Asymptotic stability of delayed fractional-order neural networks with impulsive effects

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.

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.

Non-fragile state estimation for fractional-order delayed memristive BAM neural networks

This paper deals with the non-fragile state estimation problem for a class of fractional-order memristive BAM neural networks (FMBAMNNs) with and without time delays for the first time. By means of a novel transformation and interval matrix approach, non-fragile estimators are designed and parameter mismatch problem is averted. Sufficient criteria are established to ascertain the error system is asymptotically stable based on fractional-order Lyapunov functionals and linear matrix inequalities (LMIs). Two examples are put forward to show the effectiveness of the obtained results.

Global Stability of Fractional Order Coupled Systems with Impulses via a Graphic Approach

Based on the graph theory and stability theory of dynamical system, this paper studies the stability of the trivial solution of a coupled fractional-order system. Some sufficient conditions are obtained to guarantee the global stability of the trivial solution. Finally, a comparison between fractional-order system and integer-order system ends the paper.

Stability Analysis and Application for Delayed Neural Networks Driven by Fractional Brownian Noise

This paper deals with two types of the stability problem for the delayed neural networks driven by fractional Brownian noise (FBN). The existence and the uniqueness of the solution to the main system with respect to FBN are proved via fixed point theory. Based on Hilbert-Schmidt operator theory and analytic semigroup principle, the mild solution of the stochastic neural networks is obtained. By applying the stochastic analytic technique and some well-known inequalities, the asymptotic stability criteria and the exponential stability condition are established. Both numerical example and practical application for synchronization control of multiagent system are provided to illustrate the effectiveness and potential of the proposed techniques.

Global Mittag-Leffler stability analysis of fractional-order impulsive neural networks with one-side Lipschitz condition

This paper is concerned with the stability analysis issue of fractional-order impulsive neural networks. Under the one-side Lipschitz condition or the linear growth condition of activation function, the existence of solution is analyzed respectively. In addition, the existence, uniqueness and global Mittag-Leffler stability of equilibrium point of the fractional-order impulsive neural networks with one-side Lipschitz condition are investigated by the means of contraction mapping principle and Lyapunov direct method. Finally, an example with numerical simulation is given to illustrate the validity and feasibility of the proposed results.

Lagrange α-exponential stability and α-exponential convergence for fractional-order complex-valued neural networks

This paper deals with the problem on Lagrange α-exponential stability and α-exponential convergence for a class of fractional-order complex-valued neural networks. To this end, some new fractional-order differential inequalities are established, which improve and generalize previously known criteria. By using the new inequalities and coupling with the Lyapunov method, some effective criteria are derived to guarantee Lagrange α-exponential stability and α-exponential convergence of the addressed network. Moreover, the framework of the α-exponential convergence ball is also given, where the convergence rate is related to the parameters and the order of differential of the system. These results here, which the existence and uniqueness of the equilibrium points need not to be considered, generalize and improve the earlier publications and can be applied to monostable and multistable fractional-order complex-valued neural networks. Finally, one example with numerical simulations is given to show the effectiveness of the obtained results.

Practical stability analysis of fractional-order impulsive control systems

In this paper we obtain sufficient conditions for practical stability of a nonlinear system of differential equations of fractional order subject to impulse effects. Our results provide a design method of impulsive control law which practically stabilizes the impulse free fractional-order system.