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Research article Special Issues

Fixed-time synchronization of time-delayed fuzzy memristor-based neural networks: A special exponential function method

  • This research focuses on controlling fixed-time synchronization (FixTS) in a time-delayed fuzzy memristor-based neural network (TDFMNN). To achieve FixTS and improve convergence speed, a discontinuous state feedback controller (StFC) incorporating a unique exponential function is developed for the memristor-based neural network (MNN) drive-response system (DRS). The FixTS is analyzed using the indefinite derivative Lyapunov function approach. The proposed TDFMNN incorporates multiple factors, such as memristor, time-variation of coefficients, time-delay, and fuzzy elements, making the model more practical compared to existing studies. The StFC is designed using the fixed-time stability (FixS) theory of the exponential function, which requires few parameters, thereby simplifying controller design and implementation. Numerical simulations are conducted to evaluate the performance of the proposed control strategy on two-dimensional and three-dimensional TDMNNs.

    Citation: Yan Chen, Ravie Chandren Muniyandi, Shahnorbanun Sahran, Zuowei Cai. Fixed-time synchronization of time-delayed fuzzy memristor-based neural networks: A special exponential function method[J]. Electronic Research Archive, 2025, 33(6): 3517-3542. doi: 10.3934/era.2025156

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  • This research focuses on controlling fixed-time synchronization (FixTS) in a time-delayed fuzzy memristor-based neural network (TDFMNN). To achieve FixTS and improve convergence speed, a discontinuous state feedback controller (StFC) incorporating a unique exponential function is developed for the memristor-based neural network (MNN) drive-response system (DRS). The FixTS is analyzed using the indefinite derivative Lyapunov function approach. The proposed TDFMNN incorporates multiple factors, such as memristor, time-variation of coefficients, time-delay, and fuzzy elements, making the model more practical compared to existing studies. The StFC is designed using the fixed-time stability (FixS) theory of the exponential function, which requires few parameters, thereby simplifying controller design and implementation. Numerical simulations are conducted to evaluate the performance of the proposed control strategy on two-dimensional and three-dimensional TDMNNs.



    The memristor notion was initially developed by Chua et al.[1] in a study of circuit variables symmetry and logic. However, it became popular in 2008, with the production of the first memristor component in Hewlett-Packard Labs [2]. Memristor circuits demonstrate many important characteristics, such as low power consumption, large memories, nano-scale structure, and non-volatility, making them suitable for numerous applications, including volatile memory, secure communication, analog circuits, image processing, and cryptography. Biological neural network systems inspire artificial neural networks, which are crafted to replicate and simulate the brain's neural structure and learning mechanisms [3]. As the most suitable elements to simulate neuronal synapses [4], using memristors in neural networks not only reduces hardware overhead, system size, and energy consumption, but also enhances computing speed [4]. This innovation, known as "memristor-based neural networks", outperforms traditional networks in speed, data storage, and performance across various applications [5]. MNNs are considered state-dependent nonlinear switching systems [6]. Investigating their dynamical behavior is crucial for numerous engineering applications. The majority of the current studies focus on stability and synchronization problems to facilitate the application of MNNs [7].

    The phenomenon of time delay is unavoidable when studying real network systems [5]. The occurrence of time delays is influenced by both the system's current state and its previous state, which is a more obvious feature in neural networks [8]. There are many reasons for the time delay phenomenon, including the limited signal transmission time between neurons and the insufficient speed of actuators in processing signals [9]. Time delay is one of the most difficult issues to deal with and is widely present in the system, but difficult to eliminate. Therefore, the phenomenon of time delay has attracted significant attention, and extensive research has been undertaken into the delay-dependent stability of delayed systems.[10] proposes a delay-dependent switched system to analyze the stability of delayed systems. The delay in this study [11] addresses the passivity issue in neural networks. This method allows us to incorporate more delay information to construct a novel Lyapunov–Krasovskii functional. Therefore, considering time lags is crucial when examining the dynamics of neural networks. This makes the development of time-delay memristor-based neural network (TMNN) models highly relevant for the dynamic analysis and control of MNNs[12]. The MNN model commonly includes a time-delay term to account for the signal delay due to the effect of switching and the current impedance of the circuit components [13]. The utilization of a TMNN often relies on its synchronization, multi-stability, and other associated dynamic phenomena [14]. As a nonlinear system, the TMNN exhibits complex nonlinear dynamic behaviors. The presence of time lags increases the likelihood of system instability, thereby complicating the implementation of system synchronization and rendering the synchronization process more challenging[15]. The synchronization ensures the interaction between the network nodes and the eventual convergence of their dynamical behaviors to the same state [16].

