{array}\right],\\ \xi_2^{{\rm T}}(k)&=\left[\begin{array}{ccccccc} y_k^{{\rm T}} & \delta_k^{{\rm T}} & y_{d,1}^{{\rm T}} &y_{d,2}^{{\rm T}} & g^{{\rm T}}(y_{d,1}) & g^{{\rm T}}(y_{d,2})& g^{{\rm T}}(y_k)\end{array}\right]. \end{aligned} $$

By applying Lemma 2, we have

$$ -2\xi_1^{{\rm T}}(k)M\sum_{i=k-\tau_{1}^{k}}^{k-1}(x_{i+1}-x_i)\leq \tau_0\xi_1^{{\rm T}}(k)MR_1^{-1}M^{{\rm T}}\xi_1(k)+\sum_{i=k-\tau_{1}^{k}}^{k-1}\eta_i^{{\rm T}}R_1\eta_i, $$

(31)

$$ -2\xi_1^{{\rm T}}(k)N\sum_{i=k-\tau_{2}^{k}}^{k-1}(x_{i+1}-x_i)\leq \tau_M\xi_1^{{\rm T}}(k)NR_2^{-1}N^{{\rm T}}\xi_1(k)+\sum_{i=k-\tau_{2}^{k}}^{k-1}\eta_i^{{\rm T}}R_2\eta_i, $$

(32)

$$ -2\xi_2^{{\rm T}}(k)S\sum_{i=k-d_{1}^{k}}^{k-1}(y_{i+1}-y_i)\leq d_0\xi_2^{{\rm T}}(k)SR_3^{-1}S^{{\rm T}}\xi_2(k)+\sum_{i=k-d_{1}^{k}}^{k-1}\delta_i^{{\rm T}}R_3\delta_i, $$

(33)

$$ -2\xi_2^{{\rm T}}(k)Z\sum_{i=k-d_{2}^{k}}^{k-1}(y_{i+1}-y_i)\leq d_M\xi_2^{{\rm T}}(k)ZR_4^{-1}Z^{{\rm T}}\xi_2(k)+\sum_{i=k-d_{2}^{k}}^{k-1}\delta_i^{{\rm T}}R_4\delta_i. $$

(34)

For positive definite matrices K ₁ and K ₂, it follows from the definition of η _k and δ _k that

$$ \begin{aligned} 0&=2\eta^{{\rm T}}_kK_1(x_{k+1}-x_k-\eta_k)\cr &=2\eta^{{\rm T}}_kK_1[(A-I)x_{k}+\alpha_{k}Bg(y_{d,1})+\bar{\alpha}_{k}Bg(y_{d,2})+\sigma(k,x_{k},y_{k})w_{k}-\eta_k], \end{aligned} $$

(35)

$$ \begin{aligned} 0&=2\delta^{{\rm T}}_kK_2(x_{k+1}-x_k-\delta_k)\\ &=2\delta^{{\rm T}}_kK_2[(C-I)y_{k}+\beta_{k}Dx_{\tau,1}+\bar{\beta}_{k}Dx_{\tau,2}-\delta_k]. \end{aligned} $$

(36)

Taking the expectations on both side of (35) and (36), and employing Lemma 2 yield

$$ \begin{aligned} 0=&{\mathbb{E}}\{2\eta^{{\rm T}}_kK_1[(A-I)x_{k}+\alpha_0Bg(y_{d,1})+\bar{\alpha}_0Bg(y_{d,2})-\eta_k]\}\\ =&{\mathbb{E}}\{2\eta^{{\rm T}}_kK_1(A-I)x_{k}+2\alpha_0\eta^{{\rm T}}_kK_1Bg(y_{d,1})+2\bar{\alpha}_0\eta^{{\rm T}}_kK_1Bg(y_{d,2})-2\eta^{{\rm T}}_kK_1\eta_k\}\\ \leq&{\mathbb{E}}\{2\eta^{{\rm T}}_kK_1(A-I)x_{k}+\alpha_0\eta^{{\rm T}}_kK_1\eta_k+\alpha_0g^{{\rm T}}(y_{d,1})B^{{\rm T}}K_1Bg(y_{d,1})\\ &+\bar{\alpha}_0\eta^{{\rm T}}_kK_1\eta_k+\bar{\alpha}_0g^{{\rm T}}(y_{d,2})B^{{\rm T}}K_1Bg(y_{d,2})-2\eta^{{\rm T}}_kK_1\eta_k\}\\ =&{\mathbb{E}}\{2\eta^{{\rm T}}_kK_1(A-I)x_{k}-\eta^{{\rm T}}_kK_1\eta_k \\ & +\alpha_0g^{{\rm T}}(y_{d,1})B^{{\rm T}}K_1Bg(y_{d,1}) +\bar{\alpha}_0g^{{\rm T}}(y_{d,2})B^{{\rm T}}K_1Bg(y_{d,2})\}, \end{aligned} $$

