Global dissipativity of high-order Hopfield bidirectional associative memory neural networks with mixed delays

Abstract

In this paper, the problem of the global dissipativity of high-order Hopfield bidirectional associative memory neural networks with time-varying coefficients and distributed delays is discussed. By using Lyapunov–Krasovskii functional method, inequality techniques and linear matrix inequalities, a novel set of sufficient conditions for global dissipativity and global exponential dissipativity for the addressed system is developed. Further, the estimations of the positive invariant set, globally attractive set and globally exponentially attractive set are found. Finally, two examples with numerical simulations are provided to support the feasibility of the theoretical findings.

Appendices

Appendix 1

If Assumption 2 is satisfied, we suppose that system (1) possesses an equilibrium point $(x^{*},\;y^{*})=(x_{1}^{*},\;x_{2}^{*},\ldots ,x_{n}^{*};\;y_{1}^{*},\; y_{2}^{*},\ldots ,y_{m}^{*})^{T}.$

$$\begin{aligned} \left\{ \begin{array}{ll} {\dot{x}}_{i}^{*}(t) = -a_{i}x_{i}^{*}(t)\\ \quad+\displaystyle \sum _{j=1}^{m}b_{ij}{\tilde{f}}_{j}(y_{j}^{*}(t))\\ \quad+\, \displaystyle \sum _{j=1}^{m}c_{ij}{\tilde{f}}_{j}(y_{j}^{*}(t-\tau (t)))\\ \quad +\, \displaystyle \sum _{j=1}^{m}\sum _{k=1}^{m}W_{ijk}{\tilde{f}}_{k}(y_{j}^{*}(t)){\tilde{f}}_{j}(y_{j}^{*}(t))\\ \quad +\, \displaystyle \sum _{j=1}^{m}\sum _{k=1}^{m}T_{ijk}{\tilde{f}}_{k}(y_{k}^{*}(t-\tau (t))){\tilde{f}}_{j}(y_{j}^{*}(t-\tau (t)))\\ \quad +\, \displaystyle \sum _{j=1}^{m}p_{ij}\displaystyle \int _{-\infty }^{t}K_{ij}(t-s){\tilde{f}}_{j}(y_{j}^{*}(s))\text {d}s=0,\\ {\dot{y}}_{j}^{*}(t) = -a^{1}_{j}y_{j}^{*}(t)\\ \quad+\,\displaystyle \sum _{i=1}^{n}b^{1}_{ji}{\tilde{g}}_{i}(x_{i}^{*}(t))\\ +\, \displaystyle \sum _{i=1}^{n}c^{1}_{ji}{\tilde{g}}_{i}(y_{i}^{*}(t-\sigma (t)))\\ \quad +\, \displaystyle \sum _{i=1}^{n}\sum _{k=1}^{n}W^{1}_{jik}{\tilde{g}}_{k}(x_{i}^{*}(t)) {\tilde{g}}_{i}(x_{i}^{*}(t))\\ \quad +\, \displaystyle \sum _{i=1}^{n}\sum _{k=1}^{n}T^{1}_{jik}{\tilde{g}}_{k}(x_{k}^{*}(t-\sigma (t))) {\tilde{g}}_{i}(x_{i}^{*}(t-\sigma (t)))\\ \quad +\, \displaystyle \sum _{i=1}^{n}p^{1}_{ji}\displaystyle \int _{-\infty }^{t}K_{ji}(t-s){\tilde{g}}_{i}(x_{i}^{*}(s))\text {d}s=0. \end{array} \right. \end{aligned}$$

Let $x_{i}(t)=y_{i}(t)-y^{*}_i,\;y_{j}(t)=x_{j}(t)-x^{*}_j$ for $i=1,\;2,\ldots ,n,\;j=1,\;2,\ldots ,m$ and $f_{i}(x_{i}(t))={\tilde{f}}_{i}(x_{i}(t)+y^{*})-{\tilde{f}}_{i}(y^{*}),$

$g_{j}(y_{j}(t))={\tilde{g}}_{j}(y_{j}(t)+x^{*})-{\tilde{g}}_{j}(x^{*})$ and $f_{i}(0)={\tilde{f}}_{i}(0)=0,$

$g_{i}(0)={\tilde{g}}_{i}(0)=0,\;i=1,2,\ldots ,n,\;j=1,2,\ldots ,m.$

Hence, we have:

$$\begin{aligned} \left\{ \begin{array}{ll} {\dot{x}}_{i}(t) = -\,a_{i}x_{i}(t)+\displaystyle \sum _{j=1}^{m}b_{ij}f_{j}(y_{j}(t))\\ \quad+\,\displaystyle \sum _{j=1}^{m}c_{ij}f_{j}(y_{j}(t-\tau (t)))\\ \quad +\,\displaystyle \sum _{j=1}^{m}\displaystyle \sum _{k=1}^{m}W_{ijk}f_{k}(y_{k}(t)) f_{j}(y_{j}(t))\\ \quad +\,\displaystyle \sum _{j=1}^{m}\displaystyle \sum _{k=1}^{m}T_{ijk}f_{k}(y_{k}(t-\tau (t)))f_{j}(y_{j}(t-\tau (t)))\\ \quad +\,\displaystyle \sum _{j=1}^{m}p_{ij}\displaystyle \int _{-\infty }^{t}K_{ij}(t-s)f_{j}(y_{j}(s))\text {d}s\\ \quad {\dot{y}}_{j}(t) = -a^{1}_{j}y_{j}(t)+\displaystyle \sum _{i=1}^{n}b^{1}_{ji}g_{j}(g_{i}(t))\\ \quad +\,\displaystyle \sum _{i=1}^{n}c^{1}_{ji}g_{i}(x_{i}(t-\sigma (t)))\\ \quad +\,\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{k=1}^{n}W^{1}_{jik}g_{k}(x_{k}(t)) g_{i}(x_{i}(t))\\ \quad +\,\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{k=1}^{n}T^{1}_{jik}g_{k}(x_{k}(t-\sigma (t)))g_{i}(x_{i}(t-\sigma (t))\\ \quad +\,\displaystyle \sum _{i=1}^{n}p^{1}_{ji}\displaystyle \int _{-\infty }^{t}K_{ji}(t-s)g_{i}(x_{i}(s))\text {d}s\\ \end{array} \right. \end{aligned}$$

