Appendix 1
1.1 Proof of Theorem 1
Consider the following Lyapunov-Krasovskii functional:
$$ \begin{aligned} V_1(t,e_t,i)&=e(t)^TP_ie(t)+2\sum^N_{j=1}q_{ji}\int\limits^{e_j(t)}_0\left[\beta_j(s)-\lambda_js\right]\hbox{d}s+ 2\sum^n_{j=1}r_{ji}\int\limits^{e_j(t)}_0\left[\sigma_j s-f_j(s)\right]\hbox{d}s\\ &\quad +2\sum^n_{j=1}s_{ji}\int\limits^{e_j(t)}_0\left[f_j(s)-\gamma_js\right]\hbox{d}s+\int\limits^t_{t-\tau_i(t)}\left[e(s)^TUe(s)+f(e(s))^TWf(e(s)) \right]\hbox{d}s, \end{aligned} $$
(16)
where \(Q_i=\hbox{diag}\{q_{1i},q_{2i},\ldots,q_{ni}\}, R_i=\hbox{diag}\{r_{1i},r_{2i},\ldots,r_{ni}\}, S_i=\hbox{diag}\{s_{1i},s_{2i},\ldots,s_{ni}\}\).
It can be easily verified that V
1(t, e
t
, i) is a nonnegative function over \([-\bar{\tau},+\infty)\). Evaluating the time derivative of V
1(t, e
t
, i) along the trajectory of system (9), we have that
$$ \hbox{d}V_1(t,e_t,i)=\pounds V_1(t,e_t,i)\hbox{d}t+\frac{\partial}{\partial e}V_{1}(t,e_t,i)\rho_i(t)\hbox{d}\omega(t), $$
(17)
where
$$ \begin{aligned} \pounds V_1(t,e_t,i)&= 2\left\{ {e(t)^TP_i+[\beta (t)-\Uplambda e(t)]^T}{Q_i}+{[\Upsigma e(t)-f(e(t))]^T}{R_i}+{[f(e(t))-\Upgamma e(t)]^T}{S_i}\right\} \\ &\quad\times\left[-\beta (t)+{A_i}(t)f(e(t))+{B_i}(t)f(e(t-\tau_i(t))) +{Y_{1i}}e(t)+{Y_{2i}}e(t- \tau_i(t))\right]\\ &\quad+2\sum\limits_{k = 1}^N {{\pi_{ik}}\sum\limits_{j = 1}^n \int\limits_0^{{e_j}(t)}{\left\{ {q_{jk}} {[{\beta_j}(s)-{\lambda_j}(s)] + {r_{jk}} {[\sigma_js-f_j(s)]} +{s_{jk}} {[{f_j}(s)-{\gamma_j}(s)]} } \right\}}\hbox{d}s} \\ &\quad+e{(t)^T}\left(\sum^N_{j=1}\pi_{ij}P_j+U\right) e(t)-(1-\dot{\tau}_i(t)) e{(t-\tau_i(t))^T}Ue(t-\tau_i(t))+f{(e(t))^T}Wf(e(t))\\ &\quad+\sum\limits_{j = 1}^N {{\pi_{ij}}{\tau_j}(t)} \left[e{(t-\tau_i(t))^T}Ue(t-\tau_i(t))+f{(e(t-\tau_i(t)))^T}Wf(e(t- \tau_i(t)))\right]\\ &\quad-(1-\dot{\tau}_i(t))f{(e(t-\tau_i(t)))^T} Wf(e(t-\tau_i(t)))+\frac{1}{2}\hbox{trace}\left[ {\rho_i {{(t)}^T}\frac{{{\partial ^2}}}{{\partial {e^2}}}{V_1}(t,{e_t},i)\rho_i (t)} \right]. \end{aligned} $$
(18)
From Assumptions 3 and 4, we get that
$$ \begin{aligned} &2\sum\limits_{k = 1}^N {{\pi_{ik}}
\sum\limits_{j = 1}^n \int\limits_0^{{e_j}(t)}
{\left\{ {q_{jk}} {[{\beta_j}(s)-{\lambda_j}(s)]
+ {r_{jk}} {[\sigma_js-f_j(s)]} +{s_{jk}} {[{f_j}
(s)-{\gamma_j}(s)]} } \right\}}\hbox{d}s } \\
&\le 2\sum\limits_{k = 1}^N {\pi^{\prime}_{ik}}
\sum\limits_{j = 1}^n \int\limits_0^{{e_j}(t)}
{\left\{ {q_{jk}} {({\delta_j}-{\lambda_j})s +
{r_{jk}} ({\sigma_j}-{\gamma_j})s +{s_{jk}}
({\sigma_j}-{\gamma_j})s} \right\}\hbox{d}s } \\
&= e(t)^T\sum\limits_{k = 1}^N {\pi^{\prime}_{jk}}
[{Q_k}(\Updelta-\Uplambda )+({R_k}+{S_k})(\Upsigma
-\Upgamma )]e(t). \end{aligned} $$
(19)
In addition, we derive that
