Local bilinear controllability of the Schr odinger equation. - - PowerPoint PPT Presentation

Statement of the problem Previous results Statement of the results Main steps of the proof

Local bilinear controllability of the Schr¨

• dinger 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¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Outline

1

Statement of the problem

2

Previous results Dimension 1 Dimension N ≥ 2 Results on linear controllability

3

Statement of the results Main result Case of a rectangle

4

Main steps of the proof Technical lemmas Strategy Real control Regular control

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Statement of the problem Controllability of Schr¨

• dinger equation in a neighborhood of an

eigenfunction. Control : real potential ⇒ bilinear controllability problem. Ω bounded regular open set of I

• RN. Γ = ∂Ω, T > 0.

   i ∂y

∂t + ∆y + V y = 0 in Ω × (0, T),

y = 0 on Γ × (0, T), y(0) = y0 in Ω. (Real) Eigenfunctions of Laplace operator    −∆ϕk = λkϕk in Ω, ϕk = 0 on Γ,

• Ω ϕkϕjdx = δk,j ∀ k, j = 1, · · · , +∞.

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

The problem If V = 0 (free Schr¨

• dinger equation) and y0 = ϕk the solution is

˜ ϕk(t) = e−iλktϕk. Question : Given y0 (close to ϕk), can we find V real such that y(T) = ˜ ϕk(T) = e−iλkTϕk ? V real implies Schr¨

• dinger equation preserves the L2 norm.

Therefore necessary condition

• Ω

|y0|2dx = 1. Initial data y0 will be taken on the sphere S of radius 1 in L2(Ω).

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger 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¨

• dinger 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 ∈ Lr

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¨

• dinger 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¨

• dinger 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 ∈ H1

0(Ω), ∆z ∈ H1 0(Ω)} H3(Ω) ∩ H1 0(Ω)

and µ such that ∀z ∈ H∆(Ω), µz ∈ H3(Ω) ∩ H1

0(Ω)

but in general µz / ∈ H∆(Ω). The difference comes from the boundary conditions.

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger 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¨

• dinger equation, if the right hand side is in

L2(0, T; H3(Ω) ∩ H1

0(Ω)) and y0 ∈ H∆(Ω), then the solution y belongs to

C([0, T]; H∆(Ω)). They could find a control u ∈ L2(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¨

• dinger 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 ϕk, 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¨

• dinger 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 Γ0 of Γ, for any y0 ∈ H−1(Ω) can we find g ∈ L2(0, T; L2(Γ0)) such that the solution y of        i ∂y

∂t + ∆y = 0 in Ω × (0, T),

y = g on Γ0 × (0, T), y = 0 on (Γ \ Γ0) × (0, T), y(0) = y0 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) = ϕ0 in Ω,

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger 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 ||ϕ0||2

H1

0 (Ω) ≤ C

• Γ0×(0,T)

|∂ϕ ∂ν |2dσdt, Inequality proved for any T > 0 by E.Machtyngier SICON 1994 with multiplier method when There exists x0 ∈ I RN such that Γ0 = {x ∈ Γ, (x − x0).ν > 0}. Extended by G.Lebeau JMPA 1992 when Γ0 satisfies the “geometric control condition” using micro local analysis arguments.

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger 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

• pen subset ω of Ω, for any y0 ∈ L2(Ω), can we find a control

h ∈ L2(0, T; L2(ω)) such that the solution of    i ∂y

∂t + ∆y = h.1

Iω in Ω × (0, T), y = 0 on Γ × (0, T), y(0) = y0 in Ω, satisfies y(T) = 0. This is equivalent to internal observability inequality for the adjoint state |ϕ0|2

L2(Ω) ≤ C

• ω×(0,T)

|ϕ|2dxdt.

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger 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 Γ0 is such that the boundary

• bservability is true (e.g the GCC) then the internal observability

inequality is true when ω is a neighborhood of Γ0 for example for η > 0 ωη =

• x∈Γ0

(B(x; η) ∩ Ω). 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¨

• dinger 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 Γ0 satisfying the boundary

• bservability inequality. We will consider the space of potentials

E = {V ∈ H2(Ω), V ϕk ∈ H3(Ω) ∩ H1

0(Ω)}.