We can conclude that the transformed models are equivalent to the original model only if Assumption 3 is satisfied. The system can be described as:

$$\begin{aligned} \left\{ \begin{array}{ll} {\dot{x}}_{i}(t) = -\,a_{i}x_{i}(t)+\displaystyle \sum _{j=1}^{m}b_{ij}f_{j}(y_{j}(t))\\ \quad +\, \displaystyle \sum _{j=1}^{m}c_{ij}f_{j}(y_{j}(t-\tau (t)))\\ \quad +\, \displaystyle \sum _{j=1}^{m}\displaystyle \sum _{k=1}^{m}W_{ijk}\big [\big ({\tilde{f}}_{j}(y_{j}(t))-{\tilde{f}}_{j}(y_{j}^{*}(t))\big ){\tilde{f}}_{k}(y_{k}(t))\\ \quad +\, {\tilde{f}}_{j}(y_{j}^{*}\big (f_{k}(y_{k}(t))- f_{k}(y_{k}^{*})\big )\big ]\\ \quad +\, \displaystyle \sum _{j=1}^{m} \displaystyle \sum _{k=1}^{m}T_{ijk}\big [\big ({\tilde{f}}_{j}(y_{j}(t-\tau _{j}(t)))\\ \quad - {\tilde{f}}_{j}(y_{j}^{*})\big ) {\tilde{f}}_{k}(y_{k}(t-\tau _{k}(t)))\\ \quad +\, {\tilde{f}}_{j}(y_{j}^{*})\big ({\tilde{f}}_{k}(y_{k}(t-\tau _{k}(t))-f_{k}(y_{k}^{*})\big )\big ]\\ \quad +\, \displaystyle \sum _{j=1}^{m}p_{ij} \displaystyle \int _{-\infty }^{t}K_{ij}(t-s)f_{j}(y_{j}(s))\text {d}s\\ {\dot{y}}_{j}(t) = -a^{1}_{j}y_{j}(t)+\displaystyle \sum _{i=1}^{n}b^{1}_{ji}g_{j}(x_{i}(t))\\ \quad +\, \displaystyle \sum _{i=1}^{n}c^{1}_{ji}g_{i}(x_{i}(t-\sigma (t)))\\ \quad +\, \displaystyle \sum _{i=1}^{n} \displaystyle \sum _{k=1}^{n}W^{1}_{jik}\big [\big ({\tilde{g}}_{i}(x_{i}(t))\\ \quad - {\tilde{g}}_{i}(x_{i}^{*}(t))\big ){\tilde{g}}_{k}(x_{k}(t))\\ \quad +\, {\tilde{g}}_{i}(x_{i}^{*}\big (g_{k}(x_{k}(t))- g_{k}(x_{k}^{*})\big )\big ]\\ \quad +\, \displaystyle \sum _{i=1}^{n} \displaystyle \sum _{k=1}^{n}T^{1}_{jik}\big [\big ({\tilde{g}}_{i}(x_{i}(t-\sigma _{i}(t)))-{\tilde{g}}_{i}(x_{i}^{*})\big ) {\tilde{g}}_{k}(x_{k}(t-\sigma _{k}(t)))\\ \quad +\, {\tilde{g}}_{i}(x_{i}^{*})\big ({\tilde{g}}_{k}(x_{k}(t-\sigma _{k}(t))-g_{k}(x_{k}^{*})\big )\big ]\\ \quad +\, \displaystyle \sum _{i=1}^{n}p^{1}_{ji} \displaystyle \int _{-\infty }^{t}K_{ji}(t-s)g_{i}(x_{i}(s))\text {d}s.\\ \end{array} \right. \\ \left\{ \begin{array}{ll} {\dot{x}}_{i}(t) =-\,a_{i}x_{i}(t)+\displaystyle \sum _{j=1}^{m}\big [b_{ij}\\ \quad +\, \displaystyle \sum _{k=1}^{m}\big (W_{ijk}{\tilde{f}}_{k}(y_{k}(t))+W_{ikj}{\tilde{f}}_{k}(y_{k}^{*})\big )\big ]f_{j}(y_{j}(t))\\ \quad +\, \displaystyle \sum _{j=1}^{m}\big [c_{ij}+\displaystyle \sum _{k=1}^{m}(T_{ijk}{\tilde{f}}_{k}(y_{k}(t-\tau _{k}(t)))\\ \quad + T_{ikj}{\tilde{f}}_{k}(y_{k}^{*}))\big ]f_{j}(y_{j}(t-\tau _{j}(t)))\\ \quad + \displaystyle \sum _{j=1}^{m}p_{ij}\displaystyle \int _{-\infty }^{t}K_{ij}(t-s)f_{j}(y_{j}(s))\text {d}s.\\ {\dot{y}}_{j}(t) = -a^{1}_{j}y_{j}(t)+\displaystyle \sum _{i=1}^{n}\big [b^{1}_{ji}+\displaystyle \sum _{k=1}^{n}\big (W^{1}_{jik}{\tilde{g}}_{k}(x_{k}(t))\\ \quad + W^{1}_{jki}{\tilde{g}}_{k}(x_{k}^{*})\big )\big ]g_{i}(x_{i}(t))\\ \quad + \displaystyle \sum _{i=1}^{n}\big [c^{1}_{ji}+\displaystyle \sum _{k=1}^{n}(T^{1}_{jik}{\tilde{g}}_{k}(x_{k}(t-\sigma _{k}(t)))\\ \quad + T^{1}_{jki}{\tilde{g}}_{k}(x_{k}^{*}))\big ]g_{i}(x_{i}(t-\sigma _{i}(t)))\\ \quad + \displaystyle \sum _{i=1}^{n}p^{1}_{ji} \displaystyle \int _{-\infty }^{t}K_{ji}(t-s)g_{i}(x_{i}(s))\text {d}s \end{array} \right. \\ \left\{ \begin{array}{ll} {\dot{x}}_{i}(t) = -\,a_{i}x_{i}(t) + \displaystyle \sum _{j=1}^{m}\big [b_{ij}\\ \quad + \displaystyle \sum _{k=1}^{m}(W_{ijk}+W_{ikj})\xi _{ijk}(y_{k}(t))\big ]f_{j}(y_{j}(t))+\displaystyle \sum _{j=1}^{m}\big [c_{ij}\\ \quad + \displaystyle \sum _{k=1}^{m}(T_{ijk}+T_{ikj})\zeta _{ijk}(y_{k}(t-\tau _{k}(t)))\big ]f_{j}(y_{j}(t-\tau _{j}(t)))\\ \quad + \displaystyle \sum _{j=1}^{m}p_{ij}\displaystyle \int _{-\infty }^{t}K_{ij}(t-s)f_{j}(y_{j}(s))\text {d}s,\\ {\dot{y}}_{j}(t) = -a^{1}_{j}y_{j}(t)+\displaystyle \sum _{i=1}^{n}\big [b^{1}_{ji}\\ \quad + \displaystyle \sum _{k=1}^{n}(W^{1}_{jik}+W^{1}_{jki})\xi ^{1}_{jik}(x_{k}(t))\big ]g_{i}(x_{i}(t))+\displaystyle \sum _{i=1}^{n}\big [c^{1}_{ji}\\ \quad + \displaystyle \sum _{k=1}^{n}(T^{1}_{jik}+T^{1}_{jki})\zeta ^{1}_{jik}(x_{k}(t-\sigma _{k}(t)))\big ]g_{i}(x_{i}(t-\sigma _{i}(t)))\\ \quad + \displaystyle \sum _{i=1}^{n}p^{1}_{ji}\displaystyle \int _{-\infty }^{t}K_{ji}(t-s)g_{i}(x_{j}(s))\text {d}s \end{array} \right. \\ \left\{ \begin{array}{ll} {\dot{x}}_{i}(t) = -\,a_{i}x_{i}(t)+\displaystyle \sum _{j=1}^{m}\big [b_{ij}\\ \quad + \displaystyle \sum _{k=1}^{m}\big (W_{ijk}{\tilde{f}}_{k}(y_{k}(t))\\ \quad + W_{ikj}{\tilde{f}}_{k}(y_{k}^{*})\big )\big ]f_{j}(y_{j}(t))+\displaystyle \sum _{j=1}^{m}\big [c_{ij}\\ \quad + \displaystyle \sum _{k=1}^{m}(T_{ijk}{\tilde{f}}_{k}(y_{k}(t-\tau _{k}(t)))\\ \quad + T_{ikj}{\tilde{f}}_{k}(y_{k}^{*}))\big ]f_{j}(y_{j}(t-\tau _{j}(t)))\\ \quad + \displaystyle \sum _{j=1}^{m}p_{ij} \int _{-\infty }^{t}K_{ij}(t-s)f_{j}(y_{j}(s))\text {d}s,\\ {\dot{y}}_{j}(t) = -a^{j}_{i}y_{j}(t)+\displaystyle \sum _{i=1}^{n}\big [b^{1}_{ji}\\ \quad + \displaystyle \sum _{k=1}^{n}\big (W_{jik}{\tilde{g}}_{k}(x_{k}(t))\\ \quad + W^{1}_{jki}{\tilde{g}}_{k}(x_{k}^{*})\big )\big ]g_{i}(x_{i}(t))\\ \quad + \displaystyle \sum _{i=1}^{n}\big [c^{1}_{ji}+\displaystyle \sum _{k=1}^{n}(T^{1}_{jik}{\tilde{g}}_{k}(x_{k}(t-\sigma _{k}(t)))\\ \quad + T_{jki}{\tilde{g}}_{k}(x_{k}^{*}))\big ]g_{i}(x_{i}(t-\sigma _{i}(t)))\\ \quad + \displaystyle \sum _{i=1}^{n}p^{1}_{ji} \int _{-\infty }^{t}K_{ji}(t-s)g_{i}(x_{i}(s))\text {d}s, \end{array} \right. \\ \left\{ \begin{array}{ll} {\dot{x}}_{i}(t) = -\,a_{i}x_{i}(t)+ \displaystyle \sum _{j=1}^{m}\big [b_{ij}\\ \quad + \displaystyle \sum _{k=1}^{m}(W_{ijk}+W_{ikj})\xi _{ijk}(y_{k}(t))\big ]f_{j}(y_{j}(t))\\ \quad + \displaystyle \sum _{j=1}^{m}\big [c_{ij}+\displaystyle \sum _{k=1}^{m}(T_{ijk}\\ \quad + T_{ikj})\zeta _{ijk}(y_{k}(t-\tau _{k}(t)))\big ]f_{j}(y_{j}(t-\tau _{j}(t)))\\ \quad + \displaystyle \sum _{j=1}^{n}p_{ij} \displaystyle \int _{-\infty }^{t}K_{ij}(t-s)f_{j}(x_{j}(s))\text {d}s\\ {\dot{y}}_{j}(t) = -a^{1}_{j}y_{j}(t)+\displaystyle \sum _{i=1}^{n}\big [b^{1}_{ji}\\ \quad + \displaystyle \sum _{k=1}^{n}(W^{1}_{jik}+W^{1}_{jki})\xi ^{1}_{jik}(x_{k}(t))\big ]g_{i}(x_{i}(t))\\ \quad + \displaystyle \sum _{i=1}^{n}\big [c^{1}_{ji}+\displaystyle \sum _{k=1}^{n}(T^{1}_{jik}\\ \quad + T^{1}_{jki})\zeta ^{1}_{jik}(x_{k}(t-\sigma _{k}(t)))\big ]g_{i}(x_{i}(t-\sigma _{i}(t)))\\ \quad + \displaystyle \sum _{i=1}^{n}p^{1}_{ji} \displaystyle \int _{-\infty }^{t}K_{ji}(t-s)g_{i}(x_{i}(s))\text {d}s \end{array} \right. \end{aligned}$$