$$ \begin{aligned} &\sum\limits_{j = 1}^N {{\pi_{ij}}{\tau_j}(t)} \left[e{(t-\tau_i(t))^T}Ue(t-\tau_i(t))+f{(e(t-\tau_i(t)))^T}Wf(e(t- \tau_i(t)))\right]\\ & \le {\zeta_i}{(t)^T}\sum\limits_{j = 1}^N \pi^{\prime}_{ij} {\bar{\tau }_j}\left(\varsigma_4^TU\varsigma_4+ \varsigma_5^TU\varsigma_5\right){\zeta_i}(t). \end{aligned} $$
(20)
For any \(j=1,2,\ldots,n,\) it follows from (5) that
$$ \begin{aligned} &0\leq\frac{\hbox{d}(\beta_j(e_j)-\lambda_je_j)}{\hbox{d} e_j}\leq \delta_j-\lambda_j,\\ &0\leq\frac{\hbox{d}(f_j(e_j)-\gamma_je_j)}{\hbox{d} e_j}\leq \sigma_j-\gamma_j,\\ &0\leq\frac{\hbox{d}(\sigma_je_j-f_j(e_j))}{\hbox{d} e_j}\leq \sigma_j-\gamma_j. \end{aligned} $$
Thus, we have that
$$ \begin{aligned} \frac{1}{2}\frac{{{\partial ^2}}}{{\partial {e^2}}}{V_1}(t,{e_t},i)&=P_i+Q_i\times\hbox{diag}\left\{\frac{\hbox{d}(\beta_1(e_1)-\lambda_1e_1)}{\hbox{d} e_1},\ldots,\frac{{\hbox{d}}(\beta_n(e_n)-\lambda_ne_n)}{{\hbox{d}} e_n}\right\}\\ &\quad+R_i\times\hbox{diag}\left\{\frac{{\hbox{d}}(\sigma_1e_1-f_1(e_1))}{{\hbox{d}} e_1},\ldots,\frac{{\hbox{d}}(\sigma_ne_n-f_n(e_n))}{{\hbox{d}} e_n}\right\}\\ &\quad+S_i\times\hbox{diag}\left\{\frac{{\hbox{d}}(f_1(e_1)-\gamma_1e_1)}{\hbox{d} e_1},\ldots,\frac{{\hbox{d}}(f_n(e_n)-\gamma_ne_n)}{{\hbox{d}} e_n}\right\}\\ &\leq\bar{P}_i. \end{aligned} $$
(21)
For any \(j=1,2,\ldots,n,\) from (5) we obtain that
$$ \begin{aligned} &\left(f_j(e_j(t))- \sigma_je_j(t)\right)\left(f_j(e_j(t))-\gamma_je_j(t)\right)\leq 0,\\ &\left(f_j(e_j(t-\tau(t)))- \sigma_je_j(t-\tau(t))\right)\left(f_j(e_j(t-\tau(t))) -\gamma_je_j(t-\tau(t))\right)\leq 0. \end{aligned} $$
Therefore, the following matrix inequalities hold for any positive diagonal matrices J
i
, Z
i
with compatible dimensions
$$ 0\leq -e(t)^T\Upsigma\Upgamma J_ie(t)+e(t)^TJ_i(\Upsigma+\Upgamma)f(e(t))-f(e(t))^TJ_if(e(t)), $$
(22)
$$ \begin{aligned} &0\leq -e(t-\tau(t))^T\Upsigma\Upgamma Z_i e(t-\tau_i(t))\\ &\quad+e(t-\tau_i(t))^TZ_i(\Upsigma+\Upgamma) f(e(t-\tau_i(t)))-f(e(t-\tau_i(t)))^TZ_if(e(t-\tau_i(t))). \end{aligned} $$
(23)
According to Assumption 1 and Lemma 1, for any positive scalar \(\varepsilon_i\) we have that
$$ \begin{aligned} &\frac{1}{2}\hbox{trace}\left[ {\rho_i {{(t)}^T}\frac{{{\partial ^2}}}{{\partial {e^2}}}{V_1}(t,{e_t},i)\rho_i (t)} \right]\\ &=\left[C_ie(t)+D_ie(t-\tau_i(t))+E_i\Upphi_i(t)(H_{3i}e(t) +H_{4i}e(t-\tau_i(t)))\right]^T\\ &\quad\times\bar{P_i}\left[C_ie(t)+D_ie(t-\tau_i(t)) +E_i\Upphi_i(t)(H_{3i}e(t)+H_{4i}e(t-\tau_i(t)))\right]\\ &\leq\varepsilon_i^{-1}(H_{3i}e(t)+H_{4i}e(t-\tau_i(t)))^T (H_{3i}e(t)+H_{4i}e(t-\tau_i(t)))\\ &\quad+(C_ie(t)+D_ie(t-\tau_i(t)))^T\left(\bar{P_i}^{-1}-\varepsilon_i E_iE_i^T\right)^{-1}(C_ie(t)+D_ie(t-\tau_i(t))). \end{aligned} $$
(24)
From (5), the following inequalities hold for any positive diagonal matrix M
i
with compatible dimension
$$ 0\leq2\{e(t)^TM_i\beta(e(t))-e(t)^TM_i\Upgamma e(t)\}. $$
(25)
From (18–25), we obtain that
$$ \pounds {V_1}(t,{e_t},i)\leq \xi_i(t)^T\bar{\Upomega}_i(t)\xi_i(t)+ e(t-\tau_i(t))^TK_ie(t-\tau_i(t))+ f(e(t-\tau_i(t)))^TL_if(e(t-\tau_i(t))). $$
