LMI Conditions for Global Stability of Fractional-Order Neural Networks

New delay-dependent uniform stability criteria for fractional-order BAM neural networks with discrete and distributed delays

Initially, a class of Caputo fractional-order bidirectional associative memory neural networks in two variables is developed, building upon the groundwork laid by delayed Caputo fractional system in one variable. Next, the Razumikhin-type uniform stability conditions, originally formulated for single-variable systems, are successfully extended to accommodate the complexities of delayed Caputo fractional systems in two variables. Leveraging this extension and employing a suitable Lyapunov function, the delay-dependent uniform stability criteria for the addressed fractional-order bidirectional associative memory neural networks are expressed in terms of linear matrix inequalities. Finally, the effectiveness and practicality of the theoretical findings are demonstrated through the application of two numerical examples, affirming the viability of the proposed approach.

Stability Analysis of Fractional Bidirectional Associative Memory Neural Networks With Multiple Proportional Delays and Distributed Delays

This article investigates the finite-time stability of a class of fractional-order bidirectional associative memory neural networks (FOBAMNNs) with multiple proportional and distributed delays. Different from the existing Gronwall integral inequality with single proportional delay ( ), we establish the Gronwall integral inequality with multiple proportional delays for the first time in the case of . Since the existing fractional-order single-constant delay Gronwall inequality with two different orders cannot be directly applied to the stability analysis of the aforementioned system, initially, we skillfully develop a novel one with generalized fractional multiproportional delays' Gronwall inequalities of different orders. Furthermore, combined with the newly constructed generalized inequality, the stability criteria of FOBAMNNs with fractional orders and under weaker conditions, i.e., at most linear growth and linear growth conditions rather than the global Lipschitz condition, are given respectively. Finally, numerical experiments verify the effectiveness of the proposed method.

Stability and synchronization of fractional-order reaction-diffusion inertial time-delayed neural networks with parameters perturbation

This study is centered around the dynamic behaviors observed in a class of fractional-order generalized reaction-diffusion inertial neural networks (FGRDINNs) with time delays. These networks are characterized by differential equations involving two distinct fractional derivatives of the state. The global uniform stability of FGRDINNs with time delays is explored utilizing Lyapunov comparison principles. Furthermore, global synchronization conditions for FGRDINNs with time delays are derived through the Lyapunov direct method, with consideration given to various feedback control strategies and parameter perturbations. The effectiveness of the theoretical findings is demonstrated through three numerical examples, and the impact of controller parameters on the error system is further investigated.

Finite-Time Passivity for Coupled Fractional-Order Neural Networks With Multistate or Multiderivative Couplings

This article mainly delves into the finite-time passivity (FTP) for coupled fractional-order neural networks with multistate couplings (CFNNMSCs) or coupled fractional-order neural networks with multiderivative couplings (CFNNMDCs). Distinguishing from the traditional FTP definitions, several concepts of FTP for fractional-order systems are given. On one hand, we present several sufficient conditions to ensure the FTP for CFNNMSCs by artfully designing a state-feedback controller and an adaptive state-feedback controller. On the other hand, by utilizing some inequality techniques, two sets of FTP criteria for CFNNMDCs are also established on the basis of the state-feedback and adaptive state-feedback controllers. Finally, numerical examples are used to demonstrate the validity of the derived FTP criteria.

A Comprehensive Review of Continuous-/Discontinuous-Time Fractional-Order Multidimensional Neural Networks

The dynamical study of continuous-/discontinuous-time fractional-order neural networks (FONNs) has been thoroughly explored, and several publications have been made available. This study is designed to give an exhaustive review of the dynamical studies of multidimensional FONNs in continuous/discontinuous time, including Hopfield NNs (HNNs), Cohen-Grossberg NNs, and bidirectional associative memory NNs, and similar models are considered in real ( [Formula: see text]), complex ( [Formula: see text]), quaternion ( [Formula: see text]), and octonion ( [Formula: see text]) fields. Since, in practice, delays are unavoidable, theoretical findings from multidimensional FONNs with various types of delays are thoroughly evaluated. Some required and adequate stability and synchronization requirements are also mentioned for fractional-order NNs without delays.

