${\varvec{p}}$th moment stochastic exponential anti-synchronization of delayed complex-valued neural networks

Abstract

This paper focuses on the pth moment stochastic exponential anti-synchronization problem for delayed complex-valued neural networks (CVNNs). By combining It$\hat{\mathrm{o}}$’s differential formula, Halanay inequality and some inequalities techniques, via the designed delay-dependent controller, two different criteria are presented for guaranteeing pth moment anti-synchronization exponentially of the addressed systems by constructing appropriate Lyapunov functionals. Different from the previous works, the anti-synchronization control problem is addressed for CVNNs with the simultaneous existence of stochastic disturbances and time delays. Similar results can also be obtained for delayed CVNNs without stochastic disturbances or real-valued ones. The in-depth analysis results improve and extend some in the existing literature. In the end, two simulation examples are provided to show the effectiveness of the obtained results.

Appendices

Appendix I

By combining Assumptions 1 and 2 with Lemma 1, it follows that

$$\begin{aligned}&\mathcal {L}V(t,\varpi (t)) \nonumber \\&~ \le e^{\epsilon t} \sum _{j=1}^{n}\Big [\epsilon \alpha _j-p\alpha _jd_j-p\alpha _j\xi _{1j} +\delta _je^{\epsilon \tau }\Big ]|u_j(t)|^p \nonumber \\&~ +e^{\epsilon t} \sum _{j=1}^{n}\Big [\epsilon \beta _j -p\beta _jd_j-p\beta _j\xi _{2j}+\gamma _je^{\epsilon \tau }\Big ]|v_j(t)|^p \nonumber \\&~-e^{\epsilon t}\sum _{j=1}^{n}\delta _j|u_j^{\tau }(t)|^p-e^{\epsilon t}\sum _{j=1}^{n}\gamma _j|v_j^{\tau }(t)|^p \nonumber \\&~+pe^{\epsilon t}\sum _{j=1}^{n}\alpha _j\sum _{k=1}^{n}\Big [|a_{jk}^R|(\mu _k^{RR}|u_k(t)|+\mu _k^{RI}|v_k(t)|) \nonumber \\&~+|a_{jk}^I|(\mu _k^{IR}|u_k(t)|+\mu _k^{II}|v_k(t)|)\Big ]|u_j(t)|^{p-1} \nonumber \\&~+pe^{\epsilon t}\sum _{j=1}^{n}\alpha _j\sum _{k=1}^{n}\Big [|b_{jk}^R|(\nu _k^{RR}|u_k^{\tau }(t)|+\nu _k^{RI}|v_k^{\tau }(t)|) \nonumber \\&~+|b_{jk}^I|(\nu _k^{IR}|u_k^{\tau }(t)|+\nu _k^{II}|v_k^{\tau }(t)|)\Big ]|u_j(t)|^{p-1} \nonumber \\&~+pe^{\epsilon t}\sum _{j=1}^{n} \sum _{k=1}^{n}\alpha _j\pi _{1j}|u_j(t)|^{p-1}|u_k^{\tau }(t)| \nonumber \\&~+pe^{\epsilon t}\sum _{j=1}^{n} \sum _{k=1}^{n}\beta _j\pi _{2j}|v_j(t)|^{p-1}|v_k^{\tau }(t)| \nonumber \\&~+pe^{\epsilon t}\sum _{j=1}^{n}\beta _j\sum _{k=1}^{n}[|a_{jk}^R|(\mu _k^{IR}|u_k(t)|+\mu _k^{II}|v_k(t)|) \nonumber \\&~+|a_{jk}^I|(\mu _k^{RR}|u_k(t)|+\mu _k^{RI}|v_k(t)|)]|v_j(t)|^{p-1} \nonumber \\&~+pe^{\epsilon t}\sum _{j=1}^{n}\beta _j\sum _{k=1}^{n}[|b_{jk}^R|(\nu _k^{IR}|u_k^{\tau }(t)|+\nu _k^{II}|v_k^{\tau }(t)|) \nonumber \\&~+|b_{jk}^I|(\nu _k^{RR}|u_k^{\tau }(t)|+\nu _k^{RI}|v_k^{\tau }(t)|)]|v_j(t)|^{p-1} \nonumber \\&~+\frac{1}{2}p(p-1)e^{\epsilon t} \sum _{j=1}^{n}\alpha _j|u_j(t)|^{p-2} \nonumber \\&~\times \sum _{k=1}^{n}\Big [\varphi _{1jk}|u_k(t)|^{2} +\psi _{1jk}|u_k^{\tau }(t)|^{2}\Big ] \nonumber \\&~+\frac{1}{2}p(p-1)e^{\epsilon t} \sum _{j=1}^{n}\beta _j|v_j(t)|^{p-2} \nonumber \\&~\times \sum _{k=1}^{n}\Big [\varphi _{2jk}|v_k(t)|^{2}+\psi _{2jk}|v_k^{\tau }(t)|^{2}\Big ] \nonumber \\&~\le e^{\epsilon t} \sum _{j=1}^{n}\Big [\epsilon \alpha _j-p\alpha _jd_j-p\alpha _j\xi _{1j}+\delta _je^{\epsilon \tau }\Big ]|u_j(t)|^p \nonumber \\&~ +e^{\epsilon t} \sum _{j=1}^{n}\Big [\epsilon \beta _j-p\beta _jd_j -p\beta _j\xi _{2j}+\gamma _je^{\epsilon \tau }\Big ]|v_j(t)|^p \nonumber \\&~-e^{\epsilon t}\sum _{j=1}^{n}\delta _j|u_j^{\tau }(t)|^p-e^{\epsilon t}\sum _{j=1}^{n}\gamma _j|v_j^{\tau }(t)|^p \nonumber \\&~ +e^{\epsilon t}\sum _{j=1}^{n}\sum _{k=1}^{n}\alpha _j \chi _1[(p-1)| u_j(t)|^{p}+|u_k(t)|^p] \nonumber \\&~+e^{\epsilon t}\sum _{j=1}^{n}\sum _{k=1}^{n}\alpha _j \chi _2[(p-1)| u_j(t)|^{p}+|v_k(t)|^p] \nonumber \\&~+e^{\epsilon t}\sum _{j=1}^{n}\sum _{k=1}^{n}\alpha _j (\chi _3+\pi _{1j})[(p-1)| u_j(t)|^{p}+|u_k^{\tau }(t)|^p] \nonumber \\&~+e^{\epsilon t}\sum _{j=1}^{n}\sum _{k=1}^{n}\alpha _j \chi _4[(p-1)| u_j(t)|^{p}+|v_k^{\tau }(t)|^p] \nonumber \\&~+e^{\epsilon t}\sum _{j=1}^{n}\sum _{k=1}^{n}\beta _j \zeta _1[(p-1)| v_j(t)|^{p}+|u_k(t)|^p] \nonumber \\&~+e^{\epsilon t}\sum _{j=1}^{n}\sum _{k=1}^{n}\beta _j \zeta _2[(p-1)| v_j(t)|^{p}+|v_k(t)|^p] \nonumber \\&~+e^{\epsilon t}\sum _{j=1}^{n}\sum _{k=1}^{n}\beta _j \zeta _3[(p-1)| v_j(t)|^{p}+|u_k^{\tau }(t)|^p] \nonumber \\&~+e^{\epsilon t}\sum _{j=1}^{n}\sum _{k=1}^{n}\beta _j (\zeta _4+\pi _{2j})[(p-1)| v_j(t)|^{p}+|v_k^{\tau }(t)|^p] \nonumber \\&~+\frac{1}{2}(p-1)e^{\epsilon t} \sum _{j=1}^{n}\sum _{k=1}^{n}\alpha _j\Big [\varphi _{1jk}[(p-2)|u_j(t)|^{p} \nonumber \\&~+2|u_k(t)|^{p}] +\psi _{1jk}[(p-2)|u_j(t)|^{p}+2|u_k^\tau (t)|^{p}]\Big ] \nonumber \\&~+\frac{1}{2}(p-1)e^{\epsilon t} \sum _{j=1}^{n}\sum _{k=1}^{n}\beta _j\Big [\varphi _{2jk}[(p-2)|v_j(t)|^{p} \nonumber \\&~+2|u_k(t)|^{p}] +\psi _{2jk}[(p-2)|v_j(t)|^{p}+2|v_k^\tau (t)|^{p}]\Big ],\nonumber \\ \end{aligned}$$