(26)
where
$$ \bar{\Upomega}_i(t)=\left[\begin{array}{ccccc} \psi_{1i}+\psi_{7i}&\psi_{2i}&\psi_{3i}+ \widetilde{P}_iA_i(t)&\widetilde{P}_iY_{2i} +\psi_{8i}&\widetilde{P}_iB_i(t)\\ *&-2Q_i&R_i-S_i+Q_iA_i(t)&Q_iY_{2i}&Q_iB_i(t)\\ *&*&W-J_i+\hbox{sym}\left((S_i-R_i)A_i(t)\right)&(S_i-R_i)Y_{2i}&(S_i-R_i)B_i(t)\\ *&*&*&\psi_{5i}+\psi_{9i}&\frac{1}{2}Z_i(\Upsigma+\Upgamma)\\ *&*&*&*&\psi_{6i}\\ \end{array}\right], $$
with
$$ \begin{aligned} \psi_{7i}&=C^T_i\left(\bar{P}^{-1}_i-\varepsilon_i E_iE^T_i\right)^{-1}C_i+\varepsilon^{-1}_i H^T_{3i}H_{3i},\\ \psi_{8i}&=C^T_i\left(\bar{P}^{-1}_i-\varepsilon_i E_iE^T_i\right)^{-1}D_i+\varepsilon^{-1}_i H^T_{3i}H_{4i},\\ \psi_{9i}&=D^T_i\left(\bar{P}^{-1}_i-\varepsilon_i E_iE^T_i\right)^{-1}D_i+\varepsilon^{-1}_i H^T_{4i}H_{4i}. \end{aligned} $$
Now, by (14), it is easy to see that there exists a scalar α > 1 such that
$$ \left[\begin{array}{ccccc} \tilde{\Uppsi}_i&{\mathcal{A}}_iE_i&{\mathcal{B}}_i&{\mathcal{C}}_i&0\\ *&-\epsilon_i I&0&0&0\\ *&*&-\varepsilon_i F_iF^T_i&0&I\\ *&*&*& -\varepsilon_i I&0\\ *&*&*&*&-\bar{P_i}\end{array}\right]<0, $$
(27)
where
$$ \tilde{\Uppsi}_i=\left[\begin{array}{ccccc} \alpha F_i+\psi_{1i}&\psi_{2i}&\psi_{3i}+\widetilde{P}_iA_i& \widetilde{P}_iY_{2i}+\psi_{8i}&\widetilde{P}_iB_i\\ *&-2Q_i&R_i-S_i+Q_iA_i&Q_iY_{2i}&Q_iB_i\\ *&*&W-J_i+\hbox{sym}((S_i-R_i)A_i)&(S_i-R_i)Y_{2i}&(S_i-R_i)B_i\\ *&*&*&\psi_{5i}+\psi_{9i}&\frac{1}{2}Z_i(\Upsigma+\Upgamma)\\ *&*&*&*&\psi_{6i}\\ \end{array}\right]. $$
Applying Schur complements to (27) results in
$$ \begin{aligned} &{\left[\begin{array}{ccccc} \alpha F_i+\psi_{1i}+\psi_{7i}&\psi_{2i}&\psi_{3i}+ \widetilde{P}_iA_i & \widetilde{P}_iY_{2i}+\psi_{8i}&\widetilde{P}_iB_i\\ {*}&-2Q_i&R_i-S_i+Q_iA_i&Q_iY_{2i}&Q_iB_i\\ {*}&{*}&W-J_i+\hbox{sym}((S_i-R_i)A_i)&(S_i-R_i)Y_{2i}&(S_i-R_i)B_i\\ {*}&{*}&{*}&\psi_{5i}+\psi_{9i}&\frac{1}{2}Z_i(\Upsigma+\Upgamma)\\ {*}&{*}&{*}&{*}&\psi_{6i}\\ \end{array}\right]}\\ &\quad+{\epsilon^{-1}_i\left[\begin{array}{ccccc} \widetilde{P}_iE_i\\ Q_iE_i\\ (S_i-R_i)E_i\\ 0\\ 0\\ \end{array}\right]\left[\begin{array}{cccc} \widetilde{P}_iE_i\\ Q_iE_i\\ (S_i-R_i)E_i\\ 0\\ 0\\ \end{array}\right]^T+\epsilon_i\left[\begin{array}{ccccc} 0\\ 0\\ H_{1i}^T\\ 0\\ H_{2i}^T\\ \end{array}\right]\left[\begin{array}{ccccc} 0\\ 0\\ H_{1i}^T\\ 0\\ H_{2i}^T\\ \end{array}\right]^T<0}. \end{aligned} $$
(28)
Using Assumption 1 and Lemma 1, for any positive scalar \(\epsilon_i\) we have that
