Control and stabilization of the Bloch equation - Spin Dynamics · The Bloch equation Linearized system Non exact controllability with a priori bounded controls Approximate controllability

The Bloch equationLinearized system

Non exact controllability with a priori bounded controlsApproximate controllability with unbounded controls

Explicit controls for the asymptotic exact controllabilityFeedback stabilization

Control and stabilization of the Bloch equation

Karine Beauchard (CNRS, CMLS, Ecole Polytechnique)

joint works with

Jean-Michel Coron (LJLL, Paris 6)

Pierre Rouchon (CAS, Mines de Paris)

Paulo Sergio Pereira da Silva (Escola Politecnica Sao Polo)

QUAINT, Dijon, April, 9, 2013K. Beauchard




Plan

1 The Bloch equation

2 Linearized system

3 Non exact controllability with a priori bounded controls

4 Approximate controllability with unbounded controls

5 Explicit controls for the asymptotic exact controllability

6 Feedback stabilization

K. Beauchard




The Bloch equationAn ensemble of non interacting spins, in a magnetic fieldB(t) := (u(t), v(t),B0), with dispersion in the Larmor frequencyω = γB0 ∈ (ω∗, ω

∗) (=rotation speed around z).

one spin : M(t , ω) ∈ S2

∂M∂t

(t , ω) =[u(t)e1 + v(t)e2 + ωe3

]∧M(t , ω), ω ∈ (ω∗, ω

∗)

State : M Controls : u, v

controllability of an ODE, simultaneously w.r.t. ω ∈ (ω∗, ω∗)

Li-Khaneja(06)

Applications : Nuclear Magnetic ResonanceK. Beauchard




Controllability question for the Bloch equation

∂M∂t

(t , ω) =[u(t)e1+v(t)e2+ωe3

]∧M(t , ω), (t , ω) ∈ [0,+∞)×(ω∗, ω

∗)

Ex : M0(ω) ≡ −e3, Mf (ω) ≡ +e3,But spins with different ω have different dynamics !

Goal : Use the control to compensate for the dispersion in ω.

Rk : If ω is fixed, the controllability of one ODE on S2 is trivial.

Goals of this talk :

show that the pioneer works by Li and Khaneja can becompleted by functionnal analysis methods to go further,

propose different approaches relying on Fourier analysis orfeedback stabilization.

K. Beauchard




A prototype for infinite dimensional bilinear systemswith continuous spectrum

∂M∂t

(t , ω) =[u(t)e1 + v(t)e2 + ωe3

]∧M(t , ω), ω ∈ (ω∗, ω

∗)

AM := ωe3∧M(ω) → Sp(A) = −i(ω∗, ω∗)⋃

i(ω∗, ω∗)

λ = ±iω → Mλ(ω) =

1∓i0

δω(ω)

⇒ Toy model for i∂tψ = (−∆ + V )ψ − u(t)µ(x)ψ

K. Beauchard




State of the art : bilinear control for Schrödinger PDEs

Quite well understood :exact controllability 1D

negative results : Ball-Marsden-Slemrod(82), Turinici(00),Ilner-Lange-Teismann(06), Mirrahimi-Rouchon(04)Nersesyan(10).

positive local results with discrete spectrum + gap (1D) : KB(05),KB-Laurent(09, 11, ), Morancey(12), KB-Lange-Teismann(12)....

positive global results : KB-Coron(06), Nersesyan(09).

approximate controllability with discrete spectrumChambrion-Mason-Sigalotti-Boscain-Boussaïd-Caponigro(09,...),Nersesyan(09), Ervedoza-Puel(09).

