Statement of the problem Previous results Statement of the results Main steps of the proof Local bilinear controllability of the Schr¨ odinger equation. Jean-Pierre Puel JPP : LMV, Universit´ e de Versailles St Quentin, Versailles jppuel@math.uvsq.fr JMC 60 Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
Statement of the problem Previous results Statement of the results Main steps of the proof Outline Statement of the problem 1 Previous results 2 Dimension 1 Dimension N ≥ 2 Results on linear controllability Statement of the results 3 Main result Case of a rectangle Main steps of the proof 4 Technical lemmas Strategy Real control Regular control Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
Statement of the problem Previous results Statement of the results Main steps of the proof Statement of the problem Controllability of Schr¨ odinger equation in a neighborhood of an eigenfunction. Control : real potential ⇒ bilinear controllability problem. R N . Γ = ∂ Ω, T > 0. Ω bounded regular open set of I i ∂ y ∂ t + ∆ y + V y = 0 in Ω × (0 , T ) , y = 0 on Γ × (0 , T ) , y (0) = y 0 in Ω . (Real) Eigenfunctions of Laplace operator − ∆ ϕ k = λ k ϕ k in Ω , ϕ k = 0 on Γ , � Ω ϕ k ϕ j dx = δ k , j ∀ k , j = 1 , · · · , + ∞ . Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
Statement of the problem Previous results Statement of the results Main steps of the proof The problem If V = 0 (free Schr¨ odinger equation) and y 0 = ϕ k the solution is ϕ k ( t ) = e − i λ k t ϕ k . ˜ Question : Given y 0 (close to ϕ k ), can we find V real such that ϕ k ( T ) = e − i λ k T ϕ k ? y ( T ) = ˜ odinger equation preserves the L 2 norm. V real implies Schr¨ Therefore necessary condition � | y 0 | 2 dx = 1 . Ω Initial data y 0 will be taken on the sphere S of radius 1 in L 2 (Ω). Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
Statement of the problem Previous results Statement of the results Main steps of the proof Previous results Case of dimension 1 Case of dimension 1, and V ( x , t ) = u ( t ) µ ( x ) µ : prescribed profile. Actual control : amplitude u ( . ). Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
Statement of the problem Previous results Statement of the results Main steps of the proof Negative result Bad news : Negative result by J.Ball-J.Marsden-M.Slemrod SICON 1982 : If X is the space of values of y ( t ) and if the product by µ is bounded from X to X , no hope to obtain a control u ∈ L r loc (0 , T ). They prove that the set of reachable states is contained in a countable union of compact sets, and therefore has an empty interior. Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
Statement of the problem Previous results Statement of the results Main steps of the proof Positive result Nevertheless first result by K.Beauchard JMPA 2005 using Nash Moser Theorem. Real breakthrough. Then K.Beauchard-C.Laurent JMPA 2010 gave another proof. Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
Statement of the problem Previous results Statement of the results Main steps of the proof Previous results Case of dimension 1 They used the space H ∆ (Ω) = { z ∈ H 1 0 (Ω) , ∆ z ∈ H 1 0 (Ω) } � H 3 (Ω) ∩ H 1 0 (Ω) and µ such that ∀ z ∈ H ∆ (Ω) , µ z ∈ H 3 (Ω) ∩ H 1 0 (Ω) but in general µ z / ∈ H ∆ (Ω) . The difference comes from the boundary conditions. Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
Statement of the problem Previous results Statement of the results Main steps of the proof Previous results Case of dimension 1 They proved the following regularity result in 1-dimension : In the free Schr¨ odinger equation, if the right hand side is in L 2 (0 , T ; H 3 (Ω) ∩ H 1 0 (Ω)) and y 0 ∈ H ∆ (Ω), then the solution y belongs to C ([0 , T ]; H ∆ (Ω)). They could find a control u ∈ L 2 (0 , T ) using the controllability of the linearized problem and an inverse mapping theorem. Essential action due to the boundary values. Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
