We consider an open bounded setΩ⊂ℝnand a family{K(t)}t≥0of orthogonal matrices ofℝn. SetΩt={x∈ℝn;x=K(t)y,for all y∈Ω}, whose boundary isΓt. We denote byQ^the noncylindrical domain given byQ^=∪0<t<T{Ωt×{t}}, with the regular lateral boundaryΣ^=∪0<t<T{Γt×{t}}. In this paper we investigate the boundary exact controllability for the linear Schrödinger equationu′−iΔu=finQ^(i2=−1),u=wonΣ^,u(x,0)=u0(x)inΩ0, wherewis the control.