    The synchronization behavior of TMNNs can be classified based on time taken to achieve the synchronization into, infinite-time synchronization (InfTS), finite-time synchronization (FinTS), and FixTS [17]. In the InfTS method, the time required to achieve synchronization cannot be estimated. Notable examples of InfTS synchronization include asymptotic [18] and exponential synchronization [19] of MNNs. However, in many situations, it is important to consider the control cost or time required to achieve system synchronization. FinTS refers to the stability of the error state between the DRS within a finite time. This consideration has led to the introduction of the FinTS method, wherein the duration to reach the synchronization can be estimated [20]. The finite time represents the maximum limit established by the estimated time function, which is determined by the system's initial states (InSts). It corresponds to a synchronization duration achievable within an estimated time range. Nevertheless, the FinTS method is not appropriate for systems whose initial conditions are not readily available [21]. The study by Polyakov [22] introduced a technique to determine if a system can achieve FixS. FixTS, on the other hand, indicates that the system synchronizes within a finite time, with the settling time being bounded concerning the state error, and this upper bound is independent of the InSts of the error system. This technique paved the way for a method of stability analysis that is not reliant on the system's InSts. Therefore, the FixS analysis approach can be applied to FixTS control method. The FixTS analysis method is a special form of FinTS that is not dependent on the InSts of the system and produces a bounded synchronization time (SyTi). FixTS is of great importance and has numerous practical applications. Due to its strict guarantee on convergence time, FixTS offers significant advantages in safety-critical, dynamically complex, or real-time demanding applications, such as image encryption, aerospace, smart grids, and autonomous driving. However, the issue of FixTS remains relatively immature with many queries requiring further exploration, especially the research on FixTS application in TMNNs.

    Recently, many studies considered the characteristics of time delay [23], oscillation [24], and parameter uncertainties [25] to achieve more practical applications with MNNs. However, building an accurate mathematical model representation of an actual engineering application involves a daunting task, due to the fact that ignoring the system's parameter uncertainties can produce adverse effects on the system's dynamics convergence and local stability. Fuzzy logic (FL), proposed by Lotfi Zadeh in 1965, was designed to handle problems involving uncertainty and ambiguity, making it well-suited for nonlinear systems or those that are difficult to model precisely, such as human language or complex control systems. Integrating memristors into FL systems has become a highly prominent research frontier. On one hand, memristors provide compact, energy-efficient, and highly nonlinear components that can physically implement fuzzy operators, store fuzzy rule weights, or emulate synaptic connections in fuzzy neural networks (FNNs). On the other hand, embedding FL into memristive circuits enables the realization of intelligent control strategies capable of addressing system uncertainties, parameter variations, and environmental disturbances. Therefore, this integration opens up new avenues for developing intelligent circuits and systems with enhanced learning, adaptability, and decision-making capabilities, and it is regarded as a promising research direction in the fields of neuromorphic engineering, intelligent control, and hardware-software co-design. The inclusion of fuzzy components enhances the universality and practicality of research on the robust adaptive synchronization of the MNN [26]. This is achieved by addressing the nonlinear properties of the memristor and optimizing the parameter design of the MNN. Therefore, the integration of fuzzy systems holds substantial value in exploring the dynamic analysis of MNNs in the presence of parameter perturbations. Contrary to research on the synchronization of FNNs [27,28,29] and MNNs, fewer studies have been done on fuzzy MNNs. Given the importance of synchronization behavior, further research on the synchronization of FMNNs is essential [30]. The quality of a control method can be assessed in terms of how quickly it can achieve synchronization. The SyTi of asymptotic synchronization tends to infinity, while the SyTi of FinTS has an upper bound. The FinTS control system may alter control settings in real-time according to the calculated upper limit of the SyTi, enabling the system to quickly achieve synchronization. However, since the upper bound of the SyTi required for FinTS is related to the InSts of the system, it imposes considerable limitations in practical applications.

    Compared with the FinTS technique, the convergence time upper limit of the FixTS approach is solely dependent on the control parameters of the system and is independent of InSts. This independence makes it more convenient for practical applications in production and daily life. Therefore, research on the FixTS of TDFMNNs holds more practical significance. However, there is currently limited related research in the literature [31,32,33], indicating an insufficient resolution to the TDFMNN problem. Most existing FixTS methods involve complex convergence time estimation and extensive computation. This gap motivates us to develop simpler, more effective, and diverse FixTS convergence time estimates. The FixTS control of TDFMNNs remains underexplored, prompting our research.

    The innovation of this paper lies in the proposal of TDFMNNs that consider additional objective factors, such as time-delay effects, fuzzy terms, and memristors. The FixS theory based on exponential functions is applied, involving fewer parameters, which makes the controller easier to design and implement. A special exponential function term is introduced in the modified controller, enabling easier realization of FixTS, with a simpler estimation formula for the convergence time.

    This article tries to provide novel FixTS standards for controlling TDFMNNs. The main contributions of this work are summarized as below.

    (ⅰ) Solved the FixTS problem of TDFMNNs by designing a switching type discontinuous StFC. The inclusion of a special exponential function term in the controller significantly differs from existing power function form controllers, allowing for easier implementation and achieving FixTS more effectively.

    (ⅱ) Proposed the TDFMNN model, which integrates multiple factors including nonlinearity, time-variation of coefficients, time-lag, and FL elements, thereby enhancing the model's practicality compared to previous studies [27,28,34,35].

