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

Cluster synchronization of fractional-order two-layer networks and application in image encryption/decryption

In this paper, a type of fractional-order two-layer network model is constructed, wherein each layer in the network exhibits distinct topology. Subsequently, the cluster synchronization problem of fractional-order two-layer networks is investigated through a two-step approach. The initial step involves the implementation of finite-time cluster synchronization in the first layer by utilizing a fractional-order finite-time convergence lemma. Based upon this, the second step employs a novel approach of collectively treating the nodes within the same cluster in the first layer, thereby offering a significant insight for analyzing fractional-order two-layer networks cluster synchronization. In addition, the paper proposes a novel encryption/decryption scheme based on the cluster synchronization of fractional-order two-layer networks. By leveraging the complexity of chaotic sequences generated by fractional-order two-layer networks, the security of the encryption/decryption strategy is enhanced. Furthermore, three illustrative examples are provided to validate the theoretical findings.

Quasi-synchronization for variable-order fractional complex dynamical networks with hybrid delay-dependent impulses

This paper focuses on addressing the problem of quasi-synchronization in heterogeneous variable-order fractional complex dynamical networks (VFCDNs) with hybrid delay-dependent impulses. Firstly, a mathematics model of VFCDNs with short memory is established under multi-weighted networks and mismatched parameters, which is more diverse and practical. Secondly, under the framework of variable-order fractional derivative, a novel fractional differential inequality has been proposed to handle the issue of quasi-synchronization with hybrid delay-dependent impulses. Additionally, the quasi-synchronization criterion for VFCDNs is developed using differential inclusion theory and Lyapunov method. Finally, the practicality and feasibility of this theoretical analysis are demonstrated through numerical examples.

Event-triggered hybrid impulsive control for synchronization of fractional-order multilayer signed networks under cyber attacks

In this paper, we consider the exponential bipartite synchronization (EBS) problem of fractional-order multilayer signed networks with time-varying delays (FO-MSNT) under random cyber attacks. In contrast to the existing literature, the proposed hybrid event-triggered controller combines the advantages of feedback controller and impulsive controller, and the event-triggered condition is constructed by applying the network topology and the Lyapunov function of the subsystem, rather than the state function of the subsystem. Based on the Lyapunov-Razumikhin method and the graph theory, some sufficient conditions for achieving EBS of FO-MSNT under cyber attacks which are related to the topology of networks, the event-triggered parameters, the order of fractional derivative and the signal sent by the enemy are obtained. Furthermore, fractional-order coupled Chua's circuits model and fractional-order power systems built on MSNT are established and the EBS issues under cyber attacks are analyzed. Numerical examples and simulations are provided to show the validity of our theories.

Multiple Mittag-Leffler Stability of Fractional-Order Complex-Valued Memristive Neural Networks With Delays

This article discusses the coexistence and dynamical behaviors of multiple equilibrium points (Eps) for fractional-order complex-valued memristive neural networks (FCVMNNs) with delays. First, based on the state space partition method, some sufficient conditions are proposed to guarantee that there are multiple Eps in one FCVMNN. Then, the Mittag-Leffler stability of those multiple Eps is proved by using the Lyapunov function. Simultaneously, the enlarged attraction basins are obtained to improve and extend the existing theoretical results in the previous literature. In addition, some existing stability results in the literature are special cases of a new result herein. Finally, two illustrative examples with computer simulations are presented to verify the effectiveness of theoretical analysis.

Tracking control design for fractional order systems: A passivity-based port-Hamiltonian framework

This article focuses on the design of tracking control for chaotic fractional order systems subjected to perturbations in a port-Hamiltonian framework. The fractional order systems of general form are modeled into port-controlled Hamiltonian form. Then, the extended results on the dissipativity, energy balance, and passivity of the fractional order systems are proved and presented in this paper. The port-controlled Hamiltonian form of the fractional order systems are proved to be asymptotically stable via energy balancing concept. Furthermore, a tracking controller is designed for the fractional order port-controlled Hamiltonian form by utilizing the matching conditions of the port-Hamiltonian systems. Stability of the system is established and analyzed explicitly for the closed-loop system with the help of direct Lyapunov method. Finally, an application example is solved with simulation results and discussions to prove the effectiveness of the propounded control design approach.

Stability for IT2 T-S fuzzy systems under alternate event-triggered control

In this paper, an alternate event-triggered control is proposed to achieve stability of interval type-2 Takagi-Sugeno fuzzy systems. Comparing with the existing literature, this new control strategy displays an almost complete aperiodic feature which eliminates the conservativeness caused by time-triggered property of the traditional aperiodically intermittent control. Moreover, with two events being triggered alternately in this control strategy through examining two predetermined conditions, the efficiency of control can be further improved and the resources consumption can be greatly reduced. By employing the Lyapunov function and graph theory, several stability criteria are rigorously demonstrated. In addition, Zeno behavior is excluded in our system through obtaining a positive lower bound of the time interval between two triggering points. Subsequently, the validity of the presented strategy is evidenced by single-link robot arms systems. Finally, a numerical example is given to lend insight into the feasibility of our theoretical results.

