Appendix I
1.1 Proof of Theorem 1
Proof
Consider the following Lyapunov-Krasovskii functional:
$$\begin{aligned} V(t,e_t,i)&= e(t)^TP_ie(t)+2\sum ^n_{j=1}q_{ji}\int ^{e_j(t)}_0\left[ f_j(s)-\gamma _js\right] \mathrm{d}s\nonumber \\&+2\sum ^n_{j=1}u_{ji}\int ^{e_j(t)}_0\left[ \sigma _j s\!-\! {f}_j(s)\right] \mathrm{d}s\!+\!\int ^t_{t-\tau _i(t)}\left[ e(s)^TSe(s)\!+\!f(e(s))^TRf(e(s)) \right] \mathrm{d}s,\nonumber \\ \end{aligned}$$
(15)
where \(Q_i=\hbox { diag}\{q_{1i},q_{2i},\ldots ,q_{ni}\},U_i=\hbox { diag}\{u_{1i},u_{2i},\ldots ,u_{ni}\}.\)
It can be easily verified that \(V(t,e_t,i)\) is a nonnegative function over \([-\bar{\tau },+\infty ).\) Evaluating the weak infinitesimal operator of \(V(t,e_t,i)\) along the trajectory of system (8), we have that
$$\begin{aligned} \mathrm{d}V(t,e_t,i)=\pounds V(t,e_t,i)\mathrm{d}t+V_{e}(t,e_t,i)\rho _i(t)\mathrm{d}\omega (t). \end{aligned}$$
(16)
where
$$\begin{aligned}&\pounds V(t,e_t,i)\nonumber \\&\quad = 2\left[ e{(t)^T}{P_i}+(f(e(t))-\Gamma e(t))^T{Q_i}+(\Sigma e(t)-f{{(e(t))}^T}{U_i} \right] \nonumber \\&\qquad \times \left[ -(C_i(t)-{X_i})e(t)+{Y_i}e(t-\tau _i(t))+A_i(t)f(e(t)) \right] +\sum \limits _{j=1}^N{{{\pi }_{ij}}e{(t)^T}{P_j}e(t)}\nonumber \\&\qquad +\,2\sum \limits _{k=1}^N{{{\pi }_{ik}}}\sum \limits _{j=1}^n{\left\{ {{q}_{jk}}\int _{0}^{{e_j}(t)}{\left[ {{f}_j}(s)-{{\gamma }_j}s \right] \mathrm{d}s}+{{u}_{jk}}\int _{0}^{{e_j}(t)}{\left[ {{\sigma }_j}s-{{f}_j}(s) \right] \mathrm{d}s} \right\} }\nonumber \\&\qquad -\,(1-{{{\dot{\tau }}}_i}(t))\left[ e{{(t-\tau _i(t))}^T}Se(t-\tau _i(t))+f{{(e(t-\tau _i(t)))}^T}Rf(e(t-\tau _i(t)))\right] \nonumber \\&\qquad +\,e{(t)^T}Se(t)+f{{(e(t))}^T}Rf(e(t))+\sum \limits _{j=1}^N{{{\pi }_{ij}}{\tau _j}(t)}\left[ e{{(t-\tau _i(t))}^T}Se(t-\tau _i(t))\right. \nonumber \\&\qquad \left. +\,f{{(e(t-\tau _i(t)))}^T}Rf(e(t-\tau _i(t)))\right] +\frac{1}{2}\hbox { trace}\left[ \rho _i(t)^T\frac{\partial ^2}{\partial e^2}V(t,e_t,i)\rho _i(t)\right] .\nonumber \\ \end{aligned}$$
(17)
On the other hand, it follows from (5) that
$$\begin{aligned}&2\sum \limits _{k=1}^N{{{\pi }_{ik}}}\sum \limits _{j=1}^n{{{q}_{jk}}\int _{0}^{{e_j}(t)}{\left[ {{f}_j}(s)-{{\gamma }_j}s \right] \mathrm{d}s}}&\nonumber \\&\quad \le 2\sum \limits _{k=1}^N\pi '_{ik}\sum \limits _{j=1}^n{{{q}_{jk}}\int _{0}^{{e_j}(t)}{\left[ {{f}_j}(s)-{{\gamma }_j}s \right] \mathrm{d}s}}&\nonumber \\&= 2\sum \limits _{k=1}^N\pi '_{ik}e{(t)^T}{Q_{k}}(\Sigma -\Gamma )e(t),&\end{aligned}$$
(18)
$$\begin{aligned}&2\sum \limits _{k=1}^N{{{\pi }_{ik}}}\sum \limits _{j=1}^n{{{u}_{jk}}\int _{0}^{{e_j}(t)}{\left[ {e_j}s-{{f}_j}(s) \right] \mathrm{d}s}}&\nonumber \\&\quad \le 2\sum \limits _{k=1}^N\pi '_{ik}\sum \limits _{j=1}^n{{{u}_{jk}}\int _{0}^{{e_j}(t)}{\left[ {e_j}s-{{f}_j}(s) \right] \mathrm{d}s}}&\nonumber \\&=2\sum \limits _{k=1}^N\pi '_{ik}e{(t)^T}{U_{k}}(\Sigma -\Gamma )e(t).&\end{aligned}$$
(19)
Furthermore, the following inequalities hold for any \(t>0\)
$$\begin{aligned}&\sum \limits _{j=1}^N{{{\pi }_{ij}}{\tau _j}(t)}\left[ e{{(t-\tau _i(t))}^T}Se(t-\tau _i(t)) +f{{(e(t-\tau _i(t)))}^T}Rf(e(t-\tau _i(t))) \right] \\&\quad \le \sum \limits _{j=1}^N\pi '_{ik}\bar{\tau }_j\left[ e{{(t-\tau _i(t))}^T}Se(t-\tau _i(t)) +f{{(e(t-\tau _i(t)))}^T}Rf(e(t-\tau _i(t))) \right] . \end{aligned}$$
For any \(j=1,2,\ldots ,n,\) it follows from (5) that
