A new approach for accuracy-preassigned finite-time exponential synchronization of neutral-type Cohen–Grossberg memristive neural networks involving multiple time-varying leakage delays

1.   Introduction

Neural networks (NNs) have been the subject of many studies because of their successful applications in various areas such as associative memory ^[1], autonomous navigation ^[2], data encryption ^[3], intelligent control ^[4], optimization ^[5,6], controller design ^[7], pattern recognition ^[8], spacecraft formation flying ^[9], and so on. In practice, time delays can cause NNs to oscillate, become unstable, or perform poorly ^[10,11]. Researchers have identified several types of delays, including leakage delays ^[12,13,14], distributed delays, transmission delays, and neutral delays. In recent years, a number of topics related to delayed NNs, such as stability and stabilization ^{[15,16,17,18]}, dissipativity analysis and control ^[19,20], passivity and passification ^[21], state estimation ^[22], and synchronization control ^[23,24,25], have attracted attention from many researchers.

The Cohen–Grossberg NNs (CGNNs) were proposed by Cohen and Grossberg in 1983 ^[26]. In the past few years, increasing intension in analyzing and control of CGNNs have appeared ^{[27,28,29,30]}. In 2008, Stanley R. Williams and his team found the practical memristor ^[31], which verifies Chua's prediction in 1971 ^[32]. Since the memristor mimics the forgetting and remembering processes in human brains, it has potential to be used as brain-like computers and future computers. In order to more accurately mimic the human brain via NNs, the ordinary resistor of connection weights of CGNNs is replaced by the memristor, resulting in memristive CGNNs. Therefore, memristive CGNNs have more significance in the study of human brain simulation.

Synchronization issues have garnered significant attention and research ^[33,34,35], thanks to their applications in image encryption, audio encryption, and secure communication ^[36,37,38]. In ^[39], finite-time synchronization criteria of memristive CGNNs involving time-varying delays were obtained by constructing the appropriate nonlinear transformations and employing the differential inclusion theory. For reaction-diffusion memristive CGNNs, a new definition of quasi-fixed-time synchronization was proposed in ^[40], and quasi-fixed-time synchronization theorem was investigated by designing an effective controller. In ^[28], using the reduced-order method based on the differential inclusion theory, the adaptive feedback controller was proposed to achieve global asymptotic synchronization of proportional delay inertial memristive CGNNs. In ^[41], the exponential synchronization conditions of quaternion-valued memristive CGNNs involving time-varying delays were presented by employing the differential inclusion theory, an improved one-norm method, and the set-valued map theory. For coupled memristive multi-stable CGNNs involving mixed delays, some sufficient conditions guaranteeing multi-synchronization were investigated by utilizing the M-matrix theory, the state-space decomposition, and the fixed point theory ^[42]. The synchronization issues of fractional-order delayed memristive CGNNs are addressed in ^[43,44].

In the above references about finite-time synchronization problem of NNs, the settling-time formula was derived by employed the different radial unbounded function. When the time is larger than the settling time, state variances of error system are always equal to zero in theoretically; however, due to the network noises, model approximation, DoS attacks, and so on, they realistically converge into a small domain containing the origin ^[45]. In ^[34], a parameter $\epsilon > 0$ was introduced to describe the small domain. From the angle of application, the positive number $\epsilon$ can be viewed as a preassigned accuracy, which should be satisfied whenever considering a finite-time synchronization issue. The relationship between accuracy-preassigned finite-time exponential synchronization in neutral-type Cohen–Grossberg memristive neural networks and Keller-Segel models lies in the exploration of complex dynamical systems and the synchronization phenomena in NNs, particularly under the influence of delays and spatial interactions. Specifically, the concept of accuracy-preassigned finite-time exponential synchronization in the context of neutral-type Cohen–Grossberg memristive neural networks provides a framework for understanding and analyzing synchronization behaviors in Keller-Segel models, particularly in the presence of time-varying delays. Both fields involve the study of complex systems characterized by nonlinear interactions and delays, and they share common mathematical methodologies, including Lyapunov functions, stability theory, and bifurcation analysis, for their theoretical analysis ^[46,47]. Therefore, it is great to address the accuracy-preassigned finite-time synchronization issue of NNs.

As stated, several researchers have studied the subject on the finite-time synchronization of time-delay memristive CGNNs. Nevertheless, there is no study on the finite-time synchronization of neutral-type delayed memristive CGNNs. We seek to address this gap by investigating the accuracy-preassigned finite-time exponential synchronization problem of neutral-type memristive CGNNs involving multiple time-varying leakage and transmission delays. We summarize the significance and contributions of this article below:

● Compared with the current literature, the memristive CGNN model addressed in this article is more general. The model is not only neutral-type but also involves more time-varying delays. Moreover, we provide a new idea to solve the problems related to synchronization of neutral-type memristive delayed CGNNs.