Not well understood : with continuous spectrum : Mirrahimi(09)

K. Beauchard




1st part

Control of the linearizedsystem around

(M ≡ e3,u ≡ v ≡ 0)

K. Beauchard




Linearized system around (M ≡ e3,u ≡ v ≡ 0)∂tM(t , ω) =

[u(t)e1 + v(t)e2 + ωe3

]∧M(t , ω), ω ∈ (ω∗, ω

∗)

M(0, ω) = e3

If (u, v) = ε(u, v) then M = e3 + εM + oε→0

(ε) where∂tM(t , ω) = ωe3 ∧ M +

[u(t)e1 + v(t)e2

]∧ e3

M(0, ω) = 0

The notations

M := (x , y , z), Z(t , ω) := (x + iy)(t , ω), w(t) := (v − iu)(t)

lead to the very simple ODE

dZdt

(t , ω) = iωZ(t , ω) + w(t)

K. Beauchard




Linearized system around (M ≡ e3,u ≡ v ≡ 0) :non exact controllability, approximate controllability

Z(T , ω) =

(Z0(ω) +

∫ T

0w(t)e−iωtdt

)eiωT

T > 0, the reachable set from Z0 = 0 is F [L1(−T ,0)]

the Z0 asymptotically zero controllable are F [L1(0,+∞)]

∀Z0 in that space, the control is unique∀T > 0, approximate controllability in C0[ω∗, ω

∗] withC∞c (0,T )-controls.

We will see that the NL syst has better controllability properties.

K. Beauchard




2nd part

Nonlinear system :no exact controllability with a

priori bounded controls

K. Beauchard




Whole space : structure of the reachable set

∂M∂t

(t , ω) =[u(t)e1 + v(t)e2 + ωe3

]∧M(t , ω), (t , ω) ∈ (0,T )× R

Theorem : Let T > 0 and R := 1/(8√

3T ).

∀u, v ∈ BR[L2(0,T )],∃!M = (x , y , z) solution withZ := x + iy ∈ C0([0,T ],L2(R)) ∩ C0

b ([0,T ]× R),

the image of

FT : BR[L2(0,T )]2 → L2 ∩ C0b (R)

(u, v) 7→ Z(T , .)

is a non flat submanifold of L2 ∩ C0b (R), with∞ codim.

Proof : Inverse mapping dFT (0,0).(U,V ) ∼ F(U + iV ) + 2nd orderK. Beauchard




On a bounded interval : analyticity argument

∂M∂t

(t , ω) =[u(t)e1 +v(t)e2 +ωe3

]∧M(t , ω), (t , ω) ∈ (0,T )×(ω∗, ω

∗)

T > 0,u, v ∈ L2(0,T )⇒ Z(T , .) analyticT > 0, R := 1/(8

√3T ).

There exists arbitrarily small analytic targets that cannotbe reached exactly in time T with controls in BR[L2(0,T )].

The non controllability is not a question of regularity.

K. Beauchard




3rd part

Nonlinear system :approximate controllabilitywith unbounded controls

K. Beauchard




Solutions associated to Dirac controls

∂M∂t

(t , ω) =[u(t)e1 +v(t)e2 +ωe3

]∧M(t , ω), (t , ω) ∈ (0,T )×(ω∗, ω

∗)

Classical solution for u, v ∈ L1loc(R).

If u = αδa and v = 0 then

M(a+, ω) = exp(αΩx )M(a−, ω)

→ instantaneous rotation of angle α around the x-axis, ∀ω

Rk : limit [ε→ 0] of solutions associated to u = αε 1[a,a+ε].

K. Beauchard




Approximate controllability result −∞ < ω∗ < ω∗ < +∞

∂M∂t

(t , ω) =[u(t)e1+v(t)e2+ωe3

]∧M(t , ω), (t , ω) ∈ [0,+∞)×(ω∗, ω

∗)

Theorem : Let M0 ∈ H1((ω∗, ω∗),S2). There exist

(tn)n∈N ∈ [0,+∞)N, (un)n∈N, (vn)n∈N finite sums of Diracmasses such that

U[t+n ; un, vn,M0]→ e3 weakly in H1.

Rk : Same result with u, v ∈ L∞loc[0,+∞) : αδa ← αε 1[a,a+ε]

Approximate controllability in Hs, ∀s < 1, in L∞...

K. Beauchard




First step : Li-Khaneja ’s non commutativity result

∂M∂t

(t , ω) =[u(t)e1+v(t)e2+ωe3

]∧M(t , ω), (t , ω) ∈ [0,+∞)×(ω∗, ω

∗)

Theorem : Let P,Q ∈ R[X ]. ∀ε > 0,∃τ∗ > 0 such that∀τ ∈ (0, τ∗), ∃T > 0,u, v ∼ Dirac such that∥∥∥U[T+; u, v , .]−

(I + τ [P(ω)Ωx + Q(ω)Ωy ]

)∥∥∥H1(ω∗,ω∗)

6 ετ.

