Ugo Boscain

Centre de Mathematiques Appliquees (CMAP) Ecole Polytechnique

March 20, 2010

Few remarks in my CV

40 years old, Italian

EDUCATION (multidisciplinary)

1991 Degree in Music (Conservatorio di Alessandria)

1996 Degree in Theoretical Physics (Turin University)

2000 Ph.D in Mathematics (SISSA, Trieste).

Supervisor Benedetto Piccoli

2000-2002 individual Marie Curie post-doc at Dijon with J-PGauthier

2002-2006 Permanent Researcher (Ricercatore) at SISSA

2006-2008 Charge de recherche of first class CNRS, Dijon

2007 Habilitation a diriger des recherches in Engineering

2009 ——– CR1, Centre de Mathematiques Appliquees(CMAP) Ecole Polytechnique ←

PUBLICATIONS:

Books 1Chapters of Books 2

International Journals 26←Refereed Conference Proceedings 26←

Preprints 3←Citations ∼ 185

TALKS: about 50 invited talks

ORGAN. OF CONFERENCES: about 15 conferences/sessions

ASSOCIATE EDITOR: ESAIM COCV ←

PhD STUDENTS: 2 (already defended)+3 (defense: 2009/2010/2011)

Active Grants

Grant name Title responsable organiz. total/ me

FABER Quantum Control Burgundy Region 40K/40KPost doc Region Reconstr. Images Burgundy Region 48KUlysses 09 ← Switched Systems Part. Hubert Curien 16.8 K /1.2 KESF Res. Netw. OPTPDE Eur. Sci. Foun. 486K/2.5KGDRE ← Control of PDEs CNRS/INDAM 160K/0.8K

Control theory: x = fu(x), x ∈ M, u ∈ U ⊂ Rm

Controllability: for every x0, x1, there exists u(t) such that thecorresponding trajectory of x(t) = fu(t)(x(t)) steers x0 to x1.

M

u(t)x 1

x0

Stabilization in x0: there exists u(x) such that x0 is asymptoticallystable for x(t) = fu(x(t))(x(t))

M

x 0

Optimal Control: find a control u(t) such that the correspondingtrajectory minimizes a given cost:

∫ T0

L(x(t), u(t)) dt, x(0) = x0, x(T ) = x1M

1x0 x

calculus of variations with constraints on the velocities

Optimal Controlsub−Riemannian geometryRiemannian geometry

calculus of variations

My research topics: Geometric Control Theory+ Appl.

Conjugate points

sub−RiemannianGeometry

Almost−RiemannianGeometry

Neuroscience (problems of geometry of Vision)

GEOMETRIC CONTROL THEORYcontrollability, stability, optimal control

−) Hamiltonian systems −)

−) differential geometry (Lie algebraic methods)

NMR+ Laser spectroscopy

Hypoelliptic heat equation (nonisotropic diffusion)relation SR−distance−−properties of heat kernel

Control of quantum mechanical systems(control of the bilinear Schroedinger equation)

realization of quantum gates for quantum computers reconstruction of images

−) Representation Theory−) Singularity theory

Results

I will just mention the results related to the project

How geometric control methods can be useful to study certain PDEs

→control of the Schrodinger equation(control of quantum mechanical systems)

→non-isotropic diffusion equations (heat-equation with the hypoellipticLaplacian)

Quantum control

AIM: induce transitions between energy levels of a quantumsystem (e.g. a molecule) by means of an external field

−) to use quantum states as a memory (quantum computation)

EXCITATION

External field (laser, magnetic field)

−) to induce chemical reactions(material science)

−) to measure the decay −−> images(NMR)

(atom or molecule)Quantum System

The control of the bilinear Schrodinger equation

idψ

dt= (H0 + u(t)H1)ψ, ψ ∈ S ⊂ H,

ψ ∈ S Hilbert sphere in a complex Hilbert space H;

H0,H1 self-adjoint operators on H possibly unbounded;

→Usually H = L2(Rn), H0 = −∆+ V (x), H1 = W (x)

u(·) : R → R, u(·) ∈ U (e.g. measurable bounded)

The controllability problem:

S0

Ψ1

u(t)

Ψ

Fixed ψ0 and ψ1, prove that there exists a control u(.) ∈ U that steers thesystem from ψ0 to ψ1.

can we do it exactly/approximately? Thickness of the reachable set?

can we estimate the time of the transition?

can we measure the complexity of controls? (e.g. the total variation)

can we follow a given trajectory? (tracking problem)

can we get explicit expression of controls?

Known results

controllability results with interior or boundary control (linear andsemilinear theory) [review by Zuazua 2003];

negative results (up to 2003)

non-exact controllability in S ∩H2 ∩H10

(Ball-Marsden-Slemrod, Turinici);non-controllability of the linearization (Rouchon)non-controllability of the harmonic oscillator driven byu(.)x (Mirrahimi-Rouchon)

positive results (only examples)

Control of the Eberly-Law System (Brockett-Bloch-Rangan)exact contr. in H5+(Ω) in Ω = (−1/2, 1/2)+Dirichlet B.C.,for i∂tψ(t, x) = (−∂2

x + u(t)x)ψ(t, x) (Beauchard-Coron);

Main result

Applying finite dimensional techniques on a suitable two-scale Galerkinapproximation, with my research group:

M. Sigalotti INRIA, Nancy

P. Mason, LSS, Supelec ← my ex PhD student

T. Chambrion Laboratoire, IECN, Nancy.

(λn)n∈N eigenvalues of H0 corresponding to (φn)n∈N.