(32)

where

$$\begin{aligned}&\chi _1=|a_{jk}^R|\mu _{k}^{RR}+|a_{jk}^I|\mu _{k}^{IR},~ \chi _2=|a_{jk}^R|\mu _{k}^{RI}+|a_{jk}^I|\mu _{k}^{II}, \\&\chi _3=|b_{jk}^R|\nu _{k}^{RR}+|b_{jk}^I|\nu _{k}^{IR},~ \chi _4=|b_{jk}^R|\nu _{k}^{RI}+|b_{jk}^I|\nu _{k}^{II}, \\&\zeta _1=|a_{jk}^R|\mu _{k}^{IR}+|a_{jk}^I|\mu _{k}^{RR},~ \zeta _2=|a_{jk}^R|\mu _{k}^{II}+|a_{jk}^I|\mu _{k}^{RI}, \\&\zeta _3=|b_{jk}^R|\nu _{k}^{IR}+|b_{jk}^I|\nu _{k}^{RR},~ \zeta _4=|b_{jk}^R|\nu _{k}^{II}+|b_{jk}^I|\nu _{k}^{RI}. \end{aligned}$$

Further, one can derive

$$\begin{aligned}&\mathcal {L}V(t)\le e^{\epsilon t}\sum _{j=1}^{n}\Big [\epsilon \alpha _j-p\alpha _jd_j-p\alpha _j\xi _{1j}+\delta _je^{\epsilon \tau } \nonumber \\&~+(p-1)\sum _{k=1}^{n}\alpha _j(\chi _1+\chi _2+\chi _3+\pi _{1j}+\chi _4) \nonumber \\&~+\sum _{k=1}^{n}(\alpha _k\tilde{\chi }_1+\beta _k\tilde{\zeta }_1) +\frac{1}{2}(p-1)(p-2)\alpha _j \nonumber \\&~\times \sum _{k=1}^{n}(\varphi _{1jk}+\psi _{1jk}) +(p-1)\sum _{k=1}^{n}\alpha _k\varphi _{1kj}\Big ]|u_j(t)|^p \nonumber \\&~+e^{\epsilon t}\sum _{j=1}^{n}\Big [\epsilon \beta _j-p\beta _jd_j-p\beta _j\xi _{2j}+\gamma _je^{\epsilon \tau } \nonumber \\&~+(p-1)\sum _{k=1}^{n}\beta _j(\zeta _1+\zeta _2+\zeta _3+\zeta _4+\pi _{2j}) \nonumber \\&~+\sum _{k=1}^{n}(\alpha _k\tilde{\chi }_2+\beta _k\tilde{\zeta }_2) +\frac{1}{2}(p-1)(p-2)\beta _j \nonumber \\&~\times \sum _{k=1}^{n}(\varphi _{2jk}+\psi _{2jk}) +(p-1)\sum _{k=1}^{n}\beta _k\varphi _{2kj}\Big ]|v_j(t)|^p \nonumber \\&~+e^{\epsilon t}\sum _{j=1}^{n}\Big [-\delta _j+\sum _{k=1}^{n}(\alpha _k(\tilde{\chi }_3+\pi _{1k})+\beta _k\tilde{\zeta }_3) \nonumber \\&~+(p-1)\sum _{k=1}^{n}\alpha _k\psi _{1kj}\Big ]|u_j^{\tau }(t)|^p \nonumber \\&~+e^{\epsilon t}\sum _{j=1}^{n}\Big [-\gamma _j +\sum _{k=1}^{n}(\alpha _k\tilde{\chi }_4 +\beta _k(\tilde{\zeta }_4+\pi _{2k})) \nonumber \\&~+(p-1)\sum _{k=1}^{n}\beta _k\psi _{2kj}\Big ]|v_j^{\tau }(t)|^p, \end{aligned}$$

(33)

where

$$\begin{aligned}&\tilde{\chi }_1=|a_{kj}^R|\mu _{j}^{RR}+|a_{kj}^I|\mu _{j}^{IR},~\\&\tilde{\chi }_2=|a_{kj}^R|\mu _{j}^{RI}+|a_{kj}^I|\mu _{j}^{II}, \\&\tilde{\chi }_3=|b_{kj}^R|\nu _{j}^{RR}+|b_{kj}^I|\nu _{j}^{IR},~ \tilde{\chi }_4=|b_{kj}^R|\nu _{j}^{RI}+|b_{kj}^I|\nu _{j}^{II}, \\&\tilde{\zeta }_1=|a_{kj}^R|\mu _{j}^{IR}+|a_{kj}^I|\mu _{j}^{RR},~ \tilde{\zeta }_2=|a_{kj}^R|\mu _{j}^{II}+|a_{kj}^I|\mu _{j}^{RI}, \\&\tilde{\zeta }_3=|b_{kj}^R|\nu _{j}^{IR}+|b_{kj}^I|\nu _{j}^{RR},~ \tilde{\zeta }_4=|b_{kj}^R|\nu _{j}^{II}+|b_{kj}^I|\nu _{j}^{RI}. \end{aligned}$$