Statement of the problem Previous results Statement of the results Main steps of the proof Case of dimension N ≥ 2 Regularity result has been extended to dimension N for a regular domain in J-P.P Revista Mat. Complutense 2013. But if V ( x , t ) = u ( t ) µ ( x ) the linearized problem is no longer controllable. Same argument cannot be applied. In dimension N = 2, K.Beauchard-C.Laurent (recent result to appear, hal-01333627) obtain a controllability result considering potential V satisfying Poisson equation � − ∆ V ( t ) + V ( t ) = 0 in Ω , V ( t ) = g ( t ) on Γ , and some conditions on ϕ k , the actual control being here the boundary value g ( . ). Here we will consider potentials depending on x and t concentrated near the boundary. Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
Statement of the problem Previous results Statement of the results Main steps of the proof Previous results for linear boundary controllability Boundary linear exact controllability : given a subset Γ 0 of Γ, for any y 0 ∈ H − 1 (Ω) can we find g ∈ L 2 (0 , T ; L 2 (Γ 0 )) such that the solution y of i ∂ y ∂ t + ∆ y = 0 in Ω × (0 , T ) , y = g on Γ 0 × (0 , T ) , y = 0 on (Γ \ Γ 0 ) × (0 , T ) , y (0) = y 0 in Ω , satisfies y ( T ) = 0 . Adjoint problem i ∂ϕ ∂ t + ∆ ϕ = 0 in Ω × (0 , T ) , ϕ = 0 on Γ × (0 , T ) , ϕ (0) = ϕ 0 in Ω , Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
Statement of the problem Previous results Statement of the results Main steps of the proof Previous results for linear boundary controllability Exact controllability is equivalent to boundary observability inequality � | ∂ϕ || ϕ 0 || 2 ∂ν | 2 d σ dt , 0 (Ω) ≤ C H 1 Γ 0 × (0 , T ) Inequality proved for any T > 0 by E.Machtyngier SICON 1994 with multiplier method when R N such that Γ 0 = { x ∈ Γ , ( x − x 0 ) .ν > 0 } . There exists x 0 ∈ I Extended by G.Lebeau JMPA 1992 when Γ 0 satisfies the “geometric control condition” using micro local analysis arguments. Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
Statement of the problem Previous results Statement of the results Main steps of the proof Previous results for linear distributed controllability Distributed (internal) linear exact controllability : given a non empty open subset ω of Ω, for any y 0 ∈ L 2 (Ω), can we find a control h ∈ L 2 (0 , T ; L 2 ( ω )) such that the solution of i ∂ y ∂ t + ∆ y = h . 1 I ω in Ω × (0 , T ) , y = 0 on Γ × (0 , T ) , y (0) = y 0 in Ω , satisfies y ( T ) = 0 . This is equivalent to internal observability inequality for the adjoint state � | ϕ 0 | 2 | ϕ | 2 dxdt . L 2 (Ω) ≤ C ω × (0 , T ) Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
Statement of the problem Previous results Statement of the results Main steps of the proof Previous results for linear distributed controllability E.Machtyngier proved that when Γ 0 is such that the boundary observability is true (e.g the GCC) then the internal observability inequality is true when ω is a neighborhood of Γ 0 for example for η > 0 � ω η = ( B ( x ; η ) ∩ Ω) . x ∈ Γ 0 If N = 2 and Ω is a rectangle it is proved in S.Jaffard Port. Math. 1990 that internal observability inequality is valid for any non empty open subset ω of Ω. Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
Statement of the problem Previous results Statement of the results Main steps of the proof Main result. N ≤ 3 We take potentials concentrated near Γ 0 satisfying the boundary observability inequality. We will consider the space of potentials E = { V ∈ H 2 (Ω) , V ϕ k ∈ H 3 (Ω) ∩ H 1 0 (Ω) } . Theorem N ≤ 3 , Ω of class C 3 ,α with α > 0 . Γ 0 such that the boundary observability inequality is valid, and ( λ k , ϕ k ) an eigenpair for the Laplace operator. We assume (H1) λ k is a simple eigenvalue. (H2) | ∂ϕ k ∂ν | > 0 on Γ 0 . Then there exists δ > 0 such that for every y 0 ∈ H ∆ (Ω) ∩ S with || y 0 − ϕ k || H ∆ (Ω) ≤ δ , there exists a real potential V ∈ C (0 , T ; E ) such that the corresponding solution y satisfies y ( T ) = e − i λ k T ϕ k . Jean-Pierre Puel Local bilinear controllability of the Schr¨ odinger equation
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