Theorem (Annales de l’Institut Henri Poincare, 2009)

If (λn+1 − λn)n∈N are Q-linearly independent and if 〈H1φj ,φj+1〉 )= 0 forevery j ∈ N, then the system is approximately controllable (L2 norm) witharbitrarily small controls.

Approximate controlability: if for every ψ0,ψ1 ∈ S and every ε > 0there exist u(·) and T > 0, such that

‖ψ1 − ψ(ψ0, u(·), T )‖L2 < ε.

(λn+1 − λn)n∈N are Q-linearly independent if for every N ∈ N and(q1, . . . , qN ) ∈ QN

! 0 one has∑N

n=1 qn(λn+1 − λn) )= 0.

→our result explains why in practice quantum systems are more easy tocontrol than expected (H. Rabitz, Princeton)

sub-Riemannian geometry and nonisotropic diffusion

A sub-Riemannian manifold is (M,!, g):

M : smooth manifold of dimension n ≥ 3

!: non-integrable distribution (!(x) ⊂ TxM) of rank m (2 ≤ m < n)

g: Riemannian metric on !

d(x0, x1) = infx(t)∈!(x(t))

x(0)=x0,x(T )=x1

∫ T

0

√

g(x(t), x(t))dt

The problem of finding the distance between x0 and x1 is an optimalcontrol problem (locally)

x =m∑

i=1

uiFi(x), x(0) = x0, x(T ) = x1,

∫ T

0

(

m∑

i=1

ui(t)2

) 1

2

dt → min

−) highly non−isotropic

Heisenberg

−)non−smooth even for small time

generic case

the distance is:

Diffusion in SRG

∂tψ = ∆S.R.ψ

→many results on estimates of the fundamental solutions for the sum ofsquares

∑mi=1(Fi(x))

2 (it depends on the choice of the orthonormal base)

→many results for the Heisenberg group (and STEP-2 Nilpotent Liegroups)

Open Problems

what is the intrinsic sub-Riemannian Laplacian equivalent to theLaplace Beltrami operator ?

(R. Mongomery “A Tour of sub-Riemannian Geometry”.)

does an intrinsic volume in SRG exist?

is it possible to get explicit expression of the fundamental solutionbeside the Heisenberg group (or 2-step nilpotent groups)?

Why?

Because there is a model of the functional architecture of V1 due to:

D.H. Hubel and T. Wiesel (Nobel prize 1981)

J. Petitot, G. Citti, A. Sarti

that says that the brain reconstructs an image like:

Why?

. . . solving a nonisotropic heat equation on the group ofrototranslations of the plane SE(2) ∼ R2 × S1

! = Span

cos(θ)sin(θ)

0

,

001

, ∂tψ(x, y, θ, t) = ∆S.R.ψ(x, y, θ, t)

→the initial condition is a lift of the original image→its evolution is the reconstructed image (on the lift)→projection on the plane

Up to now solutions are known only for step-2 nilpotent groups (Gaveau,1977, Beals, Greiner, Hulanicki).

A. Agrachev (SISSA, Trieste)

J.P. Gauthier (Toulon University)

F. Rossi (Bourgogne University) ← my PhD student

we obtained the following results (JFA, 2009)

we gave a intrinsic definition of sub-Riemannian Laplacian and weproved that for left-invariant structures on unimodular Lie groups it isthe sum of squares

by using noncommutative Fourier transform we got explicit expressionof the hypoelliptic heat kernel on SU(2), SO(3), SL(2) and SE(2)

The project

With my arriving at CMAP, build a group of geometric controlmethods centered on→Schrodinger equation→sub-Riemannian geometry (including non-isotropic diffusion equations)

PARIS TEAM (main):

Yacine Chitour: Professor at LSS, Supelec, Paris, teacher at CMAP(40 years old)

→geometric control methods

Frederic Jean: Professor at ENSTA, Paris (40 years old)

→sub-Riemannian geometry

Mario Sigalotti: Charge de Rechearche INRIA Nancy, at CMAPfrom ??? (33 years old)

→controllability of the Schrodinger equation

→genericity in control

post docs (6 years in the next 5 years)

PhD students (2 in the next 5 years) +2 from SISSA

Other members

Gregoire Charlot, Institut Fourier, Grenoble, France (35 years old)

→expert in quantum control

→expert in sub-Riemannian geometry

Thomas Chambrion, Institut Cartan, Nancy, France (31 years old)

→expert in quantum control

Riccardo Adami, Bicocca University, Milano, Italy (37 years old)

→expert in quantum mechanics

Jean-Paul Gauthier, Toulon University, France (senior)

→expert in automatic control

→expert in harmonic analysis on Lie groups

Andrei Agrachev, SISSA, Trieste, Italy (senior)