(9)

where

• $\xi _{ijk}(x_{k}(t))=(W_{ijk}{\tilde{f}}_{k}(y_{k}(t))+W_{ikj}{\tilde{f}}_{k}(y_{k}^{*}))/(W_{ijk}+W_{ikj})$ if it lies between ${\tilde{f}}_{k}(y_{k}(t))$ and ${\tilde{f}}_{k}(y_{k}^{*}),$

• $\xi ^{1}_{jik}(y_{k}(t))=(W^{1}_{jik}{\tilde{g}}_{k}(x_{k}(t))+W^{1}_{kij}{\tilde{g}}_{k}(x_{k}^{*}))/(W^{1}_{jik}+W^{1}_{jki})$

  if it lies between ${\tilde{g}}_{k}(x_{k}(t))$ and ${\tilde{g}}_{k}(x_{k}^{*}),$

• $\zeta _{ijk}(x_{k}(t-\tau _{k}(t)))=(T_{ijk}{\tilde{f}}_{k}(y_{k}(t-\tau _{k}(t))))+T_{ikj}{\tilde{f}}_{k}(y_{k}^{*}))/(T_{ijk}+T_{ikj})$ if it lies between ${\tilde{f}}_{k}(y_{k}(t-\tau _{k}(t))))$ and ${\tilde{f}}_{k}(y_{k}^{*}),$

• $\zeta ^{1}_{jik}(y_{k}(t-\sigma _{k}(t)))=(T^{1}_{jik}{\tilde{g}}_{k}(x_{k}(t-\sigma _{k}(t))))+T^{1}_{jki}{\tilde{g}}_{k}(x_{k}^{*}))/(T^{1}_{jik}+T^{1}_{jki})$ if it lies between ${\tilde{g}}_{k}(x_{k}(t-\sigma _{k}(t))))$ and ${\tilde{g}}_{k}(x_{k}^{*}),$

If we denote,

• $x(.)=[x_{1}(\cdot ),\;x_{2}(\cdot ),\ldots , x_{n}(\cdot )]^{T},$

• $y(.)=[y_{1}(\cdot ),\;y_{2}(\cdot ),\ldots , y_{m}(\cdot )]^{T},$

• $f(x(.))=[f_{1}(x_{1}(\cdot )),\;f_{2}(x_{2}(\cdot )),\ldots , f_{n}(x_{n}(\cdot ))]^{T},$

• $g(y(.))=[g_{1}(y_{1}(\cdot )),\;g_{2}(y_{2}(\cdot )),\ldots ,\;g_{m}(y_{m}(\cdot ))]^{T},$

• $f(x(t-\tau (t)))=[f_{1}(x_{1}(t-\tau _{1}(t))),\ldots , f_{n}(x_{n}(t-\tau _{n}(t)))]^{T},$

• $g(y(t-\sigma (t)))=[g_{1}(y_{1}(t-\sigma _{1}(t))),\ldots , g_{m}(y_{m}(t-\sigma _{m}(t)))]^{T},$

• $A=diag\{a_{1},a_{2},\ldots ,a_{n}\},$$A^{1}=diag\{a^{1}_{1}, a^{1}_{2},\ldots ,\;a_{m}\},$

• $B=(b_{ij})_{n\times n},$$B^{1}=(b^{1}_{ji})_{m\times m},$$C=(c_{ij})_{n\times n},\;C^{1}=(c^{1}_{ji})_{m\times m},$

• $P=(p_{ij})_{n\times n},$$P^{1}=(p^{1}_{ji})_{m\times m},$$W_{i}=(W_{ijk})_{n\times n},$