$$\begin{aligned} 0\le \frac{\mathrm{d}(f_j(e_j(t))-\lambda _je_j(t))}{\mathrm{d} e_j(t)}\le \sigma _j-\lambda _j, \ \ 0\le \frac{\mathrm{d}(\sigma _je_j(t)-f_j(e_j(t)))}{\mathrm{d} e_j(t)}\le \sigma _j-\lambda _j. \end{aligned}$$
Thus we have that
$$\begin{aligned}&\frac{1}{2}\frac{\partial ^2}{\partial e^2}V(t,e_t,i)&\nonumber \\&= P_i+U_i\times \hbox { diag}\left\{ \frac{d(\sigma _1e_1(t)-f_1(e_1(t)))}{d e_1(t)},\ldots ,\frac{d(\sigma _ne_n(t)-f_n(e_n(t)))}{d e_n(t)}\right\}&\nonumber \\&+Q_i\times \hbox { diag}\left\{ \frac{d(f_1(e_1(t))-\lambda _1e_1(t))}{d e_1(t)},\ldots ,\frac{d(f_n(e_n(t))-\lambda _ne_n(t))}{d e_n(t)}\right\}&\nonumber \\&\le \widetilde{P}_i.&\end{aligned}$$
(20)
In addition, from (5) the following matrix inequalities hold for any positive diagonal matrices \(W_i,Z_i\) with compatible dimensions (see [4, 22])
$$\begin{aligned}&0\le -e(t)^T\Sigma \Gamma W_ie(t)+e(t)^TW_i(\Sigma +\Gamma )f(e(t))-f(e(t))^TW_if(e(t)),&\end{aligned}$$
(21)
$$\begin{aligned}&0\le -e(t-\tau _i(t))^T\Sigma \Gamma Z_i e(t-\tau _i(t))&\nonumber \\&+e(t-\tau _i(t))^TZ_i(\Sigma +\Gamma )f(e(t-\tau _i(t)))-f(e(t-\tau _i(t)))^TZ_if(e(t-\tau _i(t))).&\nonumber \\ \end{aligned}$$
(22)
According to Assumption 1 and Lemma 1, for any positive scalar \(\varepsilon _i\) we get that
$$\begin{aligned}&\frac{1}{2}\hbox { trace}\left\{ \rho _i(t)^T\frac{\partial ^2}{\partial e^2}V(t,e_t,i)\rho _i(t)\right\} \nonumber \\&\quad = \frac{1}{2}[D_i(t)e(t)+E_i(t)e(t-\tau _i(t))]^TV_{ee}(t,e_t,i)[D_i(t)e(t)+E_i(t)e(t-\tau _i(t))]\nonumber \\&\quad \le \left[ D_ie(t)+E_ie(t-\tau _i(t))+F_iG_i(t)(K_{5i}e(t)+K_{6i}e(t-\tau _i(t)))\right] ^T\nonumber \\&\qquad \times \widetilde{P}_i\left[ D_ie(t)\!+\!E_ie(t-\tau _i(t))\!+\!F_iG_i(t)(K_{5i}e(t)\!+\!K_{6i}e(t\!-\!\tau _i(t)))\right] \nonumber \\&\quad \le [D_ie(t)+E_ie(t-\tau _i(t))]^T\widetilde{P}_i[D_ie(t)+E_ie(t-\tau _i(t))]\nonumber \\&\qquad +[D_ie(t)\!+\!E_ie(t\!-\!\tau _i(t))]^T\widetilde{P}_iF_i(\varepsilon _i I\!-\!F_i^T\widetilde{P}_iF_i)^{-1}F_i^T\widetilde{P}_i[D_ie(t)\!+\!E_ie(t\!-\!\tau _i(t))]\nonumber \\&\qquad +\varepsilon _i[K_{5i}e(t)+K_{6i}e(t-\tau _i(t))]^T[K_{5i}e(t)+K_{6i}e(t-\tau _i(t))]. \end{aligned}$$
(23)
It follows from (17)–(23) that
$$\begin{aligned} \pounds V(t,e_t,i)&\le \zeta _i(t)^T\Omega _i(t)\zeta _i(t)+e(t-\tau _i(t))^TH_ie(t-\tau _i(t))\nonumber \\&+f(e(t-\tau _i(t)))^TJ_if(e(t-\tau _i(t))), \end{aligned}$$
(24)
where
$$\begin{aligned} \zeta _i(t)= \hbox { col}\left\{ e(t),e(t-\tau _i(t)),f(e(t)),f(e(t-\tau _i(t)))\right\} , \end{aligned}$$
$$\begin{aligned} \Omega _i(t)= \left[ \begin{array}{cccc} \psi _{1i}+\Omega _{1i}(t)&{}\psi _{3i}+\Omega _{2i}(t)&{}\psi _{5i}+D_i^T\widehat{P}_iE_i&{}\bar{P}_iB_i(t)-\epsilon _i K_{1i}^TK_{3i}\\ *&{}\Omega _{3i}(t)-W_i+R&{}\widehat{Y}_i&{}\Omega _{4i}(t)\\ *&{}*&{}\psi _{7i}+E_i^T\widehat{P}_iE_i&{}\frac{1}{2}Z_i(\Sigma +\Gamma )\\ *&{}*&{}*&{}\psi _{8i}+\epsilon _i K_{3i}^TK_{3i}\\ \end{array}\right] , \end{aligned}$$
with
$$\begin{aligned} \Omega _{1i}(t)= -\hbox { sys}(\bar{P}_iC_i(t))+\epsilon _i K_{1i}^TK_{1i}^T+D_i^T\widehat{P}_iD_i,\ \widetilde{X}_i=\bar{P}_iX_i,\ \widetilde{Y}_i=\bar{P}_iY_i, \end{aligned}$$
$$\begin{aligned} \Omega _{2i}(t)= \bar{P}_iA_i(t)-C_i(t)^T(Q_i-U_i)-\epsilon _i K_{1i}^TK_{2i},\ \widehat{X}_i=(Q_i-U_i)X_i, \end{aligned}$$
$$\begin{aligned} \Omega _{3i}(t)= \hbox { sys}((Q_i-U_i)A_i(t))+\epsilon _i K_{2i}^TK_{2i},\ \ \widehat{Y}_i=(Q_i-U_i)Y_i, \end{aligned}$$