Passivity and passification of fractional-order memristive neural networks with time delays

A class of fractional-order memristive neural networks (FMNNs) with time delays is studied. At first, the original network system is converted to fractional-order uncertain one to simplify the analysis by a variable transformation. Successively, some new LMIs-based passivity criteria are derived by differential inclusions, set-valued maps, inequality techniques and linear matrix inequality approach. Furthermore, a feedback control protocol is designed to solve the passification problem for the considered system, whose feedback control effect on different neurons can be changed artificially, which can be better applied to neural networks. The obtained results include some existing ones as special cases. A numerical example is proposed to illustrate the theoretical results.

Mittag-Leffler stability of fractional-order quaternion-valued memristive neural networks with generalized piecewise constant argument

This paper studies the global Mittag-Leffler (M-L) stability problem for fractional-order quaternion-valued memristive neural networks (FQVMNNs) with generalized piecewise constant argument (GPCA). First, a novel lemma is established, which is used to investigate the dynamic behaviors of quaternion-valued memristive neural networks (QVMNNs). Second, by using the theories of differential inclusion, set-valued mapping, and Banach fixed point, several sufficient criteria are derived to ensure the existence and uniqueness (EU) of the solution and equilibrium point for the associated systems. Then, by constructing Lyapunov functions and employing some inequality techniques, a set of criteria are proposed to ensure the global M-L stability of the considered systems. The obtained results in this paper not only extends previous works, but also provides new algebraic criteria with a larger feasible range. Finally, two numerical examples are introduced to illustrate the effectiveness of the obtained results.

Synchronization Rather Than Finite-Time Synchronization Results of Fractional-Order Multi-Weighted Complex Networks

This article investigates the synchronization of fractional-order multi-weighted complex networks (FMWCNs) with order α ∈ (0,1) . A useful fractional-order inequality _t0^C D_t^α V(x(t)) ≤ -μV(x(t)) is extended to a more general form _t0^C D_t^α V(x(t)) ≤ -μV^γ(x(t)),γ ∈ (0,1] , which plays a pivotal role in studies of synchronization for FMWCNs. However, the inequality _t0^C D_t^α V(x(t)) ≤ -μV^γ(x(t)),γ ∈ (0,1) has been applied to achieve the finite-time synchronization for fractional-order systems in the absence of rigorous mathematical proofs. Based on reduction to absurdity in this article, we prove that it cannot be used to obtain finite-time synchronization results under bounded nonzero initial value conditions. Moreover, by using feedback control strategy and Lyapunov direct approach, some sufficient conditions are presented in the forms of linear matrix inequalities (LMIs) to ensure the synchronization for FMWCNs in the sense of a widely accepted definition of synchronization. Meanwhile, these proposed sufficient results cannot guarantee the finite-time synchronization of FMWCNs. Finally, two chaotic systems are given to verify the feasibility of the theoretical results.

Global Exponential Stability of Fractional Order Complex-Valued Neural Networks with Leakage Delay and Mixed Time Varying Delays

This paper investigates the global exponential stability of fractional order complex-valued neural networks with leakage delay and mixed time varying delays. By constructing a proper Lyapunov-functional we established sufficient conditions to ensure global exponential stability of the fractional order complex-valued neural networks. The stability conditions are established in terms of linear matrix inequalities. Finally, two numerical examples are given to illustrate the effectiveness of the obtained results.

