Controllability of systems of Schrödinger equations.Control problems of equations with non-regular lower order terms. Control problems of semi-linear equations. Some inverse problems

Controllability of systems of Schrodinger equations.

Alberto Mercado Saucedo.

Universidad Tecnica Federico Santa Marıa, Valparaıso Chile.

Joint work with Marcos Lopez-Garcıa (UNAM) and Luz de Teresa (UNAM).Partially supported by FONDECYT 1120610 and CONICYT ACT-1106

NUMERIWAVES Workshop EUSKADI-CHILE,BCAM, Bilbao, December 2014

Alberto Mercado (UTFSM) Control for Schrodinger. BCAM, December 2014 1 / 30

Controllability of PDE’s. Introduction.

The objective is to drive the solution of a given evolution PDE to some prescribedtarget, acting on the equation by means of a control, which could be a right hand side,boundary data, or a coefficient appearing in the equation.

Several mathematical tools:

Unique continuation properties,

Multiplier methods,

Spectral properties,

Carleman inequalities.



The objective is to drive the solution of a given evolution PDE to some prescribedtarget, acting on the equation by means of a control, which could be a right hand side,boundary data, or a coefficient appearing in the equation.

Several mathematical tools:

Unique continuation properties,

Multiplier methods,

Spectral properties,




We consider the control problem of Schrodingier equation in an open bounded setΩ ⊂ Rn with a distributed control h: :

i∂tu+ ∆u = h1ω Ω× (0, T )u = 0 Γ× (0, T )

u(0) = u0 Ω

Whereω ⊂ Ω

is the control zone,and

h ∈ L2(ω × (0, T ))

is the control.We intend to impose that u(T ) would take a prescribed value.




i∂tu+ ∆u = h1ω Ω× (0, T )u = 0 Γ× (0, T )

u(0) = u0 Ω

Whereω ⊂ Ω


h ∈ L2(ω × (0, T ))





i∂tu+ ∆u = h1ω Ω× (0, T )u = 0 Γ× (0, T )

u(0) = u0 Ω

Whereω ⊂ Ω


h ∈ L2(ω × (0, T ))





i∂tu+ ∆u = h1ω Ω× (0, T )u = 0 Γ× (0, T )

u(0) = u0 Ω

Whereω ⊂ Ω


h ∈ L2(ω × (0, T ))



The main kinds of controllability properties of evolution partial differential equations areas follows:

1 Approximate Controllability. This property means that any state can beapproximated (in a suitable norm) by a solution of the system.

2 Exact Controllability. This property means that any state can be reached by asolution of the system.

3 Null Controllability. This property states that any solution can be driven to rest.

4 Controllability to the trajectories. This property means that any non-controlledtrajectory that is a solution of the system can be reached by a controlled trajectory.This property is equivalent to null controllability for linear equations, and it isespecially important for non-linear equations.




















We can find the adjoint equation

(P ∗)

i∂tϕ+ ∆ϕ = 0 Ω× (0, T )

ϕ = 0 Γ× (0, T )ϕ(T ) = ϕT Ω


Duality controllability / observability

We have:

Theorem(P) is approximate controllable iff

ϕ = 0 in ω × (0, T ) =⇒ ϕ ≡ 0 (UCP)

(P) is exactly controllable iff

∃C > 0

∫Ω

|ϕT |2 ≤ C∫ T

0

∫ω

|ϕ|2 ∀ϕT (Obs)

(P) is null- controllable iff

∃C > 0

∫Ω

|ϕ(0)|2 ≤ C∫ T

0

∫ω

|ϕ|2 ∀ϕT (Obs-0)



We have:


ϕ = 0 in ω × (0, T ) =⇒ ϕ ≡ 0 (UCP)


∃C > 0

∫Ω

|ϕT |2 ≤ C∫ T

0

∫ω

|ϕ|2 ∀ϕT (Obs)