$$\begin{aligned} \Omega _{4i}(t)=(Q_i-U_i)B_i(t)+\epsilon _i K_{2i}^TK_{3i},\ \ \widehat{P}_i=\widetilde{P}_iF_i(\varepsilon _i I-F_i^T\widetilde{P}_iF_i)^{-1}F_i^T\widetilde{P}_i. \end{aligned}$$
Now, by (13), it is easy to see that there exists a scalar \(\lambda >1\) such that
$$\begin{aligned} \left[ \begin{array}{ccc} \widetilde{\Psi }_i&{}\mathcal {A}_iF_i&{}\mathcal {B}\widetilde{P}_iF_i\\ *&{}-\epsilon _i I&{}0\\ *&{}*&{}-\varepsilon _i I+F_i^T\widetilde{P}_iF_i\\ \end{array}\right] <0, \end{aligned}$$
(25)
where
$$\begin{aligned} \widetilde{\Psi }_i= \left[ \begin{array}{cccc} \lambda M_i+\psi _{1i}+\psi _{2i}&{}\psi _{3i}+\psi _{4i}&{}\psi _{5i}&{}\bar{P}_iB_i-\epsilon _i K_{1i}^TK_{3i}\\ *&{}\psi _{6i}-W_i+R&{}\widehat{Y}_i&{}(Q_i-U_i)B_i+\epsilon _i K_{2i}^TK_{3i}\\ *&{}*&{}\psi _{7i}&{}\frac{1}{2}Z_i(\Sigma +\Gamma )\\ *&{}*&{}*&{}\psi _{8i}+\epsilon _i K_{3i}^TK_{3i}\\ \end{array}\right] . \end{aligned}$$
Applying Schur complements to (25) results in that
$$\begin{aligned}&\left[ \begin{array}{cccc} \psi _{ai}&{}\psi _{3i}+\psi _{4i}&{}\psi _{5i}+D_i^T\widehat{P}_iE_i&{}\bar{P}_iB_i-\epsilon _i K_{1i}^TK_{3i}\\ *&{}\psi _{6i}-W_i+R&{}\widehat{Y}_i&{}(Q_i-U_i)B_i+\epsilon _i K_{2i}^TK_{3i}\\ *&{}*&{}\psi _{7i}+E_i^T\widehat{P}_iE_i&{}\frac{1}{2}Z_i(\Sigma +\Gamma )\\ *&{}*&{}*&{}\psi _{8i}+\epsilon _i K_{3i}^TK_{3i}\\ \end{array}\right] \nonumber \\&\quad +\,\epsilon _i^{-1}\left[ \begin{array}{cccc} \bar{P}_iF_i\\ (Q_i-U_i)F_i\\ 0\\ 0\\ \end{array}\right] \left[ \begin{array}{cccc} \bar{P}_iF_i\\ (Q_i-U_i)F_i\\ 0\\ 0\\ \end{array}\right] ^T+\epsilon _i\left[ \begin{array}{cccc} -K_{1i}^T\\ K_{2i}^T\\ 0\\ K_{3i}^T\\ \end{array}\right] \left[ \begin{array}{cccc} -K_{1i}^T\\ K_{2i}^T\\ 0\\ K_{3i}^T\\ \end{array}\right] ^T<0,\nonumber \\ \end{aligned}$$
(26)
where \(\psi _{ai}=\lambda M_i+\psi _{1i}+\psi _{2i}+D_i^T\widehat{P}_iD_i.\)
Using Assumption 1 and Lemma 1, for any positive scalar \(\epsilon _i\) we have that
$$\begin{aligned}&\left[ \begin{array}{cccc} -\hbox { sys}(\bar{P}_i\Delta C_i(t))&{}\bar{P}_i\Delta A_i(t)-\Delta C_i(t)^T(Q_i-U_i)&{}0&{}\bar{P}_i\Delta B_i(t)\\ *&{}\hbox { sys}((Q_i-U_i)\Delta A_i(t))&{}0&{}(Q_i-U_i)\Delta B_i(t)\\ *&{}*&{}0&{}0\\ *&{}*&{}*&{}0\\ \end{array}\right] \\&\quad = \left[ \begin{array}{cccc} \bar{P}_iF_i\\ (Q_i-U_i)F_i\\ 0\\ 0\\ \end{array}\right] G_i(t)\left[ \begin{array}{cccc} -K_{1i}^T\\ K_{2i}^T\\ 0\\ K_{3i}^T\\ \end{array}\right] ^T+\left[ \begin{array}{cccc} -K_{1i}^T\\ K_{2i}^T\\ 0\\ K_{3i}^T\\ \end{array}\right] G_i(t)^T\left[ \begin{array}{cccc} \bar{P}_iF_i\\ (Q_i-U_i)F_i\\ 0\\ 0\\ \end{array}\right] ^T\\&\quad = \left[ \begin{array}{cccc} \bar{P}_iF_i\\ (Q_i-U_i)F_i\\ 0\\ 0\\ \end{array}\right] G_i(t)\epsilon _i^{-1}G_i(t)^T\left[ \begin{array}{cccc} \bar{P}_iF_i\\ (Q_i-U_i)F_i\\ 0\\ 0\\ \end{array}\right] ^T\!\!+\!\!\left[ \begin{array}{cccc} -K_{1i}^T\\ K_{2i}^T\\ 0\\ K_{3i}^T\\ \end{array}\right] \epsilon _i\left[ \begin{array}{cccc} -K_{1i}^T\\ K_{2i}^T\\ 0\\ K_{3i}^T\\ \end{array}\right] ^T\\&\qquad -\left\{ \epsilon _i^{-1}\left[ \begin{array}{cccc} \bar{P}_iF_i\\ (Q_i-U_i)F_i\\ 0\\ 0\\ \end{array}\right] G_i(t)-\left[ \begin{array}{cccc} -K_{1i}^T\\ K_{2i}^T\\ 0\\ K_{3i}^T\\ \end{array}\right] \right\} \epsilon _i\\&\qquad \times \left\{ \epsilon _i^{-1}\left[ \begin{array}{cccc} \bar{P}_iF_i\\ (Q_i-U_i)F_i\\ 0\\ 0\\ \end{array}\right] G_i(t)-\left[ \begin{array}{cccc} -K_{1i}^T\\ K_{2i}^T\\ 0\\ K_{3i}^T\\ \end{array}\right] \right\} ^T\\&\quad \le \epsilon _i^{-1}\left[ \begin{array}{cccc} \bar{P}_iF_i\\ (Q_i-U_i)F_i\\ 0\\ 0\\ \end{array}\right] \left[ \begin{array}{cccc} \bar{P}_iF_i\\ (Q_i-U_i)F_i\\ 0\\ 0\\ \end{array}\right] ^T+\epsilon _i\left[ \begin{array}{cccc} -K_{1i}^T\\ K_{2i}^T\\ 0\\ K_{3i}^T\\ \end{array}\right] \left[ \begin{array}{cccc} -K_{1i}^T\\ K_{2i}^T\\ 0\\ K_{3i}^T\\ \end{array}\right] ^T. \end{aligned}$$
