Synchronization Control For Neural Networks With Integer-order And Fractional-order

Synchronization control of neural networks is a central topic in the investigationof the dynamics of complex networks and has been received much attention by a lotof scholars. Fractional-order calculus is a new field in mathematic study and hasbeen a hot research topic. The aim of this work is to discuss the intermittent controlof fuzzy neural networks and Cohen-Grossberg neural networks, analyze the exis-tence and uniqueness of solutions for nonlinear impulsive fractional-order equationswith boundary value conditions, investigate Mittag-Lefer stability, Mittag-Lefercomplete synchronization and globally asymptotically projective synchronization fortwo class of neural networks with fractional-order in the sense of Caputo fractionalderivative.In the first part, two class of neural networks with integer-order including fuzzyneural networks and Cohen-Grossberg neural networks are investigated based on pe-riodically intermittent control.(1) The globally exponentially lag synchronizationof fuzzy neural networks with time delays is proposed. By designing periodical-ly intermittent control to response networks, utilizing multi-parameters means andLyapunov functional as well as mathematical induction, some new and useful criteriaof lag synchronization for the addressed networks are derived in terms of p-norm anda bound of the control width is obtained. Finally, an example with simulation is giv-en to show the efectiveness of the obtained results.(2) A class of Cohen-Grossbergneural networks with time-varying delays are studied by designing a periodicallyintermittent controller. Some novel and efective exponential synchronization cri-teria based on infinity norm are derived by applying Lyapunov-Razumikhin theoryand some analysis techniques. Finally, a chaotic Cohen-Grossberg neural networkis represented to show the efectiveness and feasibility of our results. In all, the re-sults derived in this part generalize a few previous known results and remove somerestrictions on control width and time-delays.In the second part, the existence and uniqueness of solutions for a class ofboundary value problems involving nonlinear impulsive fractional diferential equa-tions are considered. By means of fractional-order diferential theory and Banach’sfixed point theorem, some criteria are derived based on Lipschitz condition to en- sure the existence and uniqueness of solutions for the fractional-order diferentialequations with impulse. Compared with some previous works, our results are moregeneral and less conservative. Particularly, the time interval for the existence of so-lutions is extended to [0, T] from [0,1]. Finally, two examples are given to illustratethe main results.In the third part, we discuss Mittag-Lefer stability, Mittag-Lefer completesynchronization and globally asymptotically projective synchronization for two classof neural networks with fractional-order in the sense of Caputo fractional derivative.(1) The stability and synchronization for a class of fractional-order neural network-s are investigated. First, a new fractional-order diferential inequality is provedby applying a known result concerning the cauchy problems for linear fractionaldiferential equations with the Caputo fractional derivative and comparison princi-ple of fractional-order diferential equations. Based on the results, Mittag-Leferstability of the networks, the existence and Mittag-Lefer stability of the equilib-rium point and Mittag-Lefer synchronization are respectively considered. Finally,several examples with numerical simulations are given to show the efectiveness ofthe obtained results.(2) The globally asymptotically projective synchronization offractional-order neural networks is investigated under hybrid control. First, somenew fractional-order diferential inequalities and a fractional-order sufcient condi-tion ensuring the monotonously nondecreasing of a continuous function are derived.Besides, by combining open loop control and fractional-order adaptive control, somenovel criteria are derived to realize globally asymptotically projective synchroniza-tion of fractional-order neural networks. As some special cases, when projectivecoefcients are selected as-1,1,0, several control strategies are given to ensure therealization of globally asymptotically complete synchronization, globally asymptot-ically anti-synchronization and the globally asymptotically stabilization of chaoticfractional-order neural networks. Finally, several examples with numerical simula-tions are given to support the obtained theoretical results.