Further results on distributed finite-time synchronization for fractional-order coupled neural networks

Given that the Newton-Leibniz formula and chain rule are not applicable to fractional calculus, the results of finite-time synchronization for integer-order systems are difficult to directly extend to fractional-order ones. In view of this, this paper is devoted to the finite-time synchronization for fractional-order coupled neural networks via a developed approach. Firstly, with the help of Laplace transform, the properties of Mittag-Leffler function and proof by contradiction, a more generalized fractional-order nonlinear differential inequality involved with finite-time convergence is established. Besides, some previous results on finite-time stability or synchronization of fractional-order systems can be further extended by this inequality, which presents a new tool for research. Under this framework, two distributed finite-time control schemes are introduced, (1) a basic distributed finite-time control scheme relying on some global information; (2) an improved fractional-order adaptive distributed finite-time control scheme, which avoids the dependence of some global information. Then, the communication topology is extended by including the synchronized state as a $(N+1)$th node in the considered networks, which eliminates the restriction on original network topology connectivity. In the end, in order to verify the feasibility of the proposed method, a numerical example of fractional-order Chua’s circuit system is provided.