Synchronization analysis of novel delayed dynamical Clifford-valued neural networks on timescales

Abstract

The aims of this research, is to examine the qualitative properties and analysis of synchronization of pseudo almost automorphic in Weyl’s sense solution of delayed Clifford-valued cellular neural networks of type shunting inhibitory on timescales. The results achieved are articulated as follows: the notion of pseudo almost automorphic in Weyl’s sense on timescales is suggested, which is a natural generalization of some recently published papers. The qualitative properties of the solution are realized using appropriate inequalities. Therefore, fixed-time synchronization was realized by considering appropriate feedback and Lyapunov functional. Two numerical examples to prove the merits of key results are presented at the end of this work.

Introduction

The synchronization of two dynamical neural networks (NNs) is one more important mathematical problem due to its interesting place in the real world application (see Aliabadi et al., ¹ Chien and Liao, ² Feki, ³ Gonzàlez-Zapata et al., ⁴ and Yu and Cao ⁵ ). Besides, the Clifford algebra is an algebraic structure that generalizes the notion of complex number and quaternion. The study of Clifford’s algebras is closely related to the theory of quadratic form, and it has important applications in geometry and theoretical physics. Their name is derived from that of the mathematician William Kingdon Clifford that introduced since 1878. The Clifford algebra on ${\mathbb{R}}^n$ is defined as

$$\mathbb{A}=\{\sum _{B\subset \{1,2,\dots ,\chi \}}{b}^B{e}_B:b^B\in \mathbb{R}\}$$

where $e_B=e_{k_1}{e}_{k_2}\cdots e_{k_v}$ , $B=k_1{k}_2\cdots k_v$ , $1\le k_1<k_2<$ $\cdot s<k_v\le \chi$ . Besides, $e_{\mathrm{\varnothing }}=e_0=1$ and $e_{k_1},e_{k_2},\dots ,e_{k_v}$ are labeled Clifford-generators and verify $e_{ij}^2+1=0$ , and $e_{ij}{e}_j+e_j{e}_{ij}=0,i\ne j$ for all $i,j=1,\dots ,\chi$ . We always assume that the product of Clifford algebra afterwards without any further comments: $e_{k_1}{e}_{k_2}\cdots e_{k_v}=e_{k_1{k}_2\cdots k_v}$ , $\mathrm{\Theta }=\{\mathrm{\varnothing },1,2,\dots ,B,\dots ,12,\dots ,\chi \}$ , then

$$\mathbb{A}=\{\sum _{B\in \mathrm{\Theta }}{b}^B{e}_B:B^B\in \mathbb{R}\}$$

For $x\in \sum _B{x}^B{e}_B\in \mathbb{A}$ , and $x{=}^t(x_1,\dots ,x_n)\in {\mathbb{A}}^n$ , we define

$$\Vert x{\Vert }_{\mathbb{A}}=\underset{B\in \mathrm{\Theta }}{max}\{|x^B|\},\Vert x{\Vert }_{{\mathbb{A}}^n}=\underset{1\le p\le n}{max}\{\Vert x_p{\Vert }_{\mathbb{A}}\}$$

The real-valued NNs^{6,10,11,9,8,7} or complex-valued NNs^12,13 and quaternion-valued NNs^14–22 are special cases of Clifford-valued NNs.

From a first part, in the environment of timescales,^23–25 the concepts of almost periodicity/automorphicity^8,7,26–30 are extended and in many ways: pseudo almost periodicity/automorphicity,^31–33 Stepanov-like pseudo almost periodicity/automorphicity^34–36 weighted pseudo almost periodicity/automorphicity,^37–40 Stepanov almost periodicity/automorphicity,^9,41 and Weyl almost periodicity/automorphicity.^42–44 From a second part, the concept of Weyl almost automophicity is not.

In this article, we study the following network:

$$\begin{array}{r}\{\begin{array}{cc}{z}_{ij}^{△}(t)& =-d_{ij}(t)z_{ij}(t)-\sum _{c_{kl}\in N_r(i,j)}^{}{b}_{ij}^{kl}(t)f_{ij}(z_{kl}(t))z_{ij}(t)\\ & -\sum _{c_{kl}\in N_s(i,j)}^{}{c}_{ij}^{kl}(t)g_{ij}(z_{kl}(t-{\beta }_{kl}(t)))z_{ij}(t)+D_{ij}(t)\end{array}\end{array}$$

(1)

where $ij\in \mathrm{\Gamma }:=\{11,\dots ,1n,\dots ,m1,\dots ,mn\}$ , $d_{ij}(.),$ $b_{ij}^{kl}(.),c_{ij}^{kl}(.),z_{ij}(.),D_{ij}(.)\in \mathbb{Q}(Quaternions)$ , ${\beta }_{kl}(.)\in {\mathbb{T}}^{+}$ satisfy $t-{\beta }_{kl}(.)\in \mathbb{T}$ for every $t\in \mathbb{T}$ and

$$\begin{array}{rl}{N}_r(i,j):=& \{c_{pq}:max(|p-i|,|q-j|)\le r,1\le p\le m,1\\ & \le q\le n\}\end{array}$$

Throughout this article, we note:

$d_{ij}(t)=\sum _B{d}_{ij}^B{e}_B\in \mathbb{A}$ , $d_{ij}^{*}(t)=\sum _{B\ne \mathrm{\varnothing }}{d}_{ij}^B{e}_B\in \mathbb{A}$ and $d_{ij}^R(t)=d_{ij}(t)-d_{ij}^{*}(t)$

$$\begin{array}{rl}& d^{-}=\underset{ij\in \mathrm{\Gamma }}{min}\{\underset{s\in \mathbb{T}}{inf}\{d_{ij}^R(s)\}\},\\ & d^{+}=\underset{ij\in \mathrm{\Gamma }}{max}\{\underset{s\in \mathbb{T}}{sup}\{d_{ij}^R(s)\}\},{\tilde{d}}^{+}=\underset{s\in \mathbb{T}}{sup}\{\Vert d_{ij}^R(s){\Vert }_{\mathbb{A}}\}\\ & b_{ij}^{k{l}^{+}}=\underset{s\in \mathbb{T}}{sup}\{b_{ij}^{kl}(s)\},c_{ij}^{k{l}^{+}}=\underset{s\in \mathbb{T}}{sup}\{c_{ij}^{kl}(s)\},\\ & {\beta }_{kl}^{+}=\underset{s\in \mathbb{T}}{sup}\{{\beta }_{kl}(s)\},{\beta }_{kl}^{\prime }=\underset{ij\in \mathrm{\Gamma }}{max}\{\underset{s\in \mathbb{T}}{sup}\{{\beta }_{kl}^{△}(s)\}\}\end{array}$$

The initial value problem of network (1) is given by

$$z_{ij}(s)={\psi }_{ij}(s),s\in [-{\beta }_{kl}^{+},0{]}_{\mathbb{T}}$$

where ${\psi }_{ij}\in C_{rd}(\mathbb{T},\mathcal{A})$ for every $ij\in \mathrm{\Gamma }$ .

The major inputs of this work are:

The notion of pseudo almost automorphic in Weyl’s sense on timescales is created, which is a natural generalization of Arbi and Tahri ¹⁴ and Li and Huang.^45,46

The qualitative properties of the network (1).

Fixed-time synchronization was achieved, with adequate feedback and a Lyapunov condidate function.

This article is structured in the following manner: In the “Preliminaries” section, the notion of pseudo almost automorphic in Weyl’s sense on timescales and the notion of synchronization are presented. In the “Main results” section, we suggest the exponential stability of system (1) and fixed-time synchronization of the control system associated of network (1). In the “Illustrative example” section, an illustrative example is presented. A short conclusion in the “Conclusion and futur problem” section.

Preliminaries

In this article, $\mathbb{E}$ equipped with the norm $\Vert .{\Vert }_{\mathbb{E}}$ is a Banach space, $\mathcal{B}\mathcal{C}(\mathbb{T},\mathbb{E})$ is the set of the continuous and bounded functions. Throughout this article, $\mathbb{T}$ is stable under translations.