Here, define

$$\begin{aligned}&\delta _j=\sum \limits _{k=1}^{n}(\alpha _k(\tilde{\chi }_3+\pi _{1k})+\beta _k\tilde{\zeta }_3) +(p-1)\sum \limits _{k=1}^{n}\alpha _k\psi _{1kj}, \nonumber \\&\gamma _j=\sum \limits _{k=1}^{n}(\alpha _k\tilde{\chi }_4+\beta _k(\tilde{\zeta }_4+\pi _{2k})) +(p-1)\sum \limits _{k=1}^{n}\beta _k\psi _{2kj}. \nonumber \\&\end{aligned}$$

(34)

By (12), there must exist a sufficiently small number $\epsilon >0$ such that

$$\begin{aligned}&\epsilon -p(d_j+\xi _{1j})+(p-1)\sum _{k=1}^{n}(\chi _1+\chi _2+\chi _3 \nonumber \\&~+\pi _{1j}+\chi _4) +\frac{\delta _je^{\epsilon \tau }}{\alpha _j} +\sum _{k=1}^{n}(\frac{\alpha _k}{\alpha _j}\tilde{\chi }_1+\frac{\beta _k}{\alpha _j}\tilde{\zeta }_1) \nonumber \\&~+\frac{1}{2}(p-1)(p-2)\sum _{k=1}^{n}(\varphi _{1jk}+\psi _{1jk}) \nonumber \\&~+(p-1)\sum _{k=1}^{n}\frac{\alpha _k}{\alpha _j}\varphi _{1kj}<0, \nonumber \\&\epsilon -p(d_j+\xi _{2j}) +(p-1)\sum _{k=1}^{n}(\zeta _1+\zeta _2+\zeta _3 \nonumber \\&~+\zeta _4+\pi _{2j})+\frac{\gamma _je^{\epsilon \tau }}{\beta _j} +\sum _{k=1}^{n}(\frac{\alpha _k}{\beta _j}\tilde{\chi }_2+\frac{\beta _k}{\beta _j}\tilde{\zeta }_2) \nonumber \\&~+\frac{1}{2}(p-1)(p-2)\sum _{k=1}^{n}(\varphi _{2jk}+\psi _{2jk}) \nonumber \\&~+(p-1)\sum _{k=1}^{n}\frac{\beta _k}{\beta _j}\varphi _{2kj}<0. \end{aligned}$$

(35)

Then, according to (33)–(35), it is easy to obtain

$$\begin{aligned}&\mathcal {L}V(t,\varpi (t))\le 0. \end{aligned}$$

(36)