    (ⅲ) The StFC controller adopted in this paper utilizes the FixS theory of the exponential function, requiring fewer parameters for controller design and implementation, thus simplifying the process.

    (ⅳ) Introduced a novel estimation formula for synchronization convergence time, which is simpler and more practical compared to existing methods, facilitating a more accurate assessment of convergence behaviour.

    The structure of the article is as follows: Section 2 delves into the DRS, which are the models under consideration. In addition, the study included the provision of definitions and assumptions, as well as the introduction of valuable lemmas. Section 3 incorporates the FixS theory, and some FixTS criteria and controller designs for TDFMNNs were proposed through rigorous mathematical proof. Section 4 introduces two numerical simulation examples. The fifth section provides a series of findings from this article and proposes some future research work.

    Notations: R represents the real field; Rn consists of all n-dimensional real vectors; sign() is the sign function; i,jN={1,2,...,n}; and ¯co[E] is the convex closure of E. Let i,jRn, (x,y) be the inner product of x and y, and V represent the generalized gradient of V.

    Consider the TDFMNNs

    dxi(t)dt=ci(t)xi(t)+nj=1aij(xi(t))Fj(xj(t))+nj=1bij(xi(t))Gj(xj(tσ))+nj=1Eijvj+nj=1Tijvj+nj=1DijFj(xj(tσ))+nj=1HijFj(xj(tσ))+nj=1Sijvj+Ii, (1)

    for i=1,2,...,n, where n represents the number of neural network units, ci(t)>0 reflects the rate by which the potentials will be reset to the equilibrium state in isolation, xi(t) represents the state variables (StVas) connected to the ith neuron, Fj(xj(t)),Gj(xj(tσ)) and Fj(xj(tσ)) denote the outputs, represent the fuzzy "AND" and fuzzy "OR" operations, Dij and Hij represent elements of the fuzzy feedback Min and Max templates, respectively, Tij and Sij denote the fuzzy feed-forward Min template and Max template, respectively, the fuzzy feed-forward template is Eij, vi and Ii are the input and bias, respectively and the memristor-based weights aij(xi(t)) and bij(xi(t)) can be defined as follows:

    aij(xi(t))={ˆaij,|xi(t)|ri,ˇaij,|xi(t)|>ri, (2)
    bij(xi(t))={ˆbij,|xi(t)|ri,ˇbij,|xi(t)|>ri, (3)

    where ri>0 is a switching jump, and ˆaij,ˇaij,ˆbij,ˇbij are constants associated with memristors.

    Lemma 2.1. (Refer to Corollary 1 of [36]). Assume that k, and let l>0, 0<q<1 be given constants. Suppose there exists a locally Lipschitz continuous and C-regular function V:RnR+ such that V(0)=0, and the following conditions are satisfied:

    dVdtlexp(Vq)V1q.

    The zero point of the equation remains stable at a constant time, and the convergence time is estimated as Γ(t0,x0)t0+Γmax,Γmax=1ql.

    Lemma 2.2. (Chain Rule [37,38]). If V:RnR is C- regular and x(t):[t0,+)Rn has absolute continuity, then

    dV(x(t))dt=<(t),dx(t)dt>,(t)V(x(t)).

    Lemma 2.3. (Refer to [39]). Let x and ˜x denote the states of system (1). Then, we can conclude that:

    |nj=1DijFj(xj)nj=1DijFj(˜xj)|nj=1|Dij||Fj(xj)Fj(˜xj)|,
    |nj=1HijFj(xj)nj=1HijFj(˜xj)|nj=1|Hij||Fj(xj)Fj(˜xj)|.

    To reach the conclusion of this article, the following conditions are considered.

    (A1) The functions Fj and Gj satisfy Fj(0)=Gj(0)=0, x,yR, and we can find Pj>0,Qj>0,j=1,2,,n satisfying

    Pj|xy||Fj(x)Fj(y)|,Qj|xy||Gj(x)Gj(y)|. (4)

    (A2) For any pair of distinct real numbers x and y, we can find Lj>0 and Mj>0 that satisfy the following inequality:

    |Fj(x)|Lj,|Gj(x)|Mj. (5)

    Utilizing the Filippov-framework theorem, it is deduced from (1) that

    dxi(t)dtci(t)xi(t)+nj=1¯co[aij(xi(t))]Fj(xj(t))+nj=1¯co[bij(xi(t))]Gj(xj(tσ))+nj=1Eijvj+nj=1Tijvj+nj=1DijFj(xj(tσ))+nj=1HijFj(xj(tσ))+nj=1Sijvj+Ii, (6)

    where

    ¯co[aij(xi(t))]={ˆaij,ri>|xi(t)|,[a_ij,ˉaij],ri=|xi(t)|,ˇaij,ri<|xi(t)|, (7)
    a_ij=min{ˆaij,ˇaij},¯aij=max{ˆaij,ˇaij},

    and

    ¯co[bij(xi(t))]={ˆbij,ri>|xi(t)|,[b_ij,ˉbij],ri=|xi(t)|,ˇbij,ri<|xi(t)|, (8)
    b_ij=min{ˆbij,ˇbij},¯bij=max{ˆbij,ˇbij}.

