On the generalized exact boundary synchronization for a coupled system of wave equations

Abstract

In this paper, we deliver a normalized synchronization transformation to study the generalized exact boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. The clear relationship among the generalized exact boundary synchronization, the exact boundary null controllability, and the generalized exactly synchronizable states is precisely obtained.This approach gives further a forthright decomposition for the generalized exact boundary synchronization problem, whereby, we gain directly the determination of generalized exactly synchronizable states. KEYWORDSgeneralized exact boundary synchronization, generalized exactly synchronizable state, normalized synchronization transformation, synchronization decomposition MSC CLASSIFICATION 35B37; 93B05; 93B07 Math Meth Appl Sci. 2019;42:7011-7029. wileyonlinelibrary.com/journal/mma

Generalized approximate boundary synchronization for a coupled system of wave equations

“…According to Wang, 9 by means of Θ p we can take an invertible transformation in which { y 1 , … , y p } and { ϵ 1 , … , ϵ p } are bi‐orthonormal: where I p is an identity matrix of order p . Under this transformation (called the generalized synchronization transformation), the state variable U of system () turns into where in which W is called the null controllable part, while V is called the synchronizable state part.…”

“…Based on the exact boundary synchronization for a coupled system of wave equations by Li and Rao, 1‐6 the corresponding generalized exact boundary synchronization was established in the literature 7‐10 . Since there will always be errors in applications, the approximate boundary synchronization was then delivered in their studies, 11,12,16 which does not demand the geometrical control condition, and can be realized by much fewer boundary controls.…”