Optimal Synchronization of Unidirectionally Coupled FO Chaotic Electromechanical Devices With the Hierarchical Neural Network

This article solves the problem of optimal synchronization, which is important but challenging for coupled fractional-order (FO) chaotic electromechanical devices composed of mechanical and electrical oscillators and electromagnetic filed by using a hierarchical neural network structure. The synchronization model of the FO electromechanical devices with capacitive and resistive couplings is built, and the phase diagrams reveal that the dynamic properties are closely related to sets of physical parameters, coupling coefficients, and FOs. To force the slave system to move from its original orbits to the orbits of the master system, an optimal synchronization policy, which includes an adaptive neural feedforward policy and an optimal neural feedback policy, is proposed. The feedforward controller is developed in the framework of FO backstepping integrated with the hierarchical neural network to estimate unknown functions of dynamic system in which the mentioned network has the formula transformation and hierarchical form to reduce the numbers of weights and membership functions. Also, an adaptive dynamic programming (ADP) policy is proposed to address the zero-sum differential game issue in the optimal neural feedback controller in which the hierarchical neural network is designed to yield solutions of the constrained Hamilton-Jacobi-Isaacs (HJI) equation online. The presented scheme not only ensures uniform ultimate boundedness of closed-loop coupled FO chaotic electromechanical devices and realizes optimal synchronization but also achieves a minimum value of cost function. Simulation results further show the validity of the presented scheme.

Artificial neural networks: a practical review of applications involving fractional calculus

In this work, a bibliographic analysis on artificial neural networks (ANNs) using fractional calculus (FC) theory has been developed to summarize the main features and applications of the ANNs. ANN is a mathematical modeling tool used in several sciences and engineering fields. FC has been mainly applied on ANNs with three different objectives, such as systems stabilization, systems synchronization, and parameters training, using optimization algorithms. FC and some control strategies have been satisfactorily employed to attain the synchronization and stabilization of ANNs. To show this fact, in this manuscript are summarized, the architecture of the systems, the control strategies, and the fractional derivatives used in each research work, also, the achieved goals are presented. Regarding the parameters training using optimization algorithms issue, in this manuscript, the systems types, the fractional derivatives involved, and the optimization algorithm employed to train the ANN parameters are also presented. In most of the works found in the literature where ANNs and FC are involved, the authors focused on controlling the systems using synchronization and stabilization. Furthermore, recent applications of ANNs with FC in several fields such as medicine, cryptographic, image processing, robotic are reviewed in detail in this manuscript. Works with applications, such as chaos analysis, functions approximation, heat transfer process, periodicity, and dissipativity, also were included. Almost to the end of the paper, several future research topics arising on ANNs involved with FC are recommended to the researchers community. From the bibliographic review, we concluded that the Caputo derivative is the most utilized derivative for solving problems with ANNs because its initial values take the same form as the differential equations of integer-order.

Event-Triggered State Estimation for Fractional-Order Neural Networks

This paper is concerned with the problem of event-triggered state estimation for a class of fractional-order neural networks. An event-triggering strategy is proposed to reduce the transmission frequency of the output measurement signals with guaranteed state estimation performance requirements. Based on the Lyapunov method and properties of fractional-order calculus, a sufficient criterion is established for deriving the Mittag–Leffler stability of the estimation error system. By making full use of the properties of Caputo operator and Mittag–Leffler function, the evolution dynamics of measured error is analyzed so as to exclude the unexpected Zeno phenomenon in the event-triggering strategy. Finally, two numerical examples and simulations are provided to show the effectiveness of the theoretical results.

Improved Results on Finite-Time Passivity and Synchronization Problem for Fractional-Order Memristor-Based Competitive Neural Networks: Interval Matrix Approach

This research paper deals with the passivity and synchronization problem of fractional-order memristor-based competitive neural networks (FOMBCNNs) for the first time. Since the FOMBCNNs’ parameters are state-dependent, FOMBCNNs may exhibit unexpected parameter mismatch when different initial conditions are chosen. Therefore, the conventional robust control scheme cannot guarantee the synchronization of FOMBCNNs. Under the framework of the Filippov solution, the drive and response FOMBCNNs are first transformed into systems with interval parameters. Then, the new sufficient criteria are obtained by linear matrix inequalities (LMIs) to ensure the passivity in finite-time criteria for FOMBCNNs with mismatched switching jumps. Further, a feedback control law is designed to ensure the finite-time synchronization of FOMBCNNs. Finally, three numerical cases are given to illustrate the usefulness of our passivity and synchronization results.