(37)

and

$$ \begin{aligned} 0=&{\mathbb{E}}\{2\delta_{k}^{{\rm T}}K_{2}[(C-I)y_{k}+\beta_{0}Dx_{\tau,1}+\bar{\beta}_{0}Dx_{\tau,2}-\delta_{k}]\}\\ =&{\mathbb{E}}\{2\delta_{k}^{{\rm T}}K_{2}(C-I)y_{k}+2\beta_{0}\delta_{k}^{{\rm T}}K_{2}Dx_{\tau,1} +2\bar{\beta}_{0}\delta_{k}^{{\rm T}}K_{2}Dx_{\tau,2}-2\delta_{k}^{{\rm T}}K_{2}\delta_{k}\}\\ \leq &{\mathbb{E}}\{2\delta_{k}^{{\rm T}}K_{2}(C-I)y_{k}+\beta_{0}\delta_{k}^{{\rm T}}K_{2}\delta_{k} +\beta_{0}x_{\tau,1}^{{\rm T}}D^{{\rm T}}K_{2}Dx_{\tau,1}\\ &+\bar{\beta}_{0}\delta_{k}^{{\rm T}}K_{2}\delta_{k}+\bar{\beta}_{0}x_{\tau,2}^{{\rm T}}D^{{\rm T}}K_{2}Dx_{\tau,2}-2\delta_{k}^{{\rm T}}K_{2}\delta_{k}\}\\ =&{\mathbb{E}}\{2\delta_{k}^{{\rm T}}K_{2}(C-I)y_{k}-\delta_{k}^{{\rm T}}K_{2}\delta_{k}+\beta_{0}x_{\tau,1}^{{\rm T}}D^{{\rm T}}K_{2}Dx_{\tau,1} +\bar{\beta}_{0}x_{\tau,2}^{{\rm T}}D^{{\rm T}}K_{2}Dx_{\tau,2}\}. \end{aligned} $$

(38)

In view of Assumption 1, we can conclude that

$$ [g_i(y_i(k))-l_{i}y_i(k)][g_i(y_i(k))-L_{i}y_i(k)]\leq0, $$

(39)

$$ [g_i(y_i(k-d_1(k)))-l_{i}y_i(k-d_1(k))][g_i(y_i(k-d_1(k)))-L_{i}y_i(k-d_1(k))]\leq0, $$

(40)

$$ [g_i(y_i(k-d_2(k)))-l_{i}y_i(k-d_2(k))][g_i(y_i(k-d_2(k)))-L_{i}y_i(k-d_2(k))]\leq0. $$

(41)

It can be deduced from (39) that there exists a diagonal matrix $\Uplambda_1=\hbox {diag}\{\lambda_{1,1},\ldots,\lambda_{1,n}\}>0$ such that

$$ \sum_{i=1}^{n}\lambda_{1,i} \left[\begin{array}{cc} y_k\\ g(y_k) \end{array}\right]^{{\rm T}} l_{i} \left[\begin{array}{cc} l_{i}L_{i}e_ie_i^{{\rm T}} & -\frac{l_{i}+L_{i}}{2}e_ie_i^{{\rm T}}\\ -\frac{l_{i}+L_{i}}{2}e_ie_i^{{\rm T}} & e_ie_i^{{\rm T}} \end{array}\right] \left[\begin{array}{cc} y_k\\ g(y_k) \end{array}\right]\\ = \left[\begin{array}{cc} y_k\\ g(y_k)\end{array}\right]^{{\rm T}} \left[\begin{array}{cc} \Uplambda_1\hat{L} & \Uplambda_1\check{L}\\ \Uplambda_1\check{L} & \Uplambda_1 \end{array}\right] \left[\begin{array}{cc} y_k\\ g(y_k)\end{array}\right] \leq 0, $$