    For i,jN, we can find αij(t)¯co[aij(xi(t))] and βij(t)¯co[bij(xi(t))] for a.e. t0 to make

    dxi(t)dt=ci(t)xi(t)+nj=1αij(t)Fj(xj(t))+nj=1βij(t)Gj(xj(tσ))+nj=1Eijvj+nj=1Tijvj+nj=1DijFj(xj(tσ))+nj=1HijFj(xj(tσ))+nj=1Sijvj+Ii. (9)

    Applying the drive-response synchronization principle and employing (1) as the drive system (DrSy), the response system (ReSy) is constructed as

    dyi(t)dt=ci(t)yi(t)+nj=1aij(yi(t))Fj(yj(t))+nj=1bij(yi(t))Gj(yj(tσ))+nj=1Eijvj+nj=1Tijvj+nj=1DijFj(yj(tσ))+nj=1HijFj(yj(tσ))+nj=1Sijvj+Ii+ui(t), (10)

    where ui(t) is the proposed controller under development, while the rest of the parameters retain their dynamic interpretations as in (1), and

    aij(yi(t))={ˆaij,|yi(t)|ri,ˇaij,|yi(t)|>ri, (11)
    bij(yi(t))={ˆbij,|yi(t)|ri,ˇbij,|yi(t)|>ri. (12)

    From Eq (10), it can be deduced that with the implementation of the aforementioned principles:

    dyi(t)dtci(t)yi(t)+nj=1¯co[aij(yi(t))]Fj(yj(t))+nj=1¯co[bij(yi(t))]Gj(yj(tσ))+nj=1Eijvj+nj=1Tijvj+nj=1DijFj(yj(tσ))+nj=1HijFj(yj(tσ))+nj=1Sijvj+Ii+ui(t), (13)

    where

    ¯co[aij(yi(t))]={ˆaij,ri>|yi(t)|,[a_ij,ˉaij],ri=|yi(t)|,ˇaij,ri<|yi(t)|, (14)
    a_ij=min{ˆaij,ˇaij},¯aij=max{ˆaij,ˇaij},

    and

    ¯co[bij(yi(t))]={ˆbij,ri>|yi(t)|,[b_ij,ˉbij],ri=|yi(t)|,ˇbij,ri<|yi(t)|, (15)
    b_ij=min{ˆbij,ˇbij},¯bij=max{ˆbij,ˇbij}.

    For i,jN, we can find ˜αij(t)¯co[aij(yi(t))] and ˜βij(t)¯co[bij(yi(t))] for a.e. t0 satisfying

    dyi(t)dt=ci(t)yi(t)+nj=1˜αij(t)Fj(yj(t))+nj=1˜βij(t)Gj(yj(tσ))+nj=1Eijvj+nj=1Tijvj+nj=1DijFj(yj(tσ))+nj=1HijFj(yj(tσ))+nj=1Sijvj+Ii+ui(t). (16)

    The initial values of the Eqs (1) and (6) are ϕ(s)C([σ,0],Rn) and Ψ(s)C([σ,0],Rn), respectively.

    Here, a criterion is constructed to ensure the FixTS of the discontinuous DrSy (1) and the discontinuous ReSy (10).

    Let ei(t)=yi(t)xi(t). If x(t) and y(t) are two arbitrary solutions to (1) and (10), respectively, the error dynamic can be obtained as follows:

    dei(t)dt=ci(t)ei(t)+nj=1αij(t)˜Fj(ej(t))+nj=1βij(t)˜Gj(ej(tσ))+nj=1(˜αij(t)αij(t))Fj(yj(t))+nj=1(˜βij(t)βij(t))Gj(yj(tσ))+nj=1DijFj(yj(tσ))nj=1DijFj(xj(tσ))+nj=1HijFj(yj(tσ))nj=1HijFj(xj(tσ))+ui(t), (17)

    where

    ˜Fj(ej(t))=Fj(yj(t))Fj(xj(t)),
    ˜Gj(ej(tσ))=Gj(yj(tσ))Gj(xj(tσ)).

    Design a control strategy

    ui(t)=δiei(t)ωisign(ei(t))nj=1kij|ej(tσ)|sign(ei(t))Γ(t,e)|ei(t)|1qsign(ei(t)), (18)

    where

    Γ(t,e)=lexp((ni=1ei(t))q),

    δi,ωi, and kij are the control parameters, and 0<q<1,l>1 are positive constants.

    Remark 3.1. In the modified controller, a specially designed exponential function term is introduced, which is noticeably different from the conventional power-function-based controllers. This optimized controller enables FixTS more readily, and the estimation of the convergence time becomes simpler. From the perspective of neural network circuit implementation, such an optimized controller is also easier to realize and entails lower economic costs.

