Some slides of the talk are available clicking on the title.
Titles and abstracts
Andrei Agrachev (SISSA) – The curvature revised.
Davide Barilari (Ecole Polytechnique) – Heat kernel small time asymptotics at the cut locus in sub-Riemannian geometry.
For a sub-Riemannian manifold provided with a smooth volume, we relate the
small time asymptotics of the heat kernel at a point y of the cut locus from
x with roughly "how much" y is conjugate to x. This is done under the
hypothesis that all minimizers connecting x to y are strongly normal, i.e.
all pieces of the trajectory are not abnormal. Our result is a refinement of
the result of Leandre $4t\log p_t(x,y)\to -d^2(x,y)$, in which only
the leading exponential term is detected. For the Grushin plane endowed with the
Euclidean volume we get the expansion $p_t(x,y)\sim t^{-5/4}\exp(-d^2(x,y)/4t)$
where y is reached from a Riemannian point x by a minimizing geodesic which
is conjugate at y.
Yuliy Baryshnikov (Illinois) – Topology of hard disk gas
Quintessential for statistical physics model of hard disks (or balls) in a box received relatively little attention as a configuration space. Yet, its topology plays an important role, say, in analysis of algorithms. In my talk I will describe some results on asymptotic topology of this space in the gaseous regime.
Ilya Bogaevsky (Moscow State University) – Shifted sub-Riemannian control systems: classification and
singularities of fronts
The admissible velocities of a shifted sub-Riemannian control system
on a three-dimensional manifold form, by definition, an ellipse lying
in an affine plane of the tangent space. We classify generic points of
such systems into two types: contactly hyperbolic and contactly
elliptic. In particular, all points of the control system naturally
defined by a contact sub-Riemannian structure are contactly elliptic.
This classification is a reformulation of the Arnold two types of
contact structures in a neighborhood of a cone. We describe the front
and its singularities of a contactly hyperbolic point for small times,
and formulate a conjecture asserting that the front of a contactly
elliptic point is the sub-Riemannian sphere, investigated by
A.A.Agrachev in 1996.
Bernard Bonnard (Dijon) – Riemannian metrics on 2D manifolds related to Euler-Ponsot rigid body
The Euler-Poinsot rigid body motion has many applications in optimal control, e.g. in the attitude control problem of a rigid spacecraft or the dynamics of spin systems and are the model for left-invariant Riemannian or Sub-Riemannian metrics on SO(3). By reduction methods they define 2D-Riemannian metrics which are analyzed in our talk. The conjugate and cut loci are described using explicit and numerical computations as tesbed of the Hampath code.
(This work is co-authored with N. Shcherbakova, INP Toulouse)
Piermarco Cannarsa (Roma) – Controllability properties of Grushin-type operators.
Grushin's operator is an important example of a degenerate elliptic operator with strong connections with almost-riemannian structures. It is also the infinitesimal generator of a strongly continuous semigroup on Lebesgue spaces with very interesting properties from the point of view of control theory. We will discuss null controllability for parabolic problems associated with Grushin-type operators on a bounded two-dimensional domain, under additive controls supported in an open set. As we shall see, such systems can be null controllable in arbitrary positive time, not controllable for any time, or even controllable only in a sufficiently large time, depending on certain parameters that measure degeneracy.
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.
The dynamics of quantum systems driven by external electric fields (lasers) can be described in first approximation by bilinear Schrödinger equations. Despite spectacular advances in the last decade by a small yet active community, many natural questions (such as controllability or design of efficient controls) remain open for conservative infinite dimensional bilinear systems.
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
In this talk we survey some results on the controllability and the stabilization of various control systems modeled by 1D nonlinear hyperbolic systems. We present applications to the stabilization of the height of the water and the flow rate in open channels, including the case where slope and friction are important.
Alexey Davydov (Vladimir State University) – Optimal stationary exploitation of size-structured population.
We analyze a model of exploitation of size-structured population when the birth, growth and mortality rates depend on the individual size and interspecies competition, while the exploitation intensity is a function of the size only. For a given exploitation intensity we establish the existence and uniqueness of nontrivial stationary state of the population under natural assumptions on these rates. Also we prove the existence of an exploitation intensity providing the maximum for selected profit functional of exploitation.
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.
It was shown by Agrachev and al. that in corank 1 SR-case, the (spherical) Hausdorf measure is C3 but not C5-smooth. This result was improved on by Barilari and al. who showed that in Corank 2, rank 4, Hausdorf measure is generically C1-smooth.
In this talk, we improve on these last results: in fact, in corank 2, any rank, spherical Hausdorf measure is generically C2-smooth. (Joint with Ugo Boscain.)
Bronislaw Jakubczyk (Warsaw) – Hamiltonian controlled and observed systems and noncommutative algebra.