Definition 1

Let $p\in [1,+\mathrm{\infty })$ . For $g\in L_{loc}^p(\mathbb{T},\mathbb{E})$ , we define the following semi-norm

$$\Vert g{\Vert }_{W^p}:=\underset{r\to +\mathrm{\infty }}{lim}\underset{\alpha \in \mathbb{T}}{sup}{\{\frac{1}{r}{\int }_{\alpha }^{\alpha +r}\Vert g{\Vert }_{\mathbb{E}}^p△\mathrm{t}\}}^{\frac{1}{p}}$$

Definition 2

A function $f\in L_{loc}^p(\mathbb{T},\mathbb{E})$ is called $W^p$ -almost automorphic if for each sequence $(s_n{)}_{n\in \mathbb{N}}\subset \mathrm{\Pi }$ , we can extract a subsequence $(s_{n_k}{)}_{k\in \mathbb{N}}$ and discover a function $\overline{g}\in L_{loc}^p(\mathbb{T},\mathbb{E})$ such that

$$\Vert f(t+s_{n_k})-\overline{g}(t){\Vert }_{W^p}⟶0,\Vert \overline{g}(t-s_{n_k})-f(t){\Vert }_{W^p}⟶0,\text{ as }k⟶+\mathrm{\infty }$$

Denote by $W^{p}AA(\mathbb{T},\mathbb{E})$ the set of all such functions.

Remark 1

$AA(\mathbb{T},\mathbb{E})\subset W^{p}AA(\mathbb{T},\mathbb{E})$ .

Definition 3

Let $h\in \mathcal{B}\mathcal{C}(\mathbb{T},\mathbb{E})$ . $h$ is called $W^p$ -ergodic if

$$\underset{r\to +\mathrm{\infty }}{lim}\frac{1}{2r}{\int }_{-r}^r{\{\underset{l\to +\mathrm{\infty }}{lim}\frac{1}{l}{\int }_{s-l}^{s+l}\Vert h(t){\Vert }^p△\mathrm{t}\}}^{\frac{1}{p}}△\mathrm{s}=0$$

(2)

The set of all functions denoted by $W^p\mathcal{E}(\mathbb{T},\mathbb{E})$ .

Definition 4

A function $f\in L_{loc}^p(\mathbb{T},\mathbb{E})$ is said $W^p$ -pseudo almost automorphic if it can be represented as

$$f=g+h$$

(3)

where $g\in W^{p}AA(\mathbb{T},\mathbb{E})$ and $h\in W^p\mathcal{E}(\mathbb{T},\mathbb{E})$ . Let $W^{p}PAA(\mathbb{T},\mathbb{E})$ the set of all such functions.

Remark 2

Unlike the almost automophic concept and their extensions like pseudo almost automophic and weighted pseudo almost automophic, the unique decomposition of the pseudo Weyl almost automophic in equation (3) functions is not guaranteed.

Assume $0_{{\mathbb{R}}^n}$ is an equilibrium point of network (1).

Definition 5

The equilibrium of network (1) is stable in a finite-time, if there exists $T_{x_0}>0$ (constant) satisfying ⁴⁷

$$\underset{t\to T_{x_0}}{lim}\Vert x_{ij}(t){\Vert }_{\mathbb{A}}=0,\text{ and }\Vert x_{ij}(t){\Vert }_{\mathbb{A}}=0,\mathrm{\forall }t>T_{x_0}$$

where $ij\in \mathrm{\Gamma }$ .

Definition 6

The equilibrium of network (1) is fixed-time stable, if satisfied ⁴⁷ :

The equilibrium of network (1) is finite-time stable

For any $x_0$ , there exists a fixed constant $T_{max}>0$ such that $T_{x_0}\le T_{max}$ .

Lemma 1

If $L:{\mathbb{A}}^n\to [0,+\mathrm{\infty })$ is continuous radially unbounded, and it verifies ⁴⁷

• *

$L(z)=0\iff z=0$ ,

*

For any nonzero solution $x(t)$ of network (1), there exists $\alpha ,\beta >0$ , $\xi \in [0,1)$ and $\eta >1$ satisfying

$$D^{+}L(x(t))\le -\alpha L^{\xi }(x(t))-\beta L^{\eta }(x(t))$$

then the equilibrium of network (1) is fixed-time stable, and

$$T_f^1=\frac{1}{\alpha (1-\xi )}+\frac{1}{\beta (\eta -1)}$$

Lemma 2

Let $a_1,a_2,\dots a_n\ge 0$ and $0<\alpha <\beta$ , then ⁴⁸

$${\left(\sum _{k=1}^n{a}_k^{\beta }\right)}^{\frac{1}{\beta }}\le {\left(\sum _{k=1}^n{a}_k^{\alpha }\right)}^{\frac{1}{\alpha }}$$

Main results

Throughout this article, $BCU(\mathbb{T},\mathbb{A})$ is the environment of bounded and uniformly continuous functions. $(BCU(\mathbb{T},{\mathbb{A}}^{mn}),\Vert .{\Vert }_{\mathrm{\infty }})$ is Banach space, where $\Vert z{\Vert }_{\mathrm{\infty }}=$ $\underset{t\in \mathbb{T}}{sup}\Vert z(t){\Vert }_{{\mathbb{A}}^{mn}}$ .

We assume the following four properties $({\mathcal{P}}_1)$ , $({\mathcal{P}}_2)$ , $({\mathcal{P}}_3)$ , and $({\mathcal{P}}_4)$ hold:

• $({\mathcal{P}}_1)$

For $ij\in \mathrm{\Gamma }$ , there exist $L_{ij}^f>0$ and $L_{ij}^g>0$ such that

$$\Vert f_{ij}(z)-f_{ij}(z^{*}){\Vert }_{\mathbb{A}}\le L_{ij}^f\Vert z-z^{*}{\Vert }_{\mathbb{A}},\Vert g_{ij}(z)-g_{ij}(z^{*}){\Vert }_{\mathbb{A}}\le L_{ij}^g\Vert z-z^{*}{\Vert }_{\mathbb{A}}$$

where $f_{ij}(0)=g_{ij}(0)=0$ for all $z,z^{*}\in \mathbb{A}$ .

$({\mathcal{P}}_2)$

For $ij\in \mathrm{\Gamma }$ , $a_{ij}^R\in AA(\mathbb{T},{\mathbb{R}}^{+})$ satisfy $-a_{ij}^R\in {\mathcal{R}}^{+}$ , ${\tilde{a}}_{ij}^{R^{+}}\in AA(\mathbb{T},\mathbb{A})$ , $b_{ij}^{kl},c_{ij}^{kl}\in AA(\mathbb{T},{\mathbb{R}}^{+})$ , $D_{ij}\in W^p$ $PAA(\mathbb{T},\mathbb{A})$ , ${\beta }_{kl}\in AA(\mathbb{T},{\mathbb{R}}^{+})\cap C_{rd}^1(\mathbb{T},\mathbb{T})$ , ${\beta }^{\prime }<1$ .

Denote ${\phi }^0=({\phi }_{11}^0,\dots ,{\phi }_{1n}^0,\dots ,{\phi }_{m1}^0,\dots ,{\phi }_{mn}^0)$ , where

$${\phi }_{ij}^0(t)={\int }_{-\mathrm{\infty }}^t{e}_{-d^{-}}(t,\sigma (s))D_{ij}(s)△\mathrm{s},ij\in \mathrm{\Gamma }$$

Using ${\phi }_{ij}^0$ is defined ⁴¹ Lemma 4.5.