(42)

where e _i denotes a column vector having “1” element on its ith row and zeros elsewhere. Similarly, by means of (40) and (41), there exist diagonal matrices $\Uplambda_{2}$ and $\Uplambda_{3}$ such that

$$ \left[\begin{array}{cc} y_{d,1} \\ g(y_{d,1}) \end{array}\right]^{{\rm T}} \left[\begin{array}{cc} \Uplambda_2\hat{L} & \Uplambda_2\check{L}\\ \Uplambda_2\check{L} & \Uplambda_2 \end{array}\right] \left[\begin{array}{cc} y_{d,1}\\ g(y_{d,1})\end{array}\right] \leq 0 $$

(43)

and

$$ \left[\begin{array}{cc} y_{d,2}\\ g(y_{d,2})\end{array}\right]^{{\rm T}} \left[\begin{array}{cc} \Uplambda_3\hat{L} & \Uplambda_3\check{L}\\ \Uplambda_3\check{L} & \Uplambda_3 \end{array}\right] \left[\begin{array}{cc} y_{d,2}\\ g(y_{d,2})\end{array}\right] \leq 0, $$

(44)

respectively.

Therefore, we have

$$ \begin{aligned} {\mathbb{E}}\{\Updelta V_k\}\leq &{\mathbb{E}}\{x_k^{{\rm T}}(2A^{{\rm T}}P_1A-P_1+(\tau_0-\tau_m+1)Q_1+(\tau_M-\tau_0)Q_2+\mu_{*}H_1)x_k\\ &+2\eta^{{\rm T}}_kK_1(A-I)x_{k}+\eta_k^{{\rm T}}(\tau_0R_1+\tau_MR_2-K_1)\eta_k\\ &+x_{\tau,1}^{{\rm T}}[\beta_0D^{{\rm T}}(2P_2+K_2)D-Q_1]x_{\tau,1} +x_{\tau,2}^{{\rm T}}[\bar{\beta}_0D^{{\rm T}}(2P_2+K_2)D-Q_2]x_{\tau,2}\\ &+y_k^{{\rm T}}(2C^{{\rm T}}P_2C-P_2+\mu_{*}H_2)y_k+2\delta^{{\rm T}}_kK_2(C-I)y_{k}+\delta_k^{{\rm T}}(d_0R_3+d_MR_4-K_2)\delta_k\\ &+g^{{\rm T}}(y_k)[(d_0-d_m+1)Q_{3}+(d_M-d_0)Q_{4}]g(y_k)\\ &+g^{{\rm T}}(y_{d,1})[\alpha_0B^{{\rm T}}(2P_1+K_1)B-Q_3]g(y_{d,1})\\ &+g^{{\rm T}}(y_{d,2})[\bar{\alpha}_0B^{{\rm T}}(2P_1+K_1)B-Q_4]g(y_{d,2})\\ &+2\xi_1^{{\rm T}}(k)M(x_k-x_{\tau,1})+2\xi_1^{{\rm T}}(k)N(x_k-x_{\tau,2})\\ &+2\xi_2^{{\rm T}}(k)S(y_k-y_{d,1})+2\xi_2^{{\rm T}}(k)Z(y_k-y_{d,2})\\ &+\xi_1^{{\rm T}}(k)(\tau_0MR_1^{-1}M^{{\rm T}}+\tau_MNR_2^{-1}N^{{\rm T}})\xi_1(k)\\ &+\xi_2^{{\rm T}}(k)(d_0SR_3^{-1}S^{{\rm T}}+d_MZR_4^{-1}Z^{{\rm T}})\xi_2(k)\\ &-\left[\begin{array}{cc} y_k \\ g(y_k)\\ \end{array}\right] ^{{\rm T}} \left[\begin{array}{cc} \Uplambda_1\hat{L} & \Uplambda_1\check{L}\\ \Uplambda_1\check{L} & \Uplambda_1\\ \end{array}\right] \left[\begin{array}{cc} y_k\\ g(y_k)\end{array}\right]\\ &-\left[\begin{array}{cc} y_{d,1}\\ g(y_{d,1})\end{array}\right]^{{\rm T}} \left[\begin{array}{cc} \Uplambda_2\hat{L} & \Uplambda_2\check{L}\\ \Uplambda_2\check{L} & \Uplambda_2 \end{array}\right] \left[\begin{array}{cc} y_{d,1}\\ g(y_{d,1})\end{array}\right]\\ &-\left[\begin{array}{cc} y_{d,2}\\ g(y_{d,2})\end{array}\right]^{{\rm T}} \left[\begin{array}{cc} \Uplambda_3\hat{L} & \Uplambda_3\check{L}\\ \Uplambda_3\check{L} & \Uplambda_3 \end{array}\right] \left[\begin{array}{cc} y_{d,2}\\ g(y_{d,2})\end{array}\right]\}\\ =&{\mathbb{E}}\{\xi_1^{{\rm T}}(k)\left[\Upomega_1+\tau_0 MR_1^{-1}M^{{\rm T}}+\tau_MNR_2^{-1}N^{{\rm T}}\right]\xi_1(k)\\ &+\xi_2^{{\rm T}}(k)\left[\Upomega_2+d_0SR_3^{-1}S^{{\rm T}}+d_MZR_4^{-1}Z^{{\rm T}}\right]\xi_2(k)\}, \end{aligned} $$