Finite-time parameter identification of fractional-order time-varying delay neural networks based on synchronization

We research the finite-time parameter identification of fractional-order time-varying delay neural networks (FTVDNNs) based on synchronization. First, based on the fractional-order Lyapunov stability theorem and feedback control idea, we construct a synchronous controller and some parameter update rules, which accomplish the synchronization of the drive-response FTVDNNs and complete the identification of uncertain parameters. Second, the theoretical analysis of the synchronization method is carried out, and the stable time is calculated. Finally, we give two examples for simulation verification. Our method can complete the synchronization of the FTVDNNs in finite time and identify uncertain parameters while synchronizing.

Exponential Stability of Fractional-Order Switched Systems With Mode-Dependent Impulses and Its Application

Most exiting results for impulsive switched systems (ISSs) are mainly built on the synchronous switching and impulses case; however, the impulses can not only occur in switched interval including switched instants but also the switched signals may exist between two impulsive points in practical instants. Under asynchronous impulses and switching signals, the main objective of this article is to study the exponential stability of fractional-order hybrid systems. In order to better characterize stability, some novel criteria are presented by adopting the mode-dependent average impulsive interval and induction method. The obtained impulsive switched criteria lead to a tradeoff between fractional-order α and impulsive strength. Especially, the impulsive effects (positive or negative) with the order α are also discussed in detail, which extends the previous integer order results. Moreover, numerical examples are given to interpret and verify the effectiveness of the obtained criteria.

Exponentially Stable Periodic Oscillation and Mittag-Leffler Stabilization for Fractional-Order Impulsive Control Neural Networks With Piecewise Caputo Derivatives

It is well known that the conventional fractional-order neural networks (FONNs) cannot generate nonconstant periodic oscillation. For this point, this article discusses a class of impulsive FONNs with piecewise Caputo derivatives (IPFONNs). By using the differential inclusion theory, the existence of the Filippov solutions for a discontinuous IPFONNs is investigated. Furthermore, some decision theorems are established for the existence and uniqueness of the (periodic) solution, global exponential stability, and impulsive control global stabilization to IPFONNs. This article achieves four key issues that were not solved in the previously existing literature: 1) the existence of at least one Filippov solution in a discontinuous IPFONN; 2) the existence and uniqueness of periodic oscillation in a nonautonomous IPFONN; 3) global exponential stability of IPFONNs; and 4) impulsive control global Mittag-Leffler stabilization for FONNs.

Synchronization of fractional-order memristive recurrent neural networks via aperiodically intermittent control

In this paper, synchronization of fractional-order memristive recurrent neural networks via aperiodically intermittent control is investigated. Considering the special properties of memristor neural network, differential inclusion theory is introduced. Similar to the aperiodically strategy of integer order, aperiodically intermittent control strategy of fractional order is proposed. Under the framework of Fillipov's solution, based on the intermittent strategy of fractional order systems and the properties Mittag-Leffler, sufficient criteria of aperiodically intermittent strategy are obtained by constructing appropriate Lyapunov functional. Some comparisons are given to demonstrate the advantages of aperiodically strategy. A simulation example is given to illustrate the derived conclusions.

Impulsive Memristive Cohen–Grossberg Neural Networks Modeled by Short Term Generalized Proportional Caputo Fractional Derivative and Synchronization Analysis

The synchronization problem for impulsive fractional-order Cohen–Grossberg neural networks with generalized proportional Caputo fractional derivatives with changeable lower limit at any point of impulse is studied. We consider the cases when the control input is acting continuously as well as when it is acting instantaneously at the impulsive times. We defined the global Mittag–Leffler synchronization as a generalization of exponential synchronization. We obtained some sufficient conditions for Mittag–Leffler synchronization. Our results are illustrated with examples.

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.

Exponential synchronization for variable-order fractional discontinuous complex dynamical networks with short memory via impulsive control

This paper considers the exponential synchronization issue for variable-order fractional complex dynamical networks (FCDNs) with short memory and derivative couplings via the impulsive control scheme, where dynamical nodes are modeled to be discontinuous. Firstly, the mathematics model with respect to variable-order fractional systems with short memory is established under the impulsive controller, in which the impulse strength is not only determined by the impulse control gain, but also the order of the control systems. Secondly, the exponential stability criterion for variable-order fractional systems with short memory is developed. Thirdly, the hybrid controller, which consists of the impulsive coupling controller and the discontinuous feedback controller, is designed to realize the synchronization objective. In addition, by constructing Lyapunov functional and applying inequality analysis techniques, the synchronization conditions are achieved in terms of linear matrix inequalities (LMIs). Finally, two simulation examples are performed to verify the effectiveness of the developed synchronization scheme and the theoretical outcomes.

Fractional-order discontinuous systems with indefinite LKFs: An application to fractional-order neural networks with time delays

In this article, we discuss bipartite fixed-time synchronization for fractional-order signed neural networks with discontinuous activation patterns. The Filippov multi-map is used to convert the fixed-time stability of the fractional-order general solution into the zero solution of the fractional-order differential inclusions. On the Caputo fractional-order derivative, Lyapunov-Krasovskii functional is proved to possess the indefinite fractional derivatives for fixed-time stability of fragmentary discontinuous systems. Furthermore, the fixed-time stability of the fractional-order discontinuous system is achieved as well as an estimate of the new settling time.. The discontinuous controller is designed for the delayed fractional-order discontinuous signed neural networks with antagonistic interactions and new conditions for permanent fixed-time synchronization of these networks with antagonistic interactions are also provided, as well as the settling time for permanent fixed-time synchronization. Two numerical simulation results are presented to demonstrate the effectiveness of the main results.

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.