Synchronization Of Coupled Partial Differential Systems Via Boundary Control

Synchronization is a natural phenomenon in nature. It can not be ignored in the complex system study and has become a hot topic. Due to the existence of information communication between subsystems, it is important to consider the factor of time-delay. In addition, many phenomena in the real world are generally modeled in spatial-temporal domain by partial differential systems(PDSs). PDSs control theory, boundary control in particular, consists of a wealth of mathematically impressive results. However, to our best knowledge, few references were reported on synchronization problem of coupled partial differential systems via boundary control. This article solves the synchronization problem via boundary control for coupled partial differential systems and coupled time-delay partial differential systems, respectively.In the first part, the synchronization problem for coupled PDSs with Dirichlet boundary conditions is considered. Firstly, according to the definition of synchronization error, the synchronization error system is obtained. By using nonsingular matrix transformation, we decouple the coupled system. The synchronization problem for the target system is turned into the problem of asymptotical stability of the decoupled system. Then, by virtue of the backstepping transformation method, effective boundary controllers are designed to make coupled PDSs achieve synchronization. Finally, two examples are given to show the effectiveness of the boundary controllers.In the second part, the synchronization problem for coupled time-delay PDSs with Neumann boundary conditions is discussed, in which the time-delay appears in dynamic nodes and coupling, respectively. It is worth noting that coupling matrix in the system can be asymmetric and reducible. By employing Lyapunov-Krasovikii functional, Schur complement lemma and linear matrix inequality, boundary controllers are designed to make coupled time-delay PDSs achieve synchronization. At last, numerical examples in which the system is with symmetric or asymmetric coupling matrix are given to demonstrate the effectiveness of the results.