● We introduce a novel method that is ground on the upper-right derivative of solutions of the error system. Especially, instead of the derivative, the upper-right derivative is employed in the theoretical derivation, which is compatible with a new controller that is different from ones in studies.

● The derived synchronization conditions are comprised of simple scalar inequalities that is convenient to implement via the common software tools. Moreover, the synchronization condition can guarantee that the states of the error system converge exponentially into a small range containing the origin under the preassigned accuracy.

The structure of the article is designed below: In Section 2, we provide preliminary results, including the drive and response neutral-type delayed memristive CGNNs model, necessary assumptions, definitions, and lemmas. In Section 3, we present the design method of controller to achieve accuracy-preassigned finite-time exponential synchronization between the drive and response neutral-type delayed memristive CGNNs. In Section 4, we validate the major results through numerical examples. Finally, the conclusions are given in Section 5.

Notations: Let $\langle n\rangle$ be the set $\{1, 2, \dots, n\}$, where $n$ is a positive integer. The symbol $\mathbb{R}$ represents the real number field. The symbols $\mathcal{C}(\mathbb{S}_1, \mathbb{S}_2)$ and $\mathcal{C}^1(\mathbb{S}_1, \mathbb{S}_2)$ denote the linear spaces over $\mathbb{R}$ of all continuous and continuously differential functions $f:\mathbb{S}_1\rightarrow \mathbb{S}_2$, respectively. The column-vectorizing operator is denoted by $\mathrm{col}(\cdot)$. The symbol $\mathcal{D}^+$ represents the upper-right derivative of functions. The Euclidean norm is denoted by $\|\cdot\|$.

2.   Preliminaries

The considered neutral-type memristive CGNN involving multiple time-varying leakage and transmission delays can be written as:

where $n$ is the number of neurons, $\varepsilon_{ij} > 0$, $\pi_{i}\in \mathcal{C}([0, +\infty), [\check{\pi}_{i}, \hat{\pi}_{i}])$ and $\eta_{ij}\in \mathcal{C}([0, +\infty), [\check{\eta}_{ij}, \hat{\eta}_{ij}])$ are the neutral, leakage, and transmission delays, respectively, $0\leq\check{\pi}_{i}$, $0\leq\check{\eta}_{ij}$, $x_{i}:[-\rho, +\infty)\rightarrow \mathbb{R}$ are the neuronal states, $\rho = \max\{\hat{\varepsilon}, \hat{\pi}, \hat{\eta}\}$, $\hat{\varepsilon} = {\max\limits_{i, j\in\langle n\rangle}}\varepsilon_{ij}$, $\hat{\pi} = {\max\limits_{i\in\langle n\rangle}}\ \hat{\pi}_{i}$, $\hat{\eta} = {\max\limits_{i, j\in\langle n\rangle}}\hat{\eta}_{ij}$, $\psi_{i}\in \mathcal{C}(\mathbb{R}, [\check{\Psi}_{i}, \hat{\Psi}_{i}])$ are the amplification functions, $0 < \check{\Psi}_{i}$, $\varphi_{i} \in \mathcal{C}(\mathbb{R}, \mathbb{R})$ are the self-signal functions, $\phi^x_{i}\in C^{1}([-\rho, 0], \mathbb{R})$ refer to the initial functions, $s_{ij}\in\mathbb{R}$, and $p_{j}, q_{j}\in \mathcal{C}(\mathbb{R}, \mathbb{R})$ represent the activation functions. In addition, the connection weights have the form as follows:

here, ${\varpi _i} > 0$ are the threshold constants, and $\hat{a}_{ij}$, $\check{a}_{ij}$, $\hat{b}_{ij}$, and $\check{b}_{ij}$ are known scalars.

Remark 1. When $s_{ij} = 0$, $\pi_{i}(\cdot)\equiv0$ and $\eta_{ij}(\cdot) = \eta_{j}(\cdot)$ for all $i, j\in\langle n\rangle$, the NN model (2.1) simplifies to memristive CGNNs with time-varying delays ^[39]. When $\pi_i(\cdot)\equiv0$, $s_{ij} = 0$, $a_{ij}(\cdot) = a_{ij}$, $b_{ij}(\cdot) = b_{ij}$ and $\eta_{ij}(\cdot) = \eta(\cdot)$ for any $i, j\in\langle n\rangle$, the NN model (2.1) reduces to CGNNs with one time-varying transmission delay ^[48].