Under the assumption $({\mathcal{P}}_2)$ , $D_{ij}\in W^{p}PAA(\mathbb{T},\mathbb{E})$ , we can choose $\alpha \ge \Vert {\phi }_{ij}^0(t){\Vert }_{\mathbb{A}}$ . Let

$$\mathrm{{\rm Y}}=\{\psi \in BCU(\mathbb{T},{\mathbb{A}}^{mn}):{\Vert \psi -{\phi }^0\Vert }_{\mathrm{\infty }}\le 2\alpha \}$$

So for each $\psi \in \mathrm{{\rm Y}}$ , we have

$${\Vert \psi \Vert }_{\mathrm{\infty }}\le {\Vert \psi -{\phi }^0\Vert }_{\mathrm{\infty }}+{\Vert {\phi }^0\Vert }_{\mathrm{\infty }}\le 2\alpha$$

• $({\mathcal{P}}_3)$
  $$K=\frac{2}{d^{-}}\underset{ij\in \mathrm{\Gamma }}{max}\{{\tilde{d}}_{ij}^{+}+\sum _{c_{kl}\in N_r(i,j)}^{}2\alpha b_{ij}^{k{l}^{+}}{L}_{ij}^f+\sum _{c_{kl}\in N_s(i,j)}^{}2\alpha c_{ij}^{k{l}^{+}}{L}_{ij}^g\}<1$$

• $({\mathcal{P}}_4)$

  First case $p>2$ :

  $$\begin{array}{rl}& \underset{ij\in \mathrm{\Gamma }}{max}\{24{\left({\displaystyle \frac{2p-4}{d^{-}p}}\right)}^{p-2}\left({\displaystyle \frac{4}{d^{-}p}}\right)\\ & [2{\left({\tilde{d}}_{ij}^{+}\right)}^p+2{\left(\sum _{c_{kl}\in N_r(i,j)}^{}2\alpha b_{ij}^{k{l}^{+}}{L}_{ij}^f\right)}^p\\ & +\left(1+{\displaystyle \frac{2e^{{\displaystyle \frac{p}{4}}}{a}^{+}{\beta }^{+}}{1-{\beta }^{\mathrm{\prime }}}}\right){\left(\sum _{c_{kl}\in N_s(i,j)}^{}2\alpha c_{ij}^{k{l}^{+}}{L}_{ij}^g\right)}^p]\}<1\end{array}$$
  Second case $p=2$ :
  $$\begin{array}{rl}& \underset{ij\in \mathrm{\Gamma }}{max}\{24{\left({\displaystyle \frac{2}{d^{-}}}\right)}^2[2{\left({\tilde{d}}_{ij}^{+}\right)}^2+2{\left(\sum _{c_{kl}\in N_r(i,j)}^{}2\alpha b_{ij}^{k{l}^{+}}{L}_{ij}^f\right)}^2\\ & +\left(1+{\displaystyle \frac{2e^{{\displaystyle \frac{1}{2}}}{d}^{+}{\beta }^{+}}{1-{\beta }^{\mathrm{\prime }}}}\right){\left(\sum _{c_{kl}\in N_s(i,j)}^{}2\alpha c_{ij}^{k{l}^{+}}{L}_{ij}^g\right)}^2]\}<1,\end{array}$$

Stability of $W^{p}PAA$ solution of (Clifford-valued shunting inhibitory cellular neural networks (CV-SICNNs)

Theorem 1

The equilibrium of network (1) is exponentially stable if properties $({\mathcal{P}}_1)-({\mathcal{P}}_3)$ are fulfilled.

Proof.

Let $k$ and $z^{*}$ be two solutions of system (1) with the initial value problem $\phi$ and $\psi$ , respectively. Denote

$$\mathrm{\forall }ij\in \mathrm{\Gamma },x_{ij}(t)=k_{ij}(t)-z_{ij}^{*}(t),{\mathrm{\Phi }}_{ij}(t)={\phi }_{ij}(t)-{\psi }_{ij}(t)$$

then we have

$$\{\begin{array}{c}{x}_{ij}^{\mathrm{△}}(t)=-a_{ij}(t)x_{ij}(t)-{\sum }_{c_{kl}\in N_r(i,j)}{b}_{ij}^{kl}(t)\\ \left[f_{ij}(k_{kl}(t))k_{ij}(t)-f_{ij}(k_{kl}^{*}(t))k_{ij}^{*}(t)\right]\\ -{\sum }_{c_{kl}\in N_r(i,j)}{c}_{ij}^{kl}(t)[g_{ij}(k_{kl}(t-{\beta }_{kl}(t)))k_{ij}(t)\\ -g_{ij}(k_{kl}^{*}(t-{\beta }_{kl}(t)))k_{ij}^{*}(t)],ij\in \mathrm{\Gamma }\end{array}$$

(4)

Let’s consider

$$\begin{array}{rl}& \mathrm{\forall }ij\in \mathrm{\Gamma },F_{ij}(\epsilon )=-\epsilon +a^{-}\\ & -\{{\tilde{a}}_{ij}^{+}+\sum _{c_{kl}\in N_r(i,j)}^{}4\alpha b_{ij}^{k{l}^{+}}{L}_{ij}^f+\sum _{c_{kl}\in N_s(i,j)}^{}2\alpha c_{ij}^{k{l}^{+}}{L}_{ij}^g\\ & +\sum _{c_{kl}\in N_s(i,j)}^{}2\alpha c_{ij}^{k{l}^{+}}{L}_{ij}^g{e}^{\epsilon {\beta }_{ij}^{+}}\}\end{array}$$

Using $({\mathcal{P}}_3)$ , we have

$$F_{ij}(0)=a^{-}-\{{\tilde{a}}_{ij}^{+}+\sum _{c_{kl}\in N_r(i,j)}^{}4\alpha b_{ij}^{k{l}^{+}}{L}_{ij}^f+\sum _{c_{kl}\in N_s(i,j)}^{}4\alpha c_{ij}^{k{l}^{+}}{L}_{ij}^g\}>0,ij\in \mathrm{\Gamma }$$

Through the continuity of $F_{ij}$ and

$$\underset{\epsilon \to +\mathrm{\infty }}{lim}{F}_{ij}(\epsilon )=-\mathrm{\infty }$$

which imposes the existence of ${\zeta }_{ij}>0$ such that $F_{ij}({\zeta }_{ij})=0$ and $F_{ij}(\epsilon )>0$ for $\epsilon \in (0,{\zeta }_{ij}),ij\in \mathrm{\Gamma }$ . Let $\zeta =\underset{ij\in \mathrm{\Gamma }}{min}\{{\zeta }_{ij}\}$ , then we have $F_{ij}(\zeta )>0,ij\in \mathrm{\Gamma }$ . We can pick $\lambda$ such that

$$0<\lambda <min\{\zeta ,a^{-}\}\mathrm{and}{F}_{ij}(\zeta )>0$$

Thus, one has

$$\begin{array}{rl}& \mathrm{\forall }ij\in \mathrm{\Gamma },{\displaystyle \frac{1}{a^{-}-\lambda }({\tilde{a}}_{ij}^{+}+\sum _{c_{kl}\in N_r(i,j)}4\alpha b_{ij}^{k{l}^{+}}{L}_{ij}^f+\sum _{c_{kl}\in N_s(i,j)}4\alpha c_{ij}^{k{l}^{+}}{L}_{ij}^g}\\ & +\sum _{c_{kl}\in N_s(i,j)}4\alpha c_{ij}^{k{l}^{+}}{L}_{ij}^g{e}^{\lambda {\beta }_{ij}^{+}})<1\end{array}$$

(5)

Choose

$$M=\underset{i\in \gamma }{max}\left\{a^{-}{({\tilde{a}}_{ij}^{+}+\sum _{c_{kl}\in N_r(i,j)}^{}4\alpha b_{ij}^{k{l}^{+}}{L}_{ij}^f+\sum _{c_{kl}\in N_s(i,j)}^{}4\alpha c_{ij}^{k{l}^{+}}{L}_{ij}^g)}^{-1}\right\}$$

Using $({\mathcal{P}}_3)$ , we have $M>1$ . Thus

$$\begin{array}{rl}& \mathrm{\forall }ij\in \mathrm{\Gamma }{\displaystyle \frac{1}{M}-{\displaystyle \frac{1}{a^{-}-\lambda }({\tilde{a}}_{ij}^{+}+\sum _{c_{kl}\in N_r(i,j)}4\alpha b_{ij}^{k{l}^{+}}{L}_{ij}^f}}\\ & +\sum _{c_{kl}\in N_s(i,j)}4\alpha c_{ij}^{k{l}^{+}}{L}_{ij}^g)<0\end{array}$$