Synchronization of Fractional Order Uncertain BAM Competitive Neural Networks

This article examines the drive-response synchronization of a class of fractional order uncertain BAM (Bidirectional Associative Memory) competitive neural networks. By using the differential inclusions theory, and constructing a proper Lyapunov-Krasovskii functional, novel sufficient conditions are obtained to achieve global asymptotic stability of fractional order uncertain BAM competitive neural networks. This novel approach is based on the linear matrix inequality (LMI) technique and the derived conditions are easy to verify via the LMI toolbox. Moreover, numerical examples are presented to show the feasibility and effectiveness of the theoretical results.

Pinning multisynchronization of delayed fractional-order memristor-based neural networks with nonlinear coupling and almost-periodic perturbations

This paper concerns the multisynchronization issue for delayed fractional-order memristor-based neural networks with nonlinear coupling and almost-periodic perturbations. First, the coexistence of multiple equilibrium states for isolated subnetwork is analyzed. By means of state-space decomposition, fractional-order Halanay inequality and Caputo derivative properties, the novel algebraic sufficient conditions are derived to ensure that the addressed networks with arbitrary activation functions have multiple locally stable almost periodic orbits or equilibrium points. Then, based on the obtained multistability results, a pinning control strategy is designed to realize the multisynchronization of the N coupled networks. By the aid of graph theory, depth first search method and pinning control law, some sufficient conditions are formulated such that the considered neural networks can possess multiple synchronization manifolds. Finally, the multistability and multisynchronization performance of the considered neural networks with different activation functions are illustrated by numerical examples.

Exponential Stability of Fractional-Order Complex Multi-Links Networks With Aperiodically Intermittent Control

In this article, the exponential stability problem for fractional-order complex multi-links networks with aperiodically intermittent control is considered. Using the graph theory and Lyapunov method, two theorems, including a Lyapunov-type theorem and a coefficient-type theorem, are given to ensure the exponential stability of the underlying networks. The theoretical results show that the exponential convergence rate is dependent on the control gain and the order of fractional derivative. To be specific, the larger control gain, the higher the exponential convergence rate. Meanwhile, when aperiodically intermittent control degenerates into periodically intermittent control, a corollary is also provided to ensure the exponential stability of the underlying networks. Furthermore, to show the practicality of theoretical results, as an application, the exponential stability of fractional-order multi-links competitive neural networks with aperiodically intermittent control is investigated and a stability criterion is established. Finally, the effectiveness and feasibility of the theoretical results are demonstrated through a numerical example.

New Criteria on Finite-Time Stability of Fractional-Order Hopfield Neural Networks With Time Delays

In this article, the finite-time stability (FTS) of fractional-order Hopfield neural networks with time delays (FHNNTDs) is studied. A widely used inequality in investigating the stability of the fractional-order neural networks is fractional-order Gronwall inequality related to the Mittag-Leffler function, which cannot be directly used to study the stability of the factional-order neural networks with time delays. In the existing works related to fractional-order Gronwall inequality with time delays, the order was divided into two cases: λ ∈ (0,0.5] and λ ∈ (0.5,+∞) . In this article, a new fractional-order Gronwall integral inequality with time delay and the unified form for all the fractional order is developed, which can be widely applied to investigate FTS of various fractional-order systems with time delays. Based on this new inequality, a new criterion for the FTS of FHNNTDs is derived. Compared with the existing criteria, in which fractional order λ ∈ (0,1) was divided into two cases, λ ∈ (0,0.5] and λ ∈ (0.5,1) , the obtained results in this article are presented in the unified form of fractional order λ ∈ (0,1) and convenient to verify. More importantly, the criteria in this article are less conservative than some existing ones. Finally, two numerical examples are given to demonstrate the validity of the proposed results.

