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Seminars 2001-2002

Variational problems with volume constraints.

Asymptotic and numerical methods for hyperbolic systems with stiff sources.

Simmetry breaking of extremal functions in weighted Sobolev inequalities.

Boundary singularities and envelopes of wave fronts.

Stability of equilibria in relaxation models.

Viscous approximation of strong shocks of systems of conservation law.

Partial symmetry and asymptotic behaviour for some elliptic variational problems.

Sign changing solutions to superlinear Schrodinger equations.

Observability of General Linear Pairs.

The K+P Problem for a Three-level Quantum System. (Part two)

The K+P Problem for a Three-level Quantum System.

We apply techniques of subriemannian geometry on Lie groups to laser-induced population transfer in a three-level quantum system. The aim is to induce transitions by two laser pulses, of arbitrary shape and frequency, minimizing the pulse energy. We prove that the Hamiltonian system given by the Pontryagin Maximum Principle is completely integrable, since this problem can be stated as a ``K+P problem'' on a semi-simple Lie group. Optimal trajectories and controls are exhausted. The main result is that optimal controls correspond to lasers that are ``in resonance''. The ``K+P problem'' is discussed in detail.

Some results on elliptic equations involving a singular potential.

Normal forms for Free nilpotent Lie Algebras.

Local properties of a control system $\dot x=f_0(x)+\sum_{j=1}^d u_jf_j(x)$ are completely determined by the structure of the Lie Algebra $\Ggi$ generated by the vector fields $f_0,f_1,\ldots,f_d$. Of particular importance, from applications point of view, are nilpotent systems. This fact motivates the efforts of the research in approximating a general system with a nilpotent one ("nilpotent approximation"). Moreover it is fundamental for an approximating nilpotent system to have a "simple" and "compact" form, i.e. canonical form. In this work we consider the class of abstract nilpotent Lie Algebras freely generated by a set of $d$ elements. We determine those nilpotent Lie Algebras stable under transformation of the generators and build the relative multiplication table. Finally we go back to nilpotent Lie Algebras of vector fields and for them we provide a canonical coordinate representation.

Concentration problems and Gamma-convergence.

Relaxation problems in Sobolev spaces with respect to a measure.

A model for quasi-state growth of Brittle fractures.

A variational approach to the Hele-Shaw with injection.

Fundamental form of rank 2 vector distribution.

Rank k vector distribution D on the manifold M is by definition a k-dimensional subbundle of the tangent bundle TM. In other words, for each point q in M a k-dimensional subspace D(q) of the tangent space T_qM is chosen and D(q) depends smoothly on q. Two vector distributions D_1 and D_2 are called locally equivalent at some point q_0 in M , if there exists a diffeomorphism F of some neighbourhood U of q_0, which transforms distribution D_1 to D_2, i.e., F_*D_1(q)=D_2 (F(q)) for all q in U. The question is when two distributions are locally equivalent? In the present talk we will restrict ourselves to the case k=2. If dim M=3 or 4, all generic germs of rank 2 distributions are equivalent (Darboux's and Engel's theorems). If dim M >= 5, the normal forms of these germs contain (dim M-4) functional parameters. Using the general theory of curves in the Lagrange Grassmannian developed in [1], we will describe the construction of basic invariant of the distribution, fundamental form, which is the obstacle to local equivalence. The case dim M=5 will be discussed in greater details. References [1] A. Agrachev, I. Zelenko, Geometry of Jacobi curves.I, J. Dynamical and Control Systems, 8(1):93-140, 2002

Nearly critical semilinear elliptic problems.

Evolution of partitions.

On the local structure of control functions corresponding to time-optimal trajectories in R^3.

In the talk it is analized the structure of a control function u(t) corresponding to an optimal trajectory for the system \dot{q}=f(q)+ug(q) in R^3 nearby a point where some nondegenericity conditions are satisfied. As we will see the control turns out to be the concatenation of some bang and some singular arcs. Studying the index of the second variation of the switching times, the number of such arcs is bounded by six.