$$ \begin{aligned} &{\left[\begin{array}{ccccc} 0&0&\widetilde{P}_i\Updelta A_i(t)&0&\widetilde{P}_i\Updelta B_i(t)\\ {*}&0&Q_i\Updelta A_i(t)&0&Q_i\Updelta B_i(t)\\ {*}&{*}&\hbox{sym}((S_i-R_i)\Updelta A_i(t))&0&(S_i-R_i) \Updelta B_i(t)\\ {*}&{*}&{*}&0&0\\ {*}&{*}&{*}&{*}&0\\ \end{array}\right]}\\ &{=\left[\begin{array}{ccccc} \widetilde{P}_iE_i\\ Q_iE_i\\ (S_i-R_i)E_i\\ 0\\ 0\\ \end{array}\right]\Upphi_i(t)\left[\begin{array}{ccccc} 0\\ 0\\ H_{1i}^T\\ 0\\ H_{2i}^T\\ \end{array}\right]^T+\left[\begin{array}{ccccc} 0\\ 0\\ H_{1i}^T\\ 0\\ H_{2i}^T\\ \end{array}\right]\Upphi_i(t)^T\left[\begin{array}{ccccc} \widetilde{P}_iE_i\\ Q_iE_i\\ (S_i-R_i)E_i\\ 0\\ 0\\ \end{array}\right]^T}\\ &{\leq\epsilon^{-1}_i\left[\begin{array}{ccccc} \widetilde{P}_iE_i\\ Q_iE_i\\ (S_i-R_i)E_i\\ 0\\ 0\\ \end{array}\right]\left[\begin{array}{ccccc} \widetilde{P}_iE_i\\ Q_iE_i\\ (S_i-R_i)E_i\\ 0\\ 0\\ \end{array}\right]^T+\epsilon_i\left[\begin{array}{ccccc} 0\\ 0\\ H_{1i}^T\\ 0\\ H_{2i}^T\\ \end{array}\right]\left[\begin{array}{ccccc} 0\\ 0\\ H_{1i}^T\\ 0\\ H_{2i}^T\\ \end{array}\right]^T.} \end{aligned} $$
This together with (28) provides that
$$ \bar{\Upomega}_i(t)+\hbox{diag}\begin{array}{ccccc} \{\alpha F_i& 0& 0& 0& 0\}<0 \end{array}. $$
By this inequality and (26), it is easy to see that
$$ \pounds V_1(t,e_t,i)<-\alpha e(t)^TF_ie(t)+e(t-\tau_i(t))^TK_ie(t-\tau_i(t)) +f(e(t-\tau_i(t)))^TL_if(e(t-\tau_i(t))). $$
Taking the mathematical expectations on both sides of (17), from above inequality we have that
$$ \begin{aligned} \hbox{d}{\mathbb{E}}\{V_1(t,e_t,i)\}&={\mathbb{E}}\pounds V_1(t,e_t,i)\hbox{d}t+{\mathbb{E}}\left\{\frac{\partial}{\partial e}V_1(t,e_t,i)\rho_i(t)\hbox{d}\omega(t)\right\}\\ &<\left[-\alpha e(t)^TF_ie(t)\hbox{d}t+e(t-\tau_i(t))^TK_ie(t-\tau_i(t))\hbox{d}t+f(e(t-\tau_i(t)))^TL_if(e(t-\tau_i(t)))\right]\hbox{d}t. \end{aligned} $$
By integrating above inequality from t − τ(t) to t, we obtain that
$$ \begin{aligned} {\mathbb{E}}\{V_1(t,e_t,i)\}-{\mathbb{E}} \{V_1(t,e_{t-\tau_i(t)},i)\}&=\int\limits_{t-\tau_i(t)}^t {\mathbb{E}}\{V_1(s,e_s,i)\}\hbox{d}s \\ &<-\alpha\int\limits_{t-\tau_i(t)}^t e(s)^TF_ie(s)\hbox{d}s+\int\limits_{t-\tau_i(t)}^t \left[e(s-\tau_i(s))^TK_ie(s-\tau_i(s))\right.\\ &\quad\left. +f(e(s-\tau_i(s)))^TL_if(e(s-\tau_i(s)))\right]\hbox{d}s. \end{aligned} $$
It follows that
$$ \begin{aligned} &{\mathbb{E}}\left[\frac{{\hbox{d}}V_1(t,e_t,i)}{\hbox{d}t} \right]+ \iota {\mathbb{E}}\left[V_1(t,e_t,i)-V_1(t,e_{t-\tau_i(t)},i)\right]\\ &\quad<-\alpha e(t)^TF_ie(t)+e(t-\tau_i(t))^TK_ie(t-\tau_i(t))\\ &+f(e(t-\tau_i(t)))^TL_if(e(t-\tau_i(t)))-\iota\alpha\int\limits_{t-\tau_i(t)}^t e(s)^TF_ie(s)\hbox{d}s\\ &+\iota\int\limits_{t-\tau(t)}^t \left[e(s-\tau_i(s))^TKe(s-\tau_i(s)) +f(e(s-\tau_i(s)))^TL_if(e(s-\tau_i(s)))\right]\hbox{d}s. \end{aligned} $$
(29)
In view of (10) and (11), we have that
$$ -e(t)^TF_ie(t)\leq -\nu e(t)^T\bar{P_i}e(t), $$
(30)
$$ -\iota\int\limits_{t-\tau_i(t)}^t e(s)^TF_ie(s)\hbox{d}s\leq -\nu\int\limits_{t-\tau_i(t)}^t \left[f(e(s))^TWf(e(s))+e(s)^TUe(s)\right]\hbox{d}s. $$
(31)
Noticing that