(6)

From (4), we have

$$\{\begin{array}{c}{x}_{ij}^{\mathrm{△}}(t)+a_{ij}^R(t)x_{ij}(t)={\tilde{a}}_{ij}(t)x_{ij}(t)+{\sum }_{c_{kl}\in N_r(i,j)}{b}_{ij}^{kl}(t)\\ \left[f_{ij}(z_{kl}(t))z_{ij}(t)-f_{ij}(z_{kl}^{*}(t))z_{ij}^{*}(t)\right]\\ +{\sum }_{c_{kl}\in N_r(i,j)}{c}_{ij}^{kl}(t)[g_{ij}(z_{kl}(t-{\beta }_{kl}(t)))z_{ij}(t)\\ -g_{ij}(z_{kl}^{*}(t-{\beta }_{kl}(t)))z_{ij}^{*}(t)],ij\in \mathrm{\Gamma }.\end{array}$$

(7)

Multiplying equation (7) by $e_{-a_{ij}^R}(t,\sigma (s))$ , and integrating over $[t_0,t]$ , where $t_0\in [-{\beta }^{+},0)$ , we have

$$\{\begin{array}{c}{x}_{ij}(t)={\mathrm{\Phi }}_{ij}(0)e_{-a_{ij}^R}(t,\sigma (s))\\ +e_{-a_{ij}^R}(t,\sigma (s))\{{\tilde{a}}_{ij}(t)x_{ij}(t)+{\sum }_{c_{kl}\in N_r(i,j)}{b}_{ij}^{kl}(t)\\ \left[f_{ij}(z_{kl}(t))z_{ij}(t)-f_{ij}(z_{kl}^{*}(t))z_{ij}^{*}(t)\right].\\ +{\sum }_{c_{kl}\in N_r(i,j)}{c}_{ij}^{kl}(t)[g_{ij}(z_{kl}(t-{\beta }_{kl}(t)))z_{ij}(t)\\ -g_{ij}(z_{kl}^{*}(t-{\beta }_{kl}(t)))z_{ij}^{*}(t)]\}\mathrm{△}\mathrm{s},ij\in \mathrm{\Gamma }.\end{array}$$

(8)

Hence, for any $\delta >0$ , it is easy to see that

$$\mathrm{\forall }t\in ]-\beta ,0[,{\Vert x(t)\Vert }_{{\mathcal{A}}^{mn}}<[{\Vert \mathrm{\Phi }\Vert }_{\beta }+\delta ]e_{-\lambda }(t,s)<M[{\Vert \mathrm{\Phi }\Vert }_{\beta }+\delta ]e_{-\lambda }(t,s)$$

Let’s assume

$$\mathrm{\forall }t\in [0,+\mathrm{\infty }[,{\Vert x(t)\Vert }_{{\mathcal{A}}^{mn}}<M[{\Vert \mathrm{\Phi }\Vert }_{\beta }+\delta ]e_{-\lambda }(t,s)$$

(9)

If not, there exists $t^{*}>0$ such that

$${\Vert x(t^{*})\Vert }_{{\mathcal{A}}^{mn}}\ge M[{\Vert \mathrm{\Phi }\Vert }_{\beta }+\delta ]e_{-\lambda }(t^{*},0)$$

(10)

and

$${\Vert x(t)\Vert }_{{\mathcal{A}}^{mn}}<M[{\Vert \mathrm{\Phi }\Vert }_{\beta }+\delta ]e_{-\lambda }(t,s),t\in [t_0,t^{*}),t_0\in [\beta ,t^{*})$$

(11)

By means (6), (7), and (11), we see

$$\begin{array}{rl}& {\Vert x_{ij}(t)\Vert }_{\mathcal{A}}\le {\Vert {\mathrm{\Phi }}_{ij}(0)\Vert }_{\mathcal{A}}{e}_{-\lambda }(t,s)\\ & +{\int }_0^t{e}_{-\lambda }(t,s)\{{\tilde{a}}_{ij}(t){\Vert x_{ij}(t)\Vert }_{\mathcal{A}}+\sum _{c_{kl}\in N_r(i,j)}{b}_{ij}^{kl}(t)\\ & \left[{\Vert f_{ij}(z_{kl}(t))z_{ij}(t)\Vert }_{\mathcal{A}}+{\Vert f_{ij}(z_{kl}^{*}(t))z_{ij}^{*}(t)\Vert }_{\mathbb{H}}\right]+\sum _{c_{kl}\in N_r(i,j)}{c}_{ij}^{kl}(t)\\ & \left[{\Vert g_{ij}(z_{kl}(t-{\beta }_{kl}(t)))z_{ij}(t)\Vert }_{\mathcal{A}}{\Vert g_{ij}(z_{kl}^{*}(t-{\beta }_{kl}(t)))z_{ij}^{*}(t)\Vert }_{\mathcal{A}}\right]\}\mathrm{△}\mathrm{s}\\ & \le \left[{\Vert \mathrm{\Phi }\Vert }_{\beta }+\delta \right]e_{-\lambda }(t,s)+M\left[{\Vert \mathrm{\Phi }\Vert }_{\beta }+\delta \right]{\int }_0^t{e}_{-\lambda }(t,s)\\ & \{{\tilde{a}}_{ij}^{-}(t)+\sum _{c_{kl}\in N_r(i,j)}4\alpha b_{ij}^{k{l}^{+}}{L}_{ij}^f\\ & +\sum _{c_{kl}\in N_s(i,j)}4\alpha c_{ij}^{k{l}^{+}}{L}_{ij}^g+\sum _{c_{kl}\in N_s(i,j)}4\alpha c_{ij}^{k{l}^{+}}{L}_{ij}^g\}{e}^{\lambda {\beta }_{ij}^{+}}\mathrm{△}\mathrm{s}\\ & \le \left[{\Vert \mathrm{\Phi }\Vert }_{\beta }+\delta \right]e_{-\lambda }(t,s),ij\in \gamma \end{array}$$

which contradicts (10). Hence, (9) holds. Let’s tender $\delta \to 0^{+}$ , inequality (9) comes

$$\mathrm{\forall }t\in [0,+\mathrm{\infty }{[}_{\mathbb{T}},{\Vert x(t)\Vert }_{{\mathcal{A}}^{mn}}\le M{\Vert \mathrm{\Phi }\Vert }_{\beta }{e}_{-\lambda }(t,s)$$

It is the exponential stability of system (1).  □

Remark 3

1. We did not use hypothesis ${\mathcal{P}}_4$ in Proof of Theorem 1.

1. In fact, the usefulness of the hypothesis is to demonstrate the existence and uniqueness, as well as the nature of the solution, (it is a type solution).

2. Similar as proof method in Theorem 3.1 by Arbi and Tahri, ¹⁴ one can prove Theorem 1, under assumptions $({\mathcal{P}}_1)-({\mathcal{P}}_4)$ .

Synchronization of delayed (CV-SICNNs)

Consider a Clifford-valued network control system

$$\{\begin{array}{cc}& y_{ij}^{△}(t)=-d_{ij}(t)y_{ij}(t)-\sum _{c_{kl}\in N_r(i,j)}^{}{b}_{ij}^{kl}(t)f_{ij}(y_{kl}(t))y_{ij}(t)\\ & -\sum _{c_{kl}\in N_s(i,j)}^{}{c}_{ij}^{kl}(t)g_{ij}(y_{kl}(t-{\beta }_{kl}(t)))y_{ij}(t)+D_{ij}(t)+S_{ij}(t)\end{array}$$

(12)

where the state is $y_{ij}\in \mathbb{A}$ , the control is $S_{ij}\in \mathbb{A}$ and $ij\in \mathrm{\Gamma }$ .

