Introduction Non-controllability result Controllability of linearized system Controllability of nonlinear system Global exact controllability in infnite time of Schrödinger equation Vahagn Nersesyan (Université de Versailles Saint-Quentin) IHP, December 9, 2010 Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Non-controllability result Controllability of linearized system Controllability of nonlinear system Introduction V. N., H. Nersisyan, Global exact controllability in infinite time of Schrödinger equation, arXiv:1006.2602, 2010. Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Non-controllability result Controllability of linearized system Controllability of nonlinear system Introduction Controlled Schrödinger equation: i ˙ z = − ∆ z + V ( x ) z + u ( t ) Q ( x ) z , x ∈ D , z | ∂ D = 0 , z ( 0 , x ) = z 0 ( x ) , where D ⋐ R d , ∂ D ∈ C ∞ , d ≥ 1, V , Q ∈ C ∞ ( D , R ) are given functions, u is the control, z is the state. Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Non-controllability result Controllability of linearized system Controllability of nonlinear system Introduction Controlled Schrödinger equation: i ˙ z = − ∆ z + V ( x ) z + u ( t ) Q ( x ) z , x ∈ D , z | ∂ D = 0 , z ( 0 , x ) = z 0 ( x ) , where D ⋐ R d , ∂ D ∈ C ∞ , d ≥ 1, V , Q ∈ C ∞ ( D , R ) are given functions, u is the control, z is the state. Let U t ( · , u ) : L 2 → L 2 , u ∈ L 1 loc ([ 0 , ∞ ) , R ) be the resolving operator, i.e. U t ( z 0 , u ) = z ( t ) . Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Non-controllability result Controllability of linearized system Controllability of nonlinear system Introduction Controlled Schrödinger equation: i ˙ z = − ∆ z + V ( x ) z + u ( t ) Q ( x ) z , x ∈ D , z | ∂ D = 0 , z ( 0 , x ) = z 0 ( x ) , where D ⋐ R d , ∂ D ∈ C ∞ , d ≥ 1, V , Q ∈ C ∞ ( D , R ) are given functions, u is the control, z is the state. Let U t ( · , u ) : L 2 → L 2 , u ∈ L 1 loc ([ 0 , ∞ ) , R ) be the resolving operator, i.e. U t ( z 0 , u ) = z ( t ) . �U t ( z 0 , u ) � L 2 = � z 0 � L 2 , t ≥ 0 . Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Non-controllability result Controllability of linearized system Controllability of nonlinear system Introduction Controlled Schrödinger equation: i ˙ z = − ∆ z + V ( x ) z + u ( t ) Q ( x ) z , x ∈ D , z | ∂ D = 0 , z ( 0 , x ) = z 0 ( x ) , where D ⋐ R d , ∂ D ∈ C ∞ , d ≥ 1, V , Q ∈ C ∞ ( D , R ) are given functions, u is the control, z is the state. Let U t ( · , u ) : L 2 → L 2 , u ∈ L 1 loc ([ 0 , ∞ ) , R ) be the resolving operator, i.e. U t ( z 0 , u ) = z ( t ) . �U t ( z 0 , u ) � L 2 = � z 0 � L 2 , t ≥ 0 . Let S := { z ∈ L 2 : � z � L 2 = 1 } . Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Non-controllability result Controllability of linearized system Controllability of nonlinear system Introduction Main result The system is globally exactly controllable in infinite time generically in V and Q. Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Non-controllability result Controllability of linearized system Controllability of nonlinear system Introduction Main result The system is globally exactly controllable in infinite time generically in V and Q. For any z 0 , z 1 ∈ S ∩ H k there is a control u ∈ H s ( R + , R ) and a sequence T n → + ∞ such U T n ( z 0 , u ) ⇀ z 1 in H k . Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Non-controllability result Controllability of linearized system Controllability of nonlinear system Plan of the talk 1 Non-controllability results 2 Controllability of linearized system 3 Controllability of nonlinear system Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Non-controllability result Controllability of linearized system Controllability of nonlinear system Previous results Ramakrishna, Salapaka, Dahleh, Rabitz, Pierce, Turinici, Altafini, Albertini, D’Alessandro, . . . Beauchard, Coron, Laurent Chambrion, Mason, Sigalotti, Boscain Mirrahimi, Beauchard, V.N. V.N. Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Non-controllability result Controllability of linearized system Controllability of nonlinear system Non-controllability result Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Non-controllability result Controllability of linearized system Controllability of nonlinear system Non-controllability Theorem (Ball, Marsden, Slemrod, 82) The Schrödinger equation is not exactly controllable in finite time in Sobolev space H 2 with controls L p loc ([ 0 , + ∞ ) , R ) , i.e., for any z 0 ∈ S the set {U t ( z 0 , u ) : t ∈ [ 0 , + ∞ ) , u ∈ L p loc ([ 0 , + ∞ ) , R ) for some p > 1 } does not contain a ball of the space H 2 . Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Non-controllability result Controllability of linearized system Controllability of nonlinear system Non-controllability Theorem The Schrödinger equation is not exactly controllable in finite time in Sobolev spaces H k , k < d with controls H 1 loc ([ 0 , + ∞ ) , R ) , i.e., for any z 0 ∈ S the set {U t ( z 0 , u ) : t ∈ [ 0 , + ∞ ) , u ∈ H 1 loc ([ 0 , + ∞ ) , R ) } does not contain a ball of the space H k . Proof is based on the ideas of Shirikyan introduced to prove non-controllability of Euler equation. Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Previous results Non-controllability result Preliminaries Controllability of linearized system Main result Controllability of nonlinear system Proof Controllability of linearized system Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Previous results Non-controllability result Preliminaries Controllability of linearized system Main result Controllability of nonlinear system Proof Previous results Let us linearize the system around trajectory U t (˜ z 0 , 0 ) : i ˙ z = − ∆ z + V ( x ) z + u ( t ) Q ( x ) U t (˜ z 0 , 0 ) , z | ∂ D = 0 , z ( 0 ) = z 0 . Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Previous results Non-controllability result Preliminaries Controllability of linearized system Main result Controllability of nonlinear system Proof Previous results Let us linearize the system around trajectory U t (˜ z 0 , 0 ) : i ˙ z = − ∆ z + V ( x ) z + u ( t ) Q ( x ) U t (˜ z 0 , 0 ) , z | ∂ D = 0 , z ( 0 ) = z 0 . Beauchard, Chitour, Kateb, Long Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Previous results Non-controllability result Preliminaries Controllability of linearized system Main result Controllability of nonlinear system Proof Preliminaries Let us linearize the system around trajectory U t (˜ z 0 , 0 ) : i ˙ z = − ∆ z + V ( x ) z + u ( t ) Q ( x ) U t (˜ z 0 , 0 ) , z | ∂ D = 0 , z ( 0 ) = z 0 . Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Previous results Non-controllability result Preliminaries Controllability of linearized system Main result Controllability of nonlinear system Proof Preliminaries Let us linearize the system around trajectory U t (˜ z 0 , 0 ) : i ˙ z = − ∆ z + V ( x ) z + u ( t ) Q ( x ) U t (˜ z 0 , 0 ) , z | ∂ D = 0 , z ( 0 ) = 0 . Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Previous results Non-controllability result Preliminaries Controllability of linearized system Main result Controllability of nonlinear system Proof Preliminaries Let us linearize the system around trajectory U t (˜ z 0 , 0 ) : i ˙ z = − ∆ z + V ( x ) z + u ( t ) Q ( x ) U t (˜ z 0 , 0 ) , z | ∂ D = 0 , z ( 0 ) = 0 . Let us rewrite this problem in the Duhamel form � t e i ( t − s )(∆ − V ) u ( s ) Q ( x ) U s (˜ z ( t ) = − i z 0 , 0 ) d s . 0 Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
Introduction Previous results Non-controllability result Preliminaries Controllability of linearized system Main result Controllability of nonlinear system Proof Preliminaries Let us linearize the system around trajectory U t (˜ z 0 , 0 ) : i ˙ z = − ∆ z + V ( x ) z + u ( t ) Q ( x ) U t (˜ z 0 , 0 ) , z | ∂ D = 0 , z ( 0 ) = 0 . Let us rewrite this problem in the Duhamel form � t e i ( t − s )(∆ − V ) u ( s ) Q ( x ) U s (˜ z ( t ) = − i z 0 , 0 ) d s . 0 Let R t be the resolving operator. Vahagn Nersesyan Exact controllability in infinite time of Schrödinger equation
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