We will consider nonlinear controlled and observed systems, with the same number of inputs and outputs. Such systems admit Hamiltonian structure (corresponding to some symplectic or Poisson structure
on the underlying manifold) if the system satisfies certain algebraic relations.
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.
The main motivation of this paper arises from the study of Carnot-Carathéodory spaces, where the class of 1-rectifiable sets does not contain smooth non-horizontal curves; therefore a new definition of rectifiable sets including non-horizontal curves is needed. This is why we introduce in any metric space a new class of curves, called continuously metric differentiable of "order" k, which are Hölder but not Lipschitz continuous when k>1. Replacing Lipschitz curves by this kind of curves we define (H^k,1)-rectifiable
sets and show a density result generalizing the corresponding one in Euclidean geometry. This theorem is a consequence of computations of Hausdorff measures along curves, for which we give an integral formula. In particular, we show that both spherical and usual Hausdorff measures along curves coincide with a class of dimensioned lengths and are related to an interpolation complexity, for which estimates have already been obtained in Carnot-Carathéodory spaces.
Velimir Jurdjevic (Toronto) – Jacobi's geodesic problem and Lie groups.
Sergei Kuksin (Ecole Polytechnique) – Around the Cauchy-Kowalewski theorem
I will discuss a general ("geometrical") approach which allows to prove
the propagation of analyticity for solutions of various quasilinear PDEs.
In particular, it implies that the assertion of the Cauchy-Kowalewski
and Ovsyannikov-Nirenberg theorems hold not only locally in time,
but till a classical solution exists.
Paul Woon Yin Lee (Hong Kong) – Optimal transportation and Curvature of Hamiltonian systems
In this talk, we will discuss curvature of Hamiltonian systems introduced by Professor Agrachev and its relations with uniform hyperbolicity of a Hamiltonian system.
Antonio Lerario (Purdue) – The topology of a random intersection of real quadrics.
The problem of computing the expected value of topological properties of random real algebraic varieties dates back to the work of Kac. In a seminal paper he proved that the expected number of real roots of a polynomial of degree d whose coefficients are i.i.n. random variables growths asimptotically as (2/Pi)log(d). This work was generalized in many directions, considering the number of real roots of a polynomial as either the volume or the Euler characteristic of its zero locus. It turns out that the most natural probability distribution on the space of the coefficients is indeed given by weighting the variances of the normals by a binomial coefficient depending on the degree of the monomial. This distribution is called the Bombieri-Weyl (or Kostlan) distribution. For instance in the case of a polynomial of degree d in one variable the coefficient of the term of degree k is assumed to be normally distributed with mean zero and variance the binomial coefficient d choose k.
With this distribution the expected number of real roots of a polynomial of degree d is exactly the square root of d. More generally the expected (normalized) volume of a real random variety of degree D is the square root of D. All these kind of results are motivated by the comparison of the volume of the real part of a variety to that of its complex part: the volume of the first is always less or equal than the second. In the case of a real univariate polynomial this is the statement that the number of real roots is less or equal than the number of complex roots. Surprisingly enough, this statement admits also a topological generalization: this is the so called Smith's inequality. It states that the sum of the Betti numbers of a real algebraic variety is always less or equal than the sum of the Betti numbers of its complex part. Following this idea we prove that in the case X is the intersection of one or two random real quadrics, then Smith's inequality is asymptotically sharp as the number of variables goes to infinity. In fact we prove that for one real random Bombieri-Weyl distributed quadric in RP^n the expected sum of its Betti numbers is asymptotically n, while for the intersection of two random real quadrics is asymptotically 2n.
As we consider the intersection of more quadrics, this offers new prospectives; e.g. the case of three quadrics has strong connections with Hilbert's 16th problem.
Daniel Liberzon (Illinois) –
Norm-controllability, or how a nonlinear system responds to large inputs
In this talk we will discuss a recently introduced notion of "norm-controllability" for nonlinear systems, which captures the system's responsiveness to control inputs. This property is formulated in terms of the ability to produce a state whose norm is large (or, more generally, such that the norm of a suitable output is large) by applying inputs of large enough magnitude for a long enough time. We will argue that this represents a reasonable nonlinear extension of the standard linear controllability, and explain how it is conceptually related to existing nonlinear system theory concepts such as input-to-state stability and norm-observability. Our basic technical result is a Lyapunov-like sufficient condition for norm-controllability, which is proved by explicitly constructing a desired control signal. Some extensions and applications will also be mentioned.
Roberto Monti (Padova) – Some new results on the regularity problem of sub-Riemannian geodesics.
We shall briefly present an explicit classification of abnormal extremals in the setting of nilpotent stratified Lie groups. Then we shall focus on the question of length minimality of abnormal extremals near singular points. The results are part of a joint research project with D. Vittone, G. P. Leonardi, and E. Le Donne.