Remark 2. In some NNs, there are a large number of synapses with different sizes and parallel paths with different lengths, which limits the space range ^[49]. Therefore, there is always a representative time delay, which is essentially different from the conventional delays, and it broadly exists in the negative feedback terms of the system, which are identified as leakage terms, named leakage delay. The leakage delay is usually incorporated in the study of network modeling. Such a type of time delay often has a tendency to destabilize the NNs and is difficult to handle. Therefore, it is of great practical significance to study the stability of NNs with leakage delays.

We take the neutral-type memristive CGNN (2.1) as the drive system, which devises a neutral-type memristive CGNN (response system) as follows:

where $y_{i}:[-\rho, +\infty)\rightarrow \mathbb{R}$ represent the neuronal states, $\phi^y_{i}\in C^{1}([-\rho, 0], \mathbb{R})$ stand for the initial functions, and $u_{i}:[0, +\infty)\rightarrow \mathbb{R}$ refer to the control inputs.

The error system related to the drive-response neutral-type memristive CGNNs (2.1) and (2.3) is expressed as:

where $\phi_{i}(s) = \phi^y_{i}(s)-\phi^x_{i}(s)$ and $e_{i}(t) = y_{i}(t)-x_{i}(t)$.

Definition 1 For any initial function $\phi(\cdot$), we say that the neutral-type memristive CGNNs (2.1) and (2.3) achieve accuracy-preassigned finite-time exponential synchronization, if for given an accuracy $\epsilon > 0$, there are $\hat{T} > 0$ and $u_{i}(t)\ (i\in\langle n\rangle)$ such that the state vector, $e(t): = \mathrm{col}\begin{pmatrix}e_{1}(t), \dots, e_{n}(t)\end{pmatrix}$ satisfies $\|e(t)\| < \epsilon$ whenever $t > \hat{T}$. The constant $\hat{T}$ is called the settling time.

The following assumptions are required.

$A_{1}$: ^[44] For every $i\in\langle n\rangle$, there are positive numbers $P_{i}$, $Q_{i}$, $\hat{P}_{i}$, and $\hat{Q}_{i}$ satisfying:

$|p_{i}(\kappa)|\le \hat{P}_{i}$, $|p_{i}(\kappa)-p_{i}(\iota)|\leq P_{i}|\kappa-\iota|$,

$|q_{i}(\kappa)|\le \hat{Q}_{i}$, $|q_{i}(\kappa)-q_{i}(\iota)|\leq Q_{i}|\kappa-\iota|$, $\kappa, \iota\in\mathbb{R}$.

$A_2$: For every $i\in\langle n\rangle$, there are positive numbers $\Psi_i$ and $\Phi_i$, satisfying:

$\left|\psi_i(\kappa_1)\varphi_{i}(\kappa_2)-\psi_i(\iota_1)\varphi_{i}(\iota_2)\right|\le \Psi_i|\kappa_1-\iota_1|+\Phi_i|\kappa_2-\iota_2|, \ \kappa_1, \kappa_2, \iota_1, \iota_2\in\mathbb{R}$.

In the case without the leakage delay, it is chosen that $\kappa_1 = \kappa_2$ and $\iota_1 = \iota_2$ ^[44].

Remark 3. In Assumptions $A_{1}$ and $A_{2}$, we require the amplification function or activation functions to be bounded continuous and satisfy Lipschitz continuity. Otherwise, the NN system may experience problems such as gradient explosion or model degradation. There are also many research results that can handle discontinuous activation functions ^[44].

We aim to design controllers $u_{i}(t)\ (i\in\langle n\rangle)$ such that the drive-response neutral-type memristive CGNNs (2.1) and (2.3) achieve accuracy-preassigned finite-time exponential synchronization. To this end, the following conclusion is needed.

Lemma 1. Set $\bar{b}_{ij} = \max\{|\check{b}_{ij}|, |\hat{b}_{ij}|\}$ and $\bar{a}_{ij} = \max\{|\check{a}_{ij}|, |\hat{a}_{ij}|\}$. Then, under assumption $A_1$, there are:

Proof. We prove only (2.5), since the other is similar. It follows from $\psi_{i}\in \mathcal{C}(\mathbb{R}, [\check{\Psi}_{i}, \hat{\Psi}_{i}])$, $\bar{a}_{ij} = \max\{|\check{a}_{ij}|\, |\hat{a}_{ij}|\}$, (2.2) and assumption $A_1$ that

□

3.   Accuracy-preassigned fixed-time exponential synchronization

Throughout this section, we assume that assumptions $A_1$ and $A_2$ hold. To realize accuracy-preassigned finite-time exponential synchronization between neutral-type memristive CGNNs (2.1) and (2.3), we design the following controllers:

where $K_{i} > 0$ represent the controller gains.

