Schrödinger equations in noncylindrical domains: exact controllability

Abstract

We consider an open bounded set Ω⊂ℝn and a family {K(t)}t≥0 of orthogonal matrices of ℝn. Set Ωt={x∈ℝn;x=K(t)y,for all y∈Ω}, whose boundary is Γt. We denote by Q^ the noncylindrical domain given by Q^=∪0<t<T{Ωt×{t}}, with the regular lateral boundary Σ^=∪0<t<T{Γt×{t}}. In this paper we investigate the boundary exact controllability for the linear Schrödinger equation u′−iΔu=f in Q^(i2=−1), u=w on Σ^, u(x,0)=u0(x) in Ω0, where w is the control.