The synchronization errors between system (1) and system (12) are defined by

$$x_{ij}(t)=z_{ij}(t)-y_{ij}(t),ij\in \mathrm{\Gamma }$$

Then the error system become

$$\{\begin{array}{cc}& x_{ij}^{△}(t)=-d_{ij}(t)x_{ij}(t)+\sum _{c_{kl}\in N_r(i,j)}^{}{b}_{ij}^{kl}(t)f_{ij}(y_{kl}(t))y_{ij}(t)\\ & -\sum _{c_{kl}\in N_r(i,j)}^{}{b}_{ij}^{kl}(t)f_{ij}(z_{kl}(t))z_{ij}(t)\\ & +\sum _{c_{kl}\in N_s(i,j)}^{}{c}_{ij}^{kl}(t)g_{ij}(y_{kl}(t-{\beta }_{kl}(t)))y_{ij}(t)\\ & -\sum _{c_{kl}\in N_s(i,j)}^{}{c}_{ij}^{kl}(t)g_{ij}(z_{kl}(t-{\beta }_{kl}(t)))z_{ij}(t)-S_{ij}(t)\end{array}$$

(13)

Let a feedback

$$\{\begin{array}{c}{S}_{ij}(t)={\sum }_{A\in \mathbb{A}}{S}_{ij}^A(t)e_A+{\sum }_{c_{kl}\in N_r(i,j)}{b}_{ij}^{kl}(t)f_{ij}(z_{kl}(t))z_{ij}(t)\\ -{\sum }_{c_{kl}\in N_r(i,j)}{b}_{ij}^{kl}(t)f_{ij}(z_{kl}(t))\\ +{\sum }_{c_{kl}\in N_s(i,j)}{c}_{ij}^{kl}(t)g_{ij}(y_{kl}(t-{\beta }_{kl}(t)))y_{ij}(t)\\ -{\sum }_{c_{kl}\in N_s(i,j)}{c}_{ij}^{kl}(t)g_{ij}(y_{kl}(t-{\beta }_{kl}(t)))\\ S_{ij}^A(t)=-M_{1ij}{x}_{ij}^A(t)-M_{2ij}{\left|z_{ij}^A(t)\right|}^{\alpha }sgn\left(x_{ij}^A(t)\right)\\ -M_{3ij}{\left|x_{ij}^A(t)\right|}^{\gamma }sgn\left(x_{ij}^A(t)\right)-M_{4ij}|\\ {\sum }_{ij\in \mathrm{\Gamma }}{x}_{ij}^A(t-{\beta }_{ij}(t))|sgn\left(x_{ij}^A(t)\right)\end{array}$$

(14)

where $ij\in \mathrm{\Gamma }$ , $A\in \mathbb{A}$ , $0<\alpha <1$ , $\gamma >1$ and $M_{1ij},M_{2ij},M_{3ij},\mathrm{and}{M}_{4ij}$ are the parameters that will be determined.

Theorem 2

Assume $({\mathcal{P}}_1)$ hold. If, in addition, $M_{1ij}$ , $M_{2ij}$ , $M_{3ij}$ , and $M_{4ij}$ of the feedback (14) satisfy

$$M_{1ij}\ge -d^{-}+\sum _{c_{kl}\in N_r(i,j)}^{}{b}_{ij}^{k{l}^{+}}{L}_{ij}^f$$

(15)

$$M_{2ij}>0$$

(16)

$$M_{3ij}>0$$

(17)

$$M_{4ij}\ge \sum _{c_{kl}\in N_s(i,j)}^{}{c}_{ij}^{k{l}^{+}}{L}_{ij}^g$$

(18)

for all $ij\in \mathrm{\Gamma }$ . By means of feedback (14), system (1), and system (12) can reach synchronization at fixed time.

Proof.

Consider

$$V(t)=\sum _{ij\in \mathrm{\Gamma }}^{}\sum _{B\in \mathbb{A}}^{}|x_{ij}^B(t)|$$

Along the network control (12), we have

$$\begin{array}{rl}& D{V}^{+}(t)=\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}sgn\left(x_{ij}^B(t)\right)(x_{ij}^B{)}^{\mathrm{△}}(t)=\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}sgn\left(x_{ij}^B(t)\right)\\ & [-d_{ij}(t)x_{ij}(t)+\sum _{c_{kl}\in N_r(i,j)}{b}_{ij}^{kl}(t)f_{ij}(y_{kl}(t))y_{ij}(t)\\ & -\sum _{c_{kl}\in N_r(i,j)}{b}_{ij}^{kl}(t)f_{ij}(z_{kl}(t))z_{ij}(t)+\sum _{c_{kl}\in N_s(i,j)}{c}_{ij}^{kl}(t)g_{ij}(y_{kl}(t-{\beta }_{kl}(t)))y_{ij}(t)\\ & -\sum _{c_{kl}\in N_s(i,j)}{c}_{ij}^{kl}(t)g_{ij}(z_{kl}(t-{\beta }_{kl}(t)))z_{ij}(t)+S_{ij}(t)]\\ & =\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}sgn\left(x_{ij}^B(t)\right)[-d_{ij}(t)x_{ij}(t)+\sum _{c_{kl}\in N_r(i,j)}{b}_{ij}^{kl}(t)f_{ij}(y_{kl}(t))\\ & -\sum _{c_{kl}\in N_r(i,j)}{b}_{ij}^{kl}(t)f_{ij}(z_{kl}(t))+\sum _{c_{kl}\in N_s(i,j)}{c}_{ij}^{kl}(t)g_{ij}(y_{kl}(t-{\beta }_{kl}(t)))\\ & -\sum _{c_{kl}\in N_s(i,j)}{c}_{ij}^{kl}(t)g_{ij}(z_{kl}(t-{\beta }_{kl}(t)))-M_{1ij}{x}_{ij}^B(t)\\ & -M_{2ij}{\left|x_{ij}^B(t)\right|}^{\alpha }sgn\left(x_{ij}^B(t)\right)-M_{3ij}{\left|x_{ij}^B(t)\right|}^{\gamma }sgn\left(x_{ij}^B(t)\right)\\ & -M_{4ij}\left|\sum _{ij\in \mathrm{\Gamma }}{x}_{ij}^B(t-{\beta }_{ij}(t))\right|sgn\left(x_{ij}^B(t)\right)]\\ & =-\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}sgn\left(x_{ij}^B(t)\right)d_{ij}(t)x_{ij}(t)\\ & +\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}sgn\left(x_{ij}^B(t)\right)\sum _{c_{kl}\in N_r(i,j)}{b}_{ij}^{kl}(t)\left(f_{ij}(y_{kl}(t))-f_{ij}(z_{kl}(t))\right)\\ & +\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}sgn\left(x_{ij}^B(t)\right)\sum _{c_{kl}\in N_s(i,j)}{c}_{ij}^{kl}(t)(g_{ij}(y_{kl}(t-{\beta }_{kl}(t)))\\ & -g_{ij}(z_{kl}(t-{\beta }_{kl}(t))))-\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}sgn\left(x_{ij}^B(t)\right)M_{1ij}{x}_{ij}^B(t)\\ & -\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}sgn\left(x_{ij}^B(t)\right)M_{2ij}{\left|x_{ij}^B(t)\right|}^{\alpha }sgn\left(x_{ij}^B(t)\right)\\ & -\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}sgn\left(x_{ij}^B(t)\right)M_{3ij}{\left|x_{ij}^B(t)\right|}^{\gamma }sgn\left(x_{ij}^B(t)\right)\\ & -\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}sgn\left(x_{ij}^B(t)\right)M_{4ij}\left|\sum _{ij\in \mathrm{\Gamma }}{x}_{ij}^B(t-{\beta }_{ij}(t))\right|sgn\left(x_{ij}^B(t)\right)\\ & =-\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}{d}_{ij}(t)\left|x_{ij}(t)\right|+\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}sgn\left(x_{ij}^B(t)\right)\\ & \sum _{c_{kl}\in N_r(i,j)}{b}_{ij}^{kl}(t)\left(f_{ij}(y_{kl}(t))-f_{ij}(z_{kl}(t))\right)+\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}sgn\left(x_{ij}^B(t)\right)\\ & \sum _{c_{kl}\in N_s(i,j)}{c}_{ij}^{kl}(t)\left(g_{ij}(y_{kl}(t-{\beta }_{kl}(t)))-g_{ij}(z_{kl}(t-{\beta }_{kl}(t)))\right)\\ & -\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}{M}_{1ij}\left|x_{ij}(t)\right|-\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}{M}_{2ij}{\left|x_{ij}^B(t)\right|}^{\alpha }\\ & -\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}{M}_{3ij}{\left|x_{ij}^B(t)\right|}^{\gamma }-\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}{M}_{4ij}\left|\sum _{ij\in \mathrm{\Gamma }}{x}_{ij}^B(t-{\beta }_{ij}(t))\right|\end{array}$$