This together with (26) provides that
$$\begin{aligned} \Omega _i(t)+\hbox { diag}\{\lambda M_i,\ 0,\ 0,\ 0\}<0. \end{aligned}$$
By this inequality and (24), we have that
$$\begin{aligned} \mathrm{d}V(t,e_t,i)&= \pounds V(t,e_t,i)\mathrm{d}t+V_{e}(t,e_t,i)[D_i(t)e(t)+E_i(t)e(t-\tau _i(t))]\mathrm{d}\omega (t)\\&< \big [-\lambda e(t)^TM_ie(t)+e(t-\tau _i(t))^TH_ie(t-\tau _i(t))+f(e(t-\tau _i(t)))^T\\&\times \, J_if(e(t\!-\!\tau _i(t)))\big ]\mathrm{d}t\!+\!V_{e}(t,e_t,i)[D_i(t)e(t)\!+\!E_i(t)e(t\!-\!\tau _i(t))]\mathrm{d}\omega (t). \end{aligned}$$
Taking the mathematical expectations on both sides of (16), from above inequality we obtain that
$$\begin{aligned} \mathrm{d}\mathbf {E}\{V(t,e_t,i)\}&= \mathbf {E}\pounds V(t,e_t,i)\mathrm{d}t\\&+\,\mathbf {E}\{V_{e}(t,e_t,i)[D_i(t)e(t)+E_i(t)e(t-\tau _i(t))]\mathrm{d}\omega (t)\}\\&< \big [-\lambda e(t)^TM_ie(t)+e(t-\tau _i(t))^TH_ie(t-\tau _i(t))\\&+f(e(t-\tau _i(t)))^TJ_if(e(t-\tau _i(t)))\big ]\mathrm{d}t. \end{aligned}$$
By integrating above inequality from \(t-\tau _i(t)\) to \(t,\) we get that
$$\begin{aligned}&\mathbf {E}\{V(t,e_t,i)\}-\mathbf {E}\{V(t,e_{t-\tau _i(t)},i)\}\\&\quad =\int _{t-\tau _i(t)}^t \mathbf {E}\{V(s,e_s,i)\}\mathrm{d}s\\&\quad <-\lambda \int _{t-\tau _i(t)}^t e(s)^TM_ie(s)\mathrm{d}s\\&\qquad +\int _{t-\tau _i(t)}^t \left[ e(s\!-\!\tau _i(s))^TH_ie(s\!-\!\tau _i(s))\!+\!f(e(s-\tau _i(s)))^TJ_if(e(s\!-\!\tau _i(s)))\right] \mathrm{d}s. \end{aligned}$$
It follows that
$$\begin{aligned}&\mathbf {E}\left[ \frac{\mathrm{d}V(t,e_t,i)}{\mathrm{d}t} \right] +\beta \mathbf {E}\left[ V(t,e_t,i)-V(t,e_{t-\tau _i(t)},i)\right] \nonumber \\&\quad < -\lambda e(t)^TM_ie(t)+e(t-\tau _i(t))^TH_ie(t-\tau _i(t))\nonumber \\&\qquad +f(e(t-\tau _i(t)))^TJ_if(e(t-\tau _i(t)))-\beta \lambda \int _{t-\tau _i(t)}^t e(s)^TM_ie(s)\mathrm{d}s\nonumber \\&\qquad +\beta \int _{t-\tau _i(t)}^t \left[ e(s\!-\!\tau _i(s))^TH_ie(s\!-\!\tau _i(s))\!+\!f(e(s\!-\!\tau _i(s)))^TJ_if(e(s\!-\!\tau _i(s)))\right] \mathrm{d}s.\nonumber \\ \end{aligned}$$
(27)
In view of (10) and (11), we derive that
$$\begin{aligned} -e(t)^TM_ie(t)\le -\alpha e(t)^T\widetilde{P}_ie(t), \end{aligned}$$
(28)
$$\begin{aligned} -\beta \int _{t-\tau _i(t)}^t e(s)^TM_ie(s)ds\le -\alpha \int _{t-\tau _i(t)}^t \left[ e(s)^TSe(s)+f(e(s))^TRf(e(s))\right] \mathrm{d}s. \end{aligned}$$
(29)
Noting that
$$\begin{aligned}&2\sum ^n_{j=1}q_{ji}\int ^{e_j(t)}_0\left[ f_j(s)-\lambda _js\right] \mathrm{d}s\\&\quad \le 2\sum ^n_{j=1}q_{ji}\int ^{e_j(t)}_0(\sigma _j-\lambda _j)s\mathrm{d}s=e(t)^TQ_i(\Sigma -\Gamma )e(t),\\&2\sum ^n_{j=1}u_{ji}\int ^{e_j(t)}_0\left[ \sigma _j s- {f}_j(s)\right] \mathrm{d}s\\&\quad \le 2\sum ^n_{j=1}u_{ji}\int ^{e_j(t)}_0(\sigma _j-\lambda _j)s\mathrm{d}s=e(t)^TU_i(\Sigma -\Gamma )e(t). \end{aligned}$$
Thus we have that
$$\begin{aligned} \mathbf {E}\{V(t,e_t,i)\}\le e(t)^T\widetilde{P}_ie(t)+\int _{t-\tau _i(t)}^t \left[ e(s)^TSe(s)+f(e(s))^TRf(e(s))\right] \mathrm{d}s. \end{aligned}$$
This together with (28)–(29) yields that