Theorem 1. Under the controller (3.1), each solution of the error system (2.4) meets the following relation:

where

$\partial_{i0}(t) = \mathrm{e}^{-K_{i}t}\|\phi\|_{\rho}\left(1+\sum_{j\in\langle n\rangle}|s_{ij}|\right)$, $\|\phi\|_{\rho} = \max\limits_{i\in\langle n\rangle}\sup\limits_{s\in[-\rho, 0]}\max\left\{|\phi_{i}(s)|, |\dot{\phi}_{i}(s)|\right\}$,

$\partial_{i1}(t) = \sum_{j\in\langle n\rangle}|s_{ij}||e_{j}(t-\varepsilon_{ij})|$,

$\partial_{i2}(t) = \Psi_{i}\int_{0}^{t}\mathrm{e}^{K_{i}(s-t)}|e_{i}(s)|\mathrm{d}s$,

$\partial_{i3}(t) = \Phi_{i}\int_{0}^{t}\mathrm{e}^{K_{i}(s-t)}|e_{i}(s-\pi_{i}(s))|\mathrm{d}s$,

$\partial_{i4}(t) = \hat{\Psi}_{i}\sum_{j\in\langle n\rangle}\int_{0}^{t}\mathrm{e}^{K_{i}(s-t)}\bar{a}_{ij}P_{j}|e_{j}(s)|\mathrm{d}s$,

$\partial_{i5}(t) = \hat{\Psi}_{i}\sum_{j\in\langle n\rangle}\int_{0}^{t}\mathrm{e}^{K_{i}(s-t)}\bar{b}_{ij}Q_{j}|e_{j}(s-\eta_{ij}(s))|\mathrm{d}s$,

$\Upsilon_i = (\hat{\Psi}_{i}-\check{\Psi}_{i})\sum\limits_{j\in\langle n\rangle}\left(\bar{a}_{ij}\hat{P}_{j}+\bar{b}_{ij}\hat{Q}_{j}\right)+\hat{\Psi}_{i}\sum\limits_{j\in\langle n\rangle}\left(|\hat{a}_{ij}-\check{a}_{ij}|\hat{P}_{j}+|\hat{b}_{ij}-\check{b}_{ij}|\hat{Q}_{j}\right)$.

Remark 4. The inequality (3.2) is different from the respective conclusions in ^[11,24,25].

Proof. It is obvious from (2.4a) that

where $E_i(t) = \mathrm{sgn}\left(e_{i}(t)-\sum_{j\in\langle n\rangle}s_{ij}e_{j}(t-\varepsilon_{ij})\right)$, and hence,

This, together with (3.1), assumption $A_2$, and Lemma 1, derives that

Taking an integration on both sides of (3.4) from $0$ to $t$, we have

and hence (3.2) holds. □

Theorem 2. For given positive numbers $\varsigma$, $\epsilon$ and $\lambda$, assume that $g = \tfrac{(1-\varsigma)\epsilon}{\sqrt{n}}$, and there are scalars $l_{i}(>1)$ and $K_{i}(>\lambda)$ such that

where $\Omega_{ij} = |s_{ij}|\mathrm{e}^{\lambda\varepsilon_{ij}}+\tfrac{\hat{\Psi}_{i}\left(\bar{a}_{ij}P_{j}+\bar{b}_{ij}Q_{j}\mathrm{e}^{\lambda\hat{\eta}_{ij}}\right)}{K_{i}-\lambda}$. Then,

$\bf (i)$ $|e_i(t)|\leq l_i\|\phi\|_{\rho}\mathrm{e}^{-\lambda t}+g$ for any $t\geq -\rho$ and $i\in\langle n\rangle$;

$\bf (ii)$ The controller (3.1) can guarantee that the neutral-type memristive CGNNs (2.1) and (2.3) achieve accuracy-preassigned finite-time exponential synchronization within the settling time:

where $L = \max\limits_{i\in \langle n\rangle}l_{i}$.

Proof. In the proof, we suppose that $\phi_{i}\in C^{1}([-\rho, 0], \mathbb{R}), \ i\in\langle n\rangle$ are finite but can be chosen arbitrarily, and $e_{i}(t)\ (i\in\langle n\rangle)$ is the corresponding solution of the error system (2.4).

(ⅰ) It is obvious when $t\in [-\rho, 0$]. We assert that (ⅰ) is correct for $t\geq-\rho$; otherwise, there exist $T > 0$ and $v\in\langle n\rangle$, satisfying

For any given $s\in[0, T]$, we can obtain from (3.7) and (3.8), $\varepsilon_{vj} > 0$, $\pi_{v}(\cdot)\in [\check{\pi}_{v}, \hat{\pi}_{v}]$ and $\eta_{vj}(\cdot)\in [\check{\eta}_{vj}, \hat{\eta}_{vj}]$ that

