Geometric Control and

sub-Riemannian Geometry

Some slides of the talk are available clicking on the title.
### Titles and abstracts

(This work is co-authored with N. Shcherbakova, INP Toulouse)

Our approach is based on the fact that, thanks to the particular geometric configuration of the space domain, null controllability is closely linked to the one-dimensional observability of the Fourier components of solutions to the adjoint system, uniformly with respect to Fourier frequency.

This talk gives a summary of the results obtained using geometric techniques in collaboration with U. Boscain, N. Boussaid, M. Caponigro P. Mason and M. Sigalotti. The first part will focus on averaging techniques for the explicit design of controls. The second part will focus on the important question of the energy needed to transfer a quantum system from a given initial state to a given target.

This work is co-authored by Anton Platov (Vladimir State University, Russia).

We will show that the best way of expressing admissibility of the Hamiltonian structure by a system is to use noncommutative algebra. Then an analytic Hamiltonian system defines an formal Lie series with finite Kirillov rank and Cauchy type estimates, an vice versa. A formal co-adjoint action will be defined. Then all analytic systems with a fixed number of inputs (outputs) can be realized as orbits of a universal coadjoint action on the space of noncommutative formal power series. This gives, in particular, a solution to the Hamiltonian realization problem.

This is a joint work with Alessio Figalli.

The talk will be based on results obtained in collaboration with A.Ardentov.

This is a work in collaboration with Y. Privat (ENS Rennes, France) and E. Zuazua (BCAM Bilbao, Spain).

A deformation is interpreted as a homeomorphism $f$ of an open domain of three-dimension Euclidean space. The Jacobi matrix $Df(x)$ is assumed to exist almost everywhere. The symmetric matrix $E(x)=\frac{1}{2}((Df(x))^t Df(x)-I)$ determines $Df(x)$ up to an orthogonal matrix. The matrix $E(x)$ is associated to the deformation or strain tensor. The notion of deformation tensor $E$ plays a key role in elasticity theory for instance: various full or partial linearization problems there are based on the assumption that the deformation tensor is sufficiently small. How can this assumption affect $f(x)$ itself? It is known that if $E(x)=0$ almost everywhere on $U$ then $f$ is a rigid motion under the condition of sufficient regularity for $f$ If $E$ is small on $U$ in some sense then what is the global difference between $f$ and a~rigid motion on the entire domain? If the difference is small globally then this property is called geometric rigidity of isometries. For similar problem on Heisenberg groups we prove sharp geometric rigidity estimates for isometries. Our main result asserts that every $(1+\varepsilon)$-quasi-isometry on a John domain of the Heisenberg group $\mathbb{H}^n$, $n>1$, is close to some isometry up to proximity order $\sqrt{\varepsilon}+\varepsilon$ in the uniform norm, and up to proximity order $\varepsilon$ in the Sobolev norm. We give examples showing the asymptotic sharpness of our results.

The talk is based on joint papers with Daria Isangulova.

Photos of the conference now available!
Some slides of the conference now available!

**Andrei Agrachev (SISSA)** – *The curvature revised.*

**Davide Barilari (Ecole Polytechnique)** – *Heat kernel small time asymptotics at the cut locus in sub-Riemannian geometry.*

**Yuliy Baryshnikov (Illinois)** – * Topology of hard disk gas*

**Ilya Bogaevsky (Moscow State University)** – *Shifted sub-Riemannian control systems: classification and
singularities of fronts*

**Bernard Bonnard (Dijon)** – *Riemannian metrics on 2D manifolds related to Euler-Ponsot rigid body*

(This work is co-authored with N. Shcherbakova, INP Toulouse)

**Piermarco Cannarsa (Roma)** – *Controllability properties of Grushin-type operators.*

Our approach is based on the fact that, thanks to the particular geometric configuration of the space domain, null controllability is closely linked to the one-dimensional observability of the Fourier components of solutions to the adjoint system, uniformly with respect to Fourier frequency.

**Italo Capuzzo Dolcetta (Roma)** – *Hamilton-Jacobi-Bellman
equations and deterministic mean field games*

**Thomas Chambrion (Nancy)** – *Geometric methods for the control of bilinear Schrödinger equations.*

This talk gives a summary of the results obtained using geometric techniques in collaboration with U. Boscain, N. Boussaid, M. Caponigro P. Mason and M. Sigalotti. The first part will focus on averaging techniques for the explicit design of controls. The second part will focus on the important question of the energy needed to transfer a quantum system from a given initial state to a given target.

**Yacine Chitour (LSS-Supelec)** – *Rolling on a space form*

**Jean-Michel Coron (Paris VI)** – *Some results on the control of hyperbolic systems*

**Alexey Davydov (Vladimir State University)** – *Optimal stationary exploitation of size-structured population.*

This work is co-authored by Anton Platov (Vladimir State University, Russia).