(45)

where

$$ \begin{aligned} \xi_1^{{\rm T}}(k)=& \left[\begin{array}{cccc}x_k^{{\rm T}} & \eta_k^{{\rm T}} & x_{\tau,1}^{{\rm T}} & x_{\tau,2}^{{\rm T}}\end{array}\right],\\ \xi_2^{{\rm T}}(k)=&\left[\begin{array}{ccccccc}y_k^{{\rm T}} & \delta_k^{{\rm T}} & y_{d,1}^{{\rm T}} &y_{d,2}^{{\rm T}} & g^{{\rm T}}(y_{d,1}) & g^{{\rm T}}(y_{d,2})& g^{{\rm T}}(y_k)\end{array}\right]. \end{aligned} $$

According to the well-known Schur complement (see, Lemma 1), one can get

$$ \begin{aligned} {\mathbb{E}}\{\Updelta V_k\}\leq & {\mathbb{E}}\{\xi_1^{{\rm T}}(k)\Upsigma_1\xi_1(k)+\xi_2^{{\rm T}}(k)\Upsigma_2\xi_2(k)\}\\ \leq& \lambda_{\max}(\Upsigma_1){\mathbb{E}}\{\|x_k\|^2+\|\eta_k\|^{2}+\|x_{\tau,1}\|^{2}+\|x_{\tau,2}\|^{2}\}\\ &+ \lambda_{\max}(\Upsigma_2){\mathbb{E}}\{\|y_k\|^2+\|\delta_k\|^{2}+\|y_{d,1}\|^{2}+\|y_{d,2}\|^{2} \\ &+\|g(y_{d,1})\|^{2}+\|g(y_{d,2})\|^{2}+\|g(y_{k})\|^{2}\}. \end{aligned} $$

In view of the conditions Σ₁ < 0 and Σ₂ < 0 in Theorem 1, it follows that

$$ {\mathbb{E}}\{\Updelta V_k\}\leq \lambda_{\max}(\Upsigma_1){\mathbb{E}}\{\|x_k\|^2\}+\lambda_{\max}(\Upsigma_2){\mathbb{E}}\{\|y_k\|^2\}\leq 0, $$

(46)

which implies that the origin of system (9) is globally asymptotically stable in the mean square sense.

Now, we are in a position to proceed with the global exponential stability analysis of the system (9). It follows from Assumption 1 that

$$ \begin{aligned} \|g(y_k)\|&\leq L_{\max}\|y_k\|,\\ \|g(y_{d,1})\|&\leq L_{\max}\|y_{d,1}\|,\\ \|g(y_{d,2})\|&\leq L_{\max}\|y_{d,2}\|, \\ \end{aligned} $$

where L _max = max{|l ₁|, ..., |l _n |, |L ₁|, ..., |L _n |}. Based upon expression of V _k , one can get

$$ \begin{aligned} {\mathbb{E}}\{V_k\}\leq &\rho_1{\mathbb{E}}\{\|x_k\|^2\}+\rho_2{\mathbb{E}}\{\|y_k\|^2\}+\rho_3\sum_{i=k-\tau_M}^{k-1}{\mathbb{E}}\{\|x_i\|^2\}\\ &+\rho_4\sum_{i=k-d_M}^{k-1}{\mathbb{E}}\{\|y_i\|^2\}+\rho_5\sum_{i=k-d_M}^{k-1}{\mathbb{E}}\{\|y_{d,1}\|^2\} +\rho_6\sum_{i=k-d_M}^{k-1}{\mathbb{E}}\{\|y_{d,2}\|^2\}\\ &+\rho_7\sum_{i=k-\tau_M}^{k-1}{\mathbb{E}}\{\|x_{\tau,1}\|^2\} +\rho_8\sum_{i=k-\tau_M}^{k-1}{\mathbb{E}}\{\|x_{\tau,2}\|^2\}, \end{aligned} $$

