Exact internal controllability and exact internal synchronization for a kind of first order hyperbolic system

¹ School of Mathematical Sciences, Shanghai Key Laboratory for Contemporary Applied Mathematics, Nonlinear Mathematical Modeling and Methods Laboratory, Fudan University, 200433 Shanghai, PR China
² School of Mathematics, Southeast University, 211189 Nanjing, PR China
³ School of Mathematical Sciences & Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, PR China

^* Corresponding author: dqli@fudan.edu.cn

Received: 1 January 2023
Accepted: 27 December 2023

Abstract

In this paper we investigate the exact controllability and exact synchronization in L² space for a 1-D first order linear hyperbolic system with internal controls located on some part of the domain. Based on the exact boundary controllability theory, we first use the constructive method to establish the exact controllability by internal controls for a time reversible system with inhomogeneous boundary conditions. The method is then adapted to prove the exact internal null controllability for a general system with homogeneous boundary conditions. These results can be then used to establish the exact internal synchronization for the system, and related subjects are further studied.