Theorem N ≤ 3, Ω of class C 3,α with α > 0. Γ0 such that the boundary

• bservability inequality is valid, and (λk, ϕk) an eigenpair for the Laplace
• perator. We assume

(H1) λk is a simple eigenvalue. (H2) |∂ϕk ∂ν | > 0 on Γ0. Then there exists δ > 0 such that for every y0 ∈ H∆(Ω) ∩ S with ||y0 − ϕk||H∆(Ω) ≤ δ, there exists a real potential V ∈ C(0, T; E) such that the corresponding solution y satisfies y(T) = e−iλkTϕk.

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Case of a rectangle or an interval In the case of a rectangle for N = 2 or an interval for N = 1 we have a similar result with simpler assumptions. Theorem N = 2 and Ω is a rectangle or N = 1 and Ω is an interval. Let (λk, ϕk) be an eigenpair for the Laplace operator with Dirichlet boundary

• conditions. We assume hypothesis (H1) in the case N = 2 and no

hypothesis if N = 1 (hypothesis (H1) is automatically satisfied). Then there exists δ > 0 such that for every y0 ∈ H∆(Ω) ∩ S with ||y0 − ϕk||H∆(Ω) ≤ δ, there exists a real potential V ∈ C([0, T]; E) such that the corresponding solution y satisfies y(T) = e−iλkTϕk.

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

• Proof. Technical lemmas

Hypothesis (H1) ⇒ ∃ǫ0 such that ϕk = 0 on ωǫ0. For ǫ and ǫ1 such that 0 < ǫ1 < ǫ < ǫ0 we define χωǫ ∈ C ∞

0 (ωǫ ∪ (∂ωǫ ∩ Γ)) such that

• 0 ≤ χωǫ ≤ 1,

χωǫ ≥ 1 Iωǫ1 . We write ω = ωǫ and χω = χωǫ. Lemma If y ∈ H3(Ω) ∩ H1

0(Ω) then χω. y

ϕk ∈ H2(Ω).

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Technical lemmas Idea of proof. The result is local. It is enough to prove it in a (small) neighborhood still denoted by ω of any point of ∂ω ∩ Γ. We can take χω = 1. Because ϕk(x) > 0 (and of course vanishes on the boundary), we can make a change of coordinates (x′, ξ) where x′ is the tangential coodinate and ξ = ϕk(x′, xN). The (local) images of Γ and ω can be taken as {ξ = 0} and {0 < ξ < ǫ 2}. We can write y(x′, xN) = ˜ y(x′, ξ) = 1 d dt ˜ y(x′, tξ)dt = 1 ∂˜ y ∂ξ (x′, tξ)ξdt. Therefore ˜ y(x′, ξ) ξ = 1 ∂˜ y ∂ξ (x′, tξ)dt ∈ H2(ω).

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Technical lemmas Lemma When N ≤ 3, for any y ∈ H∆(Ω) and V ∈ E, then χω.Vy ∈ H3(Ω) ∩ H1

0(Ω) and the mapping

(y, V ) ∈ H∆(Ω) × E → χω.Vy ∈ H3(Ω) ∩ H1

0(Ω)

is bilinear continuous. Proof. We have H2(Ω) ⊂ L∞(Ω) and H1(Ω) ⊂ L6(Ω). Easy to show that χω.Vy ∈ L2(Ω), ∂ ∂xi (χωVy) ∈ L2(Ω), ∂2 ∂xi∂xj (χωVy) ∈ L2(Ω).

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Technical lemmas Let us show that ∂3 ∂xi∂xj∂xl (χωVy) ∈ L2(Ω). The only real difficulty concerns the derivatives of the term χω ∂2V ∂xi∂xj y = ∂2V ∂xi∂xj ϕkχω y ϕk . We know that χω y ϕk ∈ H2(Ω). To obtain the result, let us show that ∂2V ∂xi∂xj ϕk ∈ H1(Ω). We can write ∂2V ∂xi∂xj ϕk = ∂2 ∂xi∂xj (V ϕk) − ∂V ∂xi ∂ϕk ∂xj − ∂V ∂xj ∂ϕk ∂xi − V ∂2ϕk ∂xi∂xj and in the right hand side, each term belongs to H1(Ω).