Laura Poggiolini (Firenze) – Local optimality and structural stability of bang--singular and bang--singular--bang extremals in the minimum time problem
We consider the minimum time problem for a single--input control--affine system. A very simple system, like the Dubins--Dodgem car problem, shows that depending on the initial and final point constraints, the optimal control can be bang--bang or bang--singular or bang--singular--bang.
Here we consider the cases when a singular arc appears, namely we assume we are given a reference normal Pontryagin extremal for the nominal problem and we assume that such reference extremal is either bang--singular--bang or bang--singular, depending on two different kinds of final point constraints. Regularity conditions on each arc and at the junction points, plus second order conditions on the singular arc only are proven to be sufficient for the strong local optimality of the given extrema. In order to prove the structural stability result we also require controllability along the singular arc. Both the local optimality and the structural stability results are proven via Hamiltonian methods.
This is a joint work with Gianna Stefani.
Ludovic Rifford (Nice) – Mañé's Conjecture from the control viewpoint.
We will briefly introduce the so-called Mañé Conjecture in weak KAM theory. We will show how geometric control ideas together with Pugh's closing techniques can lead to a partial solution to the conjecture.
This is a joint work with Alessio Figalli.
Yuri Sachkov (Pereslavl) – Exponential mapping for the nilpotent sub-Riemannian problem on the Engel group.
The nilpotent sub-Riemannian problem on the Engel group provides a homogeneous nilpotent approximation for sub-Riemannian problems with the growth vector (2,3,4). The talk will be devoted to recent results on this problem: parameterization of extremal trajectories, symmetries and Maxwell points, conjugate and cut time, global structure of the exponential mapping.
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.
We consider 2D Navier–Stokes equations in a bounded domain with smooth boundary and discuss the interconnection between controllability for the deterministic problem and mixing properties of the associated random dynamics. Namely, we first consider the problem of stabilisation of a given non-stationary solution, assuming that the control is localised in space and time and is finite-dimensional as a function of both variables. We next replace the control by a random force and prove that the resulting random dynamical system is exponentially mixing in the Kantorovich–Wasserstein distance. Some of the results of this talk are obtained in collaboration with V. Barbu and S. Rodrigues.
Hector Sussmann (Rutgers) – Universal reguarity theorems for optimal control problems
We state a general universal regularity theorem for real
analytic optimal control problems, and give a detailed proof for the
special case of minimum time problems with a scalar control taking values
in the interval [-1,1] and a dynamic equation in which the right-hand side
is linear affine in the control.
Emmanuel Trélat (Paris VI) – Optimal design problems for conservative equations.
We consider a conservative evolution equation on a given domain Omega of R^n. The purpose of this talk is to investigate some natural shape optimization problem arising in the context of mathematics, physics or engineering. Given an initial state, one may observe on a measurable subset omega of Omega with given measure the solution of the equation, or control it (by HUM) or stabilize it (by a linear damping) to rest, with a control supported on omega. In the three cases, we focus on the question to know how to determine the best possible domain omega over all subsets of Omega of fixed measure (say L=|Omega| with 0<L<1) ensuring either the best observation, or the smallest possible norm of control, or the best rate of convergence for the stabilization. These questions are first investigated with fixed initial data. We then provide relevant criteria that do not depend on the initial conditions and analyze the related shape optimization problems. In particular, we comment on the regularity of the optimal domain, which can be regular or of fractal type according to the problem under consideration. One of these problems consists of the optimization (with respect to the domain omega) of observability constants. Finally, we provide approximation procedures in order to compute numerically the best domain. In particular in dimension one efficient algorithms can be developed by using an interpretation of the problem in terms of optimal control.
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.
The following question is studied in elasticity theory: what can we say about a global deformation of a rigid body provided that local deformations are small? This question leads to the mathematical problem due to F. John.
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.
The idea to obtain invariants of vector distributions via the study of intrinsic Jacobi equations along abnormal extremals was initiated by Andrei Agrachev in the second half of ninetieth. This idea proved to be very prolific. It led not only to a new geometric-control interpretation of the classical Cartan invariant of rank 2 distributions on five dimensional manifolds, but also to a generalization of this invariant, to the construction of canonical frames for rank 2 distributions on manifolds of arbitrary dimension, and, in combination with algebraic prolongation techniques in spirit of N. Tanaka, to the construction of canonical frames for distributions of arbitrary rank. The aim of the talk is to make a survey of these developements.
Michail Zhitomirskii (Haifa) – Homogeneous subsets of the tangent bundle and abstract endowed transitive Lie algebras
A homogeneous subset of the tangent bundle is
a subset whose symmetry group acts transitively.
The talk is devoted to several basic question
on the correspondence between classification of
such subsets of the tangent bundle and classification
of abstract transitive Lie algebras, and to the
answers to these questions for 2-distributions on
5-manifolds and affine distributions on 3-manifolds
and 4-manifolds.
News
Photos of the conference now available!
Some slides of the conference now available!