• $W^{1}_{j}=(W^{1}_{jik})_{m\times m},$$W=(W_{1}+W_{1}^{T},\;W_{2}+W_{2}^{T},\ldots ,\;W_{n}+W_{n}^{T}),$

• $W^{1}=(W^{1}_{1}+(W^{1}_{1})^{T},\;W^{1}_{2}+(W^{1}_{2})^{T},\ldots ,\;W^{1}_{m}+(W^{1}_{m})^{T}),\; T_{i}=(T_{ijk})_{n\times n},$

• $T^{1}_{j}=(T^{1}_{jik})_{m\times m},\;T=(T_{1}+T_{1}^{T},\;T_{2}+T_{2}^{T},\ldots , T_{n}+T_{n}^{T}),$

• $T^{1}=(T^{1}_{1}+(T^{1}_{1})^{T},\;T^{1}_{2}+(T^{1}_{2})^{T},\ldots , T^{1}_{m}+(T^{1}_{m})^{T}),$

• $\xi =(\xi _{1},\;\xi _{2},\ldots , \xi ^{1}_{n})^{T},\;\xi ^{1}=(\xi ^{1}_{1},\;\xi _{2},\ldots ,\;\xi _{m})^{T},$

• $\varLambda =diag\{\xi _{1},\;\xi _{2},\ldots , \xi _{n}\},$$\varLambda ^{1}=diag\{\xi ^{1}_{1},\;\xi ^{1}_{2},\ldots ,\;\xi ^{1}_{m}\},$

• $\zeta =(\zeta _{1},\;\zeta _{2},\ldots ,\zeta _{n})^{T},$$\zeta ^{1}=(\zeta ^{1}_{1},\;\zeta ^{1}_{2},\ldots ,\zeta ^{1}_{m})^{T},$

• $\varGamma =diag\{\zeta _{1},\;\zeta _{2},\ldots ,\zeta _{n}\},$$\varGamma ^{1}=diag\{\zeta ^{1}_{1},\;\zeta ^{1}_{2},\ldots ,\;\zeta ^{1}_{m}\},$

• $U(.)=\big (u_{1}(\cdot ),\;u_{2}(\cdot ),\ldots ,u_{n}(\cdot )\big )^{T},{\check{V}}(\cdot )=\big (v_{1}(\cdot ),\;v_{2}(\cdot ),\ldots ,v_{n}(\cdot )\big )^{T},$

then the system (9) can be rewritten in the following vector–matrix form:

$$\begin{aligned} \left\{ \begin{array}{ll} {\dot{x}}(t) = -\,Ax(t)+Bf(y(t))+Cf(y(t-\tau (t)))\\ \quad + P\int _{-\infty }^{t}K(t-s)f(y(s))\text {d}s+\varLambda ^{T}Wf(y(t))\\ \quad + \varGamma ^{T}Tf(y(t-\tau (t)))+U(t),\\ {\dot{y}}(t) = -A^{1}y(t)+B^{1}g(x(t))+C^{1}g(x(t-\sigma (t)))\\ \quad + P^{1}\displaystyle \int _{-\infty }^{t}K(t-s)g(x(s))\text {d}s+(\varLambda ^{1})^{T}W^{1}g(x(t))\\ \quad + (\varGamma ^{1})^{T}T^{1}g(x(t-\sigma (t)))+{\check{V}}(t), \end{array} \right. \end{aligned}$$

Appendix 2 (Proof of Theorem 1)

Proof

Consider the radially unbounded and positive definite Lyapunov function:

$$\begin{aligned} V(t)= \frac{1}{2}\displaystyle \sum _{i=1}^{n}|x_{i}(t)|^{2}+\frac{1}{2}\displaystyle \sum _{j=1}^{m}|y_{j}(t)|^{2},\\ \quad D^{+}V(t)\le \displaystyle \sum _{i=1}^{n}|x_{i}(t)|\bigg [-a_{i}|x_{i}(t)|+\displaystyle \sum _{j=1}^{m}|b_{ij}||{\tilde{f}}_{j}(y_{j}(t))|\\ \quad+\displaystyle \sum _{j=1}^{m}|c_{ij}||{\tilde{f}}_{j}(y_{j}(t-\tau (t)))|\\ \quad+\displaystyle \sum _{j=1}^{m}|p_{ij}|\int _{-\infty }^{t}|K_{ij}(t-s)||{\tilde{f}}_{j}(y_{j}(s))|\text {d}s\\ \quad+\displaystyle \sum _{j=1}^{m} \displaystyle \sum _{k=1}^{m}|W_{ijk}||{\tilde{f}}_{k}(y_{k}(t))||{\tilde{f}}_{j}(y_{j}(t))|\\ \quad+\displaystyle \sum _{j=1}^{m} \displaystyle \sum _{k=1}^{m}|T_{ijk}||{\tilde{f}}_{k}(y_{k}(t-\tau (t)))||{\tilde{f}}_{j}(y_{j}(t-\tau (t)))\\ \quad+|u_{i}(t)|\bigg ]+\displaystyle \sum _{j=1}^{m}|y_{j}(t)|\bigg [-a_{j}^{1}|y_{j}(t)|\\ \quad+\displaystyle \sum _{i=1}^{n}|b_{ji}^{1}||{\tilde{g}}_{i}(x_{i}(t))|+\displaystyle \sum _{i=1}^{n}|c_{ji}^{1}||{\tilde{g}}_{i}(x_{i}(t-\sigma (t)))|\\ \quad+\displaystyle \sum _{i=1}^{n}|p_{ji}^{1}|\int _{-\infty }^{t}|K_{ji}(t-s)||{\tilde{g}}_{i}(x_{i}(s))|\text {d}s\\ \quad+\displaystyle \sum _{i=1}^{n} \displaystyle \sum _{k=1}^{n}|W_{jik}^{1}||{\tilde{g}}_{k}(x_{k}(t))||{\tilde{g}}_{i}(y_{i}(t))|\\ \quad+\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{k=1}^{n}|T_{jik}^{1}||{\tilde{g}}_{k}(x_{k}(t-\sigma (t)))||{\tilde{g}}_{i}(x_{i}(t-\sigma (t)))|+|v_{j}(t)|\bigg ]\\ \quad \le \sum _{i=1}^{n}|x_{i}(t)|\bigg [-\displaystyle \sum _{i=1}^{n}a_{i}|x_{i}(t)|\\ \quad+\bigg (\displaystyle \sum _{j=1}^{n}|b_{ij}|+\displaystyle \sum _{j=1}^{m}|c_{ij}|+\displaystyle \sum _{j=1}^{m}|p_{ij}|\\ \quad+\displaystyle \sum _{j=1}^{m}\displaystyle \sum _{k=1}^{m}|W_{ijk}|+\displaystyle \sum _{j=1}^{m}\displaystyle \sum _{k=1}^{m}|T_{ijk}|\bigg )L_{j}^{f}+|u_{i}(t)|\bigg ]\\ \quad+\displaystyle \sum _{j=1}^{m}|y_{j}(t)|\bigg [-a_{j}^{1}|y_{j}(t)|\\ \quad+\bigg (\displaystyle \sum _{i=1}^{n}|b_{ji}^{1}|+\displaystyle \sum _{i=1}^{n}|c_{ji}^{1}|+\displaystyle \sum _{i=1}^{n}|p_{ji}^{1}|\\ \quad+\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{k=1}^{n}|W_{jik}^{1}|+\displaystyle \sum _{i=1}^{n} \displaystyle \sum _{k=1}^{n}|T_{jik}^{1}|\bigg )L^{g}_{i}+|v_{j}(t)|\bigg ]\\ \quad= -\sum _{i=1}^{n}a_{i}|x_{i}(t)|^{2}+\sum _{i=1}^{n}|x_i(t)|\varGamma _{i}\\ \quad-\sum _{j=1}^{m}a^{1}_{j}|y_{i}(t)|^{2}+\sum _{j=1}^{m}|y_j(t)|{\tilde{\varGamma }}_{j}\\ \quad \le -\sum _{i=1}^{n}a_{i}|x_{i}(t)|^{2}+\frac{1}{2}\sum _{i=1}^{n}|x_i(t)|^{2}\\ \quad+\frac{1}{2}\sum _{i=1}^{n}\varGamma _{i}^{2}-\sum _{j=1}^{m}a^{1}_{j}|y_{i}(t)|^{2}\\ \quad+\frac{1}{2}\sum _{j=1}^{m}|y_j(t)|^{2}+\frac{1}{2}\sum _{j=1}^{m}{\tilde{\varGamma }}_{j}^{2}\\ \quad= \sum _{i=1}^{n}(\frac{1}{2}-a_{i})|x_{i}(t)|^{2}+\sum _{j=1}^{m}(\frac{1}{2}-a_{j}^{1})|y_j(t)|^{2}\\ \quad+(\frac{1}{2}\sum _{i=1}^{n}\varGamma _{i}^{2}+\sum _{j=1}^{m}{\tilde{\varGamma }}_j^{2})\\ \quad\le -\delta \bigg [\sum _{i=1}^{n}|x_{i}(t)|^{2}+\sum _{j=1}^{m}|y_j(t)|^{2}\bigg ]+\frac{1}{2}(\sum _{i=1}^{n}\varGamma _{i}^{2}+\sum _{j=1}^{m}{\tilde{\varGamma }}_j^{2}). \end{aligned}$$