    The key findings of this research can be demonstrated as follows:

    Theorem 3.1. Given that (A1) and (A2) hold, if the conditions

    (i) δi+ci(t)nj=1aDjiPj0,

    (ii) ωinj=1|ˆaijˇaij|Ljnj=1|ˆbijˇbij|Mj0,

    (iii) kijbDijQj|Dij|Pj|Hij|Pj0,

    are satisfied, then the ReSy (10) is synchronized with the DrSy (1) in a fixed time under control strategy (18).

    Proof. Consider the Lyapunov function

    V(t)=ni=1|ei(t)|. (19)

    The derivative of V(t) can be defined as

    dV(t)dt=ni=1dei(t)dtsign(ei(t))=ni=1[ci(t)ei(t)+nj=1αij(t)˜Fj(ej(t))+nj=1βij(t)˜Gj(ej(tσ))+nj=1(˜αij(t)αij(t))Fj(yj(t))+nj=1(˜βij(t)βij(t))Gj(yj(tσ))+nj=1DijFj(yj(tσ))nj=1DijFj(xj(tσ))+nj=1HijFj(yj(tσ))nj=1HijFj(xj(tσ))+ui(t)]sign(ei(t))ni=1ci(t)|ei(t)|+ni=1nj=1|αij(t)||˜Fj(ej(t))|+ni=1nj=1|βij(t)||˜Gj(ej(tσ))||sign(ei(t))|+ni=1nj=1|˜αij(t)αij(t)||Fj(yj(t))||sign(ei(t))|+ni=1nj=1|˜βij(t)βij(t)|Gj(yj(tσ))||sign(ei(t))|+ni=1|nj=1DijFj(yj(tσ))nj=1DijFj(xj(tσ))||sign(ei(t))|+ni=1|nj=1HijFj(yj(tσ))nj=1HijFj(xj(tσ))||sign(ei(t))|+nj=1ui(t)|sign(ei(t))|. (20)

    It is shown from (A1, A2) that

    ni=1nj=1|αij(t)||˜Fj(ej(t))|ni=1nj=1aDjiPj|ei(t)|, (21)

    where

    aDji=max{|a_ji|,|¯aji|}.
    ni=1nj=1|βij(t)||˜Gj(ej(tσ))||sign(ei(t))|ni=1nj=1bDijQj|ej(tσ)||sign(ei(t))|, (22)

    where

    bDij=max{|b_ij|,|¯bij|}.
    ni=1nj=1|˜αij(t)αij(t)||Fj(yj(t))||sign(ei(t))|ni=1nj=1|ˆaijˇaij|Lj|sign(ei(t))|. (23)
    ni=1nj=1|˜βij(t)βij(t)||Gj(yj(tσ))||sign(ei(t))|ni=1nj=1|ˆbijˇbij|Mj|sign(ei(t))|. (24)
    ni=1|nj=1DijFj(yj(tσ))nj=1DijFj(xj(tσ))||sign(ei(t))|ni=1nj=1|Dij||Fj(yj(tσ))Fj(xj(tσ))||sign(ei(t))|ni=1nj=1|Dij|Pj|(ej(tσ))||sign(ei(t))|. (25)
    ni=1|nj=1HijFj(yj(tσ))nj=1HijFj(xj(tσ))||sign(ei(t))|ni=1nj=1|Hij||Fj(yj(tσ))Fj(xj(tσ))||sign(ei(t))|ni=1nj=1|Hij|Pj|(ej(tσ))||sign(ei(t))|. (26)

    Substituting (21)–(26) into (20), we obtain

    dV(t)dtni=1[ci(t)nj=1aDjiPj]|ei(t)|+ni=1nj=1bDijQj|ej(tσ)||sign(ei(t))|+ni=1nj=1(|ˆaijˇaij|Lj+|ˆbijˇbij|Mj)|sign(ei(t))|+ni=1nj=1|Dij|Pj|(ej(tσ))||sign(ei(t))|+ni=1nj=1|Hij|Pj|(ej(tσ))||sign(ei(t))|+ni=1ui(t)|sign(ei(t))|. (27)

    Substituting (18) into (27), we obtain

    dV(t)dtni=1[ci(t)nj=1aDjiPj]|ei(t)|+ni=1nj=1bDijQj|ej(tσ)||sign(ei(t))|+ni=1nj=1(|ˆaijˇaij|Lj+|ˆbijˇbij|Mj)|sign(ei(t))|+ni=1nj=1|Dij|Pj|(ej(tσ))||sign(ei(t))|+ni=1nj=1|Hij|Pj|(ej(tσ))||sign(ei(t))|ni=1δi|ei(t)|ni=1ωi|sign(ei(t))|ni=1nj=1kij|ej(tσ)|sign(ei(t))ni=1Γ(t,e)|ei(t)|1q|sign(ei(t))|=ni=1(δi+ci(t)nj=1aDjiPj)|ei(t)|ni=1(ωinj=1|ˆaijˇaij|Ljnj=1|ˆbijˇbij|Mj)|sign(ei(t))|ni=1nj=1(kijbDijQj|Dij|Pj|Hij|Pj)|(ej(tσ)||sign(ei(t))|ni=1Γ(t,e)|ei(t)|1q|sign(ei(t))|, (28)
    Γ(t,e)=lexp((ni=1ei(t))q).