Appendix II

From (10) and Lemma 1, (24) can be bounded as

$$\begin{aligned}&\mathcal {L}V_1(t) \le \sum _{j=1}^{n}\Big \{-\tilde{\alpha }_j(d_j+\xi _{1j})|u_j(t)|^p \nonumber \\&+\,\sum _{k=1}^{n}\frac{1}{p}\tilde{\alpha }_j \big [|a^R_{jk}|\big (|\mathbb {F}_k^R(u_k(t),v_k(t))|^p \nonumber \\&+\,(p-1)|u_j(t)|^{p}\big ) \nonumber \\&+\,|a^I_{jk}|\big (|\mathbb {F}_k^I(u_k(t),v_k(t))|^p+(p-1)|u_j(t)|^{p}\big )\big ] \nonumber \\&+\,\sum _{k=1}^{n}\frac{1}{p}\tilde{\alpha }_j \big [|b^R_{jk}|\big (|\mathbb {G}_k^R(u_k(t),v_k(t))|^p \nonumber \\&+\,(p-1)|u_j(t)|^{p}\big ) \nonumber \\&+\,|b^I_{jk}|\big (|\mathbb {G}_k^I(u_k(t),v_k(t))|^p +(p-1)|u_j(t)|^{p}\big )\big ] \nonumber \\&+\,\sum _{k=1}^{n}\frac{1}{p}\pi _{1j}\tilde{\alpha }_j\big (|u_k^{\tau }(t)|^p \nonumber \\&+\,(p-1)|u_j(t)|^{p}\big )\Big \} \nonumber \\&+\,\sum _{j=1}^{n}\Big \{-\tilde{\beta }_j(d_j+\xi _{2j})|v_j(t)|^p \nonumber \\&+\,\sum _{k=1}^{n}\frac{1}{p}\tilde{\beta }_j\big [|a^R_{jk}|\big (|\mathbb {F}_k^I(u_k(t),v_k(t))|^p \nonumber \\&+\,(p-1)|v_j(t)|^{p}\big ) \nonumber \\&+\,|a^I_{jk}|\big (|\mathbb {F}_k^R(u_k(t),v_k(t))|^p+(p-1)|v_j(t)|^{p}\big )\big ] \nonumber \\&+\,\sum _{k=1}^{n}\frac{1}{p}\tilde{\alpha }_j\big [|b^R_{jk}|\big (|\mathbb {G}_k^I(u_k(t),v_k(t))|^p \nonumber \\&+\,(p-1)|u_j(t)|^{p}\big ) \nonumber \\&+\,|b^I_{jk}|\big (|\mathbb {G}_k^R(u_k(t),v_k(t))|^p +\,(p-1)|v_j(t)|^{p}\big )\big ] \nonumber \\&+\,\sum _{k=1}^{n}\frac{1}{p}\pi _{2j}\tilde{\beta }_j\big (|v_k^{\tau }(t)|^P+(p-1)|v_j(t)|^{p}\big )\Big \} \nonumber \\&+\,\frac{1}{2p}(p-1)\sum _{j=1}^{n}\sum _{k=1}^{n}\tilde{\alpha }_j \big [\varphi _{1jk}\big (2|u_k|^p\nonumber \\&+\,(p-2)|u_j(t)|^{p}\big ) \nonumber \\&+\,\psi _{1jk}\big (2|u_k^{\tau }|^p+(p-2)|u_j(t)|^{p}\big )\big ] \nonumber \\&+\,\frac{1}{2p}(p-1)\sum _{j=1}^{n}\sum _{k=1}^{n}\tilde{\beta }_j\big [\varphi _{2jk}\big (2|v_k|^p \nonumber \\&+\,(p-2)|v_j(t)|^{p}\big )+\psi _{2jk}\big (2|v_k^{\tau }|^p\nonumber \\&+\,(p-2)|v_j(t)|^{p}\big )\big ] \nonumber \\\le & {} \sum _{j=1}^{n}\Big \{\big [-\tilde{\alpha }_j(d_j+\xi _{1j}) \nonumber \\&+\,\frac{1}{2p}(p-1)(p-2) \sum _{k=1}^{n}\tilde{\alpha }_j(\varphi _{1jk}+\psi _{1jk}) \nonumber \\&+\,\frac{1}{p}(p-1)\sum _{k=1}^{n}\tilde{\alpha }_k\varphi _{1kj}\big ]|u_j(t)|^p \nonumber \\&+\,\frac{1}{p}2^{p-1}\sum _{k=1}^{n}(\tilde{\alpha }_j|a_{jk}^R|+\tilde{\beta }_j|a_{jk}^I)\big [(\mu _k^{RR})^p|u_k(t)|^p \nonumber \\&+\,(\mu _k^{RI})^p|v_k(t)|^p\big ] +\frac{1}{p}2^{p-1}\sum _{k=1}^{n}(\tilde{\alpha }_j|a_{jk}^I| \nonumber \\&+\,\tilde{\beta }_j|a_{jk}^R)\big [(\mu _k^{IR})^p|u_k(t)|^p+(\mu _k^{II})^p|v_k(t)|^p\big ] \nonumber \\&+\,\frac{1}{p}2^{p-1}\sum _{k=1}^{n}(\tilde{\alpha }_j|b_{jk}^R| +\tilde{\beta }_j|b_{jk}^I) [(\nu _k^{RR})^p|u_k^{\tau }(t)|^p \nonumber \\&+\,(\nu _k^{RI})^p|v_k^{\tau }(t)|^p] +\frac{1}{p}2^{p-1}\sum _{k=1}^{n}(\tilde{\alpha }_j|b_{jk}^I| \nonumber \\&+\,\tilde{\beta }_j|b_{jk}^R)[(\nu _k^{IR})^p|u_k^{\tau }(t)|^p+(\nu _k^{II})^p|v_k^{\tau }(t)|^p]\Big \} \nonumber \\&+\,\sum _{j=1}^{n}\Big \{\big [-\tilde{\beta }_j(d_j+\xi _{2j}) \nonumber \\&+\,\frac{1}{2p}(p-1)(p-2)\sum _{k=1}^{n}\tilde{\beta }_j(\varphi _{2jk}+\psi _{2jk}) \nonumber \\&+\,\frac{p-1}{p}\sum _{k=1}^{n}\tilde{\beta }_k\varphi _{2kj}\big ]|v_j(t)|^p +\frac{p-1}{p}\sum _{k=1}^{n}\big [\tilde{\alpha }_j(\chi _5 \nonumber \\&+\,\pi _{1j})|u_j(t)|^p+\tilde{\beta }_j(\chi _5+\pi _{2j})|v_j(t)|^p\big ] \nonumber \\&+\,sum_{k=1}^{n}\frac{1}{p}(\pi _{1j}\tilde{\alpha }_j+(p-1)\tilde{\alpha }_j\psi _{1jk})|u_k^{\tau }(t)|^p \nonumber \\&+\,\sum _{k=1}^{n}\frac{1}{p}(\pi _{2j}\tilde{\beta }_j+(p-1)\tilde{\beta }_j\psi _{2jk})|v_k^{\tau }(t)|^p\Big \} . \end{aligned}$$

