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Seminars 2000-2001

Instabilty in the restricted elliptic three body problem

The Yamabe Conjecture on CR Manifolds: the Remaining case

An extension of the Mumford and Nitzberg's variational model for image segmentation

3D-2D Asymptotyc Analysis for Micromagnetic Thin Films

Esistenza e Molteplicita' di soluzioni periodiche: metodi ed applicazioni

Analytic Properties of Sub-Riemannian Distances

Geometric Domain Variation for Neumann Problems

Strong optimality for a bang-bang trajectory

We give sufficient conditions for a bang-bang extremal to be a strong local optimum for a control problem in the Mayer form, strong means that we consider the $C^0$ topology in the state space. The controls appear linearly and take values in a polyhedron, the state space and the end points constraints are finite dimensional smooth manifolds. In the case of bang-bang extremals, the kernel of the first variation of the problem is trivial and hence the usual second variation, which is defined on the kernel of the first one, does not exist. We consider the finite dimensional subproblem generated by perturbing the switching times and we prove that the sufficient second order optimality conditions for this finite dimensional subproblem yield local strong optimality.

Optimal Control on a 3--Level Schrodinger Equation

In this talk we present how techniques of ``subriemannian geometry'' and ``singular Riemannian geometry'' can be applied to some classical problems of quantum mechanics and permit to improve some results. We study a 3--levels quantum system controlled by two laser pulses in resonance i.e. with frequencies $\omega_1=E_2-E_1,~\omega_2=E_3-E_2$, being $E_1,E_2,E_3$ the three energy levels ($\hbar=1$). Our aim is to transfer all the population from the state with energy $E_1$ to the state with energy $E_3$ minimizing the amplitude of the lasers pulses. Notice that we do not dispose of a laser with frequency $E_3-E_1$. This problem is described by a Schrodinger equation in $C^3$. We show that making a suitable unitary transformation depending on the time, this problem can be stated as a subriemannian problem on $S^5$. Then we prove that the orbit trough our initial point is a two dimensional sphere. Our problem is then reduced to the problem of finding the geodesics on the sphere, for a singular Riemannian metric. We compute the optimal synthesis and in particular the trajectory that reaches our final state. We would like to stress the fact that, since the optimal solution is invariant for time reparameterization, one can use very smooth optimal controls to go from the initial to the final state. This fact is important because smooth controls are the ones that can be realized in practice. A possible choice for the controls is presented. Finally we discuss the geometry of the manifold that one obtain by gluing all the spheres corresponding to all possible choices of (equivalent) initial conditions. Results for two level systems are also presented: this problem is simpler but, contrarily, to the 3--level system, leads to a problem in contact isoperimetric subriemannian geometry (and not singular Riemannian geometry). Moreover the ``geometric situation'' in this 2--level system is extremely nice.

Optimal and extremal minimum time synthesis in dimension two

For control systems of kind dx/dt = F(x)+ u G(x), x on a two dimensional manifold and |u| <= 1, we study the optimal synthesis and the set of extremals. A complete classification of optimal and extremal synthesis is provided, together with classifications of singularities.

About the Arnold's conjecture on generic instability of near integrable Hamiltonian systems.

Periodic Orbits for Charges on Magnetic Fields

We will discuss the problem of finding periodic orbits of charged particles under the action of an everywhere normal magnetic field on a Riemannian Surface. We will review recent results and present some new results on the limiting case of a large charge where no averaging methods are used. In particular we will demonstrate the existence of sufficient conditions on a level set of the magnetic field that ensure the existence of invariant tori bounding the motion of the particle for all times. Relations with Sub-Riemannian geometry and optimal control theory will be pointed out.