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Existence result Lemma For any y0 ∈ H∆(Ω) ∩ S and V ∈ C([0, T]; E), with V real, there exists a unique solution y ∈ C([0, T]; H∆(Ω) ∩ S) to the Schr¨

• dinger equation

   i ∂y

∂t + ∆y + χωVy = 0 in Ω × (0, T),

y = 0 on Γ × (0, T), y(0) = y0 in Ω. Proof. Take ˜ y ∈ C([0, T]; H∆(Ω)). We know that χωV ˜ y ∈ C([0, T]; H3(Ω) ∩ H1

0(Ω)). Define z as the solution of

   i ∂z

∂t + ∆z + χωV ˜

y = 0 in Ω × (0, T), z = 0 on Γ × (0, T), z(0) = y0 in Ω.

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Existence result From the regularity result (J-P.P. Rev. Compl.), we have in fact z ∈ C([0, T]; H∆(Ω)) and ||z||C([0,T];H∆(Ω)) ≤ C(||y0||H∆(Ω) + ||V ˜ y||L2(0,T;H3(Ω)∩H1

0 (Ω)))

≤ C(||y0||H∆(Ω) + √ T||V ||C([0,T];E)||˜ y||C([0,T];H∆(Ω))). Classical fixed point method for T small but independent of the initial value, then by simple iterations for any T > 0 we obtain a solution y ∈ C([0, T]; H∆(Ω)). If y0 ∈ S, then as V is real, the equation preserves the L2 norm and y(t) ∈ S for t ∈ [0, T]. We can now define the mapping Λ by Λ(y0, V ) = (y(T), y0) and Λ maps continuously H∆(Ω) ∩ S × C([0, T]; E) into (H∆(Ω) ∩ S)2. We have Λ(ϕk, 0) = ( ˜ ϕk(T), ϕk) = (e−iλkTϕk, ϕk).

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Mapping Λ Define TSk = {z ∈ L2(Ω), Re (z, ϕk)L2(Ω) = 0} Lemma The mapping Λ is differentiable at (ϕk, 0) and for any z0 ∈ H∆(Ω) ∩ TSk and W ∈ C([0, T]; E) we have Λ′(ϕk, 0)[z0, W ] = (˜ z(T), z0) where    i ∂˜

z ∂t + ∆˜

z + χωW ˜ ϕk = 0 in Ω × (0, T), ˜ z = 0 on Γ × (0, T), ˜ z(0) = z0 in Ω. Moreover, ˜ z(T) ∈ H∆(Ω) and Re (˜ z(T), ˜ ϕk(T))L2(Ω) = 0.

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Strategy In order to apply an inverse mapping theorem we want to show that Λ′(ϕk, 0) has a continuous right inverse. Writing z(t) = eiλkt ˜ z(t). it means that we have to solve the null controllability problem for    i ∂z

∂t + ∆z + λkz + χωW ϕk = 0 in Ω × (0, T),

z = 0 on Γ × (0, T), z(0) = z0 in Ω. More precisely, we want to show that for any z0 ∈ H∆(Ω) ∩ TSk, we can find a real potential (control) W ∈ C([0, T]; E) such that the solution z satisfies z(T) = 0. Two difficulties : to find a real control W and to show that W can be taken with sufficient regularity.

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Existence of a real control The following result is essentially due to K.Beauchard-C.Laurent. Proposition Let T > 0. For every z0 ∈ TSk, there exists a real control g ∈ C([0, T]; L2(Ω)) such that if z is the solution of    i ∂z

∂t + ∆z + λkz + χωg = 0 in Ω × (0, T),

z = 0 on Γ × (0, T), z(0) = z0 in Ω. then we have z(T) = 0. The proof requires several steps.

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Real control. Strategy Let us consider the adjoint equation    i ∂ψ

∂t + ∆ψ + λkψ = 0 in Ω × (0, T),

ψ = 0 on Γ × (0, T), ψ(0) = ψ0 in Ω, ψ and ˆ ψ will correspond to initial values ψ0 and ˆ ψ0.t We take T0 such that 0 < T0 < T and δ > 0 such that 4δ ≤ (T − T0) and we define a function η ∈ C ∞

0 (I

R) such that      0 ≤ η(t) ≤ 1, ∀t ∈ I R, η(t) = 1, ∀t ∈ [2δ, T − 2δ], Supp(η) = [δ, T − δ], η(t) = 0 for t ∈ (δ, T − δ).