Proof : Explicit controls→ cancel the drift term, Lie brackets.

Rk : It is not sufficient for the global approximate controllability.τωN needs TN ∼ 2Nτ

1N and more than 2N N-S.

K. Beauchard




Second step : variationnal argument

Let M0 ∈ H1((ω∗, ω∗),S2) be such that M0 6= e3.

Goal : Find U[t+n ; un, vn,M0] e3 in H1 when n→ +∞

K :=

M ; ∃U[t+n ; un, vn,M0] M in H1, [...]

m := inf‖M ′‖L2 ; M ∈ K

1) ∃e ∈ K such that m = ‖e′‖L2

2) m = 0. Otherwise, one may decrease more : ∃P,Q ∈ R[X ] st∥∥∥ ddω

[(I + τ [P(ω)Ωx + Q(ω)Ωy ]

)e]∥∥∥

L2< ‖e′‖L2

3) e3 ∈ K ∩ S2

K. Beauchard




Conclusion

Theorem : Let M0 ∈ H1((ω∗, ω∗),S2). There exist

(tn)n∈N ∈ [0,+∞)N, (un)n∈N, (vn)n∈N finite sums of Diracmasses such that

U[t+n ; un, vn,M0]→ e3 weakly in H1.

Advantages :global resultstrong convergence in Hs, ∀s < 1, L∞

Flaws : How to do ? The strategy of the proof maynot work,take a long time,cost a lot (N-S).

K. Beauchard




4th part

Nonlinear system :explicit controls for the

asymptotic exactcontrollability

K. Beauchard




Explicit controls for the asymptotic exact controllability

Notations : - (ω∗, ω∗) = (0, π), f : (0, π)→ C identified with

f : R→ R, 2π periodic symmetric, N(f ) :=∑

n∈Z |cn(f )|.- M = (x , y , z), Z := x + iy

Theorem : ∃δ > 0 / ∀M0 : (0, π)→ S2 with N[Z0] < δ andz0 > 1/2, the solution of the Bloch equation with

u(t) := πδk (t)−∑2k−1

p=1 =(

c−k+p(Z0))δk+p(t) + πδ3k (t),

v(t) := −∑2k−1

p=1 <(

c−k+p(Z0))δk+p(t),

where k = k(Z0) /∑|n|>k |cn(Z0)| < N(Z0)/4 satisfies

N[Z(3k+)] <N(Z0)

2and z(3k+) > 1/2.

K. Beauchard




Ideas of the proof

1) ’cancel’ cn(Z0) for n 6 0 with w(t) =∑N

k=0 c−kδk (t)

Z(N+, ω) ∼(Z0(ω)−

N∫0

w(t)e−iωtdt)

eiωN

∼( ∑

n∈Zcneinω −

N∑k=0

c−ke−ikω)

eiωN

2) shift to the right with u ≡ v ≡ 0,

Z(N, ω) = Z0(ω)eiNω =∑n∈Z

cnei(n+N)ω

3) reverse with u(t) = πδ0(t), M(0+) = exp(πΩx )M0

Z(0+, ω) = Z0(ω) =∑n∈Z

cne−inω

K. Beauchard




Proof

0-k-2k k 2k

t=0 c(-2k) c(-k) c(0) c(k) c(2k)

u=v=0 : shift

t=k- c(-2k) c(-k) c(0) c(k)

u=πδ(k) : reverse

t=k+ c(k) c(0) c(-k) c(-2k)

cancel

t=3k 0 0 0 c(-2k)

K. Beauchard




5th part

Nonlinear system :feedback stabilization

K. Beauchard




Goal

∂tM(t , ω) =[u(t)e1 + v(t)e2 + ωe3

]∧M(t , ω), ω ∈ (ω∗, ω

∗)

Propose explicit feedback laws u = u(M), v = v(M) that stabilizethe Bloch equation around a uniform state of spin +1/2 or −1/2.