$$\begin{aligned} -e(t)^TM_ie(t)-\beta \int _{t-\tau _i(t)}^te(s)^TM_ie(s)\mathrm{d}s\le -\alpha \mathbf {E}\{V(t,e_t,i)\}. \end{aligned}$$
(30)
Moreover, \(\mathbf {E}\{V(t,e_t,i)\}\ge e(t)^TP_ie(t),\) therefore it follows from (5) and (12) that
$$\begin{aligned}&e(t-\tau _i(t))^TH_ie(t-\tau _i(t))+f(e(t-\tau _i(t)))^TJ_if(e(t-\tau _i(t)))\nonumber \\&\quad \le e(t-\tau _i(t))^T\left[ H_i+\Theta J_i\Theta \right] e(t-\tau _i(t))\nonumber \\&\quad \le \frac{\alpha }{1+\beta \bar{\tau }}e(t-\tau _i(t))^TP_ie(t-\tau _i(t))\nonumber \\&\quad \le \frac{\alpha }{1+\beta \bar{\tau }}\mathbf {E}\{V(t,e_{t-\tau _i(t)},i)\}. \end{aligned}$$
(31)
Thus we obtain that
$$\begin{aligned}&\int _{t-\tau _i(t)}^t\left[ e(s-\tau _i(s))^TH_ie(s-\tau _i(s))+f(e(s-\tau _i(s)))^TJ_if(e(s-\tau _i(s)))\right] \mathrm{d}s\nonumber \\&\quad \le \frac{\alpha }{1+\beta \bar{\tau }}\int _{t-\tau _i(t)}^t\mathbf {E}\{V(s,e_{s-\tau _i(s)},i)\}\mathrm{d}s. \end{aligned}$$
(32)
Substituting (30)–(32) into (27) derives that
$$\begin{aligned}&\frac{\mathrm{d}\mathbf {E}\{V(t,e_t,i)\}}{\mathrm{d}t}\\&\qquad < -(\beta +\lambda \alpha )\mathbf {E}\{V(t,e_t,i)\}+\beta \mathbf {E}\{V(t,e_{t-\tau _i(t)},i)\}\\&\qquad +\frac{\alpha }{1+\beta \bar{\tau }}\left[ \mathbf {E}\{V(t,e_{t-\tau _i(t)},i)\} +\beta \int _{t-\tau _i(t)}^t\mathbf {E}\{V(s,e_{s-\tau _i(s)},i)\}\mathrm{d}s\right] \\&\quad \le -(\beta +\lambda \alpha )\mathbf {E}\{V(t,e_t,i)\}+\beta \mathbf {E}\{V(t,e_{t-\tau _i(t)},i)\}\\&\qquad +\frac{\alpha }{1+\beta \bar{\tau }}\left[ \mathbf {E}\{V(t,e_{t-\tau _i(t)},i)\}+\beta \tau _i(t)\sup _{[t-2\bar{\tau },t]} \{\mathbf {E}\{V(s,e_{s-\tau _i(s)},i)\}\}\right] \\&\quad \le -(\beta +\lambda \alpha )\mathbf {E}\{V(t,e_t,i)\}+(\beta +\alpha )\sup _{[t-2\bar{\tau },t]}\{\mathbf {E}\{V(s,e_s,i)\}\}. \end{aligned}$$
Applying Lemma 5 to above inequality results in that
$$\begin{aligned} V(t,e_t,i)\le \sup _{[-2\bar{\tau },0]}\{\mathbf {E}\{V(s,e_s,i)\}\}e^{-\varrho t}, \end{aligned}$$
where \(\varrho \) is the unique positive solution of the following equation:
$$\begin{aligned} \varrho =\beta +\lambda \alpha -(\alpha +\beta )e^{2\varrho \bar{\tau }}. \end{aligned}$$
Therefore we arrive at the conclusion that
$$\begin{aligned} \mathbf {E}\{||e(t)||^2\}\le e^{-\varrho t}\mathbf {E}\{||\varphi (t)||^2\}. \end{aligned}$$
The proof of Theorem 1 is completed. \(\square \)
Appendix II
1.1 Proof of Theorem 2
Consider the following Lyapunov-Krasovskii functional:
$$\begin{aligned} \widetilde{V}(t,{e_t},i)=V(t,{e_t},i)+\overline{V}(t,{e_t},i), \end{aligned}$$
(33)
where \(V(t,{e_t},i)\) ie defined in (15) and
$$\begin{aligned} \overline{V}(t,{e_t},i)&= \bar{\tau }_i\int _{t-\bar{\tau }_i}^t{\int _\nu ^t{\vartheta _i {{(s)}^T}{O_1}\vartheta _i (s)\mathrm{d}s\mathrm{d}\nu }}\\&+\bar{\upsilon }_i\int _{t-\bar{\upsilon }_i}^t{\int _\nu ^t{f{{(e(s))}^T}{O_2}f(e(s))\mathrm{d}s\mathrm{d}\nu }}+\int _{t-\bar{\tau }_i}^t{\int _\nu ^t{\rho _i {{(s)}^T}{O_3}\rho _i (s)\mathrm{d}s\mathrm{d}\nu }}. \end{aligned}$$
It can be easily verified that \(\widetilde{V}(t,e_t,i)\) is a nonnegative function over \([-\hat{\tau },+\infty ).\) Evaluating the weak infinitesimal operator of \(\widetilde{V}(t,e_t,i)\) along the trajectory of system (3), based on (7) we derive that