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Real control. Strategy We define w as the solution of the (backward) equation    i ∂w

∂t + ∆w + λkw + ηχωIm ψ = 0 in Ω × (0, T),

w = 0 on Γ × (0, T), w(T) = 0 in Ω. We have w ∈ C([0, T]; L2(Ω)) and w(0) ∈ TSk. Multipying by ˆ ψ and taking the imaginary part we obtain −Re (w(0), ˆ ψ0)L2(Ω) =

• ω×(0,T)

ηχωIm ψIm ˆ ψdxdt = a(ψ0, ˆ ψ0) To solve our problem we want to find ψ0 ∈ TSk such that w(0) = z0 or equivalently a solution ψ0 ∈ TSk of the variational problem a(ψ0, ˆ ψ0) = −Re

• Ω

z0 ¯ ˆ ψ0dx, ∀ ˆ ψ0 ∈ TSk

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Real control. Strategy From Lax-Milgram Theorem this will be the case If we can prove a coercivity inequality of the form ∃C > 0, ∀ψ0 ∈ TSk, |ψ0|2

L2(Ω) ≤ Ca(ψ0, ψ0) = C

• ω×(0,T)

ηχω|Im ψ|2dxdt, This will be done in two steps. Lemma There exists C > 0 such that for every ψ0 ∈ L2(Ω) |ψ0|2

L2(Ω) ≤ C

• ω×(0,T)

ηχω|Im ψ|2dxdt + C||ψ0||2

H−2(Ω).

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Real control. Proof We write 2Im ψ = ψ − ¯ ψ i |ψ|2 = 2|Im Ψ|2 + 1 2((ψ)2) + ( ¯ ψ)2). From the internal observability inequality we have |ψ0|2

L2(Ω) ≤ C

• ω×(2δ,T0+2δ)

χω|ψ|2dxdt ≤ C

• ω×(0,T)

ηχω|ψ|2dxdt ≤ 2C

• ω×(0,T)

ηχω|Im ψ|2dxdt +C 2

• ω×(0,T)

ηχω(ψ)2dxdt

• + C

2

• ω×(0,T)

ηχω( ¯ ψ)2dxdt

• .

Let us show that (analogous for ¯ ψ)

• ω×(0,T)

ηχω(ψ)2dxdt

• ≤ C||ψ0||2

H−2(Ω)

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Real control. Proof If ψ0 =

+∞

• j=1

ajϕj we have ψ(t) =

+∞

• j=1

aje−i(λj−λk)tϕj. Therefore

• ω×(0,T)

ηχω(ψ)2dxdt =

+∞

• j,l=1
• ω×(0,T)

(ηe2iλkt)e−i(λj+λl)tajalϕjϕldxdt Now we can integrate by parts 2m times the term T

0 (ηe2iλkt)e−i(λj+λl)tdt to obtain

• T

(ηe2iλkt)e−i(λj+λl)tdt

• ≤

C (λj + λl)2m

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Real control. Proof We then obtain

• ω×(0,T)

ηχω(ψ)2dxdt

• ≤ C

+∞

• j,l=1
• Ω

|aj||al| (λj + λl)2m |ϕj||ϕl|dx ≤ C

+∞

• j,l=1

|aj| λm

j

|al| λm

l

≤ C(

+∞

• j=1

|aj| λm

j

)2 ≤ C(

+∞

• j=1

|aj|2 λ2

j

)(

+∞

• j=1

1 λ(2m−2)

j

) ≤ C||ψ0||2

H−2(Ω)

from Weyl’s Theorem if we choose m large enough.