when $(x^{T}(t),\;y^T(t))^{T}\in {\mathbb {R}}^{m+n}\backslash \varUpsilon _{1},$ that is $(x^{T}(t),\;y^T(t))^{T}\notin \varUpsilon _{1},$ which implies that for $(\varphi ^{T},\;\phi ^{T})^{T}\in \varUpsilon _{1},\;t\ge t_{0};$

$\big ((x^{T}(t,\;t_{0},\;\varphi ),\;y(t,\;t_{0},\;\phi )\big )^{T}\in \varUpsilon _{1}$ holds. And for $(\varphi ^{T},\;\phi ^{T})^{T}\notin \varUpsilon _{1},$ there exists $T>0$ such that:

$\big ((x^{T}(t,\;t_{0},\;\varphi ),\;y(t,\;t_{0},\;\phi )\big )^{T}\in \varUpsilon _{1}$ holds for all $t>t_{0}+T$. From Definition 1, it is concluded that the neural network model 1 is a dissipative system and $\varUpsilon _{1}$ is a positive invariant and globally attractive set of (1) $\square$

Appendix 3 (Proof of Theorem 2)

Proof

Consider the following Lyapunov functional:

$$\begin{aligned} V(t)=e^{\alpha t}\sum _{i=1}^{n}|x_{i}(t)|+e^{\alpha t}\sum _{j=1}^{m}|y_{j}(t)| \end{aligned}$$

Calculating the upper right-hand derivative of $V(\cdot )$ along the positive half trajectory of system (1), we have

$$\begin{aligned} D^{+}V(t)&= e^{\alpha t}\bigg [\sum _{=1}^{n}(\alpha -a_{i})|x_{i}(t)|+\displaystyle \sum _{j=1}^{m}|b_{ij}||{\tilde{f}}_{j}(y_{j}(t))| \\&+\displaystyle \sum _{j=1}^{m}|c_{ij}||{\tilde{f}}_{j}(y_{j}(t-\tau (t)))| \\&+\displaystyle \sum _{j=1}^{m}\displaystyle \sum _{k=1}^{m}|W_{ijk}||{\tilde{f}}_{k}(y_{k}(t))||{\tilde{f}}_{j}(y_{j}(t))| \\&+\displaystyle \sum _{j=1}^{m}\displaystyle \sum _{k=1}^{m}|T_{ijk}||{\tilde{f}}_{k}(y_{k}(t-\tau (t)))||{\tilde{f}}_{j}(y_{j}(t-\tau (t))) \\&+\displaystyle \sum _{j=1}^{m}|p_{ij}|\int _{-\infty }^{t}|K_{ij}(t-s)||{\tilde{f}}_{j}(y_{j}(s))|\text {d}s+|u_{i}(t)|\bigg ] \\&+e^{\alpha t}[\sum _{j=1}^{m}(\alpha -a_{j}^{1})|y_{j}(t)|+\displaystyle \sum _{i=1}^{n}|b_{ji}^{1}||{\tilde{g}}_{i}(x_{i}(t))| \\&+\displaystyle \sum _{i=1}^{n}|c_{ji}^{1}||{\tilde{g}}_{i}(x_{i}(t-\sigma (t)))| \\&+\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{k=1}^{n}|W_{jik}^{1}||{\tilde{g}}_{k}(x_{k}(t))||{\tilde{g}}_{i}(y_{i}(t))| \\&+\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{k=1}^{n}|T_{jik}^{1}||{\tilde{g}}_{k}(x_{k}(t-\sigma (t)))||{\tilde{g}}_{i}(x_{i}(t-\sigma (t)))| \\&+\displaystyle \sum _{i=1}^{n}|p_{ji}^{1}|\int _{-\infty }^{t}|K_{ji}(t-s)||{\tilde{g}}_{i}(x_{i}(s))|\text {d}s+|v_{j}(t)| \\\le & e^{\alpha t}\bigg [\sum _{=1}^{n}(\alpha -a_{i})|x_{i}(t)|+\bigg (\displaystyle \sum _{j=1}^{n}|b_{ij}| \\&+\displaystyle \sum _{j=1}^{m}|c_{ij}|+\displaystyle \sum _{j=1}^{m}|p_{ij}| \\&+\big \{\displaystyle \sum _{j=1}^{m}\displaystyle \sum _{k=1}^{m}|W_{ijk}|+\displaystyle \sum _{j=1}^{m}\displaystyle \sum _{k=1}^{m}|T_{ijk}|\big \}l_{j}^{f}\bigg )l_{j}^{f}+|u_{i}(t)|\bigg ] \\&+e^{\alpha t}\bigg [\sum _{j=1}^{m}(\alpha -a_{j}^{1})|y_{j}(t)| \\&+\bigg (\displaystyle \sum _{i=1}^{n}|b_{ji}^{1}|+\displaystyle \sum _{i=1}^{n}|c_{ji}^{1}|+\displaystyle \sum _{i=1}^{n}|p_{ji}^{1}| +\big \{\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{k=1}^{n}|W_{jik}^{1}| \\&+\displaystyle \sum _{i=1}^{n} \displaystyle \sum _{k=1}^{n}|T_{jik}^{1}|\big \}l^{g}_{i}\bigg \}l^{g}_{i}+|v_{j}(t)|\bigg ]<0 \end{aligned}$$