Passive filter design for fractional-order quaternion-valued neural networks with neutral delays and external disturbance

The problem on passive filter design for fractional-order quaternion-valued neural networks (FOQVNNs) with neutral delays and external disturbance is considered in this paper. Without separating the FOQVNNs into two complex-valued neural networks (CVNNs) or the FOQVNNs into four real-valued neural networks (RVNNs), by constructing Lyapunov-Krasovskii functional and using inequality technique, the delay-independent and delay-dependent sufficient conditions presented as linear matrix inequality (LMI) to confirm the augmented filtering dynamic system to be stable and passive with an expected dissipation are derived. One numerical example with simulations is furnished to pledge the feasibility for the obtained theory results.

Necessary and Sufficient Conditions for Group Consensus of Fractional Multiagent Systems Under Fixed and Switching Topologies via Pinning Control

The group consensus problem for fractional-order multiagent systems is investigated in this paper. With the help of double-tree-form transformations, the group consensus problem of fractional-order multiagent systems is proved to be equivalent to the asymptotical stability problem of reduced-order error systems. A class of distributed control protocols and some simple LMI sufficient conditions as well as necessary and sufficient conditions are proposed in this paper to solve the group consensus problem for fractional multiagent systems. Moreover, pinning control strategy has been taken into consideration. It is shown that the system converges more rapidly when the designed pinning protocols are adopted. In addition, the case of fractional system with switching topologies is also discussed and some corresponding sufficient conditions are obtained. Finally, some simulation results are presented to illustrate the theoretical results.

Monostability and Multistability for Almost-Periodic Solutions of Fractional-Order Neural Networks With Unsaturating Piecewise Linear Activation Functions

Since the unsaturating activation function is unbounded, more complex dynamics may exist in neural networks with this kind of activation function. In this article, monostability and multistability results of almost-periodic solutions are developed for fractional-order neural networks with unsaturating piecewise linear activation functions. Some globally Mittag-Leffler attractive sets are given, and the existence of globally Mittag-Leffler stable almost-periodic solution is demonstrated by using Ascoli-Arzela theorem. In particular, some sufficient conditions are provided to ascertain the multistability of almost-periodic solutions based on locally positively invariant set. It shows that there exists an almost-periodic solution in each positively invariant set, and all trajectories converge to this periodic trajectory in that rectangular area. Two illustrative examples are provided to demonstrate the effectiveness of the proposed sufficient criteria.

Asymptotic and Finite-Time Cluster Synchronization of Coupled Fractional-Order Neural Networks With Time Delay

This article is devoted to the cluster synchronization issue of coupled fractional-order neural networks. By introducing the stability theory of fractional-order differential systems and the framework of Filippov regularization, some sufficient conditions are derived for ascertaining the asymptotic and finite-time cluster synchronization of coupled fractional-order neural networks, respectively. In addition, the upper bound of the settling time for finite-time cluster synchronization is estimated. Compared with the existing works, the results herein are applicable for fractional-order systems, which could be regarded as an extension of integer-order ones. A numerical example with different cases is presented to illustrate the validity of theoretical results.

Exponential Stability of Fractional-Order Impulsive Control Systems With Applications in Synchronization

This paper investigates exponential stability of fractional-order impulsive control systems (FICSs) and exponential synchronization of fractional-order Cohen-Grossberg neural networks (FCGNNs). First, under the framework of the generalized Caputo fractional-order derivative, some new results for fractional-order calculus are established by mainly using L'Hospital's rule and Laplace transform. Besides, FICSs are translated into impulsive differential equations with fractional-order via utilizing the definition of Dirac function, which reveals that the effect of impulsive control on fractional systems is dependent of the order of the addressed systems. Furthermore, exponential stability of FICSs is proposed and some novel criteria are obtained by applying average impulsive interval and the method of induction. As an application of the stability for FICSs, exponential synchronization of FCGNNs is considered and several synchronization conditions are established under impulsive control. Finally, several numerical examples are provided to illustrate the effectiveness of the derived results.

