International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 78192, 29 pages
doi:10.1155/IJMMS/2006/78192

Schrödinger equations in noncylindrical domains: exact controllability

G. O. Antunes,1 M. D. G. da Silva,2 and R. F. Apolaya3

1Instituto de Matemática e Estatística, Universidade do Estado do Rio de Janeiro, Rio de Janeiro 20550-900, Brazil
2Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro 21941-590, Brazil
3Instituto de Matemática, Universidade Federal Fluminense, Rio de Janeiro 24020-140, Niterói, Brazil

Received 6 July 2005; Accepted 12 March 2006

Copyright © 2006 G. O. Antunes et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.