$$\begin{aligned} \pounds \overline{V}(t,{e_t},i)&= \bar{\tau }_i^2\vartheta _i {(t)^T}{O_1}\vartheta _i (t)-\bar{\tau }_i\bigg (1-\sum \limits _{j=1}^N{{{\pi }_{ij}}\bar{\tau }_j}\bigg )\int _{t-\bar{\tau }_i}^t{\vartheta _i {{(s)}^T}{O_1}\vartheta _i(s)\mathrm{d}s}\nonumber \\&+\bar{\upsilon }_if{{(e(t))}^T}{O_2}f(e(t))-\bar{\upsilon }_i\bigg (1-\sum \limits _{j=1}^N{{{\pi }_{ij}}{{{\bar{\upsilon }}}_j}}\bigg ) \int _{t-\bar{\upsilon }_i}^t{f{{(e(s))}^T}{O_2}f(e(s))\mathrm{d}s}\nonumber \\&+\bar{\tau }_i\rho _i {(t)^T}{O_3}\rho _i (t)-\int _{t-\bar{\tau }_i}^t{\rho _i {{(s)}^T}{O_3}\rho _i (s)\mathrm{d}s}\nonumber \\&= \bar{\tau }_i^2\vartheta _i{(t)^T}{O_1}\vartheta _i(t)-\bar{\tau }_i\bigg (1-\sum \limits _{j=1}^N{{{\pi }_{ij}}\bar{\tau }_j}\bigg ) \int _{t-\bar{\tau }_i}^t{\vartheta _i {{(s)}^T}{O_1}\vartheta _i (s)\mathrm{d}s}\nonumber \\&+\bar{\upsilon }_if{{(e(t))}^T}{O_2}f(e(t))-\bar{\upsilon }_i\bigg (1-\sum \limits _{j=1}^N{{{\pi }_{ij}}{{{\bar{\upsilon }}}_j}}\bigg ) \int _{t-\bar{\upsilon }_i}^t{f{{(e(s))}^T}{O_2}f(e(s))\mathrm{d}s}\nonumber \\&+\bar{\tau }_i\rho _i {(t)^T}{O_3}\rho _i (t)-\int _{t-\tau _i(t)}^t{\rho _i {{(s)}^T}{O_3}\rho _i(s)\mathrm{d}s}-\int _{t-\bar{\tau }_i}^{t-\tau _i(t)}{\rho _i {{(s)}^T}{O_3}\rho _i (s)\mathrm{d}s}.\nonumber \\ \end{aligned}$$
(34)
Furthermore, for any \(0<\tau _i(t)<\bar{\tau }_i,\ 0<\upsilon _i(t)<\bar{\upsilon }_i,\) from Lemma 2 we obtain that
$$\begin{aligned}&-\bar{\tau }_i\int _{t-\bar{\tau }_i}^t{\vartheta _i {{(s)}^T}{O_1}\vartheta _i (s)\mathrm{d}s}\nonumber \\&\quad = -\bar{\tau }_i\int _{t-\tau _i(t)}^t{\vartheta _i {{(s)}^T}{\bar{O}_1}\vartheta _i (s)\mathrm{d}s}-\bar{\tau }_i\int _{t-\bar{\tau }_i}^{t-\tau _i(t)}{\vartheta _i {{(s)}^T}{\bar{O}_1}\vartheta _i (s)\mathrm{d}s}\nonumber \\&\quad \le -\frac{\bar{\tau }_i}{\tau _i(t)}{{\left( \int _{t-\tau _i(t)}^t{\vartheta _i (s)ds} \right) }^T}{\bar{O}_1}\left( \int _{t-\tau _i(t)}^t{\vartheta _i (s)\mathrm{d}s} \right) \nonumber \\&\qquad -\frac{\bar{\tau }_i}{\bar{\tau }_i-\tau _i(t)}{{\left( \int _{t-\bar{\tau }_i}^{t-\tau _i(t)}{\vartheta _i (s)\mathrm{d}s} \right) }^T}{\bar{O}_1}\left( \int _{t-\bar{\tau }_i}^{t-\tau _i(t)}{\vartheta _i (s)\mathrm{d}s} \right) ,\end{aligned}$$
(35)
$$\begin{aligned}&-\bar{\upsilon }_i\int _{t-\bar{\upsilon }_i}^t{f{{(e(s))}^T}{\bar{O}_2}f(e(s))\mathrm{d}s}\nonumber \\&\quad = -\bar{\upsilon }_i\int _{t-\upsilon _i(t)}^t{f{{(e(s))}^T} {\bar{O}_2}f(e(s))\mathrm{d}s}-\bar{\upsilon }_i\int _{t-\bar{\upsilon }_i}^{t-\upsilon _i(t)}{f{{(e(s))}^T}{\bar{O}_2}f(e(s))\mathrm{d}s}\nonumber \\&\quad \le -\frac{\bar{\upsilon }_i}{\upsilon _i(t)}{{\left( \int _{t-\upsilon _i(t)}^t{f(e(s))\mathrm{d}s} \right) }^T}{\bar{O}_2}\left( \int _{t-\upsilon _i(t)}^t{f(e(s))\mathrm{d}s} \right) \nonumber \\&\qquad -\frac{\bar{\upsilon }_i}{\bar{\upsilon }_i-\upsilon _i(t)}{{\left( \int _{t-\bar{\upsilon }_i}^{t-\upsilon _i(t)}{f(e(s))\mathrm{d}s} \right) }^T}{\bar{O}_2}\left( \int _{t-\bar{\upsilon }_i}^{t-\upsilon _i(t)}{f(e(s))\mathrm{d}s} \right) . \end{aligned}$$
(36)
Denoting
$$\begin{aligned} \Xi _1=\left[ \int ^t_{t-\tau _i(t)}\vartheta _i(s)\mathrm{d}s\right] ^T\bar{O}_1\left[ \int ^t_{t-\tau _i(t)}\vartheta _i(s)\mathrm{d}s\right] , \end{aligned}$$
$$\begin{aligned}&\displaystyle \Xi _2=\left[ \int ^{t-\tau _i(t)}_{t-\bar{\tau }_i}\vartheta _i(s)\mathrm{d}s\right] ^T\bar{O}_1\left[ \int ^{t-\tau _i(t)}_{t-\bar{\tau }_i}\vartheta _i(s)\mathrm{d}s\right] ,\\&\displaystyle \Xi _3=\left[ \int ^t_{t-\upsilon _i(t)}f(e(s))\mathrm{d}s\right] ^T\bar{O}_2\left[ \int ^t_{t-\upsilon _i(t)}f(e(s))\mathrm{d}s\right] ,\\&\displaystyle \Xi _4=\left[ \int ^{t-\upsilon _i(t)}_{t-\bar{\upsilon }_i}f(e(s))\mathrm{d}s\right] ^T\bar{O}_2\left[ \int ^{t-\upsilon _i(t)}_{t-\bar{\upsilon }_i}f(e(s))\mathrm{d}s\right] , \end{aligned}$$
