Periodicity and finite-time periodic synchronization of discontinuous complex-valued neural networks

This paper discusses the issue of periodicity and finite-time periodic synchronization of discontinuous complex-valued neural networks (CVNNs). Based on a modified version of Kakutani's fixed point theorem, general conditions are obtained to guarantee the periodicity of discontinuous CVNNs. Next, several criteria for finite-time periodic synchronization (FTPS) are given by using a new proposed finite-time convergence theorem. Different from the traditional convergence lemma, the estimated upper bound of the derivative of the Lyapunov function (LF) is allowed to be indefinite or even positive. In order to achieve FTPS, novel discontinuous control algorithms, including state-feedback control algorithm and generalized pinning control algorithm, are designed. In the generalized pinning control algorithm, a guideline is proposed to select neurons to pin the designed controller. Finally, two simulations are given to substantiate the main results.