A few issues in the control of quantum mechanical systems.

In this survey talk, I will review the setting commonly used for control of closed quantum mechanical systems driven by external fields (i.e. bilinear control systems on spheres and compact semisimple Lie groups) and present a few of the methods currently in use for control design in this setting.

An existence theory for Equilibrium problems.

Problems of Stochastic geometry in polymer crystallization processes

Classification of singularities of convex envelopes of apparent contours

The apparent contour of a surface S in R^3, is the set of critical values of the projection of S on a plane. The apparent contour is a collection of plane 'fronts', i.e. closed curves with double points and cusps. The convex envelope of an apparent contour is the boundary of its convex hull. We classify all singularities up to codimension 3, i.e. all singularities in generic 3-parametric families of convex envelopes of apparent contours. The study was motivated by phase transitions in thermodynamics but the problem is also intimately related to plane nonlinear optimal control problems.

Drift and diffusion in phase space

On the curvature of Hamiltonian systems and second order ODEs

Complete metrics with prescribed scalar curvature

Extremal and Optimal Controls for Chained Systems

We consider optimal control problems for 2-chained systems - a class of systems which arise naturally in applications (e.g. in robotics). Applying Pontryagin Maximum Principle and some other tools of optimal cntrol theory we study structural properties of extremal and optimal trajectories for these problems.

On the curvature of Hamiltonian systems and second order ODEs

The goal of the talk is to present a construction of the canonical connection and the curvature (generalizing Levi Civita connection and Riemannian sectional curvature) for a wide class of Hamiltonian systems and other vector fields on cotangent and tangent bundles. The construction is purely "dynamical": it involves only the flow generated by the vector field and the fiber bundle structure. The correspondent curvature enjoys properties similar to the Riemannian case, i.e. the case of the geodesic flow on the tangent bundle. In particular, it allows to state very general "Comparizon theorems". No preliminary knowledge in Riemannian geometry is required.

Levi curvature type equations

Semilinear equations on Heisenberg group

On the configuration study of the "falling-cat" problem

In this talk there will be discussed some aspects of so-called "falling cat" pro blem. Let us consider a configurational space of the mechanical system subject to the condition that the angular momentum of the system is zero.The problem is to reach the desired position of the system by changing its inner configuration without violating the condition for the angular momentum. The first part of the talk is concerned to the purely kinematic study of the co nfiguration space of the system consisted of n free particles in R^d. The result will be presented is due to A. Guichardet. In his article [1] he has shown that for any system such that the number of the particles is strictly bigger than the dimension if the space the result of any pure rotation of the initial configuration can be achieved by performing the inner configuration of the particles. For the case n <= d the particles will always remain in some less-dimensional constant subspace of R^d. In the second part of the talk there will be considered the system consisted of four equal masses such that they form a break-line of three equal segments with the fixed length. As before, the total angular momentum of the system is supposed to be zero. There will be discussed the study of the configurational space of this system in R^3. References [1] Guichardet,A. 1984. On rotation and vibration motions of molecules. Ann. Inst. H. Poincare Phys. Theor. 40(3): 329-342.

Linear equations

Preliminaries

Stabilization by Patchy Feedbacks and Robustness Properties

This talk is concerned with the problem of constructing discontinuous stabilizing feedbacks for nonlinear control system, which enjoy robustness properties with respect to external and internal perturbations. We first consider patchy vector fields, a class of discountinuous, piecewise smooth vector fields and we prove the stability of the corresponding solution set with respect to impulsive perturbations. A linear estimate of the effect produced by such perturbations is also established for a generic class of patchy vector fields in the plane, that admit discontinuities across polygonal lines.

The Inverse Problem for the Grad-Shafranov Equation with an Affine Right-hand Side

The Structure of Fibre Bundles in Optimal Chattering Sybtheses

Existence of Nonlinear Boundary Layer Solutions of Boltzmann Equation

Last modified: 18 December 2006

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