then from Lemma 3 we have that
$$\begin{aligned}&-\bar{\tau }_i\int _{t-\bar{\tau }_i}^t{\vartheta _i{{(s)}^T}{O_1}\vartheta _i(s)\mathrm{d}s}-\bar{\upsilon }_i\int _{t-\bar{\upsilon }_i}^t{f{{(e(s))}^T}{\bar{O}_2}f(e(s))\mathrm{d}s}\nonumber \\&\quad \le \max \big \{ -\Xi _1-3\Xi _2-\Xi _3-3\Xi _4,-\Xi _1-3\Xi _2-3\Xi _3-\Xi _4,\nonumber \\&\qquad -3\Xi _1-\Xi _2-\Xi _3-3\Xi _4, -3\Xi _1-\Xi _2-3\Xi _3-\Xi _4 \big \}. \end{aligned}$$
(37)
It is easy to verify that inequality (37) holds for any \(t>0\) with \(0\le \tau _i(t)\le \bar{\tau }_i,\ 0\le \upsilon _i(t)\le \bar{\upsilon }_i.\)
From [3], we obtain that
$$\begin{aligned}&\mathbf {E} \int _{t-\tau _i(t)}^t{\rho _i {{(s)}^T}{O_3}\rho _i (s)\mathrm{d}s}\nonumber \\&\quad =\mathbf {E} {{\left( \int _{t-\tau _i(t)}^t{\rho _i (s)\mathrm{d}\omega (s)} \right) }^T}{O_3}\left( \int _{t-\tau _i(t)}^t{\rho _i (s)\mathrm{d}\omega (s)} \right) ,\end{aligned}$$
(38)
$$\begin{aligned}&\mathbf {E} {{\left( \int _{t-\bar{\tau }_i}^{t-\tau _i(t)}{\rho _i {{(s)}^T}{O_3}\rho _i (s)\mathrm{d}s} \right) }^T}\nonumber \\&\quad =\mathbf {E} {{\left( \int _{t-\bar{\tau }_i}^{t-\tau _i(t)}{\rho _i (s)\mathrm{d}\omega (s)} \right) }^T}{O_3}\left( \int _{t-\bar{\tau }_i}^{t-\tau _i(t)}{\rho _i (s)\mathrm{d}\omega (s)} \right) . \end{aligned}$$
(39)
On the other hand, by the Leibniz-Newton formula, we get that
$$\begin{aligned} \int _{t-\tau _i(t)}^t{\vartheta _i(s)\mathrm{d}s}=e(t)-e(t-\tau _i(t))-\int _{t-\tau _i(t)}^t{\rho _i (s)\mathrm{d}\omega (s)}, \end{aligned}$$
(40)
$$\begin{aligned} \int _{t-\bar{\tau }_i}^{t-\tau _i(t)}{\vartheta _i(s)\mathrm{d}s}=e(t-\tau _i(t))-e(t-\bar{\tau }_i)-\int _{t-\bar{\tau }_i}^{t-\tau _i(t)}{\rho _i (s)\mathrm{d}\omega (s)}. \end{aligned}$$
(41)
Similar to the proof of Theorem 1, from (17)–(23), (34)–(41) and Lemma 1 we have that
$$\begin{aligned}&\mathbf {E}\pounds \widetilde{V}(t,{e_t},i)\\&\quad \le \xi _i{(t)^T}\left( \Omega _i+{{\mathbf {A}}_i}G_i(t)\mathbf {C}_i^T+{{\mathbf {C}}_i}G_i(t)\mathbf {A}_i^T\right) \xi _i(t) +\bar{\tau }_i^2\vartheta _i{(t)^T}{O_1}\vartheta _i(t)\\&\qquad +{{(D_ie(t)+{{E}_i}e(t-\tau _i(t)))}^T}\widehat{P}_iF_i{{(\varepsilon _iI -F_i^T\widehat{P}_iF_i)}^{-1}}F_i^T\widehat{P}_i(D_ie(t)\\&\qquad +{{E}_i}e(t-\tau _i(t)))+2\max \left\{ -\Xi _1-\Xi _3,-\Xi _1-\Xi _4,-\Xi _2-\Xi _3,-\Xi _2-\Xi _4 \right\} \\&\quad \le \xi _i{(t)^T}\left( \Omega _i+{{\epsilon }_i}{{\mathbf {C}}_i}G_i{(t)^T}G_i(t)\mathbf {C}_i^T +\epsilon _i^{-1}{{\mathbf {A}}_i}{{\mathbf {A}}_i}^T+{{\mathbf {B}}_i}\widehat{P}_iF_i{{(\varepsilon _iI -F_i^T\widehat{P}_iF_i)}^{-1}}\right. \\&\qquad \left. \times \, F_i^T\widehat{P}_i\mathbf {B}_i^T\right) \xi _i(t)+\bar{\tau }_i^2{{\left[ {\chi _{1i}}(t)+F_iG_i(t){\chi _{2i}}(t) \right] }^T}{O_1}\left[ {\chi _{1i}}(t)+F_iG_i(t){\chi _{2i}}(t) \right] \\&\qquad +\,2\max \left\{ -\Xi _1-\Xi _3,-\Xi _1-\Xi _4,-\Xi _2-\Xi _3,-\Xi _2-\Xi _4 \right\} , \end{aligned}$$
where
$$\begin{aligned} \xi _i(t)&= \hbox { col}\left\{ {\zeta _i}(t),\int _{t-\upsilon _i(t)}^t{f(e(s))ds},e(t-\bar{\tau }_i),\right. \\&\qquad \qquad \left. \int _{t-\bar{\upsilon }_i}^{t-\upsilon _i(t)}{f(e(s))\mathrm{d}s},\int _{t-\tau _i(t)}^t{\rho _i (s)\mathrm{d}\omega (s),}\int _{t-\bar{\tau }_i}^{t-\tau _i(t)}{\rho _i (s)\mathrm{d}\omega (s)} \right\} ,\\ {\chi _{1i}}(t)&= -(C_i-{X_i})e(t)+{Y_i}e(t-\tau _i(t))+A_if(e(t))\\&+B_if(e(t-\tau _i(t)))+{L_i} \int _{t-\upsilon _i(t)}^t{f(e(s))\mathrm{d}s}, \end{aligned}$$