Rigidity of nonregular abnormal extremals of 2-distribution

The talk is devoted to the nonregular abnormal extremals of completely nonholonomic 2-distribution D on n-dimensional manifold M. Nonregularity means that such extremals do not satisfy the Strong Generalized Legendre-Clebsch condition. First, we discuss the question of their existence by analysis of stationary points of some special characteristic vector field on the annihilator (D^2)^\perp\subset T^*M of D^2. Secondly, we consider the question of their rigidity (i.e., W_\infty^1-isolatedness). Introducing the notion of diagonal form of the second variation, we generalize some results of [1] about rigidity of regular abnormal extremals to the nonregular case. In order to reduce the second variation to the diagonal form, we construct a special curve of Lagrangian subspaces, a Jacobi curve. We show that certain geometric properties of this curve (like simplicity) imply the rigidity of the corresponding abnormal extremal. In particular we prove a local rigidity of the nonregular extremals, i.e., the rigidity of their restriction to the sufficiently small time interval. The results of the talk can be found in [2]. [1] A.A. Agrachev, A.V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity, Ann.Inst.Henri Poincare, Analyse non lineaire, vol. 13, n. 6, 1996, p. 635-690. [2] I. Zelenko, Nonregular abnormal extremals of 2-distribution: existence, second variation and rigidity, J. Dynamical and Control Systems, 5 (1999), No. 3, 347-383.

Non-Lipschitzian Minimizers and the Lavrentiev Phenomenon in Variational Problems

We study regularity properties of minimizers for various problems of the Calculus of Variations such as Basic Problem ${\cal J}(x)= \int_{0}^{1}L(t,x(t),\dot{x}(t))dt\rightarrow \inf $, the problem with high-order derivatives and the Lagrange problem of optimal control.

On the controllability and trajectories generation of rolling surfaces

We address the issues of controllability and motion planning for the control system ${\cal {S}}_R$ that results from the rolling without slipping nor spinning of a two dimensional Riemmanian manifold $M_1$ onto another one $M_2$. In a first part, we study ${\cal {S}}_R$ and its Lie algebraic structure. This leads to a recovery of the result of Agrachev and Sachkov, who provided a necessary and sufficient condition on the manifolds for complete controllability of ${\cal {S}}_R$. In a second part, we present two procedures to tackle the motion planning problem when $M_1$ is a plane and $M_2$ a convex surface. The first approach is based on differential algebra. The second technic consists of the use of a homotopy continuation method. Finally we conclude by remarks regarding the numerical implementation of the continuation method.

Global bifurcation for quasilinear ellpitic equations on Rn

Soluzioni omocline per l'equazione di Sine-Gordon

Caustic for Sub-Riemannian (3,6)-problem

We study the properties of a control system \dot x=u, x,u \in E^n \dot y=x\wedge u, y\in E^n\wedge E^n, \int u^2dt \rightarrow \min x(0)=0, y(0)=0. For $n=3$ we describe the caustic of the corresponding Lagrangian map -- the set where the extremals (in the sense of Pontriagin maximum principle) loose their optimality. We also formulate a hypothesis on the form of the first caustic for an arbitrary $n$.

On the nilpotent sub-Riemannian (2,3,5) problem

We consider the nilpotent left-invariant sub-Riemannian problem with the growth vector (2, 3, 5), i.e., the following optimal control problem: dq/dt = u_1 X_1(q) + u_2 X_2(q), u_1, u_2 \in R, q \in G, q(0) = q_0, q(T) = q_1, l(q(.)) = \int_0^T \sqrt{u_1^2 + u_2^2} dt -> min, where G is the connected, simply connected 5-dimensional nilpotent Lie group with the Lie algebra spanned by the vector fields X_1, X_2, X_3 = [X_1, X_2], X_4 = [X_1, X_3], X_5 = [X_2, X_3]. This problem is a local nilpotent approximation to an arbitrary sub-Riemannian problem with the growth vector (2, 3, 5), e.g. to the following ones: (1) a pair of bodies rolling one on another without slipping and twisting, (2) a car with 2 off-hooked trailers. A characteristic feature of the problem is the presence of abnormal minimizers. For the nilpotent problem, we study geodesics and the exponential mapping. In the neighborhood of abnormal extremals, we analyze the conjugate locus. We state and discuss a recent conjecture on the structure of the cut locus.

Semiclassical stationary states of Nonlinear Schroedinger equations with elelectromagnetic field.

Geometry of Jacobi Curves

Jacobi curves are far going generalizations of the spaces of ``Jacobi fields'' along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians (the set of all Lagrangian subspaces of some symplectic space). Differential geometry of these curves provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. In the present talk we would like to discuss two principal invariants: the generalized Ricci curvature, which is an invariant of the parametrized curve in Lagrange Grassamnnian providing the curve with a natural projective structure, and a fundamental form, which is a well-defined 4-order differential on the curve. The arguments will be given that this fundamental form has to play very important role in the equivalence problem of the vector distributions. We also want to discuss the question, what are flat curves in the Lagrange Grassmannian. This question leads to a very interesting conjecture that we succeeded to prove in some particular cases. The talk is based on the joint work with Prof. Andrei Agrachev.