$$ \begin{aligned} 2\sum^n_{j=1}q_{ji}\int\limits^{e_j(t)}_0\left[\beta_j(s)-\lambda_js\right]\hbox{d}s&\leq 2\sum^n_{j=1}q_{ji}\int\limits^{e_j(t)}_0(\delta_j-\gamma_j)s\hbox{d}s=e(t)^TQ_i(\Updelta-\Uplambda)e(t),\\ 2\sum^n_{j=1}r_{ji}\int\limits^{e_j(t)}_0\left[\sigma_j s-f_j(s)\right]\hbox{d}s&\leq 2\sum^n_{j=1}r_{ji}\int\limits^{e_j(t)}_0(\sigma_j-\gamma_j)s\hbox{d}s=e(t)^TR_i(\Upsigma-\Upgamma)e(t),\\ 2\sum^n_{j=1}s_{ji}\int\limits^{e_j(t)}_0\left[f_j(s)-\gamma_js\right]\hbox{d}s&\leq 2\sum^n_{j=1}s_{ji}\int\limits^{e_j(t)}_0(\sigma_j-\gamma_j)s\hbox{d}s=e(t)^TS_i(\Upsigma-\Upgamma)e(t). \end{aligned} $$
Therefore, from (16) we have that
$$ {\mathbb{E}}\{V_1(t,e_t,i)\}\leq e(t)^T\bar{P_i}e(t)+\int\limits_{t-\tau_i(t)}^t \left[e(s)^TUe(s)+f(e(s))^TWf(e(s))\right]\hbox{d}s. $$
This together with (30–31) yields that
$$ -e(t)^TF_ie(t)-\iota\int\limits_{t-\tau_i(t)}^t \left[e(s)^TF_ie(s)\right]\hbox{d}s\leq-\nu {\mathbb{E}}\{V_1(t,e_t,i)\}. $$
(32)
Moreover, \({\mathbb{E}\{V_1(t,e_t,i)\}\geq e(t)^TP_ie(t)}\), therefore, it follows from (5) and (12) that
$$ \begin{aligned} &e(t-\tau_i(t))^TK_ie(t-\tau_i(t)) +f(e(t-\tau_i(t)))^TL_if(e(t-\tau_i(t)))\\ &\leq e(t-\tau_i(t))^T\left(K_i+\Uptheta L_i\Uptheta\right)e(t-\tau_i(t))\\ &\leq \frac{\nu}{1+\iota\bar{\tau}_i}e(t-\tau_i(t))^TP_ie(t-\tau_i(t))\\ &\leq \frac{\nu}{1+\iota\bar{\tau}_i}{\mathbb{E}}\{V_1(t,e_{t-\tau_i(t)},i)\}. \end{aligned} $$
(33)
Thus, we obtain that
$$ \begin{aligned} &\int\limits_{t-\tau_i(t)}^t\left[e(s-\tau_i(s))^TK_ie(s-\tau_i(s)) +f(e(s-\tau_i(s)))^TL_if(e(s-\tau_i(s)))\right]\hbox{d}s\\ &\leq \frac{\nu}{1+\iota\bar{\tau}_i} \int\limits_{t-\tau_i(t)}^t{\mathbb{E}}\{V_1(s,e_{s-\tau_i(s)},i)\}\hbox{d}s. \end{aligned} $$
(34)
Substituting (32–34) into (29) derives that
$$ \begin{aligned} &\frac{{\hbox{d}}{\mathbb{E}}\{V_1(t,e_t,i)\}}{\hbox{d}t}<-(\iota+\alpha\nu){\mathbb{E}}\{V_1(t,e_t,i)\}+\iota {\mathbb{E}}\{V_1(t,e_{t-\tau_i(t)},i)\} \\ &\quad+\frac{\nu}{1+\iota\bar{\tau}_i} \left[{\mathbb{E}}\{V_1(t,e_{t-\tau_i(t)},i)\}+\iota\int\limits_{t-\tau_i(t)}^t {\mathbb{E}}\{V_1(s,e_{s-\tau_i(s)})\}\hbox{d}s\right]\\ &\leq-(\iota+\alpha\nu){\mathbb{E}}\{V_1(t,e_t,i)\}+\iota {\mathbb{E}}\{V_1(t,e_{t-\tau_i(t)},i)\}\\ &\quad+\frac{\nu}{1+\iota\bar{\tau}_i}\left[{\mathbb{E}} \{V_1(t,e_{t-\tau_i(t)},i)\}+\iota\tau_i(t)\sup_{[t-2\bar{\tau},t]} {\mathbb{E}}\{V_1(s,e_{s-\tau_i(s)})\}\right]\\ &\leq-(\iota+\alpha\nu){\mathbb{E}}\{V_1(t,e_t,i)\} +(\iota+\nu)\sup_{[t-2\bar{\tau},t]}{\mathbb{E}}\{V_1(s,e_s,i)\}. \end{aligned} $$
Noting that α > 1, applying Lemma 5 to above inequality results in
$$ {\mathbb{E}}\{V_1(t,e_t,i)\}\leq \sup_{[-2\bar{\tau},0]}{\mathbb{E}}\{V_1(s,e_s,i)\}e^{-\kappa t}, $$
where κ is the unique positive solution of the following equation:
$$ \kappa =\iota+\alpha\nu-(\nu+\iota)e^{2\kappa \bar{\tau}}. $$
Therefore, we arrive at the conclusion that
$$ {\mathbb{E}}\{||e(t)||^2\}\leq e^{-\kappa t}{\mathbb{E}}\{||\varphi(t)||^2\}. $$
The proof is completed.