(37)

Consequently, we deduce

$$\begin{aligned}&\mathcal {L}V_1(t) \le \sum _{j=1}^{n}\Big [-\tilde{\alpha }_j(d_j+\xi _{1j}) \nonumber \\&\quad +\frac{1}{2p}(p-1)(p-2)\sum _{k=1}^{n}\tilde{\alpha }_j(\varphi _{1jk}+\psi _{1jk}) \nonumber \\&\quad +\frac{1}{p}(p-1)\sum _{k=1}^{n}\tilde{\alpha }_k\varphi _{1kj} +\sum _{k=1}^{n}\frac{1}{p}(\tilde{\alpha }_k\hat{\chi }_1+\tilde{\beta }_k\hat{\zeta }_1) \nonumber \\&\quad +\sum _{k=1}^{n}\frac{p-1}{p}\tilde{\alpha }_j(\chi _5+\pi _{1j})\Big ]|u_j(t)|^p \nonumber \\&\quad +\sum _{j=1}^{n}\Big [-\tilde{\beta }_j(d_j+\xi _{2j}) \nonumber \\&\quad +\frac{1}{2p}(p-1)(p-2)\sum _{k=1}^{n}\tilde{\beta }_j(\varphi _{2jk}+\psi _{2jk}) \nonumber \\&\quad +\frac{1}{p}(p-1)\sum _{k=1}^{n}\tilde{\beta }_k\varphi _{2kj} +\sum _{k=1}^{n}\frac{1}{p}(\tilde{\alpha }_k\hat{\chi }_2+\tilde{\beta }_k\hat{\zeta }_2) \nonumber \\&\quad +\sum _{k=1}^{n}\frac{p-1}{p}\tilde{\beta }_j(\chi _5+\pi _{2j})\Big ]|v_j(t)|^p \nonumber \\&\quad +\sum _{j=1}^{n}\Big [\sum _{k=1}^{n}\frac{1}{p}(\tilde{\alpha }_k\hat{\chi }_3+\tilde{\beta }_k\hat{\zeta }_3) \nonumber \\&\quad +\sum _{k=1}^{n}\frac{1}{p}\tilde{\alpha }_k(\pi _{1k}+(p-1)\psi _{1kj})\Big ]|u_j^{\tau }(t)|^p \nonumber \\&\quad +\sum _{j=1}^{n}\Big [\sum _{k=1}^{n}\frac{1}{p}(\tilde{\alpha }_k\hat{\chi }_4+\tilde{\beta }_k\hat{\zeta }_4) \nonumber \\&\quad +\sum _{k=1}^{n}\frac{1}{p}\tilde{\beta }_k(\pi _{2k}+(p-1)\psi _{2kj})\Big ]|v_j^{\tau }(t)|^p \nonumber \\&=-\varOmega _1\sum _{j=1}^{n}\frac{1}{p}\tilde{\alpha }_j|u_j(t)|^p -\varOmega _2\sum _{j=1}^{n}\frac{1}{p}\tilde{\beta }_j|v_j(t)|^p \nonumber \\&\quad +\varOmega _3\sum _{j=1}^{n}\frac{1}{p}\tilde{\alpha }_j|u_j^{\tau }(t)|^p +\varOmega _4\sum _{j=1}^{n}\frac{1}{p}\tilde{\beta }_j|v_j^{\tau }(t)|^p \nonumber \\&\le -\varOmega V_1(t)+\varOmega ^{\prime }\sup _{t-\tau \le s\le t}V_1(s), \end{aligned}$$

