Introduction Main results Idea of proof Other results Bilinear control of nonlinear Schrödinger and wave equation Camille Laurent (in collaboration with K. Beauchard) CMAP , Ecole Polytechnique
Introduction Main results Idea of proof Other results Bilinear control Model system i ∂ t ψ ( t , x )+ ∂ 2 x ψ ( t , x ) = − u ( t ) µ ( x ) ψ ( t , x ) . (1) u (the control) and µ the real valued potential . So, at each time t , the available control u ( t ) is only the amplitude and not a distributed fonction. 2/20
Introduction Main results Idea of proof Other results Bilinear control Model system i ∂ t ψ ( t , x )+ ∂ 2 x ψ ( t , x ) = − u ( t ) µ ( x ) ψ ( t , x ) . (1) u (the control) and µ the real valued potential . So, at each time t , the available control u ( t ) is only the amplitude and not a distributed fonction. Aim : local control by perturbation 2/20
Introduction Main results Idea of proof Other results Bilinear control Model system i ∂ t ψ ( t , x )+ ∂ 2 x ψ ( t , x ) = − u ( t ) µ ( x ) ψ ( t , x ) . (1) u (the control) and µ the real valued potential . So, at each time t , the available control u ( t ) is only the amplitude and not a distributed fonction. Aim : local control by perturbation Other results : nonlinear Schrödinger and nonlinear wave equation 2/20
Introduction Main results Idea of proof Other results Bibliography Exact controllability • Negative result : Ball-Marsden-Slemrod (82) • Positive result : Local exact controllability in 1D : in H 7 , in large time Beauchard (05), Coron(06) : T min > 0, controllability in 1D between eigenstates : Beauchard and Coron (06) 3/20
Introduction Main results Idea of proof Other results Bibliography Approximate controllability • By Gallerkin approximation and finite dimensional methods Chambrion-Mason-Sigalotti-Boscain(09) • By stabilization Nersesyan (09) • Exact controllability "at T = ∞ " Nersesyan-Nersisyan (10) 4/20
Introduction Main results Idea of proof Other results First obstruction Ball-Marsden-Slemrod Theorem (Ball-Marsden-Slemrod 82) If the multiplication by µ is bounded on the functional space X, then the set of reachable states is a countable union of compact sets of X ⇒ no controllability in X. 5/20
Introduction Main results Idea of proof Other results First obstruction Ball-Marsden-Slemrod Theorem (Ball-Marsden-Slemrod 82) If the multiplication by µ is bounded on the functional space X, then the set of reachable states is a countable union of compact sets of X ⇒ no controllability in X. Once the functional space X is chosen, we must chose a potentiel µ enough regular to be able to do a perturbation theory, but not too much otherwise Ball-Marsden-Slemrod applies. 5/20
Introduction Main results Idea of proof Other results First obstruction Ball-Marsden-Slemrod Theorem (Ball-Marsden-Slemrod 82) If the multiplication by µ is bounded on the functional space X, then the set of reachable states is a countable union of compact sets of X ⇒ no controllability in X. Once the functional space X is chosen, we must chose a potentiel µ enough regular to be able to do a perturbation theory, but not too much otherwise Ball-Marsden-Slemrod applies. First solution given by K. Beauchard : use of Nash-Moser theorem. Improved method (with K. Beauchard) : prove directly that the system can be well posed even if the potential is "bad" ⇒ optimal with respect to regularity and time of control ; easier proof that can be extended to other cases. 5/20
Introduction Main results Idea of proof Other results Main results Denote ϕ k the eigenfunctions of the Dirichlet Laplacian operator. We control near the ground eigenstate ϕ 1 with solution ψ 1 ( t ) = e − i λ 1 t ϕ 1 . S is the unit sphere of L 2 (] 0 , 1 [ x ) . Theorem (with K. Beauchard) Let T > 0 and µ ∈ H 3 (] 0 , 1 [ , R ) be such that ∃ c > 0 such that c k 3 � |� µ ϕ 1 , ϕ k �| , ∀ k ∈ N ∗ . (2) There exists δ > 0 such that for any ψ f ∈ S ∩ H 3 ( 0 ) (] 0 , 1 [ , C ) with � ψ f − ψ 1 ( T ) � H 3 < δ there exists a control u ∈ L 2 (] 0 , T [ , R ) s.t. the solution of (1) with initial condition ψ ( 0 ) = ϕ 1 and control u satisfies ψ ( T ) = ψ f . 6/20
Introduction Main results Idea of proof Other results Remarks about assumption (2) √ � 1 4 [( − 1 ) k + 1 µ ′ ( 1 ) − µ ′ ( 0 )] 2 0 ( µ ϕ 1 ) ′′′ ( x ) cos ( k π x ) � µ ϕ 1 , ϕ k � L 2 = − k 3 π 2 ( k π ) 3 x 4 [( − 1 ) k + 1 µ ′ ( 1 ) − µ ′ ( 0 )] + ℓ 2 sequence = . k 3 π 2 k 3 and we can prove that assumption (2) is generic in H 3 (] 0 , 1 [) . 7/20