$$\begin{array}{rl}& \sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}-d^{-}|x_{ij}(t)|+\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}\sum _{c_{kl}\in N_r(i,j)}{b}_{ij}^{k{l}^{+}}{L}_{ij}^f|x_{ij}(t)|\\ & +\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}\sum _{c_{kl}\in N_s(i,j)}{c}_{ij}^{k{l}^{+}}{L}_{ij}^g|x_{ij}^B(t-{\beta }_{ij}(t))|\\ & -\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}{M}_{1ij}|x_{ij}(t)|-\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}{M}_{2ij}{|x_{ij}^B(t)|}^{\alpha }\\ & -\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}{M}_{3ij}{|x_{ij}^B(t)|}^{\gamma }\\ & -\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}{M}_{4ij}|\sum _{ij\in \mathrm{\Gamma }}{x}_{ij}^A(t-{\beta }_{ij}(t))|\\ & \le \sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}\{-d^{-}-M_{1ij}+\sum _{c_{kl}\in N_r(i,j)}{b}_{ij}^{k{l}^{+}}{L}_{ij}^f\}|x_{ij}(t)|\\ & +\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}\{-M_{4ij}+\sum _{c_{kl}\in N_s(i,j)}{c}_{ij}^{k{l}^{+}}{L}_{ij}^g\}|x_{ij}^B(t-{\beta }_{ij}(t))|\\ & -\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}{M}_{2ij}{|x_{ij}^B(t)|}^{\alpha }-\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}{M}_{3ij}{|x_{ij}^B(t)|}^{\gamma }\\ & \le -\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}{M}_{2ij}{|x_{ij}^B(t)|}^{\alpha }-\sum _{ij\in \mathrm{\Gamma }}\sum _{B\in \mathbb{A}}{M}_{3ij}{|x_{ij}^B(t)|}^{\gamma }\end{array}$$

(19)

From Lemma 2, we obtain

$$\sum _{ij\in \mathrm{\Gamma }}^{}\sum _{B\in \mathbb{A}}^{}{M}_{2ij}{|x_{ij}^B(t)|}^{\alpha }\le {\left(\sum _{ij\in \mathrm{\Gamma }}^{}\sum _{B\in \mathbb{A}}^{}{M}_{2ij}|x_{ij}^B(t)|\right)}^{\alpha }$$

(20)

$$\sum _{ij\in \mathrm{\Gamma }}^{}\sum _{B\in \mathbb{A}}^{}{M}_{3ij}{|x_{ij}^B(t)|}^{\gamma }\le {\left(\sum _{ij\in \mathrm{\Gamma }}^{}\sum _{B\in \mathbb{A}}^{}{M}_{3ij}|x_{ij}^B(t)|\right)}^{\gamma }$$

(21)

From (20) and (21), the inequality (19) becomes

$$D{V}^{+}(t)\le -\underset{ij\in \mathrm{\Gamma }}{min}\left\{M_{2ij}\right\}{\left(\sum _{ij\in \mathrm{\Gamma }}^{}\sum _{B\in \mathbb{A}}^{}|x_{ij}^B(t)|\right)}^{\alpha }-\underset{ij\in \mathrm{\Gamma }}{min}\left\{M_{3ij}\right\}{\left(\sum _{ij\in \mathrm{\Gamma }}^{}\sum _{B\in \mathbb{A}}^{}|x_{ij}^B(t)|\right)}^{\gamma }$$

According to Lemma 1, error network (1) and system (12) synchronized in fixed time under feedback (14).  □

In particular, for $\sum _{c_{kl}\in N_r(i,j)}^{}{b}_{ij}^{k{l}^{+}}{L}_{ij}^f\ge d^{-}$ , we have the following corollary

Corollary 1

Assume $({\mathcal{P}}_1)$ hold. If in addition, $M_{1ij}$ , $M_{2ij}$ , $M_{3ij}$ , and $M_{4ij}$ of feedback (14) satisfy

$$M_{1ij}\ge 0$$

(22)

$$M_{2ij}>0$$

(23)

$$M_{3ij}>0$$

(24)

$$M_{4ij}\ge \sum _{c_{kl}\in N_s(i,j)}^{}{c}_{ij}^{k{l}^{+}}{L}_{ij}^g$$

(25)

for all $ij\in \mathrm{\Gamma }$ . By means of feedback (14), system (1), and system (12) can reach synchronization at fixed time.

Illustrative example

Example 1

In network (1), let $n=2$ , $r=s=1$ , where $\mathrm{\Theta }=\{0,1,2,12\}$ and

$$\begin{array}{rl}{z}_{ij}(t)& =z_{ij}^0(t)e_0+z_{ij}^1(t)e_1+z_{ij}^2(t)e_2+z_{ij}^{12}(t)e_{12}\\ d_{11}(t)& =32{\cos }^2(1.71t)e_0+0.28\sin (2t)e_1-0.4\cos (2.23t)e_2+0.001{\sin }^2(3t)e_{12}\\ d_{21}(t)& =39|\sin (1.41t)|e_0-0.5\cos (2t)e_1-0.33{\cos }^2(2.64t)e_2+0.17\sin (t)e_{12}\\ d_{12}(t)& =33{\sin }^2(t)e_0-i0.28\cos (5t)e_1-0.25{\sin }^3(2.23t)e_2+1.66{\sin }^2(3t)e_{12}\\ d_{22}(t)& =37{\cos }^4(1.71t)e_0+0.25\sin (7t)e_1-0.33|\cos (3.31t)|e_2+0.12{\sin }^2(5t)e_{12}\\ f_{11}(z)=f_{12}(z)& =0.4\sin (0.25z^0+0.28z^2)e_0-0.4|1.05z^1+0.34z^2|e_1+0.25\cos (0.2z^0)e_2+0.25\sin (0.2z^0)e_{12}\\ f_{21}(z)=f_{22}(z)& =0.25|0.47z^1+0.24z^{12}|e_0-0.17\sin (0.2z^0+0.28z^{12})e_1-0.4|1.05z^1+0.34z^2|e_2+0.47\sin (0.2z^1)e_{12}\\ g_{11}(z)=g_{12}(z)& =0.05*\sin (6.34z^0)e_0-0.04*\sin (1.4z^{12})e_1+0.07*\sin (0.33z^2+0.33z^1)e_2+0.04*\sin (5.64z^0)e_{12}\\ g_{21}(z)=g_{22}(z)& =0.05*|0.66z^1+0.33z^2|e_0-0.04*\sin (1.4z^{12}+z^0)e_1+0.04*\sin (5.64z^0)e_2-0.04*\sin (1.4z^{12})e_{12}\\ D_{11}(t)=D_{12}(t)& =1.41*\cos (t)e_0+1.33e^{-|t|}{e}_1+0.47*\sin (0.5t)e_2-0.2*\sin (2t)e_{12}\\ D_{21}(t)=D_{22}(t)& =0.47*\sin (0.5t)e_0+0.47*\cos (1.41t)e_1-0.2*\sin (2t)e_2+e^{-|t|}{e}_{12}\end{array}$$