(47)

where

$$ \begin{aligned} \rho_1&=\lambda_{\max}(P_1),\rho_2=\lambda_{\max}(P_2),\\ \rho_3&=(\tau_0-\tau_m+1)\lambda_{\max}(Q_1)+(\tau_M-\tau_0)\lambda_{\max}(Q_2) +[\tau_0\lambda_{\max}(R_1)+\tau_M\lambda_{\max}(R_2)]\|A-I\|,\\ \rho_4&=[(d_0-d_m+1)\lambda_{\max}(Q_3)+(d_M-d_0)\lambda_{\max}(Q_4)]L_{\max} +[d_0\lambda_{\max}(R_3)+d_M\lambda_{\max}(R_4)]\|C-I\|,\\ \rho_5&=[\tau_0\lambda_{\max}(R_1)+\tau_M\lambda_{\max}(R_2)]\alpha_0\|B\|L_{\max},\\ \rho_6&=[\tau_0\lambda_{\max}(R_1)+\tau_M\lambda_{\max}(R_2)]\bar{\alpha}_0\|B\|L_{\max},\\ \rho_7&=[d_0\lambda_{\max}(R_3)+d_M\lambda_{\max}(R_4)]\beta_0\|D\|,\\ \rho_8&=[d_0\lambda_{\max}(R_3)+d_M\lambda_{\max}(R_4)]\bar{\beta}_0\|D\|. \\ \end{aligned} $$

For any scalar μ > 1, the above inequality (47), together with (46), implies that

$$ \begin{aligned} \mu^{k+1}{\mathbb{E}}\{V_{k+1}\}-\mu^k{\mathbb{E}}\{V_k\}=&\mu^{k+1}{\mathbb{E}}\{\Updelta V_{k}\}+\mu^k(\mu-1){\mathbb{E}}\{V_k\}\\ \leq&\psi_1(\mu)\mu^k{\mathbb{E}}\{\|x_k\|^2\}+\psi_2(\mu)\mu^k{\mathbb{E}}\{\|y_k\|^2\} +\psi_3(\mu)\sum_{i=k-\tau_M}^{k-1}\mu^k{\mathbb{E}}\{\|x_i\|^2\}\\ &+\psi_4(\mu)\sum_{i=k-d_M}^{k-1}\mu^k{\mathbb{E}}\{\|y_i\|^2\}+\psi_5(\mu)\sum_{i=k-d_M}^{k-1}\mu^k{\mathbb{E}}\{\|y_{d,1}\|^2\} +\psi_6(\mu)\sum_{i=k-d_M}^{k-1}\mu^k{\mathbb{E}}\{\|y_{d,2}\|^2\}\\ &+\psi_7(\mu)\sum_{i=k-\tau_M}^{k-1}\mu^k{\mathbb{E}}\{\|x_{\tau,1}\|^2\} +\psi_8(\mu)\sum_{i=k-\tau_M}^{k-1}\mu^k{\mathbb{E}}\{\|x_{\tau,2}\|^2\}, \end{aligned} $$

(48)

where $\psi_i(\mu)=\mu\lambda_{\max}(\Upsigma_i)+(\mu-1)\rho_i, i=1,2$ and ψ _j (μ) = (μ − 1)ρ _j , j = 3, ..., 8.

Furthermore, for any integer N ≥ 1, summing up both sides of (48) from 0 to N − 1 with respect to k, yields

$$ \begin{aligned} \mu^{N}{\mathbb{E}}\{V_{N}\}-{\mathbb{E}}\{V_0\}\leq&\psi_1(\mu)\sum_{i=0}^{N-1}\mu^i{\mathbb{E}}\{\|x_i\|^2\} +\psi_2(\mu)\sum_{i=0}^{N-1}\mu^i{\mathbb{E}}\{\|y_i\|^2\}\\ &+\psi_3(\mu)\sum_{k=0}^{N-1}\sum_{i=k-\tau_M}^{k-1}\mu^k{\mathbb{E}}\{\|x_i\|^2\} +\psi_4(\mu)\sum_{k=0}^{N-1}\sum_{i=k-d_M}^{k-1}\mu^k{\mathbb{E}}\{\|y_i\|^2\}\\ &+\psi_5(\mu)\sum_{k=0}^{N-1}\sum_{i=k-d_M}^{k-1}\mu^k{\mathbb{E}}\{\|y_{i-d_1^i}\|^2\} +\psi_6(\mu)\sum_{k=0}^{N-1}\sum_{i=k-d_M}^{k-1}\mu^k{\mathbb{E}}\{\|y_{i-d_2^i}\|^2\}\\ &+\psi_7(\mu)\sum_{k=0}^{N-1}\sum_{i=k-\tau_M}^{k-1}\mu^k{\mathbb{E}}\{\|x_{i-\tau_1^i}\|^2\} +\psi_8(\mu)\sum_{k=0}^{N-1}\sum_{i=k-\tau_M}^{k-1}\mu^k{\mathbb{E}}\{\|x_{i-\tau_2^i}\|^2\}. \end{aligned} $$

