Control Problems for Schrödinger Equations and Hartree ......Jean-Pierre Puel LMV, Universit e de Versailles St Quentin, Versailles, France [email protected] Bilbao, July 14, 2015

Physical motivation Mathematical model I Control for Model I Mathematical Model II Control for Model II

Control Problems for Schrodinger Equations andHartree-Fock Systems

Jean-Pierre Puel

LMV, Universite de Versailles St Quentin, Versailles, [email protected]

Bilbao, July 14, 2015Bastille day !

Jean-Pierre Puel Control Problems for Schrodinger Equations and Hartree-Fock Systems


Outline

1 Physical motivation

2 Mathematical model I

3 Control for Model I

4 Mathematical Model II

5 Control for Model II



Motivation. Control

We can act on a closed quantum system via an electric field derivingfrom a potential performed for example by laser beams or a magneticfield (which will not be considered here).Control problems for these systems ask the following questions : can wechoose the external (electric) potential in order to obtain a “desired”behavior of the solution?This has to be made precise.



Motivation. Control

Schrodinger equation (all constants are taken equal to one){i∂tψ + ∆ψ + V0ψ + V1ψ + ... = 0, t ∈ (0,T )ψ(t = 0) = ψ0,

V0 internal potential : Coulombian attraction, harmonic oscillator,....V1 external potential (control).... : nonlinear terms etc.



Exact Controllability

Given an initial state ψ0 and a target state ψ1, can we find a potential V1

such thatψ(0) = ψ0, and ψ(T ) = ψ1

Many mathematical difficulties. Very few results up to now.One of the main directions of research. Hope for better results soon.....K.Beauchard, C.Laurent, J-M.Coron, J.P.P., ....Strange remark (for bounded domains) : only values of the potential nearthe boundary seem important.....Approximate controllability using finite dimensional approximations : U.Boscain and collaborators.



Quantum Information Theory

Trapped ions or cubits (one or two) : two level system stabilized by aharmonic oscillator. Very rapidly oscillating potential with phasedepending on the position.How to drive a given initial data to a prescribed final state ?Creation of quantum logic gates, etc.Some work done on approximate controllability (P.Rouchon, S.Ervedoza-J.P.P.).



Quantum Chemistry

Breaking of molecules to obtain desired states, etc.Many groups working on Optimal Control in these directions includingnumerical treatment.H. Rabitz and collaborators.C. Le Bris, E. Cances and collaborators.G. Turinici and collaborators.J.P.P.-L.Beaudouin for mathematical analysis.Optimal Control : To minimize a functional in which enters the distancefrom the solution to the target at time T .Here : case of a hydrogenoid atom subject to an external electric field.First step : linearized model concerning the wave function of theelectron, the position of the nucleus being prescribed, even if not fixed.Second step ; coupled system electron-nucleus in the Hartree-Fockapproximation.



Mathematical model I

Position of the nucleus : a(t) given function.Linear Schrodinger equation with Coulombian attraction from the nucleusand external electric potential.

{i∂tu + ∆u +

u

|x − a(t)|+ V1(x , t)u = 0, (x , t) ∈ R3 × (0,T )

u(x , 0) = u0(x), x ∈ R3

where the external electric potential V1 takes its values in R.This equation preserves the L2 norm for the solution.



Mathematical model I

Remarks :-V1 can be unbounded, for example if the electric field is constant....-Even if a(.) is very regular but not zero, if we take the derivative in timewe have a very singular potential.....We define

H1 =

{v ∈ L2(R3),

∫R3

(1 + |x |2)|v(x)|2 dx < +∞}

H2 =

{v ∈ L2(R3),

∫R3

(1 + |x |2)2|v(x)|2 dx < +∞}.

One can notice that H1 and H2 are respectively the images of H1 and H2

under the Fourier transform.