    When conditions (ⅰ)–(ⅲ) are met, then

    dV(t)dtni=1Γ(t,e)|ei(t)|1q|sign(ei(t))|ni=1lexp((ni=1ei(t))q)|ei(t)|1q|sign(ei(t))|ni=1lexp((ni=1ei(t))q)|ei(t)|1q, (29)
    ni=1|ei(t)|1q(ni=1|ei(t)|)1q=V1q,

    and, therefore,

    dV(t)dtlexp(Vq)V1q,
    Γ(t0,e0)t0+Γmax2,Γmax2=1ql.

    Hence, ReSy (10) is synchronized with DrSy (1) in a fixed time under control strategy (18). The proof is complete.

    Remark 3.2. FinTS and FixTS can both achieve synchronization between DRS within a finite time, but they have fundamental differences in theory and application. The convergence time for FinTS depends on InSts, ensuring synchronization within a finite time, but the convergence time cannot be uniformly estimated. In contrast, FixTS completes synchronization within a uniform upper bound that is independent of InSts. FinTS is suitable for systems with known or controllable InSts, where controller design is relatively simple. However, it has unstable convergence times when faced with disturbances or frequent resets, making it less feasible in engineering applications. On the other hand, FixTS is more suitable for high-demand scenarios like aerospace and smart grids, where time and safety are critical, despite its more complex controller design.

    Remark 3.3. In this work, a switched discontinuous StFC is designed, which facilitates the achievement of FixTS. The inclusion of switched discontinuous terms effectively addresses the variability and uncertainty of Filippov solutions in the DRS. Naturally, switched discontinuous controllers also have drawbacks: they are prone to inducing oscillations, where even minor parameter perturbations may trigger such oscillatory behavior, which is detrimental to synchronization performance. This issue can be mitigated using optimization algorithms or by approximating the discontinuous functions with continuous ones.

    Remark 3.4. Reference [40] investigates the predetermined-time stability of memristive chaotic systems, while our study focuses on the FixTS problem of neural networks. In the controller design presented in [40], intermittent control is used, whereas our approach employs a switched StFC. Additionally, the fuzzy method applied in the two studies differs: Reference [40] utilizes the T-S fuzzy method, whereas we adopt the fuzzy 'AND' and fuzzy 'OR' methods.

    Remark 3.5. In Reference [41], the FixTS of fuzzy complex networks incorporating reaction-diffusion factors is studied, where intermittent control is used to achieve synchronization. The method employed involves power functions and sign functions, whereas our approach utilizes a special exponential function method to achieve FixTS.

    This section presents two numerical examples to validate the viability of the FixTS strategy introduced in the preceding section.

    Example 4.1. Consider the 2-D TDFMNNs (1) as the DrSy, and system (10) as the ReSy with parameters: c1(t)=c2(t)=1,I1=I2=sin(8t),D11=D12=D21=D22=H11=H12=H21=H22=2, σ=1,Lj=12,Mj=14,Pj=12,Qj=14.

    The neuron activations are defined as

    Fj(x)=0.5tanh(x),Gj(x)=0.25tanh(x),0.5|xy||Fj(x)Fj(y)|,0.25|xy||Gj(x)Gj(y)|.

    The InSts of the DrSy of (1) is given as

    ϕ(s)=(2,3)T,s[1,0].

    The InSts of the ReSy of (10) is given as

    Ψ(s)=(1,0.6)T,s[1,0].

    The memristor-based weights are given as follows:

    a11(x1(t))={2,|x1(t)|0.8,1,|x1(t)|>0.8,a12(x1(t))={1,|x1(t)|0.8,1.5,|x1(t)|>0.8,
    a21(x2(t))={1.2,|x2(t)|0.8,0.5,|x2(t)|>0.8,a22(x2(t))={1.5,|x2(t)|0.8,0.5,|x2(t)|>0.8.
    b11(x1(t))={3,|x1(t)|0.8,2,|x1(t)|>0.8,b12(x1(t))={2,|x1(t)|0.8,1,|x1(t)|>0.8,
    b21(x2(t))={2.5,|x2(t)|0.8,1,|x2(t)|>0.8.b22(x2(t))={0.5,|x2(t)|0.8,1,|x2(t)|>0.8.

    Choosing δ1=2,δ2=2,ω1=5,ω2=4,k11=k12=k21=k22=4,l=1,q=0.5, and ui(t) gives the following design:

    δ1+c1(t)2j=1aDj1Pj=1.40,δ2+c2(t)2j=1aDj2Pj=1.50,ω12j=1|ˆa1jˇa1j|Lj2j=1|ˆb1jˇb1j|Mj=3.750,ω22j=1|ˆa2jˇa2j|Lj2j=1|ˆb2jˇb2j|Mj=2.650,k11bD11Qj|D11|Pj|H11|Pj=1.250,k21bD21Qj|D21|Pj|H21|Pj=1.3750,k12bD12Qj|D12|Pj|H12|Pj=1.50,k22bD22Qj|D22|Pj|H22|Pj=1.750.