(49)

Note that for τ _M  ≥ 1. It follows that

$$ \begin{aligned} \sum_{k=0}^{N-1}\sum_{i=k-\tau_M}^{k-1}\mu^k{\mathbb{E}}\{\|x_i\|^2\}\leq&\left(\sum_{i=-\tau_M}^{-1}\sum_{k=0}^{i+\tau_M}+\sum_{i=0}^{N-1-\tau_M}\sum_{k=i+1}^{i+\tau_M} +\sum_{i=N-\tau_M}^{N-1}\sum_{k=i+1}^{N-1}\right)\mu^k{\mathbb{E}}\{\|x_i\|^2\}\\ \leq&\tau_M\sum_{i=-\tau_M}^{-1}\mu^{i+\tau_M}{\mathbb{E}}\{\|x_i\|^2\} +\tau_M\sum_{i=0}^{N-1-\tau_M}\mu^{i+\tau_M}{\mathbb{E}}\{\|x_i\|^2\} +\tau_M\sum_{i=N-1-\tau_M}^{N-1}\mu^{i+\tau_M}{\mathbb{E}}\{\|x_i\|^2\}\\ \leq&\tau_M\mu^{\tau_M}\max_{-\tau_M\leq i\leq0}{\mathbb{E}}\{\|x_i\|^2\} +\tau_M\mu^{\tau_M}\sum_{i=0}^{N-1}\mu^{i}{\mathbb{E}}\{\|x_i\|^2\}. \end{aligned} $$

(50)

Hence, Eq. (49) can be written as

$$ \begin{aligned} \mu^{N}{\mathbb{E}}\{V_{N}\}\leq&{\mathbb{E}}\{V_0\}+ [\psi_3(\mu)+\psi_7(\mu)+\psi_8(\mu)]\tau_M\mu^{\tau_M}\max_{-\tau_M\leq i\leq0}{\mathbb{E}}\{\|x_i\|^2\}\\ &+[\psi_1(\mu)+\tau_M\mu^{\tau_M}\psi_3(\mu)]\sum_{i=0}^{N-1}\mu^{i}{\mathbb{E}}\{\|x_i\|^2\} +\tau_M\mu^{\tau_M}\psi_7(\mu)\sum_{i=0}^{N-1}\mu^{i}{\mathbb{E}}\{\|x_{i-\tau_1^i}\|^2\}\\ &+\tau_M\mu^{\tau_M}\psi_8(\mu)\sum_{i=0}^{N-1}\mu^{i}{\mathbb{E}}\{\|x_{i-\tau_2^i}\|^2\} +[\psi_4(\mu)+\psi_5(\mu)+\psi_6(\mu)]d_M\mu^{d_M}\max_{-d_M\leq i\leq0}{\mathbb{E}}\{\|y_i\|^2\}\\ &+[\psi_2(\mu)+d_M\mu^{d_M}\psi_4(\mu)]\sum_{i=0}^{N-1}\mu^{i}{\mathbb{E}}\{\|y_i\|^2\} +d_M\mu^{d_M}\psi_5(\mu)\sum_{i=0}^{N-1}\mu^{i}{\mathbb{E}}\{\|y_{i-d_1^i}\|^2\}\\ &+d_M\mu^{d_M}\psi_6(\mu)\sum_{i=0}^{N-1}\mu^{i}{\mathbb{E}}\{\|y_{i-d_2^i}\|^2\}. \end{aligned} $$

(51)