Result for Model I

We need a result of existence uniqueness and regularity for this equation(Beaudouin-Kavian-Puel)

Theorem

Let T > 0 be an arbitrary time and let a and the potential V1 satisfy

a ∈W 2,1(0,T ),(1 + |x |2)−1V1 ∈ L∞((0,T )× R3),(1 + |x |2)−1∂tV1 ∈ L1(0,T ; L∞) and(1 + |x |2)−1∇V1 ∈ L1(0,T ; L∞).

for any u0 ∈ H2 ∩ H2, there exists a unique solution u with

u ∈ L∞(0,T ; H2 ∩ H2) and ∂tu ∈ L∞(0,T ; L2)

In fact the solution is continuous with values in H1 ∩ H1 and weaklycontinuous with values in H2 ∩ H2.



Estimate :If for some α > 0 and ρ > 0,∣∣∣a∣∣∣

W 2,1(0,T )≤ α and∣∣∣(1 + |x |2)−1V1

∣∣∣W 1,1(0,T ,L∞)

+∣∣∣(1 + |x |2)−1∇V1

∣∣∣L1(0,T ,L∞)

≤ ρ

then there exists a non negative constant CT ,α,ρ depending on T , α andρ such that

‖u‖L∞(0,T ;H2∩H2) + ‖∂tu‖L∞(0,T ;L2) ≤ CT ,α,ρ‖u0‖H2∩H2.



Model I. Control

For every potential V1 in a suitable class we have a solution.Can we choose the potential such that the solution at time T has adesired property, for example to be as close as possible to a prescribedtarget u1?Exact controllability : can we obtain u(T ) = u1?Very few results in this sense and only for the simplest cases : 1-d andwithout Coulombian potential (Beauchard, Beauchard-Laurent). This is amajor direction of research.Approximate controllability : can we obtain u(T )− u1 as small as wewant? Some results without singular potentials using Galerkinapproximation and finite dimensional control(Boscain-Chambrion-Sigalotti and collaborators). Some results forsystems modeling trapped ions with very rapidly oscillating potentialsusing approximate equations (Ervedoza-Puel).For the present case, no result of controllability.



Model I. Bilinear Optimal Control

The electric potential V1 is the control, and if u1 ∈ L2 is a given target,the problem reads:

Find a minimizer V1 ∈ H for

inf {J(V1), V1 ∈ H}

where

H :={

V ,(1 + |x |2

)− 12 V ∈ H1(0,T ; W )

}(1)

with W an Hilbert space such that W ↪→W 1,∞,

J(V ) =1

2

∫R3

|u(T )− u1|2 dx +r

2‖V ‖2

H with r > 0,

Also find an optimality condition which can be interpreted..



Theorem

There exists an optimal control V1 such that

J(V1) = infV∈H

J(V )

and it satisfies the optimality condition:

∀δV ∈ H, r〈V1, δV 〉H = Im

∫ T

0

∫R3

δV (x , t)u(x , t)p(x , t) dxdt

where u is solution of the state equation and p is solution of the adjointproblem{

i∂tp + ∆p +p

|x − a|+ V1p = 0 in R3 × (0,T )

p(T ) = u(T )− u1 in R3.



Remark.The previous Theorem relies essentially on the regularity result which hasbeen obtained before. Also, we show that the mapping

V1 ∈ H → u(T ) ∈ L2

is differentiable.If we consider the mapping with values in H1 for example it is not clearthat it is still differentiable....Of course the proof starts by considering a minimizing sequence and thenone has to prove “good” convergences in order to pass to the limit.



Interpretation

in the particular case when W = H3(R3):

H ={

V ,(1 + |x |2

)− 12 V ∈ H1(0,T ; H3)

}.

Optimality condition:

∀δV ∈ H, r〈V1, δV 〉H = Im

∫ T

0

∫R3

δVup dxdt

In this particular case, if V ∈ H, there exists X ∈ H1(0,T ; H3) such that

V =(1 + |x |2

) 12 X . Moreover, X = (I −∆)−1Y with Y ∈ H1(0,T ; H1).