∃C > 0

∫Ω

|ϕ(0)|2 ≤ C∫ T

0

∫ω

|ϕ|2 ∀ϕT (Obs-0)



We have:


ϕ = 0 in ω × (0, T ) =⇒ ϕ ≡ 0 (UCP)


∃C > 0

∫Ω

|ϕT |2 ≤ C∫ T

0

∫ω

|ϕ|2 ∀ϕT (Obs)


∃C > 0

∫Ω

|ϕ(0)|2 ≤ C∫ T

0

∫ω

|ϕ|2 ∀ϕT (Obs-0)


Dualidad controlabilidad / observabilidad

Boundary controllability:i∂tu+ ∆u = 0 Ω× (0, T )

u = h1Γ0 Γ× (0, T )u(0) = u0 Ω

whereΓ0 ⊂ ∂Ω

is the control zone,

(P) is controlable iff: ∫Ω

|ϕ(0)|2 ≤ C∫∫

Γ0

∣∣∣∣∂ϕ∂ν∣∣∣∣2 (1)


Dualidad controlabilidad / observabilidad

Boundary controllability:i∂tu+ ∆u = 0 Ω× (0, T )

u = h1Γ0 Γ× (0, T )u(0) = u0 Ω

whereΓ0 ⊂ ∂Ω

is the control zone,

(P) is controlable iff: ∫Ω

|ϕ(0)|2 ≤ C∫∫

Γ0

∣∣∣∣∂ϕ∂ν∣∣∣∣2 (1)


Observability properties.

To prove observability inequalities:

Multiplier methods

Spectral properties

Microlocal Analysis

Carleman inequalities


Observability properties.

To prove observability inequalities:

Multiplier methods

Spectral properties

Microlocal Analysis

Carleman inequalities


Geometric control. Wave Equation.

DefinitionWe say Γ0 ⊂ ∂Ω satisfies GC hypothesis in Ω at time T , if every ray in Ω, with the lawsof the Geometric Optics, intersects Γ0 in some time 0 ≤ t ≤ T .

We have:

Theorem(Bardos-Lebeau-Rauch 1992)The GC hypothesis for Γ0 is equivalent to the controllability of Wave Equation withcontrol acting in Γ0.


Geometric control. Wave Equation.

DefinitionWe say Γ0 ⊂ ∂Ω satisfies GC hypothesis in Ω at time T , if every ray in Ω, with the lawsof the Geometric Optics, intersects Γ0 in some time 0 ≤ t ≤ T .

We have:

Theorem(Bardos-Lebeau-Rauch 1992)The GC hypothesis for Γ0 is equivalent to the controllability of Wave Equation withcontrol acting in Γ0.


Controllability of Schrodinger equation.

Jaffad (1990) Interior controllability in a rectangle with observations in an opensubdomain ω ⊂ Ω ⊂ R2.

Lebeau (1992) If Γ0 ⊂ ∂Ω satisfies GC hypothesis for some T0 > 0, thenSchrodinger equation is controllable with controls acting on Γ0, for all T > 0.

Machtyngier (1994). Boundary controllability in Ω (boundary C3), withobservations in Γ0 = x ∈ ∂Ω : n(x) · (x− x0) ≥ 0.Komornik 1994. Generalization of Jaffad for Rn.

Phung (2001) Controllability in L2.

Rosier, Zhang (2009) Controllability in L2 of the nonlinear equation.

Ramdani, Takahashi, Tenenbaum, Tucsnak (2005), and Tenenbaum, Tucsnak(2009), Boundary controllability in a rectangle, with observations in any open Γ0

containing a corner.

See survey Zuazua (2003), Remarks on the controllability of the Schrodingerequation.

Recall that the GC hypothesis in Lebeau result are not necessary.



Carleman inequalities were introduced by Trosten Carleman in 1939 in the study ofuniqueness for some PDE’s.Since then, this tool has been widely used in the study of unique continuationproperties, controllability:

Lebeau-Robianno (1995), Fursikov-Imanuvilov (1996), Tataru (1996).