Appendix 2
1.1 Proof of Theorem 2
Define the following Lyapunov-Krasovskii functional:
$$ V(t,{e_t},i) = \sum\limits_{j = 1}^2 {{V_j}(t,{e_t},i)}, $$
where V
1(t, e
t
, i) ie defined in (16) and
$$ \begin{aligned} {V_2}(t,{e_t},i) &= {\bar{\tau }_i}\int\limits_{t-{\bar{\tau }_i}}^t {\int\limits_v^t {\chi_i {{(s)}^T}{T_1}\chi_i (s)\hbox{d}s\hbox{d}v} } +{\bar{\varrho}_i}\int\limits_{t- {\bar{\varrho}_i}}^t {\int\limits_v^t {f{{(e(s))}^T}{T_2}f(e(s))\hbox{d}s\hbox{d}v} } \\ &\quad+\int\limits_{t-{\bar{\tau }_i}}^t {\int\limits_v^t {\rho {{(s)}^T}{T_3}\rho (s)\hbox{d}s\hbox{d}v} }. \end{aligned} $$
It can be easily verified that V(t, e
t
, i) is a nonnegative function over \([-\hat{\tau},+\infty)\). Evaluating the time derivative of V(t, e
t
, i) along the trajectory of system (3), we have that
$$ \hbox{d}V(t,e_t,i)=\pounds V(t,e_t,i)\hbox{d}t+\frac{\partial}{\partial e}V(t,e_t,i)\rho_i(t)\hbox{d}\omega(t), $$
(35)
where
$$ \begin{aligned} \pounds V_1(t,e_t,i)&\leq 2\left\{ {e(t)^TP_i+[\beta (t)-\Uplambda e(t)]^T}{Q_i}+{[\Upsigma e(t)-f(e(t))]^T}{R_i}+{[f(e(t))-\Upgamma e(t)]^T}{S_i}\right\}\\ &\quad\times\left[-\beta (t)+{A_i}(t)f(e(t))+{B_i}(t)f(e(t-\tau_i(t)))+{G_i}(t)\int\limits_{t-{\varrho_i}(t)}^t {f(e(s))} \hbox{d}s \right.\\ &\left.\quad+{Y_{1i}}e(t)+{Y_{2i}}e(t- \tau_i(t))\right]+e(t)^T\sum\limits_{k = 1}^N {\pi^{\prime}_{jk}} [{Q_k}(\Updelta -\Uplambda )+({R_k}+{S_k})(\Upsigma -\Upgamma )]e(t)\\ &\quad+e{(t)^T}\left(\sum^N_{j=1}\pi_{ij}P_j+U\right)e(t)-(1-\dot{\tau}_i(t)) e{(t-\tau_i(t))^T}Ue(t-\tau_i(t)) \\ &\quad+\sum\limits_{j = 1}^N \pi^{\prime}_{ij} {\bar{\tau }_j} \left[e{(t-\tau_i(t))^T}Ue(t-\tau_i(t))+f{(e(t-\tau_i(t)))^T}Wf(e(t- \tau_i(t)))\right]\\ &\quad+f{(e(t))^T}Wf(e(t))- (1-\dot{\tau}_i(t))f{(e(t-\tau_i(t)))^T}Wf(e(t-\tau_i(t)))+\rho_i (t)^T\widetilde{P}_i\rho_i (t), \end{aligned} $$
(36)
$$ \begin{aligned} \pounds{V_2}(t,{e_t},i) &= \bar \tau_i^2\chi_i {(t)^T}{T_1}\chi_i (t)-{\bar{\tau }_i}\left( {1-\sum\limits_{j = 1}^N {{\pi_{ij}}{\bar{\tau }_j}} } \right)\int\limits_{t-{\bar{\tau }_i}}^t {\chi_i {{(s)}^T}{T_1}\chi_i (s)\hbox{d}s}\\ &\quad+\bar\varrho_i^2f{(e(t))^T}{T_2}f(e(t))-{{\bar\varrho}_i}\left( {1-\sum\limits_{j = 1}^N {{\pi_{ij}}{{\bar\varrho}_j}} } \right)\int\limits_{t-{{\bar\varrho}_i}}^t {f{{(e(s))}^T}{T_2}f(e(s))\hbox{d}s} \\ &\quad+{\bar{\tau }_i}\rho_i {(t)^T}{T_3}\rho_i (t)-\left( {1-\sum\limits_{j = 1}^N {{\pi_{ij}}{\bar{\tau }_j}} } \right)\int\limits_{t-\bar \tau_i }^t {\rho_i {{(s)}^T}{T_3}\rho_i (s)\hbox{d}s}. \end{aligned} $$
(37)
For any t with \(0<\tau_i(t)<\bar\tau_i\) and \(0<\varrho_i(t)<\bar\varrho_i\), from Lemma 2 we have the following inequalities
