Finite-time and fixed-time synchronization of discontinuous complex networks: A unified control framework design

This paper is concerned with the finite-time and fixed-time synchronization of complex networks with discontinuous nodes dynamics. Firstly, under the framework of Filippov solution, a new theorem of finite-time and fixed-time stability is established for nonlinear systems with discontinuous right-hand sides by using mainly reduction to absurdity. Furthermore, for a class of discontinuous complex networks, a general control law is firstly designed. Under the unified control framework and the same conditions, the considered networks are ensured to achieve finite-time or fixed-time synchronization by only adjusting the value of a key control parameter. Based on the similar discussion, a unified control strategy is also provided to realize respectively asymptotical, exponential and finite-time synchronization of the addressed networks. Finally, the derived theoretical results are supported by an example with numerical simulations.