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Ugo Boscain

  1. Adiabatic control of the Schr\"odinger equation via conical intersections of the eigenvalues.

    Wed, 02/16/2011 - 16:01 — SomNambul
    Authors: Ugo Boscain, Mario Sigalotti, Paolo Mason, Francesca Chittaro
    Subjects: Optimization and Control
    Abstract

    In this paper we present a constructive method to control the bilinear
    Schr\"odinger equation via two controls. The method is based on adiabatic
    techniques and works if the spectrum of the Hamiltonian admits eigenvalue
    intersections, and if the latter are conical (as it happens generically). We
    provide sharp estimates of the relation between the error and the
    controllability time.

  2. Lipschitz classification of almost-Riemannian distances on compact oriented surfaces.

    Fri, 03/26/2010 - 16:01 — SomNambul
    Authors: Ugo Boscain, Grégoire Charlot, Roberta Ghezzi, Mario Sigalotti
    Subjects: Optimization and Control
    Abstract

    Two-dimensional almost-Riemannian structures are generalized Riemannian
    structures on surfaces for which a local orthonormal frame is given by a Lie
    bracket generating pair of vector fields that can become collinear. We consider
    the Carnot--Caratheodory distance canonically associated with an
    almost-Riemannian structure and study the problem of Lipschitz equivalence
    between two such distances on the same compact oriented surface.

  3. Existence of planar curves minimizing length and curvature.

    Wed, 02/24/2010 - 16:01 — SomNambul
    Authors: Ugo Boscain, Grégoire Charlot, Francesco Rossi
    Subjects: Differential Geometry
    Abstract

    In this paper we consider the problem of reconstructing a curve that is
    partially hidden or corrupted by minimizing the functional $\int
    \sqrt{1+K_\gamma^2} ds$, depending both on length and curvature $K$. We fix
    starting and ending points as well as initial and final directions.

    For this functional we discuss the problem of existence of minimizers on
    various functional spaces. We find non-existence of minimizers in cases in
    which initial and final directions are considered with orientation. In this
    case, minimizing sequences of trajectories can converge to curves with angles.

  4. Hypoelliptic heat kernel on 3-step nilpotent Lie groups.

    Thu, 02/04/2010 - 16:01 — SomNambul
    Authors: Ugo Boscain, Jean-Paul Gauthier, Francesco Rossi
    Subjects: Analysis of PDEs
    Abstract

    In this paper we provide explicitly the connection between the hypoelliptic
    heat kernel for some 3-step sub-Riemannian manifolds and the quartic
    oscillator. We study the left-invariant sub-Riemannian structure on two
    nilpotent Lie groups, namely the (2,3,4) group (called the Engel group) and the
    (2,3,5) group (called the Cartan group or the generalized Dido problem). Our
    main technique is noncommutative Fourier analysis that permits to transform the
    hypoelliptic heat equation in a one dimensional heat equation with a quartic
    potential.

  5. Two-Dimensional Almost-Riemannian Structures with Tangency Points.

    Wed, 08/19/2009 - 19:30 — SomNambul
    Authors: Andrei Agrachev, Ugo Boscain, Grégoire Charlot, Roberta Ghezzi, Mario Sigalotti
    Subjects: Differential Geometry
    Abstract

    Two-dimensional almost-Riemannian structures are generalized Riemannian
    structures on surfaces for which a local orthonormal frame is given by a Lie
    bracket generating pair of vector ?elds that can become collinear. We study the
    relation between the topological invariants of an almost-Riemannian structure
    on a compact oriented surface and the rank-two vector bundle over the surface
    which de?nes the structure. We analyse the generic case including the presence
    of tangency points, i.e. points where two generators of the distribution and
    their Lie bracket are linearly dependent.

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