Noting that

we have from (3.9) that

$\partial_{v1}(T)\leq\sum\limits_{j\in\langle n \rangle}|s_{vj}|\mathrm{e}^{\lambda\varepsilon_{vj}}l_{j}\|\phi\|_{\rho}\mathrm{e}^{-\lambda T}+g\sum\limits_{j\in\langle n \rangle}|s_{vj}|$,

$\partial_{v2}(T)\leq\Psi_{v}\tfrac{1}{K_{v}-\lambda}l_{v}\|\phi\|_{\rho}\mathrm{e}^{-\lambda T}+g\Psi_{v}\tfrac{1}{K_{v}}$,

$\partial_{v3}(T)\leq\Phi_{v}\tfrac{\mathrm{e}^{\lambda\hat{\pi}_{v}}}{K_{v}-\lambda}l_{v}\|\phi\|_{\rho}\mathrm{e}^{-\lambda T}+g\Phi_{v}\tfrac{1}{K_{v}}$,

$\partial_{v4}(T)\leq\hat{\Psi}_{v}\sum\limits_{j\in\langle n \rangle}\tfrac{\bar{a}_{vj}P_{j}}{K_{v}-\lambda}l_{j}\|\phi\|_{\rho}\mathrm{e}^{-\lambda T}+g\hat{\Psi}_{v}\sum\limits_{j\in\langle n \rangle}\bar{a}_{vj}P_{j}\tfrac{1}{K_{v}}$,

$\partial_{v5}(T)\leq\hat{\Psi}_{v}\sum\limits_{j\in\langle n \rangle}\tfrac{\bar{b}_{vj}Q_{j}\mathrm{e}^{\lambda\hat{\eta}_{vj}}}{K_{v}-\lambda}l_{j}\|\phi\|_{\rho}\mathrm{e}^{-\lambda T}+g\hat{\Psi}_{v}\sum\limits_{j\in\langle n \rangle}\bar{b}_{vj}Q_{j}\tfrac{1}{K_{v}}$.

It follows from (3.2) that

This, combined with (3.5c), means that $|e_{v}(T)| < l_{v}\|\phi\|_{\rho}\mathrm{e}^{-\lambda T}+g$, which contradicts (3.8). Therefore, (ⅰ) holds.

(ⅱ) When $t > \hat{T}$, one has from (ⅰ) and (3.6) that

hence,

In accordance with Definition 1, the controller (3.1) can guarantee that the neutral-type memristive CGNNs (2.1) and (2.3) achieve accuracy-preassigned finite-time exponential synchronization within the settling time given by (3.6). □

Remark 5. Theorem 2 presents the existence and design approach of a novel controller that can be used to establish accuracy-preassigned finite-time exponential synchronization between the drive-response neutral-type memristive CGNNs (2.1) and (2.3).

Remark 6. The error system (2.4) contains multiple delays and some coupling terms of two or three functions, which becomes hard for synchronizing CGNNs (2.1) and (2.3). To overcome this difficult problem, we design controller (3.1) to deal with the neutral delays and introduce the parameter $\varsigma$ to eliminate the couplings among amplification functions, self-signal functions, and activation functions.

Remark 7. As mentioned in the Introduction section, the positive number $\epsilon$ is viewed as a preassigned accuracy, which has to be satisfied considering a finite-time synchronization issue. Thus, $\epsilon$ is usually a small positive number, which implies that $\ln\tfrac{\varsigma\epsilon}{\sqrt{n}} < 0$, and hence for given initial functions, the feasibility of (3.5a) can be easily guaranteed. Then, (3.5b) and (3.5c) can be satisfied by choosing appropriately the positive numbers $l_i$ and $K_i$, $i\in\langle n\rangle$.

Noting that the nonlinear inequalities in (3.5) are not convenient to solve, we provide the following conclusion:

Corollary 1. For given positive constants $\varsigma$, $\epsilon$, and $\lambda$, assume that there are positive numbers $\tau$, $r_{i}$, and $h_{i}$ ($i\in\langle n\rangle$) such that

Then, the controller (3.1) with $K_{i} = h_{i}r_{i}^{-1}\ (i\in\langle n\rangle)$ can ensure that the neutral-type memristive CGNNs (2.1) and (2.3) achieve accuracy-preassigned finite-time exponential synchronization within the settling time given in (3.6).