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Real control. Proof The second step requires ψ0 ∈ TSk. Lemma There exists C > 0 such that for every ψ0 ∈ TSk, |ψ0|2

L2(Ω) ≤ C

• ω×(0,T)

ηχω|Im ψ|2dxdt = Ca(ψ0, ψ0). This is done using a compactness uniqueness argument. We have to show that the set K = {ψ0 ∈ TSk, a(ψ0, ψ0) = 0} is reduced to {0}. K is finite dimensional and the operator −i(∆ + λkI) maps K into K and is antisymmetric. It can be diagonalized and then we can show that all eigenfunctions have to be equal to 0.

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

Regularity of control Now we want to show that we have regularity on our control. This is done following the lines of S.Ervedoza-E.Zuazua DCDS 2010. Lemma If z0 ∈ H∆(Ω) ∩ TSk, then the solution ψ0 satisfies ψ0 ∈ H∆(Ω) ∩ TSk. This implies that ηχωIm ψ ∈ C([0, T]; H3(Ω) ∩ H1

0(Ω)).

if A = −(∆ + λkI), D(A) = H2(Ω) ∩ H1

0(Ω) and H∆(Ω) = D(A

3 2 ).

If z0 ∈ D(A), taking for τ > 0 small enough ˆ ψ0 = ψ(τ) − 2ψ0 + ψ(−τ) τ 2 we have ˆ ψ(t) = ψ(t + τ) − 2ψ(t) + ψ(t − τ) τ 2 .

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

• Regularity. Proof

We consider H defined by H =

• ω×(0,T)

ηχωIm ψIm ˆ ψdxdt = a(ψ0, ˆ ψ0). On the one hand we have H = Re (z0, ψ(τ) − 2ψ0 + ψ(−τ) τ 2 )L2(Ω) = Re (z0, ψ(τ) − ψ0 τ 2 )L2(Ω) − Re (z0, ψ0 − ψ(−τ) τ 2 )L2(Ω). This implies H ≤ C|Az0|L2(Ω)|ψ(τ) − ψ0 τ |L2(Ω). On the other hand, using the observability inequality, it can be shown that |ψ(τ) − ψ0 τ |2

L2(Ω) ≤ C|z0|L2(Ω)|ψ(τ) − ψ0

τ |L2(Ω) − CH.

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

• Regularity. Proof

Therefore ∀τ > 0 small enough, |ψ(τ) − ψ0 τ |2

L2(Ω) ≤ C||z0||2 D(A)

which implies ψ0 ∈ D(A) and ||ψ0||D(A) ≤ C||z0||D(A). We can iterate this process exactly in the same way to show that when z0 ∈ D(A2) ∩ TSk, then ψ0 ∈ D(A2) and ||ψ0||D(A2) ≤ C||z0||D(A2).

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

• Regularity. Proof

By interpolation we see that when z0 ∈ D(A

3 2 ) ∩ TSk then ψ0 ∈ D(A 3 2 )

and ||ψ0||D(A

3 2 ) ≤ C||z0||D(A 3 2 ).

As H∆(Ω) = D(A

3 2 ) we see that z0 ∈ H∆(Ω) ∩ TSk implies ψ0 ∈ H∆(Ω)

which itself implies ψ ∈ C([0, T]; H∆(Ω)) and therefore ηχω Im ψ ∈ C([0, T]; H3(Ω) ∩ H1

0(Ω)).

We can then write the control in the form ηχω Im ψ = ηχωW ϕk, with W ∈ C([0, T]; E).

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

End of proof this proves that the derivative of the mapping Λ defined in (22) at the point (ϕk, 0) has a continuous right inverse. Using an inverse mapping theorem, we can find a neighborhood U0 of 0 in C([0, T]; E) and a neighborhood U1 of (e−iλTϕk, ϕk) in H∆(Ω)2 such that for any (y0, y1) ∈ U1, there exists V ∈ U0 such that the solution y of    i ∂y

∂t + ∆y + ηχωVy = 0 in Ω × (0, T),

y = 0 on Γ × (0, T), y(0) = y0 in Ω (1) satisfies y(T) = y1.

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation

Statement of the problem Previous results Statement of the results Main steps of the proof

THANK YOU FOR YOUR ATTENTION. HAPPY BIRTHDAY JEAN-MICHEL ! WELCOME TO THE VERSION 6.0 OF LIFE !

Jean-Pierre Puel Local bilinear controllability of the Schr¨

• dinger equation