    As of right now, it is easily obtained that all circumstances (ⅰ)–(ⅲ) in Theorem 3.1 are satisfied.

    Due to this, in a fixed time, Eqs (1) and (10) can synchronize robustly. Moreover, from Theorem 3.1, Γmax can be determined as follows:

    Γ(t0,e0)t0+Γmax2,Γmax2=1ql=10.5=2.

    A numerical simulation is performed utilizing the MATLAB 2020B software to validate the theories. The phase-plane behavior of the TDFMNNs without the controller is shown in Figure 1 for the system's StVas, x1(t) and x2(t). Figures 2 and 3 illustrate the states of x1(t),y1(t) and x2(t),y2(t), respectively, under the controller's action for the controlled TDFMNNs system. Furthermore, Figure 4 illustrates the trajectories of the synchronization errors between the DrSy and the ReSy. It is evident from Figure 4 that the state of the ReSy, propelled by the synchronous controller, converges to that of the DrSy, with the convergence error remaining zero after Γmax. This observation confirms the effectiveness of the proposed controller in driving the system to achieve FixTS.

    Figure 1.  Phase plane of x1(t) and x2(t) for DrSy (1) in Example 4.1.
    Figure 2.  System states x1(t), y1(t) with control protocol in Example 4.1.
    Figure 3.  System state x2(t), y2(t) with control protocol in Example 4.1.
    Figure 4.  Synchronization error trajectories of DrSy (1) and corresponding ReSy(10) under the control strategy(18) in Example 4.1.

    Example 4.2. Consider a 3-D TDFMNNs described by system (1) as the DrSy and system (10) as the ReSy with parameters c1(t)=c2(t)=c3(t)=1,I1=I2=I3=cos(12t),Dij=Hij=110,σ=1,Lj=35,Mj=35,Pj=15,Qj=15.

    The neuron activations are defined as

    Fj(x)=15tanh(x)+25,0.2|xy||Fj(x)Fj(y)|,Gj(x)=15tanh(x)+25,0.2|xy||Gj(x)Gj(y)|.

    The InSts of the DrSy of (1) is given as

    ϕ(s)=(52,32,1)T,s[1,0].

    The InSts of the ReSy of (10) is given as

    Ψ(s)=(2,3,3)T,s[1,0].

    The memristor-based weights are as follows:

    a11(x1(t))={0.1,|x1(t)|0.3,0.3,|x1(t)|>0.3,a12(x1(t))={0.2,|x1(t)|0.3,0.3,|x1(t)|>0.3,a13(x1(t))={0.4,|x1(t)|0.30.3,|x1(t)|>0.3
    a21(x2(t))={0.7,|x2(t)|0.3,0.2,|x2(t)|>0.3,a22(x2(t))={0.7,|x2(t)|0.3,0.4,|x2(t)|>0.3,a23(x2(t))={0.6,|x2(t)|0.3,0.3,|x2(t)|>0.3,
    a31(x3(t))={0.4,|x3(t)|0.3,0.3,|x3(t)|>0.3,a32(x3(t))={0.5,|x3(t)|0.3,0.2,|x3(t)|>0.3,a33(x3(t))={0.3,|x3(t)|0.3,0.8,|x3(t)|>0.3,
    b11(x1(t))={0.3,|x1(t)|0.3,0.2,|x1(t)|>0.3,b12(x1(t))={0.5,|x1(t)|0.3,0.1,|x1(t)|>0.3,b13(x1(t))={0.2,|x1(t)|0.3,0.8,|x1(t)|>0.3,
    b21(x2(t))={0.2,|x2(t)|0.3,0.6,|x2(t)|>0.3,b22(x2(t))={0.3,|x2(t)|0.3,0.5,|x2(t)|>0.3,b23(x2(t))={0.4,|x2(t)|0.3,0.2,|x2(t)|>0.3,
    b31(x3(t))={0.5,|x3(t)|0.3,0.3,|x3(t)|>0.3,b32(x3(t))={0.4,|x3(t)|0.3,0.2,|x3(t)|>0.3,b33(x3(t))={0.5,|x3(t)|0.3,1.5,|x3(t)|>0.3.