Proof. For any $i\in\langle n\rangle$, it follows from (3.10) and $K_{i} = h_{i}r_{i}^{-1}$ that (3.5c) holds, and

and hence,

that is,

Furthermore,

that is,

Thus, there exist $l_{i} > 1$ ($i\in\langle n\rangle$) such that (3.5a) and (3.5b) hold. By Theorem 2, the controller (3.1) with $K_{i} = h_{i}r_{i}^{-1}\ (i\in\langle n\rangle)$ can guarantee that the neutral-type memristive CGNNs (2.1) and (2.3) achieve accuracy-preassigned finite-time exponential synchronization within the settling time given in (3.6). □

Remark 8. From Theorem 2 and Corollary 1, one can conclude that the positive numbers $l_{i}$ ($i\in\langle n\rangle$) in the settling time can be obtained based on the following steps:

$\bf Step\; 1.$ Take appropriate values of positive numbers $\varsigma$, $\epsilon$, and $\lambda$;

$\bf Step\; 2.$ Find positive numbers $\tau$, $r_{i}$ and $h_{i}$ ($i\in\langle n\rangle$) such that the linear scalar inequalities in (3.10) are feasible;

$\bf Step\; 3.$ Calculate the controller gains $K_i$ by $K_{i} = h_{i}r_{i}^{-1}$, $i\in\langle n\rangle$;

$\bf Step\; 4.$ Find positive numbers $l_{i}$ ($i\in\langle n\rangle$) such that the scalar inequalities in (3.5a) and (3.5b) are feasible.

When $s_{ij} = 0$, $\pi_{i}(t)\equiv0$, and $\eta_{ij}(t) = \eta_{j}(t)$ for all $t\ge 0$ and $i, j\in\langle n\rangle$, the NN models (2.1) and (2.3) simplify to memristive CGNNs with time-varying delays:

and

respectively. Furthermore, the error system is as follows:

where

and $K_{i} > 0$ stand for the controller gains.

Now, we can obtain the following conclusion.

Corollary 2. For given positive constants $\varsigma$, $\epsilon$, and $\lambda$, assume that there are positive numbers $\tilde{l}_i$ and $\tilde{h}_i$, such that

Then, the controller (3.15) with $K_{i} = \tilde{h}_i\tilde{l}_i^{-1}\ (i\in\langle n\rangle)$ can guarantee that the memristive CGNNs (3.12) and (3.13) achieve accuracy-preassigned finite-time exponential synchronization within the settling time given in (3.6) with $L = \max_{i\in\langle n\rangle}(\tilde{l}_i+1)$.

Proof. In light of (3.16a) and $K_{i} = \tilde{h}_i\tilde{l}_i^{-1}\ (i\in\langle n\rangle)$, we have

Set $l_i = \tilde{l}_i+1$, $i\in\langle n\rangle$. Then,

that is,

where $\tilde{\Omega}_{ij} = \tfrac{\hat{\Psi}_{i}\left(\bar{a}_{ij}P_{j}+\bar{b}_{ij}Q_{j}\mathrm{e}^{\lambda\hat{\eta}_{j}}\right)}{K_{i}-\lambda}$. By Theorem 2, (3.16b) and (3.16c), the controller (3.15) with $K_{i} = \tilde{h}_i\tilde{l}_i^{-1}\ (i\in\langle n\rangle)$ can guarantee that the memristive CGNNs (3.12) and (3.13) achieve accuracy-preassigned finite-time exponential synchronization within the settling time given in (3.6) with $L = \max_{i\in\langle n\rangle}(\tilde{l}_i+1)$. □

Remark 9. In ^[39], finite-time synchronization criteria of memristive CGNNs (3.12) and (3.13) are derived by employing the differential inclusion theory and constructing the appropriate nonlinear transformations. Compared with the results obtained in ^[39], Corollary 2 provides a simpler synchronization criterion. In addition, our method can cause the states of the error system to move directly to the preassigned range containing the origin, which avoids the two-stage procedures required in ^[39].

4.   Numerical examples

To verify the applicability of the results obtained in this article, the following two examples are provided.

Example 1. In neutral-type memristive CGNNs (2.1) and (2.3), we choose $n = 2$, and

$\psi_{1}(\cdot) = 0.01\sin^2(\cdot)+0.01$, $\psi_{2}(\cdot) = 0.01\cos^2(\cdot)+0.02$,

$\varphi_{1}(\cdot) = 0.5+0.1\sin(\cdot)$, $\varphi_{2}(\cdot) = 0.7+0.1\cos(\cdot)$,

$[\check{a}_{ij}]_{2\times 2} = \begin{bmatrix}2.5 & -0.1\\ -1.9 & 2\end{bmatrix}$, $[\hat{a}_{ij}]_{2\times 2} = \begin{bmatrix}2.3 & -0.13\\ -1.7 & 1.8\end{bmatrix}$,

$[\check{b}_{ij}]_{2\times 2} = \begin{bmatrix}-1.1 & 0.22\\ 1.15 & -3.5\end{bmatrix}$, $[\hat{b}_{ij}]_{2\times 2} = \begin{bmatrix}-1.3 & 0.3\\ 1.2 & -3.7\end{bmatrix}$,