M(t , ω) −−−−→t→+∞

e3 uniformly wrt ω ∈ (ω∗, ω∗)

Interest : less sensible to random perturbations than open loopcontrols

K. Beauchard




Strategy

Feedback design tool : control Lyapunov function

Convergence for ODEs : LaSalle invariance principle

Convergence for PDEs : several adaptions- approximate stabilization : with discrete [KB-Mirrahimi(09)] orcontinuous spectrum [Mirrahimi(09)]- weak stabilization : under a strong compactness assumption[Ball-Slemrod(79)] or without [here, KB-Nersesyan(10), Morancey(12)]- strong stabilization : with compact trajectories[d’Andréa-Novel-Coron(98)] or strict Lyapunov functions[Coron-d’Andréa-Novel-Bastin(07)]

K. Beauchard




The impulse train structure controlIn view of the previous results, it is natural to consider

u = usmooth +∞∑

k=1

πδ(t − kT )

(x , y , z)(kT+) = (x ,−y ,−z)(kT−)

With ε(t) = (−1)E(t/T ), the change of variables

(x , y , z)← (x , ε(t)y , ε(t)z), u ← u+∞∑

k=1

πδ(t−kT ), v ← ε(t)v

transforms the Bloch equation into

∂M∂t

(t , ω) =[u(t)e1 + v(t)e2 + ε(t)ωe3

]∧M(t , ω)

K. Beauchard




The impulse train structure reduces the dispersion

e1M(0,.)

M(T,.)M(2T,.)

e2

e3

Initial free system

∂M∂t

(t , ω) = ωe3 ∧M(t , ω), M(0, ω) = e1K. Beauchard




The impulse train structure reduces the dispersion

e1

M(T,.)e2

e3

M(0,.)=M(2T,.)

New free system

∂M∂t

(t , ω) = ε(t)ωe3 ∧M(t , ω), M(0, ω) = e1K. Beauchard




Driftless form

∂M∂t

(t , ω) =[u(t)e1 + v(t)e2 + ε(t)ωe3

]∧M(t , ω)

M(t , ω)←− Rz

(−ω

∫ t

0ε(s)ds

)M(t , ω)

∂M∂t

(t , ω) =[u(t)W1(t , ω) + v(t)W2(t , ω)

]∧M(t , ω)

Interest : Any fonction may be a (controlled) Lyapunov function.

K. Beauchard




Control design : control Lyapunov function

L(t) :=

∫ ω∗

ω∗

[∥∥∥∥∂M∂ω

(t , ω)

∥∥∥∥2

+ z(t , ω)

]dω

dLdt

(t) = < [Ω(t)H(t)]

where

H(t) :=

∫ ω∗

ω∗

[iζ(t)[Zz ′ −Z ′z]−Z(t , ω)

]e−iωζ(t)dω

So we take

Ω(t) := −H(t) thendLdt

(t) = −|Ω(t)|2

K. Beauchard




Local stabilizationTheorem : There exists δ > 0 such that, for everyM0 ∈ H1((ω∗, ω

∗),S2) with ‖M0 + e3‖H1 < δ, the solution of the closedloop system satisfies

M(t) −e3 in H1(ω∗, ω∗) when t → +∞.

Rk : M(t , ω)→ −e3 uniformly with respect to ω ∈ (ω∗, ω∗).

Proof : 1. Invariant set = −e3 locally.2. Ω(t)→ 0 a.e.3. −e3 is the only possible weak H1-limit :

If M(tn)→ M0∞ weakly in H1 and strongly in H1/2 then

M(tn + τ)→ M∞(τ) strongly in H1/2, ∀τ > 0, thusΩ[M(tn + τ)]→ Ω[M∞(τ)]. Therefore Ω[M∞] ≡ 0.

Key point : Ω(M) is well defined for M only in H1/2

K. Beauchard




No global stabilization

Topological obstructions : H1((ω∗, ω∗),S2) cannot be

continuously deformed to one point.

Actually, there is an infinite number of invariant solutions, thatmay be expressed explicitly...

