Controllability problems for the wave equation on a half-plane and modified Sobolev spaces

Abstract

The 2-d wave equation wtt=Δw, t∈(0,T), on the half-plane x1>0 controlled by the Dirichlet boundary condition w(0,x2,t)=δ(x2)u(t) is considered in Sobolev spaces, where T>0 is a constant and u∈L∞(0,T) is a control. This control system is transformed to a control system for the 1-d wave equation in modified Sobolev spaces. These spaces play an important role in the study. Necessary and sufficient conditions of (approximate) L∞-controllability are obtained for the 1-d control problem. It is also proved that the 2-d control system replicates the controllability properties of the 1-d control system and vice versa. Finally, necessary and sufficient conditions of (approximate) L∞-controllability are obtained for the original 2-d control problem.