and $k,l=1,2\in N_1(i,j)$ for any $i=j=1,2$ ,

$$\begin{array}{rl}{b}_{ij}^{11}(t)& =b_{ij}^{21}(t)=0.33|\sin (1.41t)|e_0+0.16\cos (4t)e_1+(0.75\cos (2t)+1)e_2+0.25{\cos }^2(2.23t)e_{12}\\ b_{ij}^{12}(t)& =b_{ij}^{22}(t)=0.3*|\cos (t)|e_0+0.17*\sin (3t)e_1+0.25*\cos (2t)e_2+0.21*\sin (2t)e_{12}\\ c_{ij}^{11}(t)=c_{ij}^{21}(t)& =(\sin (1.73t)+3)e_0+|\cos (2.23t)|e_1+2{\sin }^2(2.64t)e_2+0.25\cos (2t)e_{12}\\ c_{ij}^{12}(t)& =c_{ij}^{22}(t)=\cos (1.5t)e_0+\sin (2t)e_1+{\cos }^3(2.64t)e_2+0.2\sin (2t)e_{12}\\ & b_{ij}^{kl}:\left(\begin{array}{ccc}{b}_{ij}^{11}& b_{ij}^{12}& b_{ij}^{13}\\ b_{ij}^{21}& b_{ij}^{22}& b_{ij}^{23}\\ b_{ij}^{31}& b_{ij}^{32}& b_{ij}^{33}\end{array}\right)\\ & c_{ij}^{kl}:\left(\begin{array}{ccc}{c}_{ij}^{11}& c_{ij}^{12}& c_{ij}^{13}\\ c_{ij}^{21}& c_{ij}^{22}& c_{ij}^{23}\\ c_{ij}^{31}& c_{ij}^{32}& c_{ij}^{33}\end{array}\right)\\ {\beta }_{11}(t)& =0.04{\cos }^8(1.41t),{\beta }_{12}(t)=0.06{\sin }^2(1.41t)\\ {\beta }_{21}(t)& =0.02{\cos }^2(0.28t),{\beta }_{22}(t)=0.03{\cos }^4(0.24t)\\ M_{1ij}& =10,M_{2ij}=5,M_{3ij}=6,M_{4ij}=12\end{array}$$

After all calculations, we have

$$\begin{array}{rl}& L_{11}^f=L_{12}^f=0.4,L_{21}^f=L_{22}^f=0.47\\ & L_{11}^g=L_{12}^g=0.07,L_{21}^g=L_{22}^g=0.05\\ & {\tilde{d}}_{11}^{+}=0.4,{\tilde{d}}_{12}^{+}=1.66,{\tilde{d}}_{21}^{+}=0.5,{\tilde{d}}_{22}^{+}=0.33\\ & \sum _{c_{kl}\in N_1(1,1)}^{}{b}_{11}^{k{l}^{+}}=\sum _{c_{kl}\in N_1(1,2)}^{}{b}_{12}^{k{l}^{+}}=\sum _{c_{kl}\in N_1(2,1)}^{}{b}_{21}^{k{l}^{+}}=\sum _{c_{kl}\in N_1(2,2)}^{}{b}_{22}^{k{l}^{+}}=4.1\\ & \sum _{c_{kl}\in N_1(1,1)}^{}{c}_{11}^{k{l}^{+}}=\sum _{c_{kl}\in N_1(1,2)}^{}{c}_{12}^{k{l}^{+}}=\sum _{c_{kl}\in N_1(2,1)}^{}{c}_{21}^{k{l}^{+}}=\sum _{c_{kl}\in N_1(2,2)}^{}{c}_{22}^{k{l}^{+}}=10\\ & d^{-}=32,d^{+}=39,\beta =0.16,{\beta }^{\prime }=0.5\end{array}$$

Take $\alpha =0.06\ge \Vert {\mathrm{\Phi }}^0\Vert$ . For $1\le i,j\le 2$ , we have

$$K=\frac{2}{d^{-}}\underset{ij\in \mathrm{\Gamma }}{max}[{\tilde{d}}_{ij}^{+}+\sum _{c_{kl}\in N_r(i,j)}^{}0.12b_{ij}^{k{l}^{+}}{L}_{ij}^f+\sum _{c_{kl}\in N_s(i,j)}^{}0.12c_{ij}^{k{l}^{+}}{L}_{ij}^g]=0.0495<1$$

For $i,j\in \{1,2\}$ , choose $p=3$ , we have

$$\begin{array}{rl}& \underset{ij\in \mathrm{\Gamma }}{max}\{24\left(\frac{2}{3d^{-}}\right)\left(\frac{4}{3d^{-}}\right)[2{\left({\tilde{d}}_{ij}^{+}\right)}^3+2{\left(\sum _{c_{kl}\in N_r(i,j)}^{}0.12b_{ij}^{k{l}^{+}}{L}_{ij}^f\right)}^3\\ & +(1+\frac{2e^{\frac{3}{4}}{d}^{+}\beta }{1-{\beta }^{\prime }}){\left(\sum _{c_{kl}\in N_s(i,j)}^{}0.12c_{ij}^{k{l}^{+}}{L}_{ij}^g\right)}^3]\}=0.1761<1\end{array}$$

and, for $p=2$ , we obtain

$$\begin{array}{rl}& \underset{ij\in \mathrm{\Gamma }}{max}\{24{\left(\frac{2}{d^{-}}\right)}^2[2{\left({\tilde{d}}_{ij}^{+}\right)}^2+2{\left(\sum _{c_{kl}\in N_r(i,j)}^{}0.12b_{ij}^{k{l}^{+}}{L}_{ij}^f\right)}^2\\ & +(1+\frac{2e^{\frac{1}{2}}{d}^{+}\beta }{1-{\beta }^{\prime }}){\left(\sum _{c_{kl}\in N_s(i,j)}^{}0.12c_{ij}^{k{l}^{+}}{L}_{ij}^g\right)}^2]\}=0.5422<1\end{array}$$

Which proves assumptions $({\mathcal{P}}_1)-({\mathcal{P}}_4)$ , for example, $i=j=1$ , $k,l=1,2\in N_1(1,1)$ ,

$$\begin{array}{rl}& \sum _{c_{kl}\in N_1(1,1)}^{}{b}_{11}^{kl}(t)f_{11}(z_{kl}(t))z_{11}(t)\\ & =b_{11}^{11}(t)f_{11}(z_{11}(t))z_{11}(t)+b_{11}^{12}(t)f_{11}(z_{12}(t))z_{11}(t)+b_{11}^{21}(t)f_{11}(z_{21}(t))z_{11}(t)+b_{11}^{22}(t)f_{11}(z_{22}(t))z_{11}(t)\end{array}$$

$M_{1ij}=10,M_{2ij}=5,M_{3ij}=6,M_{4ij}=12\in \mathbb{R}$

Figures 1 to 4 show the trajectories with different initial conditions, coincide quickly (at exponential speed), which illustrates the exponential stability of the solutions of network (1).

Figure 1.

Curves of $z_{11}^0(t)$ ,   $z_{11}^1(t)$ ,   $z_{11}^2(t)$ , and $z_{11}^{12}(t)$ of (1) with different initial values.

Figure 2.

Trajectory of $z_{12}^0(t)$ ,   $z_{12}^1(t)$ ,   $z_{12}^2(t)$ , and $z_{12}^{12}(t)$ .

Figure 3.

Trajectory of $z_{21}^0(t)$ ,   $z_{21}^1(t)$ ,   $z_{21}^2(t)$ , and $z_{21}^{12}(t)$ .

Figure 4.

Trajectory of $z_{22}^0(t)$ ,   $z_{22}^1(t)$ ,   $z_{22}^2(t)$ , and $z_{22}^{12}(t)$ .