(38)

where

$$\begin{aligned}&\varOmega _1=p(d_j+\xi _{1j}) -\frac{1}{2}(p-1)(p-2)\sum _{k=1}^{n}(\varphi _{1jk}+\psi _{1jk}) \\&\quad -(p-1)\sum _{k=1}^{n}\frac{\tilde{\alpha }_k}{\tilde{\alpha }_j}\varphi _{1kj} -\sum _{k=1}^{n}\frac{2^{p-1}}{\tilde{\alpha }_j}(\tilde{\alpha }_k\hat{\chi }_1+\tilde{\beta }_k\hat{\zeta }_1) \\&\quad -\sum _{k=1}^{n}(p-1)(\chi _5+\pi _{1j}), \\&\varOmega _2=p(d_j+\xi _{2j}) -\frac{1}{2}(p-1)(p-2)\sum _{k=1}^{n}(\varphi _{2jk}+\psi _{2jk}) \\&\quad -(p-1)\sum _{k=1}^{n}\frac{\tilde{\beta }_k}{\tilde{\beta }_j}\varphi _{2kj} -\sum _{k=1}^{n}\frac{2^{p-1}}{\tilde{\beta }_j}(\tilde{\alpha }_k\hat{\chi }_2+\tilde{\beta }_k\hat{\zeta }_2) \\&\quad -\sum _{k=1}^{n}(p-1)(\chi _5+\pi _{2j}), \\&\varOmega _3=\sum _{k=1}^{n}\frac{2^{p-1}}{\tilde{\alpha }_j}(\tilde{\alpha }_k\hat{\chi }_3+\tilde{\beta }_k\hat{\zeta }_3) +\sum _{k=1}^{n}\frac{\tilde{\alpha }_k}{\tilde{\alpha }_j}(\pi _{1k} \\&\quad +(p-1)\psi _{1kj}), \\&\varOmega _4=\sum _{k=1}^{n}\frac{2^{p-1}}{\tilde{\beta }_j}(\tilde{\alpha }_k\hat{\chi }_4+\tilde{\beta }_k\hat{\zeta }_4) +\sum _{k=1}^{n}\frac{\tilde{\beta }_k}{\tilde{\beta }_j}(\pi _{2k} \\&\quad +(p-1)\psi _{2kj}), \\&\hat{\chi }_1=|a_{kj}^R|(\mu _{j}^{RR})^p+|a_{kj}^I|(\mu _{j}^{IR})^p,~ \hat{\chi }_2=|a_{kj}^R|(\mu _{j}^{RI})^p \\&\quad +|a_{kj}^I|(\mu _{j}^{II})^p,~ \hat{\chi }_3=|b_{kj}^R|(\nu _{j}^{RR})^p+|b_{kj}^I|(\nu _{j}^{IR}+\pi _{1j})^p, \\&\hat{\chi }_4=|b_{kj}^R|(\nu _{j}^{RI})^p+|b_{kj}^I|(\nu _{j}^{II})^p,~ \hat{\zeta }_1=|a_{kj}^R|(\mu _{j}^{IR})^p \\&\quad +|a_{kj}^I|(\mu _{j}^{RR})^p,~ \hat{\zeta }_2=|a_{kj}^R|(\mu _{j}^{II})^p+|a_{kj}^I|(\mu _{j}^{RI})^p, \\&\hat{\zeta }_3=|b_{kj}^R|(\nu _{j}^{IR})^p+|b_{kj}^I|(\nu _{j}^{RR})^p,~ \hat{\zeta }_4=|b_{kj}^R|(\nu _{j}^{II})^p \\&\quad +|b_{kj}^I|(\nu _{j}^{RI}+\pi _{2j})^p,~ \chi _5=|a_{jk}^R|+|a_{jk}^I|+|b_{jk}^R|+|b_{jk}^I|. \end{aligned}$$