$$ \begin{aligned} &-{\bar{\tau }_i}\left( {1-\sum\limits_{j = 1}^N {{\pi_{ij}}{\bar{\tau }_j}} } \right)\int\limits_{t-{\bar{\tau }_i}}^t {\chi_i {{(s)}^T}{T_1}\chi_i (s)\hbox{d}s}\\ &\quad=- {\bar{\tau }_i}\int\limits_{t-\tau_i(t)}^t {\chi_i {{(s)}^T}{\bar{T}_1}\chi_i (s)\hbox{d}s} -{\bar{\tau }_i}\int\limits_{t-{\bar{\tau }_i}}^{t- \tau_i(t)} {\chi_i {{(s)}^T}{\bar{T}_1}\chi_i (s)\hbox{d}s}\\ &\quad\leq- \frac{\bar\tau_i}{\tau_i(t)}\left(\int\limits_{t-\tau_i(t)}^t \chi_i(s)\hbox{d}s\right)^T\bar{T}_1\left(\int\limits_{t-\tau_i(t)}^t \chi_i(s)\hbox{d}s\right)\\ &\quad-\frac{\bar\tau_i}{\bar\tau_i-\tau_i(t)} \left(\int\limits_{t-\bar\tau_i}^{t-\tau_i(t)} \chi_i(s)\hbox{d}s\right)^T\bar T_1\left(\int\limits_{t-\bar\tau_i}^{t-\tau_i(t)} \chi_i(s)\hbox{d}s\right), \end{aligned} $$
(38)
$$ \begin{aligned} &-{{\bar\varrho}_i}\left( {1-\sum\limits_{j = 1}^N {{\pi_{ij}}{{\bar\varrho}_j}} } \right)\int\limits_{t-{{\bar\varrho}_i}}^t {f{{(e(s))}^T}{T_2}f(e(s))\hbox{d}s} \\ &\quad= -{{\bar\varrho}_i}\int\limits_{t-{\varrho_i}(t)}^t {f{{(e(s))}^T}{\bar{T}_2}f(e(s))\hbox{d}s} -{{\bar\varrho}_i}\int\limits_{t-{{\bar\varrho}_i}}^{t-{\varrho_i}(t)} {f{{(e(s))}^T}{\bar{T}_2}f(e(s))\hbox{d}s} \\ &\quad\leq- \frac{\bar\varrho_i}{{\varrho_i}(t)}\left(\int\limits_{t-{\varrho_i}(t)}^t f(e(s))\hbox{d}s\right)^T\bar{T}_2\left(\int\limits_{t-{\varrho_i}(t)}^t f(e(s))\hbox{d}s\right) \\ &\quad- \frac{\bar\varrho_i}{\bar\varrho_i-{\varrho_i}(t)} \left(\int\limits_{t-\bar\varrho_i}^{t-\varrho_i(t)} f(e(s))\hbox{d}s\right)^T\bar{T}_2\left(\int\limits_{t-\bar\varrho_i}^{t-\varrho_i(t)} f(e(s))\hbox{d}s\right). \end{aligned} $$
(39)
Set λ
j
 = 1, μ
j
 = 3, based on Lemma 2 we get from (38–39) that
$$ \begin{aligned} &- {\bar{\tau }_i}\int\limits_{t-{\bar{\tau }_i}}^t {\chi_i{{(s)}^T}{\bar{T}_1}\chi_i(s)\hbox{d}s} -{{\bar\varrho}_i}\int\limits_{t-\bar\varrho_i}^{t} {f{{(e(s))}^T}{\bar{T}_2}f(e(s))\hbox{d}s} \\ &\quad\le \max \left\{-\varsigma_7^T{\bar{T}_1}{\varsigma_7}-3\varsigma_8^T {\bar{T}_1}{\varsigma_8}-\varsigma_6^T{\bar{T}_2}{\varsigma_6}- 3\varsigma_{11}^T{\bar{T}_2}{\varsigma_{11}}, -\varsigma_7^T{\bar{T}_1}{\varsigma_7}-3 \varsigma_8^T{\bar{T}_1}{\varsigma_8}-3\varsigma_6^T {\bar{T}_2}{\varsigma_6}- \varsigma_{11}^T{\bar{T}_2}{\varsigma_{11}},\right.\\ &\left.\quad\quad\quad\quad-3\varsigma_7^T{\bar{T}_1}{\varsigma_7} -\varsigma_8^T{\bar{T}_1}{\varsigma_8}-3 \varsigma_6^T{\bar{T}_2}{\varsigma_6}- \varsigma_{11}^T{\bar{T}_2}{\varsigma_{11}} ,-3\varsigma_7^T{\bar{T}_1}{\varsigma_7} -\varsigma_8^T{\bar{T}_1}{\varsigma_8}-\varsigma_6^T {\bar{T}_2}{\varsigma_6}- 3\varsigma_{11}^T{\bar{T}_2}{\varsigma_{11}}\right\}. \end{aligned} $$
(40)
It is easy to verify that Eq. (40) holds for any t with \(0\leq\tau_i(t)\leq\bar\tau_i\) and \(0\leq\varrho_i(t)\leq\bar\varrho_i.\)
From [4, 17], we have that
$$ {\mathbb{E}}\left(\int\limits_{t-\tau_i(t)}^t {\rho_i {{(s)}^T}{T_3}\rho_i (s)} \hbox{d}s\right) ={\mathbb{E}}\left\{\left( \int\limits_{t-\tau_i(t)}^t \rho_i (s)\hbox{d}\omega(s) \right)^TT_3\left( \int\limits_{t-\tau_i(t)}^t \rho_i (s)\hbox{d}\omega(s) \right)\right\}, $$
(41)