(10)

when $(x^{T}(t),\;y^{T}(t))^{T}\in {\mathbb {R}}^{m+n}{\setminus} {\tilde{\varUpsilon }}_{2}$. Intergrating two sides of inequality (10) between 0 and $t>0$, it follows that:

$$\begin{aligned} |x(t)|+|y(t)|<e^{-\alpha t}(|x(0)|+|y(0)|). \end{aligned}$$

Thus, it is concluded that ${\tilde{\varUpsilon }}_{2}$ is a globally exponentially attractive set and also the neural network model (1) is a globally exponentially dissipative system. $\square$

Appendix 4 (Proof of Theorem 3)

Proof

We choose the following Lyapunov–Krasovskii functional:

$$\begin{aligned} V(t)=V_{1}(t)+V_{2}(t) \end{aligned}$$

where:

$$\begin{aligned} V_{1}(t)&= 2\sum _{i=1}^{n}\int _{0}^{x_{i}(t)}g_{i}(s)\text {d}s+\sum _{i=1}^{n}\int _{t-\sigma _{i}(t)}^{t}g_{i}^{2}(x_{i}(s))\text {d}s\\&+\sum _{j=1}^{m}\sum _{i=1}^{n}|p_{ji}|\int _{0}^{+\infty }K_{ji}(s)\int _{t-s}^{t}g_{i}^{2}(x_{i}(s_{1}))\text {d}s_{1}\text {d}s\\&+\frac{\varepsilon _{4}}{1-\sigma }\int _{t-\sigma (t)}^{t}g^{T}(x(s))T^{1}(T^{1})^{T}g(x(s))\text {d}s\\ V_{2}(t)&= 2\sum _{j=1}^{m}\int _{0}^{y_{j}(t)}f_{j}(s)\text {d}s+\sum _{j=1}^{m}\int _{t-\tau _{j}(t)}^{t}f_{j}^{2}(y_{j}(s))\text {d}s\\&+\sum _{i=1}^{n}\sum _{j=1}^{m}|p^{1}_{ij}|\int _{0}^{+\infty }K_{ij}(s)\int _{t-s}^{t}f_{j}^{2}(y_{j}(s_{1}))\text {d}s_{1}\text {d}s\\&+\frac{\varepsilon _{2}}{1-\tau }\int _{t-\tau (t)}^{t}f^{T}(y(s))TT^{T}f(y(s))\text {d}s \end{aligned}$$

Calculating the time derivative of $V(\cdot )$ along any trajectory of system (1), it yields that:

$$\begin{aligned} D^{+}V_{1}(t)\le & 2g^{T}(x(t))\big [-Ax(t)+(B+\varLambda ^{T}W)f(y(t))\\&+(C+\varGamma ^{T}T)f(y(t-\tau (t)))\\&+P\displaystyle \int _{-\infty }^{t}K(t-s)f(y(s))\text {d}s+U(t)\big ]\\&+g^{2}(x(t))-(1-\sigma )g^{2}(x(t-\sigma (t)))\\&+\displaystyle \sum _{j=1}^{m}\sum _{i=1}^{n}|p_{ji}|\displaystyle \int _{0}^{+\infty }K_{ji}(s)[g_{i}^{2}(x_{i}(t))-g_{i}^{2}(x_{i}(t-s))]\text {d}s\\&+\frac{\varepsilon _{4}}{1-\sigma }g^{T}(x(t))(T^{1})^{T}T^{1}g(x(t))\\&-\varepsilon _{4}g^{T}(x(t-\sigma (t)))T^{1}(T^{1})^{T}g(x(t-\sigma (t)))\\\le & -2\frac{A}{L^{g}}g^{2}(x(t))+2g^{T}(x(t))(B+\varLambda ^{T}W)f(y(t))\\&+2g^{T}(x(t))(C+\varGamma ^{T}T)f(y(t-\tau (t)))\\&+2g^{T}(x(t))P\displaystyle \int _{-\infty }^{t}K(t-s)f(y(s))\text {d}s\\&+2|g(x(t))||U(t)|+g^{2}(x(t))\\&-(1-\sigma )g^{2}(x(t-\sigma (t)))\\&+\displaystyle \sum _{j=1}^{m}\sum _{i=1}^{n}|p_{ji}|\displaystyle \int _{0}^{+\infty }K_{ji}(s)[g_{i}^{2}(y_{i}(t))\\&-g_{i}^{2}(y_{i}(t-s))]\text {d}s+\frac{\varepsilon _{4}}{1-\sigma }g^{T}(x(t))T^{1}(T^{1})^{T}g(x(t))\\&-\varepsilon _{4}g^{T}(x(t-\sigma (t)))T^{1}(T^{1})^{T}g(x(t-\sigma (t)))\\ D^{+}V_{2}(t)\le & 2f(y(t))\bigg [-A^{1}y(t)+(B^{1}+(\varLambda ^{1})^{T}W^{1})g(x(t))\\&+(C^{1}+(\varGamma ^{1})^{T}T^{1})g(x(t-\sigma (t)))\\&+P^{1}\displaystyle \int _{-\infty }^{t}K(t-s)g(x(s))\text {d}s+{\check{V}}(t)\bigg ]\\&+f^{2}(y(t))-(1-\tau )f^{2}(y(t-\tau (t)))\\&+\sum _{i=1}^{n}\sum _{j=1}^{m}|p^{1}_{ij}|\int _{0}^{+\infty }K_{ij}(s)[f_{j}^{2}(y_{j}(t))-f_{j}^{2}(y_{j}(t-s))]\text {d}s\\&+\frac{\varepsilon _{2}}{1-\tau }f^{T}(y(t))TT^{T}f(y(t))\\&-\varepsilon _{2}f^{T}(y(t-\tau (t)))TT^{T}f(y(t-\tau (t)))\\\le & -2\frac{A^{1}}{L^{f}}f^{2}(y(t))+2f^{T}(y(t))(B^{1}+(\varLambda ^{1})^{T}W^{1})g(x(t))\\&+2f^{T}(y(t))(C^{1}+(\varGamma ^{1})^{T}T^{1})g(x(t-\sigma (t)))\\&+2f^{T}(y(t))P^{1}\displaystyle \int _{-\infty }^{t}K(t-s)g(x(s))\text {d}s\\&+2|f(y(t))||{\check{V}}(t)|+f^{2}(y(t))-(1-\tau )f^{2}(y(t-\tau (t)))\\&+\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{m}|p^{1}_{ij}|\displaystyle \int _{0}^{+\infty }K_{ij}(s)[f_{j}^{2}(y_{j}(t))-f_{j}^{2}(y_{j}(t-s))]\text {d}s\\&+\frac{\varepsilon _{2}}{1-\tau }f^{T}(y(t))TT^{T}f(y(t))\\&-\varepsilon _{2}f^{T}(y(t-\tau (t)))TT^{T}f(y(t-\tau (t))) \end{aligned}$$