Carleman inequalities are used to study:

Control problems of equations with non-regular lower order terms.

Control problems of semi-linear equations.

Some inverse problems.



Carleman inequalities were introduced by Trosten Carleman in 1939 in the study ofuniqueness for some PDE’s.Since then, this tool has been widely used in the study of unique continuationproperties, controllability:

Lebeau-Robianno (1995), Fursikov-Imanuvilov (1996), Tataru (1996).

Carleman inequalities are used to study:

Control problems of equations with non-regular lower order terms.

Control problems of semi-linear equations.

Some inverse problems.


Carleman estimates. An example.

Consider L = ∆ for functions w ∈ C∞c (Ω).We define

Lφw = e−λφL(eλφw)

∆(eλφw) = eλφ(λ2|∇φ|2w + λ∆φw + 2λ∇φ · ∇w + ∆w

)If φ(x) = α · x with α ∈ Rn \ 0 then:

Lφw = λ2|α|2w + ∆w + 2λα · ∇w







Lφw = λ2|α|2w + ∆w + 2λα · ∇w







Lφw = λ2|α|2w + ∆w + 2λα · ∇w







Lφw = λ2|α|2w + ∆w + 2λα · ∇w



Consider L = ∆ for functions w ∈ Cc(Ω).

∆(eλφw) = eλφ(λ2|∇φ|2w + λ∆φw + 2λ∇φ · ∇w + ∆w)

If φ(x) = α · x with α ∈ Rn \ 0 then:

Lφw = λ2|α|2w + ∆w︸︷︷︸Aw

+ 2λα · ∇w︸︷︷︸Bw

A is self-adjoint.B is anti-adjoint.



Consider L = ∆ for functions w ∈ Cc(Ω).

∆(eλφw) = eλφ(λ2|∇φ|2w + λ∆φw + 2λ∇φ · ∇w + ∆w)

If φ(x) = α · x with α ∈ Rn \ 0 then:

Lφw = λ2|α|2w + ∆w︸︷︷︸Aw

+ 2λα · ∇w︸︷︷︸Bw

A is self-adjoint.B is anti-adjoint.



Lφw = λ2|α|2w + ∆w︸︷︷︸Aw

+ 2λα · ∇w︸︷︷︸Bw

We have

‖Lφw‖2L2 = ‖Aw‖2L2 + ‖Bw‖2L2 + 2 〈Aw,Bw〉L2 (2)

A is self-adjoint and B is anti-adjoint (and both have constant coefficients), we get

2 〈Aw,Bw〉L2 = 〈[A,B]w,w〉L2 = 0, ∀ w ∈ C∞c (Ω)

Thus

‖Lφw‖L2 ≥ 2λ‖α · ∇w‖L2 (3)

≥ λδ‖w‖L2 (4)

Which means that

λ‖e−λφu‖L2 ≤ C‖e−λφ∆u‖L2 (5)



Lφw = λ2|α|2w + ∆w︸︷︷︸Aw

+ 2λα · ∇w︸︷︷︸Bw

We have



2 〈Aw,Bw〉L2 = 〈[A,B]w,w〉L2 = 0, ∀ w ∈ C∞c (Ω)

Thus

‖Lφw‖L2 ≥ 2λ‖α · ∇w‖L2 (3)

≥ λδ‖w‖L2 (4)

Which means that




Lφw = λ2|α|2w + ∆w︸︷︷︸Aw

+ 2λα · ∇w︸︷︷︸Bw

We have



2 〈Aw,Bw〉L2 = 〈[A,B]w,w〉L2 = 0, ∀ w ∈ C∞c (Ω)

Thus

‖Lφw‖L2 ≥ 2λ‖α · ∇w‖L2 (3)

≥ λδ‖w‖L2 (4)

Which means that




Lφw = λ2|α|2w + ∆w︸︷︷︸Aw

+ 2λα · ∇w︸︷︷︸Bw

We have



2 〈Aw,Bw〉L2 = 〈[A,B]w,w〉L2 = 0, ∀ w ∈ C∞c (Ω)