Introduction Main results Idea of proof Other results Remarks about assumption (2) √ � 1 4 [( − 1 ) k + 1 µ ′ ( 1 ) − µ ′ ( 0 )] 2 0 ( µ ϕ 1 ) ′′′ ( x ) cos ( k π x ) � µ ϕ 1 , ϕ k � L 2 = − k 3 π 2 ( k π ) 3 x 4 [( − 1 ) k + 1 µ ′ ( 1 ) − µ ′ ( 0 )] + ℓ 2 sequence = . k 3 π 2 k 3 and we can prove that assumption (2) is generic in H 3 (] 0 , 1 [) . Such assumption implies that multiplication by µ does not map H 3 ( 0 ) into itself. 7/20
Introduction Main results Idea of proof Other results Remarks about assumption (2) √ � 1 4 [( − 1 ) k + 1 µ ′ ( 1 ) − µ ′ ( 0 )] 2 0 ( µ ϕ 1 ) ′′′ ( x ) cos ( k π x ) � µ ϕ 1 , ϕ k � L 2 = − k 3 π 2 ( k π ) 3 x 4 [( − 1 ) k + 1 µ ′ ( 1 ) − µ ′ ( 0 )] + ℓ 2 sequence = . k 3 π 2 k 3 and we can prove that assumption (2) is generic in H 3 (] 0 , 1 [) . Such assumption implies that multiplication by µ does not map H 3 ( 0 ) into itself. Rk : there are some cases where assumption (2) is not fufilled but Beauchard and Coron manage to prove the controllability with additional techniques : return method or power series expansions. 7/20
Introduction Main results Idea of proof Other results "Regularizing" effect � ( − ∆ Dirichlet ) 3 / 2 � H 3 = D ( 0 ) � u ( 0 ) = u ( 1 ) = 0 = u ′′ ( 0 ) = u ′′ ( 1 ) � u ∈ H 3 � � = Proposition (with K. Beauchard) Let f ∈ L 2 (( 0 , T ) , H 3 ∩ H 1 0 ) (not necessarily H 3 ( 0 ) ). Then, the solution ψ of i ∂ t ψ ( t , x )+ ∂ 2 � x ψ ( t , x ) = f Ψ( 0 ) = 0 belongs to C 0 ([ 0 , T ] , H 3 ( 0 ) ) 8/20
Introduction Main results Idea of proof Other results Method of proof • Prove that the linearized problem is controlable by Ingham Theorem. • Use classical inverse mapping theorem thanks to our "regularity result". 9/20
Introduction Main results Idea of proof Other results Method of proof • Prove that the linearized problem is controlable by Ingham Theorem. • Use classical inverse mapping theorem thanks to our "regularity result". Rk : In certain cases treated by Beauchard and Coron, we can get controllability even if the linearized system is not controllable (use return method and quasi-static transformation or expansion to higher order). Our result should improve the regularity in these results. 9/20
Introduction Main results Idea of proof Other results Controllability of the linearized system We linearize around the trajectory ψ 1 ( t , x ) = e − i λ 1 t ϕ 1 . � i ∂ t Ψ( t , x )+ ∂ 2 x Ψ( t , x ) = − v ( t ) µ ( x ) ψ 1 ( t , x ) ψ ( 0 , x ) = 0 . 10/20
Introduction Main results Idea of proof Other results Controllability of the linearized system We linearize around the trajectory ψ 1 ( t , x ) = e − i λ 1 t ϕ 1 . � i ∂ t Ψ( t , x )+ ∂ 2 x Ψ( t , x ) = − v ( t ) µ ( x ) ψ 1 ( t , x ) ψ ( 0 , x ) = 0 . � � T ∞ � v ( t ) e i ( λ k − λ 1 ) t dt e − i λ k T ϕ k . ∑ Ψ( T ) = i � µ ϕ 1 , ϕ k � 0 k = 1 10/20
Introduction Main results Idea of proof Other results Controllability of the linearized system We linearize around the trajectory ψ 1 ( t , x ) = e − i λ 1 t ϕ 1 . � i ∂ t Ψ( t , x )+ ∂ 2 x Ψ( t , x ) = − v ( t ) µ ( x ) ψ 1 ( t , x ) ψ ( 0 , x ) = 0 . � � T ∞ � v ( t ) e i ( λ k − λ 1 ) t dt e − i λ k T ϕ k . ∑ Ψ( T ) = i � µ ϕ 1 , ϕ k � 0 k = 1 Ψ( T ) = Ψ f is equivalent to the trigonometric moment problem � T v ( t ) e i ( λ k − λ 1 ) t dt = d k − 1 (Ψ f ) := � Ψ f , ϕ k � e i λ k T i � µ ϕ 1 , ϕ k � , ∀ k ∈ N ∗ . (3) 0 By Ingham theorem : ∀ T > 0 ;Ψ f ∈ H 3 ( 0 ) (] 0 , 1 [ there exists one v ∈ L 2 (] 0 , T [) solution. (if T = 2 / π , it is only Fourier series in time) 10/20
Introduction Main results Idea of proof Other results Ingham Theorem Theorem (Ingham, Haraux) Let N ∈ N , ( ω k ) k ∈ Z be an increasing sequence of real numbers such that ω k + 1 − ω k � γ > 0 , ∀ k ∈ Z , | k | � N , ω k + 1 − ω k � ρ > 0 , ∀ k ∈ Z , and T > 2 π / γ . The map F := Clos L 2 (] 0 , T [) ( Span { e i ω k t ; k ∈ Z } ) l 2 ( Z , C ) J : → � � T � 0 v ( t ) e i ω k t dt �→ v k ∈ Z is an isomorphism. This is a kind of Fourier decomposition for "not exactly orthogonal basis" (Riesz basis). 11/20
Introduction Main results Idea of proof Other results Proof of the "regularizing" effect � t � � t ∞ ∞ � e − i ∂ 2 x e i λ k s ds x s f ( s ) ds = ∑ ∑ 0 � f ( s ) , ϕ k � L 2 ϕ k = x k ( t ) ϕ k . 0 k = 1 k = 1 12/20
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