Let σ₁ = max{ρ₃, ρ₇, ρ₈}, ζ₁(μ) = (μ − 1)σ₁, σ₂ = max{ρ₄, ρ₅, ρ₆} and ζ₂(μ) = (μ − 1)σ₂. It follows from (51) that

$$ \begin{aligned} \mu^{N}{\mathbb{E}}\{V_{N}\}\leq&{\mathbb{E}}\{V_0\}+ 3\zeta_1(\mu)\tau_M\mu^{\tau_M}\max_{-\tau_M\leq i\leq0}{\mathbb{E}}\{\|x_i\|^2\}\\ &+[\psi_1(\mu)+\tau_M\mu^{\tau_M}\zeta_1(\mu)]\sum_{i=0}^{N-1}\mu^{i} \left({\mathbb{E}}\{\|x_i\|^2\}+{\mathbb{E}}\{\|x_{\tau,1}\|^2\}+{\mathbb{E}}\{\|x_{\tau,2}\|^2\}\right)\\ &+3\zeta_2(\mu)d_M\mu^{d_M}\max_{-d_M\leq i\leq0}{\mathbb{E}}\{\|y_i\|^2\}\\ &+[\psi_2(\mu)+d_M\mu^{d_M}\zeta_2(\mu)]\sum_{i=0}^{N-1}\mu^{i} \left({\mathbb{E}}\{\|y_i\|^2\}+{\mathbb{E}}\{\|y_{d,1}\|^2\}+{\mathbb{E}}\{\|y_{d,2}\|^2\}\right). \end{aligned} $$

(52)

Define $\Upphi(\mu)=\max\{\psi_1(\mu)+\tau_M\mu^{\tau_M}\zeta_1(\mu), \psi_2(\mu)+d_M\mu^{d_M}\zeta_2(\mu)\}.$ Note that it can be verified that there exists a scalar θ > 1 such that Φ(θ) = 0. Therefore, for such a scalar θ, we have

$$ \begin{aligned} \theta^{N}{\mathbb{E}}\{V_{N}\} \leq& {\mathbb{E}}\{V_0\}+ 3\zeta_1(\theta)\tau_M\theta^{\tau_M}\max_{-\tau_M\leq i\leq0}{\mathbb{E}}\{\|x_i\|^2\}\\ &+3\zeta_2(\theta)d_M\theta^{d_M}\max_{-d_M\leq i\leq0}{\mathbb{E}}\{\|y_i\|^2\}. \end{aligned} $$

(53)

Meanwhile, one can derive from (47) that

$$ {\mathbb{E}}\{V_{0}\}\leq(\rho_1+3\sigma_1\tau_M)\max_{-\tau_M\leq i\leq0}{\mathbb{E}}\{\|x_i\|^2\} +(\rho_2+3\sigma_2d_M)\max_{-d_M\leq i\leq0}{\mathbb{E}}\{\|y_i\|^2\}. $$

(54)

Substituting (54) into (53) yields

$$ \begin{aligned} \theta^{N}{\mathbb{E}}\{V_{N}\} \leq& (\rho_1+3\sigma_1\tau_M+3\zeta_1(\theta)\tau_M\theta^{\tau_M})\max_{-\tau_M\leq i\leq 0}{\mathbb{E}}\{\|x_i\|^2\}\\ &+ (\rho_2+3\sigma_2d_M+3\zeta_2(\theta)d_M\theta^{d_M})\max_{-d_M\leq i\leq 0}{\mathbb{E}}\{\|y_i\|^2\}. \end{aligned} $$

(55)

On the other hand, from (16), it is easy to obtain

$$ \begin{aligned} {\mathbb{E}}\{V_{N}\}&\geq\lambda_{\min}(P_1){\mathbb{E}}\{\|x_N\|^2\}+\lambda_{\min}(P_2){\mathbb{E}}\{\|y_N\|^2\}\\ &\geq\lambda_{m}{\mathbb{E}}\{\|x_N\|^2+\|y_N\|^2\}. \end{aligned} $$

(56)

where $\lambda_{m}=\min\{\lambda_{\min}(P_1),\lambda_{\min}(P_2)\}.$

Combining (55) and (56), one can get

$$ \begin{aligned} \theta^{N}\lambda_{m}{\mathbb{E}}\{\|x_N\|^2+\|y_N\|^2\}\leq&\left[\rho_1+3\sigma_1\tau_M+3\tau_M\theta^{\tau_M}\zeta_1(\theta)\right]\max_{-\tau_M\leq i\leq0}{\mathbb{E}}\{\|x_i\|^2\}\\ &+\left[\rho_2+3\sigma_2d_M+3d_M\theta^{d_M}\zeta_2(\theta)\right]\max_{-d_M\leq i\leq0}{\mathbb{E}}\{\|y_i\|^2\}, \end{aligned} $$

(57)

yielding

$$ {\mathbb{E}}\{\|x_N\|^2+\|y_N\|^2\} \leq\rho_{*}\left(\frac{1} {\theta}\right)^{N}{\mathbb{E}}\left\{ \max_{-\tau_M\leq i\leq0}\|x_i\|^2+\max_{-d_M\leq i\leq0}\|y_i\|^2\right\}, $$