$$\begin{aligned} {\chi _{2i}}(t)=-{K_{1i}}e(t)+{K_{2i}}f(e(t))+{K_{3i}}f(e(t-\tau _i(t)))+{K_{4i}}\int _{t-\upsilon _i(t)}^t{f(e(s))\mathrm{d}s}. \end{aligned}$$
Moreover, it follows from Assumption 2 and Lemma 1 that
$$\begin{aligned}&{{\left[ {\chi _{1i}}(t)+F_iG_i(t){\chi _{2i}}(t) \right] }^T}{O_1}\left[ {\chi _{1i}}(t)+F_iG_i(t){\chi _{2i}}(t) \right] \le {\chi _{1i}}{(t)^T}{O_1}{\chi _{1i}}(t)\\&\quad +{\chi _{1i}}{(t)^T}{O_1}F_i{{\left( \iota _iI -F_i^T{O_1}F_i\right) }^{-1}}F_i^T{O_1}{\chi _{1i}}(t)+\iota _i{\chi _{2i}}{(t)^T}{\chi _{2i}}(t). \end{aligned}$$
Therefore, we derive that
$$\begin{aligned} \mathbf {E}\pounds \widetilde{V}(t,{e_t},i)\le \xi _i{(t)^T}\widetilde{\Omega }_i\xi _i(t), \end{aligned}$$
where
$$\begin{aligned} \widetilde{\Omega }_i&= \Omega _i +{{\epsilon }_i}{{\mathbf {C}}_i}\mathbf {C}_i^T+\epsilon _i^{-1}{{\mathbf {A}}_i}^T{{\mathbf {A}}_i} +{{\mathbf {B}}_i}\widehat{P}_iF_i{{\left( \varepsilon _iI-F_i^T\widehat{P}_iF_i\right) }^{-1}}F_i^T\widehat{P}_i\mathbf {B}_i^T\\&+\bar{\tau }_i^2\left[ \mathbf {D}_i{O_1}\mathbf {D}_i^T+\mathbf {D}_i{O_1}F_i{{\left( \iota _iI -F_i^T{O_1}F_i\right) }^{-1}}F_i^T{O_1}\mathbf {D}_i^T+\iota _i\mathbf {C}_i\mathbf {C}_i^T\right] \\&+2\max \left\{ -\widehat{\Xi }_1-\widehat{\Xi }_3,-\widehat{\Xi }_1-\widehat{\Xi }_4,-\widehat{\Xi }_2-\widehat{\Xi }_3,-\widehat{\Xi }_2-\widehat{\Xi }_4 \right\} , \end{aligned}$$
with
$$\begin{aligned} \widehat{\Xi }_1= \widehat{\mathcal {I}}_1^T{\bar{O}_1}{\widehat{\mathcal {I}}_1},\ \ \widehat{\Xi }_2=\widehat{\mathcal {I}}_2^T\bar{O}_1\widehat{\mathcal {I}}_2,\ \ \widehat{\Xi }_3=\widehat{\mathcal {I}}_3^T\bar{O}_2\widehat{\mathcal {I}}_3,\ \ \widehat{\Xi }_4=\widehat{\mathcal {I}}_4^T\bar{O}_2\widehat{\mathcal {I}}_4, \end{aligned}$$
$$\begin{aligned} \widehat{\mathcal {I}}_1= \hbox { col}\left\{ -{I_n},\ {0_n},\ {I_n},\ {0_{4n\times n}},\ {I_n},\ 0_{n} \right\} , \end{aligned}$$
$$\begin{aligned} \widehat{\mathcal {I}}_2= \hbox { col}\left\{ 0_{2n\times n},\ -{I_n},\ {0_{2n\times n}},\ {I_n},\ {0_{2n\times n}},\ {I_n} \right\} , \end{aligned}$$
$$\begin{aligned} \widehat{\mathcal {I}}_3= \hbox { col}\left\{ 0_{7n\times n},\ {I_n},\ {0_{ n}} \right\} , \widehat{\mathcal {I}}_4=\hbox { col}\left\{ 0_{8n\times n},\ {I_n}\right\} . \end{aligned}$$
From the well-known Schur complement, it is easy to see that \(\mathbf {E}\pounds \widetilde{V}(t,{e_t},i)<0\) if and only if the LMIs (14) hold.
Setting \(\varsigma =\min _{i\in \mathcal {N}}\big \{\lambda _m\big (-\widetilde{\Omega }_i\big )\big \},\) it follows that \(\varsigma >0.\) For any \(t>0,\) we achieve that
$$\begin{aligned} \mathbf {E}\pounds \widetilde{V}(t,{e_t},i)\le -\varsigma \xi _i(t)^T\xi _i(t)\le -\varsigma e(t)^Te(t). \end{aligned}$$
By Dynkin’s formula, the following inequality holds
$$\begin{aligned} \mathbf {E}\widetilde{V}(t,e_t,\eta (t))-\mathbf {E}\widetilde{V}(0,e_0,\eta _0)\le -\varsigma \mathbf {E}\left\{ \int ^t_0e(s)^Te(s)\mathrm{d}s\right\} , \end{aligned}$$
and hence
$$\begin{aligned} \mathbf {E}\left\{ \int ^t_0e(s)^Te(s)\mathrm{d}s\right\} \le \frac{1}{\varsigma }\mathbf {E}\widetilde{V}(0,e_0,\eta _0), \end{aligned}$$
which implies that (3) is globally asymptotically stable in the mean square. This completes the proof of Theorem 2.