Applying Lemma1 to the above inequality,

$$\begin{aligned}&2g^{T}(x(t))(B+B^{1})f(y(t))\le g^{T}(x(t))(B+B^{1})Q^{-1}(B+B^{1})^{T}g(x(t))\\&\quad +f^{T}(y(t))Qf(y(t))\\&\quad 2g^{T}(x(t))Cf(y(t-\tau (t)))\le \frac{1}{1-\tau }g^{T}(x(t))CC^{T}g(x(t))\\&\quad +(1-\tau )f^{T}(y(t-\tau (t)))f(y(t-\tau (t)))\\&\quad 2f^{T}(y(t))C^{1}g(x(t-\sigma (t)))\le \frac{1}{1-\sigma }f^{T}(y(t))C^{1}(C^{1})^{T}f(y(t))\\&\quad +(1-\sigma )g^{T}(x(t-\sigma (t)))g(x(t-\sigma (t))) \end{aligned}$$

It can be verified that (for more details see [46]):

$$\begin{aligned}&\varGamma ^{T}\varGamma =\zeta ^{T} \zeta I=\displaystyle \sum _{i=1}^{n}(\zeta _{i})^{2}I\le \displaystyle \sum _{i=1}^{n}\delta _{0i}^{2}I=\delta _{0}I,\\&\varLambda ^{T}\varLambda =\xi ^{T} \xi I=\displaystyle \sum _{i=1}^{n}(\xi _{i})^{2}I\le \displaystyle \sum _{i=1}^{n}\delta _{1i}^{2}I=\delta _{1}I,\\&(\varGamma ^{1})^{T}\varGamma ^{1}=(\zeta ^{1})^{T}\zeta ^{1} I=\displaystyle \sum _{i=1}^{n}(\zeta ^{1}_{i})^{2}I\le \displaystyle \sum _{i=1}^{n}\delta _{2i}^{2}I=\delta _{2}I,\\&(\varLambda ^{1})^{T}\varLambda ^{1}=(\xi ^{1})^{T}\xi ^{1} I=\displaystyle \sum _{i=1}^{n}(\xi ^{1}_{i})^{2}I\le \displaystyle \sum _{i=1}^{n}\delta _{3i}^{2}I=\delta _{3}I \end{aligned}$$

Therefore, by Lemma 2, we have:

$$\begin{aligned}&2g^{T}(x(t))\varLambda ^{T}Wf(y(t))\le \varepsilon _{1}^{-1}g^{T}(x(t))\varLambda ^{T}\varLambda g(x(t))\\&\qquad +\varepsilon _{1}f^{T}(y(t))TT^{T}f(y(t))\\&\quad \le \delta _{1}\varepsilon _{1}^{-1}g^{T}(x(t))g(x(t))\\&\qquad +\varepsilon _{1}f^{T}(y(t))TT^{T}f(y(t))\\&2g^{T}(x(t))\varGamma ^{T}Tf(y(t-\tau (t)))\le \varepsilon _{2}^{-1}g^{T}(x(t))\varGamma ^{T}\varGamma g(x(t))\\&\qquad +\varepsilon _{2}f^{T}(y(t-\tau (t)))TT^{T}f(y(t-\tau (t)))\\&\quad \le \delta _{0}\varepsilon _{2}^{-1}g^{T}(x(t))g(x(t))\\&\qquad +\varepsilon _{2}f^{T}(y(t-\tau (t)))TT^{T}f(y(t-\tau (t)))\\&2f^{T}(y(t))(\varLambda ^{1})^{T}W^{1}g(x(t))\le \varepsilon _{3}^{-1}f^{T}(y(t))(\varLambda ^{1})^{T}\varLambda ^{1} f^{T}(y(t))\\&\qquad +\varepsilon _{3}g^{T}(x(t))W^{1}(W^{1})^{T}g(x(t))\\&\quad \le \delta _{3}\varepsilon _{3}^{-1}f^{T}(y(t))f^{T}(y(t))\\&\qquad +\varepsilon _{3}g^{T}(x(t))W^{1}(W^{1})^{T}g(x(t))\\&2f^{T}(y(t))(\varGamma ^{1})^{T}T^{1})g(x(t-\sigma (t)))\le \varepsilon _{4}^{-1}f^{T}(y(t))(\varGamma ^{1})^{T}\varGamma ^{1} f^{T}(y(t))\\&\qquad +\varepsilon _{4}g^{T}(x(t-\sigma (t)))T^{1}(T^{1})^{T}g(x(t-\sigma (t)))\\&\quad \le \delta _{2}\varepsilon _{4}^{-1}f^{T}(y(t))f^{T}(y(t))\\&\qquad +\varepsilon _{4}g^{T}(x(t-\sigma (t)))T^{1}(T^{1})^{T}g(x(t-\sigma (t)))\\&D^{+}V(t)\le -2\frac{A}{L^{g}}g^{2}(x(t))\\&\qquad +g^{T}(x(t))(B+B^{1})Q^{-1}(B+B^{1})^{T}g(x(t))\\&\qquad +f^{T}(y(t))Qf(y(t))+\delta _{0}\varepsilon _{2}^{-1}g^{T}(x(t))g(x(t))\\&\qquad +\varepsilon _{2}f^{T}(y(t-\tau (t)))T^{T}Tf(y(t-\tau (t)))\\&\qquad +\frac{1}{1-\tau }g^{T}(x(t))CC^{T}g(x(t))\\&\qquad +(1-\tau )f^{T}(y(t-\tau (t)))f(y(t-\tau (t)))\\&\qquad +\delta _{1}\varepsilon _{1}^{-1}g^{T}(x(t))g(x(t))+\varepsilon _{1}f^{T}(y(t))TT^{T}f(y(t))\\&\qquad +2g^{T}(x(t))P\displaystyle \int _{-\infty }^{t}K(t-s)f(y(s))\text {d}s\\&\qquad +2|g(x(t))||U(t)|+g^{2}(x(t))-(1-\sigma )g^{2}(x(t-\sigma (t)))\\&\qquad +\displaystyle \sum _{j=1}^{m}\sum _{i=1}^{n}|p_{ji}|\displaystyle \int _{0}^{+\infty }K_{ji}(s)[g_{i}^{2}(x_{i}(t))-g_{i}^{2}(x_{i}(t-s))]\text {d}s\\&\qquad -2\frac{A^{1}}{L^{f}}f^{2}(y(t))+\delta _{3}\varepsilon _{3}^{-1}f^{T}(y(t))f^{T}(y(t))\\&\qquad +\varepsilon _{3}g^{T}(x(t))W^{1}(W^{1})^{T}g(x(t))\\&\qquad +\frac{1}{1-\sigma }f^{T}(y(t))C^{1}(C^{1})^{T}f(y(t))\\&\qquad +(1-\sigma )g^{T}(x(t-\sigma (t)))g(x(t-\sigma (t)))\\&\qquad +\delta _{2}\varepsilon _{4}^{-1}f^{T}(y(t))f(y(t))\\&\qquad +\varepsilon _{4}g^{T}(x(t-\sigma (t)))T^{1}(T^{1})^{T}g(x(t-\sigma (t)))\\&\qquad +2f^{T}(y(t))P^{1}\displaystyle \int _{-\infty }^{t}K(t-s)g(x(s))\text {d}s\\&\qquad +\frac{\varepsilon _{4}}{1-\sigma }g^{T}(x(t))T^{1}(T^{1})^{T}g(x(t))\\&\qquad -\varepsilon _{4}g^{T}(x(t-\sigma (t)))T^{1}(T^{1})^{T}g(x(t-\sigma (t)))\\&\qquad +2|f(y(t))||{\check{V}}(t)|+f^{2}(y(t))-(1-\tau )f^{2}(y(t-\tau (t)))\\&\qquad +\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{m}|p^{1}_{ij}|\displaystyle \int _{0}^{+\infty }K_{ij}(s)[f_{j}^{2}(y_{j}(t))-f_{j}^{2}(y_{j}(t-s))]\text {d}s\\&\qquad +\frac{\varepsilon _{4}}{1-\sigma }g^{T}(x(t))T^{1}(T^{1})^{T}g(x(t))\\&\qquad -\varepsilon _{4}g^{T}(x(t-\sigma (t)))T^{1}(T^{1})^{T}g(x(t-\sigma (t)))\\&\quad =-\,2\frac{A}{L^{g}}g^{2}(x(t))+g^{T}(x(t))\bigg [I+(B+B^{1})Q^{-1}(B+B^{1})^{T}\\&\qquad +\delta _{0}\varepsilon _{2}^{-1}+\frac{1}{1-\tau }CC^{T}+\varepsilon _{3}W^{1}(W^{1})^{T}\\&\qquad +\delta _{1}\varepsilon _{1}^{-1}+\frac{\varepsilon _{4}}{1-\sigma }T^{1}(T^{1})^{T}\bigg ]g(x(t))\\&\qquad +2g^{T}(x(t))P\displaystyle \int _{-\infty }^{t}K(t-s)f(y(s))\text {d}s+2|g(x(t))||U(t)|\\&\qquad +\sum _{j=1}^{m}\sum _{i=1}^{n}|p_{ji}|\int _{0}^{+\infty }K_{ji}(s)[g_{i}^{2}(x_{i}(t))-g_{i}^{2}(x_{i}(t-s))]\text {d}s\\&\qquad -2\frac{A^{1}}{L^{f}}f^{2}(y(t))+f^{T}(y(t))\bigg [Q+I+\varepsilon _{1}TT^{T}\\&\qquad +\delta _{3}\varepsilon _{3}^{-1}C^{1}(C^{1})^{T}+\frac{1}{1-\sigma }C^{1}(C^{1})^{T}\\&\qquad +\delta _{2}\varepsilon _{4}^{-1}+\frac{\varepsilon _{2}}{1-\tau }TT^{T}\bigg ]f(y(t))\\&\qquad +2f^{T}(y(t))P^{1}\displaystyle \int _{-\infty }^{t}K(t-s)g(x(s))\text {d}s+2|f(y(t))||{\check{V}}(t)|\\&\qquad +\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{m}|p^{1}_{ij}|\displaystyle \int _{0}^{+\infty }K_{ij}(s)[f_{j}^{2}(y_{j}(t))-f_{j}^{2}(y_{j}(t-s))]\text {d}s \end{aligned}$$