Edge-Based Fractional-Order Adaptive Strategies for Synchronization of Fractional-Order Coupled Networks With Reaction-Diffusion Terms

In this paper, spatial diffusions are introduced to fractional-order coupled networks and the problem of synchronization is investigated for fractional-order coupled neural networks with reaction-diffusion terms. First, a new fractional-order inequality is established based on the Caputo partial fractional derivative. To realize asymptotical synchronization, two types of adaptive coupling weights are considered, namely: 1) coupling weights only related to time and 2) coupling weights dependent on both time and space. For each type of coupling weights, based on local information of the node's dynamics, an edge-based fractional-order adaptive law and an edge-based fractional-order pinning adaptive scheme are proposed. Furthermore, some new analytical tools, including the method of contradiction, L'Hopital rule, and Barbalat lemma are developed to establish adaptive synchronization criteria of the addressed networks. Finally, an example with numerical simulations is provided to illustrate the validity and effectiveness of the theoretical results.

Design and Practical Stability of a New Class of Impulsive Fractional-Like Neural Networks

In this paper, a new class of impulsive neural networks with fractional-like derivatives is defined, and the practical stability properties of the solutions are investigated. The stability analysis exploits a new type of Lyapunov-like functions and their derivatives. Furthermore, the obtained results are applied to a bidirectional associative memory (BAM) neural network model with fractional-like derivatives. Some new results for the introduced neural network models with uncertain values of the parameters are also obtained.

Global Stabilization of Fractional-Order Memristor-Based Neural Networks With Time Delay

This paper addresses the global stabilization of fractional-order memristor-based neural networks (FMNNs) with time delay. The voltage threshold type memristor model is considered, and the FMNNs are represented by fractional-order differential equations with discontinuous right-hand sides. Then, the problem is addressed based on fractional-order differential inclusions and set-valued maps, together with the aid of Lyapunov functions and the comparison principle. Two types of control laws (delayed state feedback control and coupling state feedback control) are designed. Accordingly, two types of stabilization criteria [algebraic form and linear matrix inequality (LMI) form] are established. There are two groups of adjustable parameters included in the delayed state feedback control, which can be selected flexibly to achieve the desired global asymptotic stabilization or global Mittag-Leffler stabilization. Since the existing LMI-based stability analysis techniques for fractional-order systems are not applicable to delayed fractional-order nonlinear systems, a fractional-order differential inequality is established to overcome this difficulty. Based on the coupling state feedback control, some LMI stabilization criteria are developed for the first time with the help of the newly established fractional-order differential inequality. The obtained LMI results provide new insights into the research of delayed fractional-order nonlinear systems. Finally, three numerical examples are presented to illustrate the effectiveness of the proposed theoretical results.

Global Nonfragile Synchronization in Finite Time for Fractional-Order Discontinuous Neural Networks With Nonlinear Growth Activations

This paper is concerned with the global nonfragile Mittag-Leffler synchronization and the global synchronization in finite time for fractional-order discontinuous neural networks, where activation functions are discontinuous at 0, or modeled as a local Hölder functions with the nonlinear growth property in a neighborhood of 0. First, two lemmas concerned with the convergence with respect to an absolutely continuous function are developed. Second, a new property, which introduces an inequality of the fractional derivative for the variable upper limit integral with respect to the nonsmooth integrable function, is presented and applied in the synchronization results' analysis. In addition, under the fractional Filippov differential inclusion framework, by utilizing the Lur'e Postnikov-type Lyapunov functional, nonsmooth analysis method, and the convergence properties developed in this paper, the synchronization conditions are derived in the form of linear matrix inequalities. Moreover, the upper bound of the setting time for the global nonfragile synchronization in finite time is calculated accurately. Finally, two illustrations are presented to verify the correctness of the theoretical results.

Adaptive Synchronization of Fractional-Order Complex-Valued Neural Networks with Discrete and Distributed Delays

In this paper, the synchronization problem of fractional-order complex-valued neural networks with discrete and distributed delays is investigated. Based on the adaptive control and Lyapunov function theory, some sufficient conditions are derived to ensure the states of two fractional-order complex-valued neural networks with discrete and distributed delays achieve complete synchronization rapidly. Finally, numerical simulations are given to illustrate the effectiveness and feasibility of the theoretical results.