Therefore,

〈V1, δV 〉H = 〈X1, δX 〉H1(0,T ;H2) = 〈Y1, δY 〉H1(0,T ;H1)

and on the one hand,

〈Y1, δY 〉H1(0,T ;L2) =

∫ T

0

∫R3

(I − ∂2t )Y1δY

+

∫R3

(∂tY1(T )δY (T )− ∂tY1(0)δY (0)) .

while on the other hand,

〈∇Y1,∇δY 〉H1(0,T ;L2) =

∫ T

0

∫R3

−(I − ∂2t )∆Y1δY

+

∫R3

(∂t∆Y1(0)δY (0)− ∂t∆Y1(T )δY (T )) .



We obtain

〈Y1, δY 〉H1(0,T ;H1) =

∫ T

0

∫R3

(I − ∂2t )(I −∆)Y1δY

+

∫R3

(∂t(Y1 −∆Y1)(T )δY (T )− ∂t(Y1 −∆Y1)(0)δY (0)

).

The optimality condition becomes:

∀δY ∈ H1(0,T ; L2),

r

∫ T

0

∫R3

(I − ∂2t )(I −∆)Y1δY dxdt + r

∫R3

∂t(Y1 −∆Y1)(T )δY (T ) dx

− r

∫R3

∂t(Y1 −∆Y1)(0)δY (0) dx

= Im

∫ T

0

∫R3

up√

1 + |x |2 (I −∆)−1 δY dxdt



and from an integration by parts, we obtain for all δY in H1(0,T ; L2),

r

∫ T

0

∫R3

(I − ∂2t )(I −∆)Y1δY dxdt

+ r

∫R3

(∂t(Y1 −∆Y1)(T )δY (T )− ∂t(Y1 −∆Y1)(0)δY (0)

)dxs

= Im

∫ T

0

∫R3

δY (I −∆)−1(

up√

1 + |x |2)

dxdt.

It can be noticed that by some regularity arguments the right hand sidehas a meaning.



Thus, the optimality condition corresponds to the system:{r(I − ∂2

t )(I −∆)Y1 = (I −∆)−1(

Im(up)√

1 + |x |2)

in R3 × (0,T )

∂t(Y1 −∆Y1)(T ) = ∂t(Y1 −∆Y1)(0) = 0 in R3.

where

V1 =(1 + |x |2

) 12 (I −∆)−1 Y1.



Mathematical Model II

On the one hand, since the nucleus is much heavier than the electrons,we consider it as a point particle which moves according to the Newtondynamics in the external electric field and in the electric potential createdby the electronic density (nucleus-electron attraction of Hellman-Feynmantype). We obtain a second order in time ordinary differential equationsolved by the position a(t) of the nucleus (of mass m).



On the other hand, under the restricted Hartree-Fock formalism, wedescribe the behavior of the electrons by a wave function, solution of atime dependent Hartree-Fock equation. We can define it as aSchrodinger equation with a coulombian potential due to the nucleus,singular at finite distance, an electric potential corresponding to theexternal electric field, possibly unbounded at infinity (for exampleV1 = I (t).x), and a nonlinearity of Hartree type in the right hand side.We want precisely to study the optimal control of the wave function ofthe electrons only, the control being performed by the electric potential.



We are in fact considering the following coupled system :

i∂tu + ∆u +1

|x − a|u + V1u = (|u|2 ? 1

|x |)u, in R3 × (0,T )

u(0) = u0, in R3

md2a

dt2=

∫R3

−|u(x)|2∇ 1

|x − a|dx −∇V1(a), in (0,T )

a(0) = a0,da

dt(0) = v0

(2)

where V1 is the external electric potential which takes its values in R



We need a result of existence and regularity for this problem.

Theorem

We assume that T is a positive arbitrary time and the same hypothesison V1 plus

∇V1 ∈ L2(0,T ; W 1,∞loc

(R3)).

(We can take V1 ∈ H).If u0 ∈ H2 ∩ H2, a0, v0 ∈ R, then the system has at least a solution

(u, a) ∈(W 1,∞(0,T ; L2) ∩ L∞(0,T ; H2 ∩ H2)

)×W 2,1(0,T ).

RemarkWe have no proof of uniqueness up to now....



The proof of the existence and regularity result is not straightforward andrequires first of all a result of local existence and in a second step a lot ofestimates in order to show the global existence and regularity result.



