Fixed-Time Synchronization of Complex Networks With Impulsive Effects via Nonchattering Control

Dealing with impulsive effects is one of the most challenging problems in the field of fixed-time control. In this paper, we solve this challenging problem by considering fixed-time synchronization of complex networks (CNs) with impulsive effects. By designing a new Lyapunov function and constructing comparison systems, a sufficient condition formulated by matrix inequalities is given to ensure that all the dynamical subsystems in the CNs are synchronized with an isolated system in a settling time, which is independent of the initial values of both the CNs and the isolated system. Then, by partitioning impulse interval and using the convex combination technique, sufficient conditions in terms of linear matrix inequalities are provided. Our synchronization criteria unify synchronizing and desynchronizing impulses. Compared with the existing controllers for fixed-time and finite-time techniques, the designed controller is continuous and does not include any sign function, and hence, the chattering phenomenon in most of the existing results is overcome. An optimal algorithm is proposed for the estimation of the settling time. Numerical examples are given to show the effectiveness of our new results.