Exact Controllability in L 2(Ω) of the Schrödinger Equation in a Riemannian Manifold with L 2(Σ1)-Neumann Boundary Control

2007 ◽  
pp. 613-636 ◽  
Author(s):  
Roberto Triggiani
Keyword(s):  
Riemannian Manifold ◽  
Schrödinger Equation ◽  
Boundary Control ◽  
Schrodinger Equation ◽  
Exact Controllability ◽  
Neumann Boundary

scholarly journals CONTROL AND STABILIZATION OF THE NONLINEAR SCHRÖDINGER EQUATION ON RECTANGLES

2010 ◽  
Vol 20 (12) ◽  
pp. 2293-2347 ◽  
Author(s):  
LIONEL ROSIER ◽  
BING-YU ZHANG
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Control Region ◽  
Internal Control ◽  
Schrodinger Equation ◽  
Exact Controllability ◽  
Neumann Boundary ◽  
Local Controllability ◽  
Boundary Controls ◽  
Semilinear Schrödinger Equation

This paper studies the local exact controllability and the local stabilization of the semilinear Schrödinger equation posed on a product of n intervals (n ≥ 1). Both internal and boundary controls are considered, and the results are given with periodic (resp. Dirichlet or Neumann) boundary conditions. In the case of internal control, we obtain local controllability results which are sharp as far as the localization of the control region and the smoothness of the state space are concerned. It is also proved that for the linear Schrödinger equation with Dirichlet control, the exact controllability holds in H-1(Ω) whenever the control region contains a neighborhood of a vertex.

scholarly journals Schrödinger equations in noncylindrical domains: exact controllability

2006 ◽  
Vol 2006 ◽  
pp. 1-29
Author(s):  
G. O. Antunes ◽  
M. D. G. da Silva ◽  
R. F. Apolaya
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Exact Controllability ◽  
Schrodinger Equations ◽  
Lateral Boundary ◽  
Schrödinger Equations ◽  
Orthogonal Matrices ◽  
Noncylindrical Domain ◽  
Bounded Set

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.

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