Controllability and Stabilizability of a Higher Order Wave Equation on a Periodic Domain

Abstract

This article studies internal control of the sixth order Boussinesq equation $u_{tt}-u_{xx}+\beta u_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=f $ posed on a periodic domain $\mathbb{T}$ with the internal control input $f(\cdot,t)$ acting on an arbitrarily small open subset of the domain $\mathbb{T}$. It is shown that the system is locally exactly controllable and exponentially stabilizable in the classic Sobolev space $H^{s+3}(\mathbb{T})\times H^s(\mathbb{T})$ for any $s\geq 0$.