The controllability problem for the previous system is completely out ofreach at the moment and we will give a result of optimal control.In the case where a(.) is given and we only have a nonlinear Schrodingerequation, the result is similar to the one for the first model and we alsohave an interpretation of the optimality condition using and adjoint state.In the case of the full coupled system, we willl give a result of existencefor an optimal control but the optimality system is formal...Let us formulate the optimal control problem.



if u1 ∈ L2 is a given target, find a minimizer V1 ∈ H for

inf{J(V1), V1 ∈ H} (3)

where the cost functionnal J is defined by

J(V1) =1

2

∫R3

|u(T )− u1|2 dx +r

2

∣∣∣V1

∣∣∣2H, r > 0, (4)

where,

H ={

V ,(1 + |x |2

)− 12 V ∈ H1(0,T ; W )

}and W is an Hilbert space such that W ↪→W 1,∞(R3) .



We obtain the following result of existence for an optimal control.

Theorem

There exists an optimal control V1 ∈ H such that

J(V1) = inf{J(V ), V ∈ H}

and for all δV in H,

In general we don’t have a rigorous result for an optimality condition butthis result can be proved if the system is decoupled and a(.) is given.



Formal Optimality System

A first step is to consider the functional

Φ : H −→ L∞(0,T ; L2(R3))× L∞(0,T )

V1 7−→ (u(V1), a(V1)))

Assuming that we have uniqueness of the solution of our system andassuming that Φ is differentiable, if we set DΦ(δV1) = (z , b), then(z(t, x), b(t)) has to satisfy the following coupled system set in

R3 × (0,T ) with V0 =1

|x − a|

i∂tz + ∆z + V0z + V1z = −∂V0

∂a· b u − δV1u

+(|u|2 ? 1

|x |)z + 2(Re(uz) ?

1

|x |)u,

md2b

dt2= −

∫R3

|u|2∇∂V0

∂a· b − 2

∫R3

Re(uz)∇V0 −∇δV1(a)

−∇(∇V1) · b(a)

with initial conditions z(0) = 0, b(0) = 0,db

dt(0) = 0 and .



we can deduce the candidate adjoint system:

i∂tp + ∆p + V0p + V1p = (|u|2 ? 1

|x |)p

+2i (Im(up) ?1

|x |)u − 2iu ∇V0 · %,

p(T ) = u(T )− u1,

md2%

dt2= −

∫R3

∂V0

∂aIm(up)− 2

∫R3

Re(u∇u)∂V0

∂a· %−∇(∇V1)(a) · %,

%(T ) = 0,d%

dt(T ) = 0.



Therefore, the bilinear optimal control V1 is the solution of a partialdifferential equation defined by variational formulation in the followingway:

r〈V1, δV1〉H =

∫ T

0

∫R3

Im (u(t, x)p(t, x)) δV1(t, x) dxdt

−∫ T

0

⟨%(t) · ∇δa(t), δV1(t)

⟩dt.

As for the first model by choosing W = H3 we can give an interpretationof the previous condition in terms of PDE for Y1 such that

V1(x , t) =(1 + |x |2

) 12 (I −∆)−1 Y1(x , t) + E1(t)

....



Thus, the optimality condition corresponds to the system:

r[(I − ∂2

t ) + (I −∆)](I −∆)Y1 = (I −∆)−1G in R3 × (0,T )

∂t(Y1 −∆Y1)(T ) = ∂t(Y1 −∆Y1)(0) = 0 in R3

r

(E1 −

d2E1

dt2

)=∫R3 G (x) dx in (0,T )

dE1

dt(T ) =

dE1

dt(0) = 0

where

G (x , t) = Im(

u(x , t)p(x , t))√

1 + |x |2

− %(t) · ∇δa(t)(x)√

1 + |a(t)|2 +a(t) · %(t)√1 + |a(t)|2

δa(t)(x)

and

V1(x , t) =(1 + |x |2

) 12((I −∆)−1 Y1(x , t) + E1(t)

).


Documents

Control Problems for Schrödinger Equations and Hartree ......Jean-Pierre Puel LMV, Universit e de Versailles St Quentin, Versailles, France [email protected] Bilbao, July 14, 2015