$[s_{ij}]_{2\times2} = \begin{bmatrix} 0.1 & 0.02 \\ -0.03 & 01 \end{bmatrix}$, $\varpi_1 = 0.3$, $\varpi_2 = 0.2$,

$\varepsilon_{11} = 0.01$, $\varepsilon_{12} = 0$, $\varepsilon_{21} = 0.02$, $\varepsilon_{22} = 0.03$,

$\pi_1(\cdot) = 0.1|\sin(\cdot)|$, $\pi_2(\cdot) = 0.1|\cos(\cdot)|$,

$\eta_{11}(t) = \tfrac{\mathrm{e}^t}{1+10\mathrm{e}^t}$, $\eta_{12}(t) = \tfrac{\mathrm{e}^t}{1+20\mathrm{e}^t}$, $\eta_{21}(t) = \tfrac{\mathrm{e}^t}{1+30\mathrm{e}^t}$, $\eta_{22}(t) = \tfrac{\mathrm{e}^t}{1+40\mathrm{e}^t}$,

$p_{j}(v) = q_{j}(v) = \tfrac{0.1}{1+\mathrm{e}^{-v}}$, $j = 1, 2$, $t\ge 0$, $v\in\mathbb{R}$,

$\phi_1^x(s)\equiv0.6$, $\phi_2^x(s)\equiv-0.4$, $\phi_1^y(s)\equiv-0.6$, $\phi_2^y(s)\equiv-0.4$, $s\in[-0.1, 0]$.

It is clear that $\hat{\varepsilon} = 0.03$, $\rho = \hat{\pi} = \hat{\eta} = 0.1$, $\hat{\Psi}_{1} = 0.02$, $\check{\Psi}_{1} = 0.01$, $\hat{\Psi}_{2} = 0.03$, $\check{\Psi}_{2} = 0.02$, $\bar{a}_{11} = 2.5$, $\bar{a}_{12} = 0.13$, $\bar{a}_{21} = 1.9$, $\bar{a}_{22} = 2$, $\bar{b}_{11} = 1.3$, $\bar{b}_{12} = 0.3$, $\bar{b}_{21} = 1.2$, $\bar{b}_{22} = 3.7$, and $\|\phi\|_\rho = 1.2$. Set $\Phi_1 = 0.002$, $\Phi_2 = 0.003$, $\Psi_1 = 0.006$, $\Psi_2 = 0.008$, and $P_{i} = \hat{P}_{i} = Q_{i} = \hat{Q}_{i} = 0.1$, $i\in\langle 2\rangle$. Then, assumptions $A_{1}$ and $A_{2}$ are satisfied. For $\varsigma = 0.001$, $\lambda = 1.5$, and $\epsilon = 0.04$, using the software tool YALMIP, one can gain the following feasible solution to the inequalities in (3.10):

hence, $K_{1} = 1.9494$ and $K_{2} = 1.6471$. Then, it can be obtained that $l_1 = 1.5299$ and $l_2 = 1.6353$ by solving the inequalities in (3.5a) and (3.5b), hence, $L = 1.6353$ and $\hat{T} = 7.4316$. By Corollary 1, the controller (3.1) can guarantee that the neutral-type memristive CGNNs (2.1) and (2.3) achieve accuracy-preassigned finite-time exponential synchronization within the settling time $\hat{T}$.

Under the designed controller (3.1), the state trajectories of the considered neutral-type memristive CGNNs are presented in Figures 1 and 2, and the corresponding errors and their norms are given in Figures 3 and 4. From these four figures, it can be observed that the errors with the controller exhibit a gradual convergence towards zero. In addition, without the controller, Figures 5 and 6 illustrate the state trajectories of the considered neutral-type memristive CGNNs. Consequently, to resolve the synchronization issue, it is of paramount importance to design an appropriate controller to ensure that the states of considered error system gradually reduce the origin.

It is seen from (3.6) that the settling time $\hat{T}$ is affected by parameter $\varsigma$, the preassigned accuracy $\epsilon$, the norm $\|\phi\|_\rho$ of initial function, and the decay rate $\lambda$, which is explained by Tables 1–4. From which, it is concluded that the settling time $\hat{T}$ is positively dependent on the norm $\|\phi\|_\rho$ of initial function, and negatively dependent on parameter $\varsigma$, the preassigned accuracy $\epsilon$, or the decay rate $\lambda$.