Example 2

In network (1), let $n=2$ , $r=s=1$ , where $\mathrm{\Theta }=\{0,1,2,12\}$ and

$$\begin{array}{rl}{z}_{ij}(t)& =z_{ij}^0(t)e_0+z_{ij}^1(t)e_1+z_{ij}^2(t)e_2+z_{ij}^{12}(t)e_{12}\\ d_{11}(t)& =32{\cos }^2(1.71t)e_0+0.28\sin (2t)e_1-0.4\cos (2.23t)e_2+0.001{\sin }^2(3t)e_{12}\\ d_{21}(t)& =39|\sin (1.41t)|e_0-14.1\cos (2t)e_1-0.33{\cos }^2(2.64t)e_2+0.17\sin (t)e_{12}\\ d_{12}(t)& =33{\sin }^2(t)e_0-i0.28\cos (5t)e_1-0.25{\sin }^3(2.23t)e_2+1.66{\sin }^2(3t)e_{12}\\ d_{22}(t)& =37{\cos }^4(1.71t)e_0+0.25\sin (7t)e_1-0.33|\cos (3.31t)|e_2+0.12{\sin }^2(5t)e_{12}\\ f_{11}(z)& =f_{12}(z)=0.4\sin (0.25z^0+0.28z^2)e_0-0.4|1.05z^1+0.34z^2|e_1+0.25\sin (0.2z^0)e_{12}\\ f_{21}(z)& =f_{22}(z)=0.25|0.47z^1+0.24z^{12}|e_0-0.17\sin (0.2z^0+0.28z^{12})e_1+0.47\sin (0.2z^1)e_{12}\\ g_{11}(z)& =g_{12}(z)=0.05\sin (6.34z^0)e_0-0.04\sin (1.4z^{12})e_1+0.07\sin (0.33z^2+0.33z^1)e_2\\ g_{21}(z)& =g_{22}(z)=0.05|0.66z^1+0.33z^2|e_0-0.04\sin (1.4z^{12}+z^0)e_1+0.04\sin (5.64z^0)e_2\\ D_{11}(t)& =D_{12}(t)=1.41\cos (t)e_0+1.33e^{-|t|}{e}_1+0.47\sin (0.5t)e_2\\ D_{21}(t)& =D_{22}(t)=0.47\cos (1.41t)e_1-0.2\sin (2t)e_2+e^{-|t|}{e}_{12}\end{array}$$

and

$$\begin{array}{rl}{b}_{11}(t)& =0.33|\sin (1.41t)|,b_{12}(t)=0.16\cos (4t)\\ b_{21}(t)& =0.75\cos (2t)+1,b_{22}(t)=0.25{\cos }^2(2.23*t)\\ c_{11}(t)& =\sin (1.73*t)+3,c_{12(t)=|\cos (2.23*t)|}\\ c_{21}(t)& =2{\sin }^2(2.64*t),c_{22}(t)=0.25\cos (2t)+1.75\\ {\beta }_{11}(t)& =0.04{\cos }^8(1.41*t),{\beta }_{12}(t)=0.06{\sin }^2(1.41*t)\\ {\beta }_{21}(t)& =0.02{\cos }^2(0.28*t),{\beta }_{22}(t)=0.03{\cos }^4(0.24*t)\end{array}$$

After all calculations, we have

$$\begin{array}{rl}{L}_{11}^f& =L_{12}^f=0.4,L_{21}^f=L_{22}^f=0.47\\ L_{11}^g& =L_{12}^g=0.07,L_{21}^g=L_{22}^g=0.05\\ {\tilde{d}}_{11}^{+}& =0.4,{\tilde{d}}_{12}^{+}=1.66,{\tilde{d}}_{21}^{+}=0.5,{\tilde{d}}_{22}^{+}=0.33\\ b_{11}^{k{l}^{+}}& =0.33,b_{12}^{k{l}^{+}}=0.16,b_{21}^{k{l}^{+}}=1.75,b_{22}^{k{l}^{+}}=0.25\\ c_{11}^{k{l}^{+}}& =4,c_{12}^{k{l}^{+}}=1,c_{21}^{k{l}^{+}}=2,c_{22}^{k{l}^{+}}=2\\ d^{-}& =32,d^{+}=39,\beta =0.16,{\beta }^{\prime }=0.5\end{array}$$

$M_{1ij}=10,M_{2ij}=5,M_{3ij}=6,M_{4ij}=12\in \mathbb{R}$

Take $\alpha =0.06\ge \Vert {\mathrm{\Phi }}^0\Vert$ . For $1\le i,j\le 2$ we have

$$K=\frac{2}{d^{-}}\underset{ij\in \mathrm{\Gamma }}{max}[{\tilde{d}}_{ij}^{+}+\sum _{c_{kl}\in N_r(i,j)}^{}0.12b_{ij}^{k{l}^{+}}{L}_{ij}^f+\sum _{c_{kl}\in N_s(i,j)}^{}0.12c_{ij}^{k{l}^{+}}{L}_{ij}^g]=0.4559<1$$

For $i,j\in \{1,2\}$ , choose $p=3$ we have

$$\begin{array}{r}\underset{ij\in \mathrm{\Gamma }}{max}\{24\left(\frac{2}{3d^{-}}\right)\left(\frac{4}{3d^{-}}\right)[2{\left({\tilde{d}}_{ij}^{+}\right)}^3+2{\left(\sum _{c_{kl}\in N_r(i,j)}^{}0.12b_{ij}^{k{l}^{+}}{L}_{ij}^f\right)}^3\\ +(1+\frac{2e^{\frac{3}{4}}{d}^{+}\beta }{1-{\beta }^{\prime }}){\left(\sum _{c_{kl}\in N_s(i,j)}^{}0.12c_{ij}^{k{l}^{+}}{L}_{ij}^g\right)}^3]\}=0.1978<1\end{array}$$

and, for $p=2$ we obtain

$$\begin{array}{r}\underset{ij\in \mathrm{\Gamma }}{max}\{24{\left(\frac{2}{d^{-}}\right)}^2\left(\frac{4}{2d^{-}}\right)[2{\left({\tilde{d}}_{ij}^{+}\right)}^2+2{\left(\sum _{c_{kl}\in N_r(i,j)}^{}0.12b_{ij}^{k{l}^{+}}{L}_{ij}^f\right)}^2\\ +(1+\frac{2e^{\frac{1}{2}}{a}^{+}\beta }{1-{\beta }^{\prime }}){\left(\sum _{c_{kl}\in N_s(i,j)}^{}0.12c_{ij}^{k{l}^{+}}{L}_{ij}^g\right)}^2]\}=0.5179<1.\end{array}$$

Which proves assumptions $({\mathcal{P}}_1)-({\mathcal{P}}_4)$ .

Figures 5 to 8 are the curves of system (1)

Figure 5.

Curves of $z_{11}^0(n)$ ,   $z_{11}^1(n)$ ,   $z_{11}^2(n)$ and $z_{11}^{12}(n)$ of (1) with different initial values.

Figure 6.

Curves of $z_{12}^0(n)$ ,   $z_{12}^1(n)$ ,   $z_{12}^2(n)$ , and $z_{12}^{12}(n)$ of (1) with different initial values.

Figure 7.

Curves of $z_{21}^0(n)$ ,   $z_{21}^1(n)$ ,   $z_{21}^2(n)$ , and $z_{21}^{12}(n)$ of (1) with different initial values.

Figure 8.

Curves of $z_{22}^0(n)$ ,   $z_{22}^1(n)$ ,   $z_{22}^2(n)$ , and $z_{22}^{12}(n)$ of (1) with different initial values.

Conclusion and futur problem

In this article, we have defined the pseudo almost automorphic in Weyl’s sense functions on timescales, which is a natural generalization of results by Arbi and Tahri ¹⁴ and Li and Huang.^45,46 Furthermore, the qualitative properties of the network (1) is realized. Therefore, fixed-time synchronization was realized by considering appropriate feedback and Lyapunov functional. The outcomes of this article illustrated with a simulated example. An idea may be interesting for scholars is to try to study other neuronal models like inertial NNs, BAM NNs, Cohen-Grossberg. It is possible, based on this work, to generalize our space $W^{p}PAA(\mathbb{T},\mathbb{E})$ in a way analogous to those of the spaces weighted pseudo almost periodic/automorphic^37–39 and $v$ -pseudo almost periodic/automorphic.^49,50