International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Issue 6, Pages 78192, 29 p.
doi:10.1155/IJMMS/2006/78192
Schrödinger equations in noncylindrical domains: exact controllability
G.O. Antunes1
, M.D.G. Da Silva2
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
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.