Example 2. In memristive CGNNs (3.12) and (3.13), we choose $n = 2$, and

$\psi_{1}(\cdot) = 0.15-0.01\sin(\cdot)$, $\psi_{2}(\cdot) = 0.03\cos^2(\cdot)$,

$\varphi_{1}(\cdot) = 0.2+0.1\sin(\cdot)$, $\varphi_{2}(\cdot) = 0.2+0.1\cos(\cdot)$,

$\varpi_1 = 0.3$, $\varpi_2 = 0.2$,

$[\check{a}_{ij}]_{2\times 2} = \begin{bmatrix}2.2 & -0.12\\ -1.9 & 2\end{bmatrix}$, $[\hat{a}_{ij}]_{2\times 2} = \begin{bmatrix}2.0 & -0.15\\ -2.1 & 1.8\end{bmatrix}$,

$[\check{b}_{ij}]_{2\times 2} = \begin{bmatrix}-1.15 & 0.24\\ 1.05 & -2.5\end{bmatrix}$, $[\hat{b}_{ij}]_{2\times 2} = \begin{bmatrix}-0.95 & 0.2\\ 0.95 & -2.7\end{bmatrix}$,

$p_{j}(\cdot) = q_{j}(\cdot) = 0.1\tanh(\cdot)$, $\eta_{j}(t) = \tfrac{\mathrm{e}^t}{1+10\mathrm{e}^t}$, $j = 1, 2$, $t\ge 0$,

$\phi_1^x(s)\equiv6$, $\phi_2^x(s)\equiv-4$, $\phi_1^y(s)\equiv-6$, $\phi_2^y(s)\equiv-4$, $s\in[-0.1, 0]$.

It is clear that $\rho = \hat{\eta}_{j} = 0.1$, $\hat{\Psi}_{1} = 0.16$, $\check{\Psi}_{1} = 0.14$, $\hat{\Psi}_{2} = 0.03$, $\check{\Psi}_{2} = 0$, $\bar{a}_{11} = 2.2$, $\bar{a}_{12} = 0.15$, $\bar{a}_{21} = 2.1$, $\bar{a}_{22} = 2.0$, $\bar{b}_{11} = 1.15$, $\bar{b}_{12} = 0.24$, $\bar{b}_{21} = 1.05$, $\bar{b}_{22} = 2.7$, and $\|\phi\|_\rho = 12$. Set $\Phi_1 = 0.016$, $\Phi_2 = \Psi_1 = 0.003$, $\Psi_2 = 0.009$ and $P_{i} = \hat{P}_{i} = Q_{i} = \hat{Q}_{i} = 0.1$, $i\in\langle 2\rangle$. Then, assumptions $A_{1}$ and $A_{2}$ are satisfied. For $\varsigma = 0.001$, $\lambda = 1.4$, and $\epsilon = 0.04$, using the software tool YALMIP, one can gain the following feasible solution to the inequalities in Corollary 2:

hence, $K_{1} = 1.6536$, $K_{2} = 1.9809$, $L = 0.8581$, and $\hat{T} = 9.6983$. By Corollary 2, the controller (3.15) can guarantee that the considered memristive CGNNs achieve accuracy-preassigned finite-time exponential synchronization within the settling time $\hat{T}$.

Under the designed controller (3.15), the state trajectories of the considered memristive CGNNs are presented in Figures 7 and 8, and the corresponding errors and their norms are given in Figures 9 and 10. From these four figures, it can be observed that the errors with the controller exhibit a gradual convergence towards zero. In addition, without the controller, Figures 11 and 12 illustrate the state trajectories of the considered memristive CGNNs. Consequently, to resolve the synchronization issue, it is of paramount importance to design an appropriate controller to ensure that the states of the considered error system gradually reduce the origin.

5.   Conclusions

For a class of neutral-type memristive CGNNs with time-varying multiple leakage and transmission delays, we examine the problem of accuracy-preassigned finite-time exponential synchronization control. By proposing an upper-right derivative-based direct method, a novel controller that is related to neutral delays is presented to achieve accuracy-preassigned finite-time exponential synchronization of the drive and response neutral-type memristive CGNNs. The investigated synchronization conditions are composed of several scalar inequalities that can be checked via the standard software tools. Finally, two numerical examples present the effectiveness of the designed controller. This study fills the gap by presenting sufficient conditions guaranteeing the accuracy-preassigned finite-time exponential synchronization of neutral-type memristive CGNNs with multiple time-varying leakage delays and transmission delays.

Author contributions

Er-Yong Cong: Writing-review & editing, writing-original draft, validation, investigation, conceptualization; Yantao Wang: Writing-review & editing, resources, methodology, conceptualization; Xian Zhang: Writing-review & editing, resources, methodology, conceptualization; Li Zhu: Writing-review & editing, visualization, software, investigation, funding acquisition. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported in part by the Natural Science Foundation of Heilongjiang Province (No. LH2022F046, No. YQ2022F015, and No. PL2024G011), the Project for Research Foundation of Young Doctor of Harbin University (No. HUDF2022112), and Harbin Science and Technology Plan Project (No. 2023ZCZJCG045 and No. 2023ZCZJCG046).

Conflict of interest

All authors declare no conflicts of interest in this paper.