(58)

where $\rho_{*}=\max\{\rho_1+3\sigma_1\tau_M+3\zeta_1(\theta)\tau_M\theta^{\tau_M}, \rho_2+3\sigma_2d_M+3\zeta_2(\theta)d_M\theta^{d_M}\}/\lambda_{m}.$

Since N is an any positive integer, it can be concluded from Definition 1 that the origin of system (9) is globally exponentially stable in the mean square. This completes the proof of the theorem. □

Appendix B: Proof of Theorem 2

Consider the same Lyapunov–Krasovskii functional as that in the proof of Theorem 1. Then replace A, B, C and D in Theorem 1 by A + HF(k)E ₁, B + HF(k)E ₂, C + HF(k)E ₃ and D + HF(k)E ₄, respectively.

Since F ^T(k)F(k) ≤ I, it follows that

$$ \begin{aligned} &k_1x_k^{{\rm T}}E_1^{{\rm T}}E_1x_k-k_1(F(k)E_1x_k)^{{\rm T}}(F(k)E_1x_k)\geq0,\\ &k_2x_{\tau,1}^{{\rm T}}E_4^{{\rm T}}E_4x_{\tau,1}-k_2(F(k)E_4x_{\tau,1})^{{\rm T}}(F(k)E_4x_{\tau,1})\geq0,\\ &k_3x_{\tau,2}^{{\rm T}}E_4^{{\rm T}}E_4x_{\tau,2}-k_3(F(k)E_4x_{\tau,2})^{{\rm T}}(F(k)E_4x_{\tau,2})\geq0,\\ &k_4y_k^{{\rm T}}E_3^{{\rm T}}E_3y_k-k_4(F(k)E_3y_k)^{{\rm T}}(F(k)E_3y_k)\geq0,\\ &k_5g(y_{d,1})^{{\rm T}}E_2^{{\rm T}}E_2g(y_{d,1})-k_5(F(k)E_2g(y_{d,1}))^{{\rm T}}(F(k)E_2g(y_{d,1}))\geq0,\\ &k_6g(y_{d,2})^{{\rm T}}E_2^{{\rm T}}E_2g(y_{d,2})-k_6(F(k)E_2g(y_{d,2}))^{{\rm T}}(F(k)E_2g(y_{d,2}))\geq0. \end{aligned} $$

(59)

Calculating ${\mathbb{E}}\{\Updelta V_k\}$ together with the above inequalities, we obtain

$$ \begin{aligned} {\mathbb{E}}\{\Updelta V_k\}\leq &{\mathbb{E}} \left\{\tilde{\xi}_1^{{\rm T}}(k)\left[\Upomega_1^*+\tau_0M^*R_1^{-1}M^{*{\rm T}} +\tau_MN^*R_2^{-1}N^{*{\rm T}}\right]\tilde{\xi}_1(k)\right.\\ &\left.+\tilde{\xi}_2^{{\rm T}}(k)\left[\Upomega_2^*+d_0S^*R_3^{-1}S^{*{\rm T}}+d_MZ^*R_4^{-1}Z^{*{\rm T}}\right]\tilde{\xi}_2(k)\right\}. \end{aligned} $$

where

$$ \begin{aligned} \tilde{\xi}_1^{{\rm T}}(k)=&\left[ \begin{array}{ccccccc} x_k^{{\rm T}} & \eta_k^{{\rm T}} & x_{\tau,1}^{{\rm T}} & x_{\tau,2}^{{\rm T}} & (F(k)E_1x_k)^{{\rm T}} & (F(k)E_4x_{\tau,1})^{{\rm T}} & (F(k)E_4x_{\tau,2})^{{\rm T}}\end{array}\right],\\ \tilde{\xi}_2^{{\rm T}}(k)=&\left[\begin{array}{cccccccccc} y_k^{{\rm T}} & \delta_k^{{\rm T}} & y_{d,1}^{{\rm T}} & y_{d,2}^{{\rm T}} & g^{{\rm T}}(y_{d,1}) & g^{{\rm T}}(y_{d,2}) & g^{{\rm T}}(y_k)& (F(k)E_3y_k)^{{\rm T}} & (F(k)E_2g(y_{d,1}))^{{\rm T}} & (F(k)E_2g(y_{d,2}))^{{\rm T}} \end{array}\right], \end{aligned} $$

The remaining proof for global robust exponential stability is similar to those in the proof of Theorem 1. For the sake of simplicity, we omit it here. □