**Gianni Dal Maso (SISSA)** – *Generalized functions of bounded deformations*

**Bruno Franchi (Bologna)** – *Maxwell's equations in Carnot groups*

**Revaz Gamkrelidze (Moscow)** – *Differential-geometric and invariance properties of the equations of Maximum Principle.*

**Nicola Garofalo (Purdue and Padova)** – * Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds*

**Jean-Paul Gauthier (Toulon)** – *The Hausdorff measure in sub-Riemannian geometry.*

**Bronislaw Jakubczyk (Warsaw)** – *Hamiltonian controlled and observed systems and noncommutative algebra.*

We will show that the best way of expressing admissibility of the Hamiltonian structure by a system is to use noncommutative algebra. Then an analytic Hamiltonian system defines an formal Lie series with finite Kirillov rank and Cauchy type estimates, an vice versa. A formal co-adjoint action will be defined. Then all analytic systems with a fixed number of inputs (outputs) can be realized as orbits of a universal coadjoint action on the space of noncommutative formal power series. This gives, in particular, a solution to the Hamiltonian realization problem.

**Frédéric Jean (ENSTA)** – *A new class of (H^k,1)-rectifiable subsets of metric
spaces.*

**Velimir Jurdjevic (Toronto)** – *Jacobi's geodesic problem and Lie groups.*

** Sergei Kuksin (Ecole Polytechnique) ** – *Around the Cauchy-Kowalewski theorem*

**Paul Woon Yin Lee (Hong Kong)** – *Optimal transportation and Curvature of Hamiltonian systems*

**Antonio Lerario (Purdue)** – *The topology of a random intersection of real quadrics.*

** Daniel Liberzon (Illinois)** – *
Norm-controllability, or how a nonlinear system responds to large inputs*

**Roberto Monti (Padova)** – *Some new results on the regularity problem of sub-Riemannian geodesics.*

** Laura Poggiolini (Firenze) ** – *Local optimality and structural stability of bang--singular and bang--singular--bang extremals in the minimum time problem*

**Ludovic Rifford (Nice)** – *Mañé's Conjecture from the control viewpoint.*

This is a joint work with Alessio Figalli.

**Yuri Sachkov (Pereslavl)** – *Exponential mapping for the nilpotent sub-Riemannian problem on the Engel group.*

The talk will be based on results obtained in collaboration with A.Ardentov.

**Armen Shirikyan (Cergy-Pontoise)** – *Control and mixing for 2D Navier–Stokes equations with space-time localised force.*

**Hector Sussmann (Rutgers)** – *Universal reguarity theorems for optimal control problems*

**Emmanuel Trélat (Paris VI)** – *Optimal design problems for conservative equations.*

This is a work in collaboration with Y. Privat (ENS Rennes, France) and E. Zuazua (BCAM Bilbao, Spain).

**Sergey Vodopyanov (Sobolev Institute)** – *Sharp Geometric Rigidity of Isometries on Heisenberg Groups.*

A deformation is interpreted as a homeomorphism $f$ of an open domain of three-dimension Euclidean space. The Jacobi matrix $Df(x)$ is assumed to exist almost everywhere. The symmetric matrix $E(x)=\frac{1}{2}((Df(x))^t Df(x)-I)$ determines $Df(x)$ up to an orthogonal matrix. The matrix $E(x)$ is associated to the deformation or strain tensor. The notion of deformation tensor $E$ plays a key role in elasticity theory for instance: various full or partial linearization problems there are based on the assumption that the deformation tensor is sufficiently small. How can this assumption affect $f(x)$ itself? It is known that if $E(x)=0$ almost everywhere on $U$ then $f$ is a rigid motion under the condition of sufficient regularity for $f$ If $E$ is small on $U$ in some sense then what is the global difference between $f$ and a~rigid motion on the entire domain? If the difference is small globally then this property is called geometric rigidity of isometries. For similar problem on Heisenberg groups we prove sharp geometric rigidity estimates for isometries. Our main result asserts that every $(1+\varepsilon)$-quasi-isometry on a John domain of the Heisenberg group $\mathbb{H}^n$, $n>1$, is close to some isometry up to proximity order $\sqrt{\varepsilon}+\varepsilon$ in the uniform norm, and up to proximity order $\varepsilon$ in the Sobolev norm. We give examples showing the asymptotic sharpness of our results.

The talk is based on joint papers with Daria Isangulova.

**Igor Zelenko (Texas A&M)** – *The Role of Fields of Abnormal Extremals in Geometry of
Distributions.*

**Michail Zhitomirskii (Haifa)** – *Homogeneous subsets of the tangent bundle and abstract endowed transitive Lie algebras*

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