Thus

‖Lφw‖L2 ≥ 2λ‖α · ∇w‖L2 (3)

≥ λδ‖w‖L2 (4)

Which means that




In other cases:

Lφw = λ2|∇φ|2w + ∆w︸︷︷︸Aw

+ 2λ∇φ · ∇w︸︷︷︸Bw

We have


2 〈Aw,Bw〉L2 = 〈[A,B]w,w〉L2 = lower order + boundary terms, ∀ w ∈ C∞c (Ω)

Thus

‖Lφw‖L2 ≥ 2λ‖α · ∇w‖L2 (7)

≥ λδ‖w‖L2 (8)

Which means that




Given a differential operator P and a smooth function φ, we define

Pφ = eλφPe−λφ

Remark that Pφ = p(x,D + iλ∇φ)

Theorem (Carleman inequalities)If φ is pseudoconvex with respect to P then

‖v‖Hmλ≤ C‖Pφv‖L2

for λ large enough.

For instance, φ is pseudoconvex if:

For P = ∂t −∆ if |∇φ| 6= 0

For P = ∂2t −∆ if φ is convex.

For P = i∂t −∆ si φ is convex.

Boundary condition: Usually is required∂φ

∂ν< 0 in ∂Ω.










For P = ∂t −∆ if |∇φ| 6= 0




∂ν< 0 in ∂Ω.










For P = ∂t −∆ if |∇φ| 6= 0




∂ν< 0 in ∂Ω.










For P = ∂t −∆ if |∇φ| 6= 0




∂ν< 0 in ∂Ω.



If the previous properties are not satisfied in a set ω ⊂ Ω or ω ⊂ ∂Ω , then



Theorem (Carleman inequalities)If φ is pseudoconvex with respect to P in Ω \ ω then

‖v‖Hmλ≤ C‖Pφv‖L2 + ‖v‖Hm(ω)



For P = ∂t −∆ if |∇φ| 6= 0




∂ν< 0 in ∂Ω.



If the previous properties are not satisfied in a set ω ⊂ Ω or ω ⊂ ∂Ω , then


Theorem (Carleman inequalities)If φ is pseudoconvex with respect to P in Ω \ ω then

‖v‖Hmλ≤ C‖Pφv‖L2 + ‖v‖Hm(ω)


In the original variable, we get:

‖e−λφw‖Hm ≤ C‖e−λφPw‖L2 + ‖e−λφw‖Hm(ω)︸︷︷︸observation


Carleman for Schrodinger Equation.

Theorem (Baudouin, Puel 2002.)If exists an x0 ∈ Rn such that

Γ0 ⊃ x ∈ ∂Ω : (x− x0) · n(x) > 0.

Then ∫ T

0

∫Ω

(ρ|∇w|2 + ρ3|w|2

)dxdt ≤ C

∫ T

0

∫Γ0

ρ∣∣∂w∂ν

∣∣2dxdt (10)

The weight function is

ψ(x) = |x− x0|2.


Carleman for Schrodinger Equation.

Theorem (Baudouin, Puel 2002.)If exists an x0 ∈ Rn such that

Γ0 ⊃ x ∈ ∂Ω : (x− x0) · n(x) > 0.

Then ∫ T

0

∫Ω

(ρ|∇w|2 + ρ3|w|2

)dxdt ≤ C

∫ T

0

∫Γ0

ρ∣∣∂w∂ν

∣∣2dxdt (10)

The weight function is

ψ(x) = |x− x0|2.


Schrodinger Equation. Carleman inequality.

Remark: The precise hypothesis:

1 λ|∇ψ. ξ|2 +D2ψ(ξ, ξ)≥ ε |ξ|2 for all ξ ∈ Cn

2 Γ0 ⊃ x ∈ ∂Ω : ∇ψ · n(x) > 0

1 From the Carleman estimate, we can deduce an observability estimate in H1.2 This implies a controllability result in H−1

3 Used to prove stability of some inverse problems.


Schrodinger Equation. Carleman inequality.