K. Beauchard




Numerical simulations

Parameters : (ω∗, ω∗) = (0,1), T = 2π, G := 1/(2T 2) x0(ω)

y0(ω)z0(ω)

:=

cos(π, ω)√

1− z0(ω)2

sin(π, ω)√

1− z0(ω)2

0.8− 0.1 sin(4πω)

.

Simulation until Tf = 50T

Conclusion : The convergence speed is rapid at the beginningbut decreases at the end.

K. Beauchard





0 50 100 150 200 250 300 3500.1

0.2

0.3

0.4

0.5

Time

Lyap

ouno

v fu

nctio

n

0 50 100 150 200 250 300 350−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

Time

Re(Ω)Im(Ω)

K. Beauchard




Adaptation to stabilize a variable profile

Target : Mf (ω)

Find R(ω) ∈ SO3 such that R(ω)Mf (ω) = −e3.Force N(t , ω) := R(ω)M(t , ω)→ −e3 when t → +∞(same strategy).

Proof OK forMf ∈ H1((ω∗, ω

∗), S2) with 〈Mf (ω),e3〉 6= 0, ∀ω ∈ (ω∗, ω∗),

‖M0 −Mf‖H1 small enough.

K. Beauchard





Parameters : ω∗ = 0, ω∗ = 1, T = 2π x0(ω)y0(ω)z0(ω)

:=

−0.35 sin(0.9πω − π/2)−0.35 cos(πω − π/2)√

1− x0(ω)2 − y0(ω)2

xf (ω)

yf (ω)zf (ω)

:=

−0.79 sin(πω − π/2)−0.79 cos(0.9πω − π/2)

−√

1− xf (ω)2 − yf (ω)2

Simulation until Tf = 80T

K. Beauchard





−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1

y

Profile Stabilization on The Bloch Sphere

z

x

M0

Mf

M(Tf)

North/South PolesBloch Sphere

K. Beauchard





0 100 200 300 400 500 6000

1

2

3

Time

Lyap

unov

Fun

ctio

n

0 1 2 3 4 5 6 7−4

−2

0

2

Time (t< 6.3 s)

u(t)

and

v(t

) u(t)v(t)

0 100 200 300 400 500 600−0.5

0

0.5

Time (t> 6.3 s)

u(t)

and

v(t

) u(t)v(t)

K. Beauchard





0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

ω

x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

0

1

ω

y

M0

M(Tf)

Mf

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

ω

z

K. Beauchard




Conclusion of the talk : Controllability

Linearized system :

non exact controllability, L1 controls : F [L1(−T ,0)]

non asymptotic zero controllabilityuniqueness of the controlapproximate controllability, unbounded controls

Nonlinear system :

non exact controllability, BR[L2(0,T )]-controls : manifoldapproximate controllability in Hs, s < 1, unboundedcontrols : non commutativity + variationnal methodexplicit controls for the (local) asymptotic exactcontrollability to e3 : Fourier method, many controls work

The nonlinearity allows to recover controllability.K. Beauchard




Conclusion of the talk : Stabilization

impulse train controldriftless formcontrol Lyapunov function : H1-distance to the targetexplicit damping feedback lawsweak H1 local stabilization

K. Beauchard




Open problems, perspectives

exact controllability in finite time with unboundedcontrols ?strong stabilization with the same feedback laws ?explicit feedbacks for the semi-global stabilizationconvergence rates ? arbitrarily fast stabilization ?

K. Beauchard




References

1 K. Beauchard, J.-M. Coron and P. Rouchon. Controllabilityissues for continuous-spectrum systems and ensemblecontrollability of Bloch equations. CMP, 296, 2, June 2010,p.525-557.

2 K. Beauchard, P. Pereira da Silva and P. Rouchon. Stabilizationand motion planning for an ensemble of half spin systems.Automatica 48, pp.68-76, 2012.

3 K. Beauchard, P. Pereira da Silva and P. Rouchon. Stabilizationof an arbitrary profile for an ensemble of half-spin systems.Automatica (to appear).

K. Beauchard

Documents

Control and stabilization of the Bloch equation - Spin Dynamics · The Bloch equation Linearized system Non exact controllability with a priori bounded controls Approximate controllability