$$ {\mathbb{E}}\left(\int\limits_{t-{\bar{\tau }_i}}^{t-\tau_i(t)} {\rho_i {{(s)}^T}{T_3}\rho_i (s)} \hbox{d}s\right) = {\mathbb{E}}\left\{ {\left( {\int\limits_{t-{\bar{\tau }_i}}^{t-\tau_i(t)} {\rho_i (s)} \hbox{d}\omega(s)} \right)^T}{T_3}\left( {\int\limits_{t-{\bar{\tau }_i}}^{t-\tau_i(t)} {\rho_i (s)} \hbox{d}\omega(s)} \right)\right\}. $$
(42)
On the other hand, by the Leibniz-Newton formula, we get that
$$ \int\limits_{t-\tau_i(t)}^t {\chi_i(s)} \hbox{d}s = e(t)-e(t-\tau_i(t))-\int\limits_{t-\tau_i(t)}^t {\rho_i (s)} \hbox{d}\omega(s). $$
Therefore, the following equalities hold for any real matrices X
ji
(j = 1, 2, 3) with compatible dimensions
$$ \begin{aligned} &2\chi_i{(t)^T}{X^T_{1i}}\left\{\vphantom{\int\limits_{t-{\varrho_i}(t)}^t }-\chi_i(t)-\beta(e(t)) +{A_i}(t)f(e(t))+{B_i}(t)f(e(t-\tau_i(t)))\right.\\ &\left.\quad\quad\quad+{G_i}(t)\int\limits_{t-{\varrho_i}(t)}^t {f(e(s))} \hbox{d}s+{Y_{1i}}e(t)+{Y_{2i}}e(t-\tau_i(t))\right\} = 0, \end{aligned} $$
(43)
$$ 2\left({X_{2i}}e(t)+{X_{3i}}e(t-\tau_i(t))\right)^T\left\{e(t)-e(t-\tau_i(t))-\int\limits_{t-\tau_i(t)}^t {\rho_i (s)} \hbox{d}\omega(s)-\int\limits_{t-\tau_i(t)}^t {\chi_i(s)} \hbox{d}s\,\right\} = 0. $$
(44)
From Lemma 1, the following matrix inequalities hold for any positive scalar \(\epsilon_i\)
$$ \begin{aligned} &2{\zeta_i}{(t)^T}\psi_{ai}^T(\Updelta{A_i}(t){\varsigma_3} +\Updelta{B_i}(t){\varsigma_5}+ \Updelta{G_i}(t){\varsigma_6}){\zeta_i}(t)\\ &\quad= 2{\zeta_i}{(t)^T}\psi_{ai}^T{E_i}\Upphi_i(t)({H_{1i}}{\varsigma_3}+{H_{2i}}{\varsigma_5}+ {H_{3i}}{\varsigma_6}){\zeta_i}(t)\\ &\quad\le {\zeta_i}{(t)^T}\left\{ {\epsilon_i^{-1}\psi_{ai}^T{E_i}E_i^T\psi_{ai}^T}+{\epsilon_i} {{({H_{1i}}{\varsigma_3}+{H_{2i}}{\varsigma_5}+{H_{3i}}{\varsigma_6})}^T} ({H_{1i}}{\varsigma_3}+{H_{2i}}{\varsigma_5}+ {H_{3i}}{\varsigma_6}) \right\}{\zeta_i}(t). \end{aligned} $$
(45)
By (22–25), (36–37) and (40–45), and taking the mathematical expectations on both sides of (35), we obtain that
$$ \begin{aligned} \hbox{d}{\mathbb{E}}\{V(t,e_t,i)\}&={\mathbb{E}}\pounds V(t,e_t,i)\hbox{d}t+{\mathbb{E}}\left\{\frac{\partial}{\partial e}V(t,e_t,i)\rho_i(t)\hbox{d}\omega(t)\right\}\\ &\leq\zeta_i(t)^T\left(\widetilde{\Upomega}_i+2\max \left\{ -\varsigma_8^T{\bar{T}_1}{\varsigma_8}-\varsigma_{11}^T{\bar{T}_2} {\varsigma_{11}},-\varsigma_8^T{\bar{T}_1}{\varsigma_8}-\varsigma_6^T {\bar{T}_2}{\varsigma_6},\right.\right.\\ &\left.\left.\quad\quad-\varsigma_7^T{\bar{T}_1}{\varsigma_7}-\varsigma_6^T {\bar{T}_2}{\varsigma_6},-\varsigma_7^T{\bar{T}_1} {\varsigma_7}-\varsigma_{11}^T{\bar{T}_2}{\varsigma_{11}}\right\}\right) \zeta_i(t). \end{aligned} $$
From (15), there exists a positive scalar α
0 such that
$$ \hbox{d}{\mathbb{E}}\{V(t,e_t,i)\}<-\alpha_0{\mathbb{E}}||e(t)||^2. $$
Similar to the proof of Theorem 1 in [27], it implies that the error system (3) is globally exponentially stable. This completes the proof.