Remark: The precise hypothesis:

1 λ|∇ψ. ξ|2 +D2ψ(ξ, ξ)≥ ε |ξ|2 for all ξ ∈ Cn

2 Γ0 ⊃ x ∈ ∂Ω : ∇ψ · n(x) > 0

1 From the Carleman estimate, we can deduce an observability estimate in H1.2 This implies a controllability result in H−1

3 Used to prove stability of some inverse problems.


Schrodinger Equation. More general hypothesis.

In the search of stability without the GC hypothesis:

Theorem (M, Osses, Rosier 2008)If ψ > 0, ∇ψ 6= 0 is such that

λ|∇ψ. ξ|2 +D2ψ(ξ, ξ)≥ 0

then we have the Carleman estimate∫ T

0

∫Ω

(ρ|∇w · ∇ψ|2 + ρ3|w|2

)dxdt ≤ C

∫ T

0

∫Γ0

ρ∣∣∂w∂ν

∣∣2dxdt (11)


Schrodinger equation.

Figure : Observation regions in a stadium for the Schrodinger equation using degenerateCarleman weights with spatial dependence of the type x · e1.

For ψ = ξ · x we get

Theorem (M, Osses, Rosier 2008)Interior observability:∫ T

0

∫Ω

ρ3|w|2dxdt ≤ C∫ T

0

∫ω

ρ(|w|2 + |∇w|2)dxdt (12)

where ω ⊂ Ω is such that is intersected by any right line in direction ξ.


Schrodinger equation.

Figure : Observation regions in a stadium for the Schrodinger equation using degenerateCarleman weights with spatial dependence of the type x · e1.

For ψ = ξ · x we get

Theorem (M, Osses, Rosier 2008)Interior observability:∫ T

0

∫Ω

ρ3|w|2dxdt ≤ C∫ T

0

∫ω

ρ(|w|2 + |∇w|2)dxdt (12)

where ω ⊂ Ω is such that is intersected by any right line in direction ξ.



This implies a result for the controllability of a Schrodinger equation:i∂tu+ ∆u = h1ω Ω× (0, T )

u = 0 Γ× (0, T )u(0) = u0 Ω

Corolary (from the Carleman of M - Osses - Rosier (2008))

For each u0 ∈ L2(Ω) existsh ∈ L2(0, T ;H−1(Ω))

supported in ω such thatu(T ) = 0

Remark: In the Carleman estimate, we have only observation of ∇ξw := ∇w · ξ.




u = 0 Γ× (0, T )u(0) = u0 Ω








u = 0 Γ× (0, T )u(0) = u0 Ω








u = 0 Γ× (0, T )u(0) = u0 Ω






System of Schrodinger equations.

Let Ω ⊂ RN be a bounded, open set with C2 boundary, N ≥ 1. Let ω and O be twononempty open subsets of Ω.We consider the following cascade system of Schrodinger equations:

ipt + ∆p = h1ω in Q,p = 0 on Σ,p (x, 0) = p0 (x) in Ω,

(13)

iut + ∆u = p1O in Q,u = 0 on Σ,u (x, 0) = u0 (x) in Ω,

(14)

where p0, u0 are given, h is a control with support in ω × (0, T ) .

In this system, the problem is to prove controllability with:1 Less controls than equations.2 No geometric hypothesis on ω.3 General case ω ∩ O = ∅.



Let Ω ⊂ RN be a bounded, open set with C2 boundary, N ≥ 1. Let ω and O be twononempty open subsets of Ω.We consider the following cascade system of Schrodinger equations:

ipt + ∆p = h1ω in Q,p = 0 on Σ,p (x, 0) = p0 (x) in Ω,

(13)

iut + ∆u = p1O in Q,u = 0 on Σ,u (x, 0) = u0 (x) in Ω,

(14)

where p0, u0 are given, h is a control with support in ω × (0, T ) .

