Lyapunov approach to manifolds stability for impulsive Cohen-Grossberg-type conformable neural network models

In this paper, motivated by the advantages of the generalized conformable derivatives, an impulsive conformable Cohen-Grossberg-type neural network model is introduced. The impulses, which can be also considered as a control strategy, are at fixed instants of time. We define the notion of practical stability with respect to manifolds. A Lyapunov-based analysis is conducted, and new criteria are proposed. The case of bidirectional associative memory (BAM) network model is also investigated. Examples are given to demonstrate the effectiveness of the established results.

On the Analysis of Regularized Fuzzy Systems of Uncertain Differential Equations

This article analyzes a regularized set of fuzzy differential equations with respect to an uncertain parameter. We provide sufficient conditions for the correctness of a new regularization scheme. For the resulting family of regularized fuzzy differential equations, the following properties are analyzed, and efficient criteria are proposed: successive approximations, continuity, global existence of solutions, existence of approximate solutions, existence of solutions in the autonomous case. In addition, we develop stability criteria for the regularized family of fuzzy differential equations on the basis of the comparison technique and the method of nonlinear integral inequalities. We expect that the derived results will inspire future research work in this direction.

Impulsive Reaction-Diffusion Delayed Models in Biology: Integral Manifolds Approach

In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single state notion and has important applications in the study of multi-stable systems. By means of an extension of the Lyapunov method integral manifolds' existence, results are established. Based on the Lyapunov functions technique combined with a Poincarè-type inequality qualitative criteria related to boundedness, permanence, and stability of the integral manifolds are also presented. The application of the proposed impulsive control model is closely related to a most important problems in the mathematical biology-the problem of optimal control of epidemic models. The considered impulsive effects can be used by epidemiologists as a very effective therapy control strategy. In addition, since the integral manifolds approach is relevant in various contexts, our results can be applied in the qualitative investigations of many problems in the epidemiology of diverse interest.

Almost Periodicity in Impulsive Fractional-Order Reaction-Diffusion Neural Networks With Time-Varying Delays

A neural-network model of fractional order with impulsive perturbations, time-varying delays, and reaction-diffusion terms is investigated in this article. The focus is on investigating qualitative properties of the states and developing new almost periodicity and stability criteria. The uncertain case is also considered. Examples are established and the effectiveness of the obtained criteria is demonstrated.

Fractional order controllers increase the robustness of closed-loop deep brain stimulation systems

We studied the effects of using fractional order proportional, integral, and derivative (PID) controllers in a closed-loop mathematical model of deep brain stimulation. The objective of the controller was to dampen oscillations from a neural network model of Parkinson's disease. We varied intrinsic parameters, such as the gain of the controller, and extrinsic variables, such as the excitability of the network. We found that in most cases, fractional order components increased the robustness of the model multi-fold to changes in the gains of the controller. Similarly, the controller could be set to a fixed set of gains and remain stable to a much larger range, than for the classical PID case, of changes in synaptic weights that otherwise would cause oscillatory activity. The increase in robustness is a consequence of the properties of fractional order derivatives that provide an intrinsic memory trace of past activity, which works as a negative feedback system. Fractional order PID controllers could provide a platform to develop stand-alone closed-loop deep brain stimulation systems.

Fractional Lotka-Volterra-Type Cooperation Models: Impulsive Control on Their Stability Behavior

We present a biological fractional n-species delayed cooperation model of Lotka-Volterra type. The considered fractional derivatives are in the Caputo sense. Impulsive control strategies are applied for several stability properties of the states, namely Mittag-Leffler stability, practical stability and stability with respect to sets. The proposed results extend the existing stability results for integer-order n-species delayed Lotka-Volterra cooperation models to the fractional-order case under impulsive control.

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.

Mittag-Leffler synchronization of fractional neural networks with time-varying delays and reaction-diffusion terms using impulsive and linear controllers

In this paper, we propose a fractional-order neural network system with time-varying delays and reaction-diffusion terms. We first develop a new Mittag-Leffler synchronization strategy for the controlled nodes via impulsive controllers. Using the fractional Lyapunov method sufficient conditions are given. We also study the global Mittag-Leffler synchronization of two identical fractional impulsive reaction-diffusion neural networks using linear controllers, which was an open problem even for integer-order models. Since the Mittag-Leffler stability notion is a generalization of the exponential stability concept for fractional-order systems, our results extend and improve the exponential impulsive control theory of neural network system with time-varying delays and reaction-diffusion terms to the fractional-order case. The fractional-order derivatives allow us to model the long-term memory in the neural networks, and thus the present research provides with a conceptually straightforward mathematical representation of rather complex processes. Illustrative examples are presented to show the validity of the obtained results. We show that by means of appropriate impulsive controllers we can realize the stability goal and to control the qualitative behavior of the states. An image encryption scheme is extended using fractional derivatives.

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.