Constructive Necessary and Sufficient Conditions for Strict Triangularizability of Control Lie Algebras

We will analyse the problem of representing a Lie Algebra G of smooth vector fields on a manifold M in strictly triangular form. Necessary and sufficient (checkable) conditions are given in a coordinate free setting. These conditions, if satisfied, lead to a constructive procedure for building the reparametrization of the manifold which transforms the vector fields of G in a strictly triangular form. If we consider a control system an the Lie Algebra generated by the control vector fields, above conditions are the conditions for which the control system is state equivalent to a strictly triangular form system. The importance of this result for the integration and steering of control systems will be highlighted.

Towards the Sub-Riemannian Geometry in the Large

Sub-Riemannian or Carnot--Caratheodory distance between two points of the Riemannian manifold is the infimum of the lengths of the paths in the manifold connecting the points and satisfying certain nonholonomic constraints. In this talk we deal with a three-dimensional manifold endowed with an oriented contact structure. Admissible paths are integral curves of the contact structure (i.e. the "Legendrian curves"); the sub-Riemannian distance is the infimum of the lengths of the admissible paths connecting given points. We give a sufficient condition for a complete sub-Riemannian manifold to be compact and present an upper bound for its diameter explicitely expressed via fundamental differential invariants of the sub-Riemannian metric. We also present an integral formula for the Euler class of the contact structure: a sub-Riemannian analog of the Gauss--Bonnet formula. No preliminary knowledge on sub-Riemannian geometry is assumed.

Planning and control of general nonholonomic systems via nilpotent approximations

In this talk we will discuss the problem of motion planning and control for those nonholonomic systems which are not exactly nilpotentizable nor flat (general nonholonomic systems). In this case, recently developed steering and stabilizing techniques based on the aforesaid properties are not useful. The synthesis of approximate steering techniques is a first step towards the solution of both the steering and the stabilization problem for general nonholonomic systems. To this end, it is possible to use nilpotent approximations, which retain the controllability properties of the original system and at the same time allow the computation of steering controls in view of their triangular and polynomial structure. If the approximation is accurate, such approach may already represent a practical solution to the steering problem. More interestingly, under suitable assumptions, feedback stabilization to a point can be obtained by iterated application of approximate steering controls. One difficulty with the above stabilization paradigm is that nilpotent approximations are intrinsically local: not only the values of the approximation vector fields depend on the point, but even their structure will be altered if the growth vector changes around the point (singular points). We will propose a method for dealing with both regular and singular points in a unified framework, obtaining a nonhomogeneous nilpotent approximation of global validity. We will also show how this nonhomogeneous approximation induces a uniform estimation of the control distance. Other issues deserving further investigation will be discussed; among these, we mention the possibility of computing effective bounds for the approximation error as well the extension to nonholonomic systems with drift. Throughout the discussion, several physical systems will be considered as case studies.

Two isssues in thin films analysis: Scaling and debonding

Invariants of distributions and abnormal curves

We shall discuss geometry of tangent distributions defined as subbundles of the tangent bundle, $D\subset TM$. If $D=span\{X_1,\dots,X_k\}$, with $X_1,\dots,X_k$ smooth vector fields on a manifold $M$, the distribution $D$ defines (uniquely, up to feedback equivalence) a control system $$ \dot x=\sum_i u_i X_i(x).$$ It turns out that among invariants of distributions a special role is played by special trajectories of the above control system called abnormal curves, characteristic curves or singular curves (special curves satisfying the Pontryagin maximum principle). We will discuss, as one of the problems, the problem when the characteristic curves form a complete set of invariants of a distribution.

An introduction to Paneitz type operators

To the uniqueness problem for nonlinear elliptic and parabolic equations

Homogenization of Elasticity problems on singular structures

Regularity results under nonstandard grouth conditions

General double porosity model

Quasiconformal mappings and composites

On multibump solutions for a Shroedinger equation

On a fourth order equation involving Sobolev exponent

Last modified: 18 December 2006

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