→expert in sub-Riemannian geometry

→expert in controllability of PDEs by geometric methods

Quantum Control [TASK 3]

Exploit all consequences of our controllability method, in relation with realapplications

Genericity of our hypotheses [TASK 3.4, R=5]

Tracking [TASK 3.1, R=2]1ψψ

0

controlling several (slightly different) systems with the same control[TASK 3.2, R=3]

→very important in practical applications

→when one is controlling several identical systems, the coupling is notthe same

→some parameters of the system are not know precisely

evaluate the steering time and the complexity of controls [TASK 3.3,R=5]

explicit algorithm to control the orientation of a molecule [TASK 4,R=2-3]

→these are the true systems with discrete spectrum

→at the moment no other methods permit to get results of this type

Nonisotropic diffusion in Sub-Riemannian geometry[TASK 1 and 2]

With our methods coming from noncommutative Fourier analysis and usingthe knowledge we have about singularity of the distance in SRG we plan to:

find explicitly the heat kernel for the Nilpotent Lie groups (234)-Engeland (235)-Cartan [TASK 1.1, R=1]

→first examples of heat kernels for STEP 3

get an asymptotc expansion of the kernel in the 3D contact case[TASK 1.2, R=3]

In Riemannian geometry

Pt(x, y)t→0∼ (4πt)(−n/2) exp

(

−d(x, y)2

4t

)

(u0(x, y) + u1(x, y)t+ . . .)

recover the cut locus from the heat-kernel [TASK 1.3, R=5]

In Riemannian geometry setting Et(x, y) = −t log pt(x, y) , we have:y ∈ Cut(x) iff lim

t→0‖HessyEt(x, y)‖ = ∞, where Cut(x) is the cut

locus starting from x. (Neel-Stroock)

In SRG there are several volume forms (e.g. the Popp measure andthe Hausdorff measure) →several intrinsic Laplacians (with the samesymbol). Exploit the difference between them [TASK 2, R=1-2-5]

Why at CMAP?

→because there is an interest of the Laboratory in building a young andstrong group of geometric control(I was asked to come from Dijon to Paris for this reason)

→at Ecole Polytechnique there are among the best students(I am already teaching at Ecole Polytechnique a course of Control Theoryto attract students)

→the long-term project is to build an INRIA project (4+4 years)

at the beginning our team will be financed by ERC

after by INRIA+ French grants (DIGITEO/ ANR) + EU (I will tryto build a network in 4 years)

→numerical methods (Groups of Allaire, Alouges, Chambolle, Bonnans,etc...)

The End

Post-Docs and PhD

Post Docs

1) Heat-kernels on higher order nilpotent Lie groups [TASK 1.1] (Engel,Cartan) years X0000

2) Relation between the sub-Riemannian distance and the heat-kernel[TASKS 1.2,1.3] →stochastic experience years 0XX00

3) Simultaneous Controllability and estimation of the steering time [TASKS3.2, 3.3] →numerical experience years 00XX0

4) Generic Controllability [TASK 3.4] years 0000X

PhD

1) Control of molecular orientation [TASK 4] (control of the Schrodingerequation on S1, S2, SO(3)) years XXX00

2) Relation between the Popp and the Hausdorff volumes insub-Riemannian geometry [TASK 2] years 00XXX

Financial table

Cost Category year 1 year 2 year 3 year 4 year 5 total

D¯ir. Costs: Personnel:

PI 66.1K 66.1K 66.1K 66.1K 66.1K 330.5Kpost-docs 48K 48K 96K 48K 48K 288K

PhD 33.5K 33.5K 67K 33.5K 33.5K 201KTotal Personnel: 147.6K 147.6K 229.1K 147.6K 147.6K 819.5K

Other Direct CostsEquipement 8K 8K 8K 8K 8K 40K

Travels for members 25K 25K 25K 25K 25K 125KInvitations 10K 10K 10K 10K 10K 50K

org. School/Worksh. 0 25K 40K 25K 0 90KTot. other dir. costs: 43K 68K 83K 68K 43K 305K

Tot. dir. costs: 190.6K 215.6K 312.1K 215.6K 190.6K 1124.5K

Ind. Costs: 38.1 K 43.1K 62.4K 43.1K 38.1K 224.8K

Tot. Costs: 1349.3KReq. Grant: 1349.3K

→my salary 82.6k by year→travels 2.270K by person (9+ 1 post doc+1 PhD), by year

Workshops and School

Worshop 1: (25 K)Title: Geometry and Analysis on sub-Riemannian manifoldsWhere: ToulonWhen: Second YearParticipants: 30

Workshop 2: (25 K)Title: Geometry and Analysis on sub-Riemannian manifoldsWhere: Ecole Polytechnique Paris or SISSA, TriesteWhen: Fourth yearParticipants: 30

School: (40 K)Title: Control of the Schrodinger equationWhere: Ecole Polytechnique, Paris or SISSA, TriesteWhen: Third yearTeachers: Agrachev, Coron, Gauthier, Kuksin, Sarychev, Shirikyan,Zuazua,Participants: 30