Schrödinger equations in noncylindrical domains: exact controllability

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 ̂ ∑ = ⋃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.