    Choosing δ1=3,δ2=3,δ3=3,ω1=5,ω2=5,ω3=5,kij=4,l=1,q=0.5, and ui(t) gives the following design:

    δ1+c1(t)3j=1aDj1Pj=3.720,δ2+c2(t)3j=1aDj2Pj=3.70,δ3+c3(t)3j=1aDj3Pj=3.640,ω13j=1|ˆa1jˇa1j|Lj3j=1|ˆb1jˇb1j|Mj=4.10,ω23j=1|ˆa2jˇa2j|Lj3j=1|ˆb2jˇb2j|Mj=3.860,ω33j=1|ˆa3jˇa3j|Lj3j=1|ˆb3jˇb3j|Mj=3.620,k11bD11Qj|D11|Pj|H11|Pj=3.90,k12bD12Qj|D12|Pj|H12|Pj=3.860,k13bD13Qj|D13|Pj|H13|Pj=3.80,k21bD21Qj|D21|Pj|H21|Pj=3.840,k22bD22Qj|D22|Pj|H22|Pj=3.860,k23bD23Qj|D23|Pj|H23|Pj=3.880,k31bD31Qj|D31|Pj|H31|Pj=3.860,k32bD32Qj|D32|Pj|H32|Pj=3.880,k33bD33Qj|D33|Pj|H33|Pj=3.660.

    As of right now, it is easily obtained that Theorem 3.1's requirements (conditions (ⅰ)–(ⅲ)) are all met.

    Consequently, DrSy (1) in a fixed time effectively accomplished robust synchronization with ReSy (10). Additionally, Theorem 3.1 provides a means to determine Γmax as

    Γ(t0,e0)t0+Γmax2,Γmax2=1ql=10.5=2.

    Figure 5 illustrates the 3-D phase plane characteristics of the StVas x1(t),x2(t), and x3(t) for the uncontrolled TDFMNNs. Similarly, the trajectories of the controlled TDFMNNs system states are shown in Figures 68, respectively. The controlled system's trajectories show that the suggested control approach effectively achieves synchronization between DrSy and ReSy. Figure 9 illustrates the controller's performance, demonstrating how the system driven by the synchronous controller reaches the state of DrSy and maintains a convergence error of zero after Γmax.

    Figure 5.  Phase plane of x1(t),x2(t), and x3(t) for DrSy (1) in Example 4.2.
    Figure 6.  States x1(t) and y1(t), with control protocol in Example 4.2.
    Figure 7.  States x2(t) and y2(t), with control protocol in Example 4.2.
    Figure 8.  States x3(t) and y3(t), with control protocol in Example 4.2.
    Figure 9.  Synchronization error trajectories of DrSy (1) and corresponding ReSy(10) under the control strategy(18) in Example 4.2.

    The overall results demonstrate the controller's ability to readily achieve FixTS. Moreover, the simplicity of requiring only three design conditions for obtaining the control rate parameters suggests that the design and estimation of convergence time are less complex, thus affirming the practical feasibility of this optimized controller. In terms of neural network circuit implementation, this optimized controller is also straightforward to implement and incurs lower economic costs. As far as the authors are aware, the neural networks in this paper are constructed considering more objective factors, including memristors, time-variation of coefficients, time-delay, and FL. While previous studies such as [27] investigated FinTS of delayed FNNs with discontinuous activation, and [28] explored FinTS and FixTS, these models do not incorporate memristors. Additionally, although [35] addresses the FixTS problem for MNN, the system in [35] lacks time delay and FL components. Moreover, this paper introduces a modified controller with a special exponential function term, distinct from existing power function form controllers. By employing specially designed exponential functions and leveraging the indefinite derivative Lyapunov function method, a discontinuous StFC is developed to effectively achieve FixTS control of the memristor-based FNNs. Moreover, this approach provides a diversified method for estimating the convergence time, involving very few parameters and offering a simpler formulation for convergence time estimation.

    This research examines FixTS problems concerning TDFMNNs. To solve these problems, a method that makes use of a special exponential function and Lyapunov function method was adopted. Additionally, a discontinuous StFC was designed, which was able to achieve FixTS while also significantly increasing the convergence rate. In contrast to earlier published literature, FixTS offers several advantages. First, compared to traditional exponential and asymptotic synchronization, FixTS achieves faster convergence. In contrast, the error states in [12,18,42] synchronize over an infinite time horizon, while the FixTS in this paper is achieved within a finite time. Second, the FixTS proposed here also outperforms FinTS. In [43,44], the convergence time of FinTS heavily affected by the InSts of the system, whereas the convergence time of our FixTS is independent of any InSts. The third advantage lies in the different methods used to achieve FixTS. In [41], fixed-time control is implemented using power and sign functions, while our approach employs a special exponential function to achieve fixed-time control. The control designs in these two approaches are fundamentally different, with our controller having the advantage of fewer control parameters. Although FixTS control can shorten a system's convergence time, they have limitations when the control objective must be reached within a specified, predefined-time frame, independent of the InSts or system factors [45]. For instance, in special engineering applications such as predefined-time altitude control of spacecraft, DC microgrids, aircraft docking, and secure communication, systems must frequently attain specified objectives within a predetermined timeframe by actual requirements [46]. Inspired by [40,41], predefined-time pinning control of FNNs with reaction-diffusion terms is also a highly significant research topic. We plan to dedicate more time and effort to this area in our future studies.

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

    This work was supported by Humanity and Social Science Youth foundation of Ministry of Education of China (No. 22YJCZH009), Scientific Research Key Project of Hunan Provincial Education Department (No. 23A0684).

    The authors declare there is no conflicts of interest.



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