By the inequality $2ab\le a^{2}+b^{2}$ for any $a,\;b\in {\mathbb {R}},$ we obtain:

$$\begin{aligned}&D^{+}V(t)=D^{+}V_{1}(t)+D^{+}V_{2}(t)\\&\quad \le -2\frac{A(\xi _{1}+\xi _{2})}{L^{g}}g^{2}(x(t))+g^{T}(x(t))\bigg [(1+\Vert P\Vert _{1}\\&\qquad +\Vert P^{1}\Vert _{\infty })I+(B+B^{1})Q^{-1}(B+B^{1})^{T}+\delta _{0}\varepsilon _{2}^{-1}\\&\qquad +\frac{1}{1-\tau }CC^{T}+\varepsilon _{3}W^{1}(W^{1})^{T}\\&\qquad +\delta _{1}\varepsilon _{1}^{-1}+\frac{\varepsilon _{4}}{1-\sigma }T^{1}(T^{1})^{T}\bigg ]g(x(t))+2|g(x(t))||U(t)|\\&\qquad -2\frac{A^{1}(\zeta _{1}+\zeta _{2})}{L^{f}}f^{2}(y(t))+f^{T}(y(t))\bigg [Q+(1+\Vert P^{1}\Vert _{1}\\&\qquad +\Vert P\Vert _{\infty })I+\varepsilon _{1}TT^{T}+\delta _{3}\varepsilon _{3}^{-1}C^{1}(C^{1})^{T}\\&\qquad +\frac{1}{1-\sigma }C^{1}(C^{1})^{T}+\delta _{2}\varepsilon _{4}^{-1}\\&\qquad +\frac{\varepsilon _{2}}{1-\tau }TT^{T}\bigg ]f(y(t))+2|f(y(t))||{\check{V}}(t)|\\&\quad \le g^{T}(x(t))\chi g(x(t))-2\frac{A\xi _{1}}{L^{g}}g(x(t))(g(x(t))\\&\qquad -\frac{L^{g}|u_i(t)|}{A\xi _{1}})+f^{T}(y(t)){\bar{\chi }}f(y(t))\\&\qquad -2\frac{A^{1}\zeta _{1}}{L^{f}}f(y(t))(f(y(t))-\frac{L^{f}|{\check{V}}(t)|}{\zeta _{1}A^{1}}). \end{aligned}$$

where

$$\begin{aligned}&\chi =N+(1+\Vert P\Vert _{1}+\Vert P^{1}\Vert _{\infty })I\\&\qquad +(B+B^{1})Q^{-1}(B+B^{1})^{T}+\delta _{0}\varepsilon _{2}^{-1}+\frac{1}{1-\tau }CC^{T}\\&\qquad +\varepsilon _{3}W^{1}(W^{1})^{T}+\delta _{1}\varepsilon _{1}^{-1}+\frac{\varepsilon _{4}}{1-\sigma }T^{1}(T^{1})^{T},\\&{\bar{\chi }}={\bar{N}}+Q+(1+\Vert P^{1}\Vert _{1}+\Vert P\Vert _{\infty })I+\varepsilon _{1}TT^{T}\\&\qquad +\delta _{3}\varepsilon _{3}^{-1}C^{1}(C^{1})^{T}+\frac{1}{1-\sigma }C^{1}(C^{1})^{T}\\&\qquad +\delta _{2}\varepsilon _{4}^{-1}+\frac{\varepsilon _{2}}{1-\tau }TT^{T} \end{aligned}$$

With the use of Lemma (3) and Eq. (4), we get:

$$\begin{aligned} \chi<0,\;{\bar{\chi }}<0 \end{aligned}$$

(11)

In accordance with Eqs. (4) and (11), we obtain:

$$\begin{aligned}&D^{+}V(t)\le -2\frac{A\xi _{1}}{L^{g}}g^{T}(x(t))\bigg (g(x(t))-\frac{L^{g}|U(t)|}{A\xi _{1}}\bigg ) \\&\quad -2\frac{A^{1}\zeta _{1}}{L^{f}}f^{T}(y(t))\bigg (f(y(t))-\frac{L^{f}|{\check{V}}(t)|}{A^{1}\zeta _{1}}\bigg )<0. \end{aligned}$$

(12)

When $(x^{T}(t),\;y^{T}(t))\in {\mathbb {R}}^{n+m}\setminus \varUpsilon _{3}$, it follows that Eq. (12) concludes that the neural network model (1) is a dissipative system, and $\varUpsilon _{3}$ is a positive invariant and globally attractive set of (1). $\square$