In this system, the problem is to prove controllability with:1 Less controls than equations.2 No geometric hypothesis on ω.3 General case ω ∩ O = ∅.


Some references.

Rosier, de Teresa (2011). Periodic boundary conditions, ω ∩ O = ∅.Alabau, Leautaud (2013). Both ω and O satisfies GC hypothesis

Alabau, (2013). Systems of N equations.

Dehman, Le-Rousseau, Leatud (2014). Equations in manifolds.

We are able to deal with 2 of 3 stated objectives.


Some references.

Rosier, de Teresa (2011). Periodic boundary conditions, ω ∩ O = ∅.Alabau, Leautaud (2013). Both ω and O satisfies GC hypothesis

Alabau, (2013). Systems of N equations.

Dehman, Le-Rousseau, Leatud (2014). Equations in manifolds.

We are able to deal with 2 of 3 stated objectives.



For the system, our main controllability result is:

Theorem (Lopez-Garcıa - M - de Teresa (preprint 2014))

Let X := H2 ∩H10 (Ω). Suppose there exists an open set

ω ⊂ ω ∩ O

a observation set for the degenerate Carleman inequality, and such that

ξ · n(x) = 0 for each x ∈ ∂ω ∩ ∂Ω.

Then for each (p0, u0) ∈ L2(Ω)2 there exists a controlh ∈ L2 (0, T ;X ′) supported in ω × (0, T ) such that

(p(T ), u(T )) = (0, 0).


Idea of the proof

We consider the adjoint systemizt + ∆z = 0 in Q,z = 0 on Σ,z (x, T ) = z0 (x) in Ω,

(15)

iqt + ∆q = zρO in Q,q = 0 on Σ,q (x, T ) = q0 (x) in Ω,

(16)

Adding the Carleman inequalities for the adjoint system, we get∫ T

0

∫Ω

ρ3(|z|2 + |q|2)dxdt ≤ C∫ T

0

∫ω

ρ(|z|2 + |∇ξz|2 + |q|2 + |∇ξq|

2)dxdt (17)


Idea of the proof

We consider the adjoint systemizt + ∆z = 0 in Q,z = 0 on Σ,z (x, T ) = z0 (x) in Ω,

(15)

iqt + ∆q = zρO in Q,q = 0 on Σ,q (x, T ) = q0 (x) in Ω,

(16)

Adding the Carleman inequalities for the adjoint system, we get∫ T

0

∫Ω

ρ3(|z|2 + |q|2)dxdt ≤ C∫ T

0

∫ω

ρ(|z|2 + |∇ξz|2 + |q|2 + |∇ξq|

2)dxdt (17)


Idea of the proof

We can eliminate one observation:∫ T

0

∫ω

ρ|z|2 =

∫ T

0

∫ω

ρz(Pq)

=

∫ T

0

∫ω

ρPzq + Lower order terms

≤ ε∫ T

0

∫ω

ρ|z|2 +

∫ T

0

∫ω

(|q|2 + |∇q|2)

(18)

and ∫ T

0

∫ω

ρ|∇ξz|2 =

∫ T

0

∫ω

ρ∇ξz∇ξ (Pq)

=

∫ T

0

∫ω

ρ∇ξPz∇ξq + Lower order terms

≤ ε∫ T

0

∫ω

ρ(|∇ξz|2) + C‖q‖H2(ω)

(19)

We have used the hypothesis on the boundary in order to avoid boundary terms.


Ongoing work

1 Control of H1 initial conditions with L2 controls.2 Remove hypothesis on ∂(ω ∩ O).3 The case ω ∩ O = ∅.


Thanks!Eskerrak!

¡Muchas gracias!


Documents

Controllability of systems of Schrödinger equations.Control problems of equations with non-regular lower order terms. Control problems of semi-linear equations. Some inverse problems