arXiv subjects

Ugo Boscain

Publications and source records attributed to Ugo Boscain.

At least 73 records · Page 4Linked to original sources

arXiv2011

The Laplace-Beltrami operator in almost-Riemannian Geometry

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. Generically, the singular set is an embedded one dimensional manifold and there are three type of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. Generically tangency points are isolated. In this paper we study the Laplace-Beltrami operator on such a structure. In the case of a compact orientable surface without tangency points, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence a quantum particle in such a structure cannot cross the singular set and the heat cannot flow through the singularity. This is an interesting phenomenon since when approaching the singular set (i.e. where the vector fields become collinear), all Riemannian quantities explode, but geodesics are still well defined and can cross the singular set without singularities. This phenomenon appears also in sub-Riemannian structure which are not equiregular i.e. in which the grow vector depends on the point. We show this fact by analyzing the Martinet case.

math.SP↗

arXiv2011

On the Hausdorff volume in sub-Riemannian geometry

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4 on every smooth curve) but in general not C^5. These results answer to a question addressed by Montgomery about the relation between two intrinsic volumes that can be defined in a sub-Riemannian manifold, namely the Popp and the Hausdorff volume. If the nilpotent approximation depends on the point (that may happen starting from dimension 5), then they are not proportional, in general.

math.DG↗

arXiv2011

Adiabatic control of the Schrödinger equation via conical intersections of the eigenvalues

In this paper we present a constructive method to control the bilinear Schrödinger 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.

math.OC↗

arXiv2010

Lipschitz classification of almost-Riemannian distances on compact oriented surfaces

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. We analyse the generic case, allowing in particular for the presence of tangency points, i.e., points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a characterization of the Lipschitz equivalence class of an almost-Riemannian distance in terms of a labelled graph associated with it.

math.OC↗

arXiv2010

Existence of planar curves minimizing length and curvature

In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $\int \sqrt{1+K_γ^2} ds$, 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. We instead prove existence of minimizers for the "time-reparameterized" functional $$\int \| \dotγ(t) \|\sqrt{1+K_\ga^2} dt$$ for all boundary conditions if initial and final directions are considered regardless to orientation. In this case, minimizers can present cusps (at most two) but not angles.

math.DG↗

arXiv2010

Hypoelliptic heat kernel on 3-step nilpotent Lie groups

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.

math.AP↗

arXiv2009

Two-Dimensional Almost-Riemannian Structures with Tangency Points

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 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 defines 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. The main result of the paper provides a classification of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss?Bonnet formula for almost-Riemannian structures with tangency points.

math.DG↗

arXiv2008

The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp's volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares. We then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane. Our study is motivated by some recent results about the cut and conjugate loci on these sub-Riemannian manifolds. The perspective is to understand how singularities of the sub-Riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation.

math.AP↗

arXiv2008

A note on stability conditions for planar switched systems

This paper is concerned with the stability problem for the planar linear switched system $\dot x(t)=u(t)A_1x(t)+(1-u(t))A_2x(t)$, where the real matrices $A_1,A_2\in \R^{2\times 2}$ are Hurwitz and $u(\cdot) [0,\infty[\to\{0,1\}$ is a measurable function. We give coordinate-invariant necessary and sufficient conditions on $A_1$ and $A_2$ under which the system is asymptotically stable for arbitrary switching functions $u(\cdot)$. The new conditions unify those given in previous papers and are simpler to be verified since we are reduced to study 4 cases instead of 20. Most of the cases are analyzed in terms of the function $Γ(A_1,A_2)={1/2}(\tr(A_1) \tr(A_2)- \tr(A_1A_2))$.

math.OC↗

arXiv2008

Controllability of the discrete-spectrum Schrodinger equation driven by an external field

We prove approximate controllability of the bilinear Schrödinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the Galerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential, both controlled by suitable potentials.

math.OC↗

arXiv2008

Projective Reeds-Shepp car on $S^2$ with quadratic cost

Fix two points $x,\bar{x}\in S^2$ and two directions (without orientation) $η,\barη$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $$ J[γ]=\int_0^T g_{γ(t)}(\dotγ(t),\dotγ(t))+ K^2_{γ(t)}g_{γ(t)}(\dotγ(t),\dotγ(t)) ~dt$$ along all smooth curves starting from $x$ with direction $η$ and ending in $\bar{x}$ with direction $\barη$. Here $g$ is the standard Riemannian metric on $S^2$ and $K_γ$ is the corresponding geodesic curvature. The interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-Riemannian problem on the lens space L(4,1). We compute the global solution for this problem: an interesting feature is that some optimal geodesics present cusps. The cut locus is a stratification with non trivial topology.

math.OC↗

arXiv2008

Invariant Carnot-Caratheodory metrics on $S^3$, $SO(3)$, $SL(2)$ and lens spaces

In this paper we study the invariant Carnot-Caratheodory metrics on $SU(2)\simeq S^3$, $SO(3)$ and $SL(2)$ induced by their Cartan decomposition and by the Killing form. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory distance as the inverse of an elementary function. We then prove that the metric given on $SU(2)$ projects on the so called lens spaces $L(p,q)$. Also for lens spaces, we compute the cut loci (globally). For $SU(2)$ the cut locus is a maximal circle without one point. In all other cases the cut locus is a stratified set. To our knowledge, this is the first explicit computation of the whole cut locus in sub-Riemannian geometry, except for the trivial case of the Heisenberg group.

math.DG↗

arXiv2006

Limit Time Optimal Synthesis for a Control-Affine System on $S^2$

For $α\in(0,π/2)$, let $(Σ)_α$ be the control system $\dot{x}=(F+uG)x$, where $x$ belongs to the two-dimensional unit sphere $S^2$, $u\in [-1,1]$ and $F,G$ are $3\times3$ skew-symmetric matrices generating rotations with perpendicular axes of respective length $\cos(α)$ and $\sin(α)$. In this paper, we study the time optimal synthesis (TOS) from the north pole $(0,0,1)^T$ associated to $(Σ)_α$, as the parameter $α$ tends to zero. We first prove that the TOS is characterized by a ``two-snakes'' configuration on the whole $S^2$, except for a neighborhood $U_α$ of the south pole $(0,0,-1)^T$ of diameter at most $Ø(α)$. We next show that, inside $U_α$, the TOS depends on the relationship between $r(α):=π/2α-[π/2α]$ and$α$. More precisely, we characterize three main relationships, by considering sequences $(α_k)_{k\geq 0}$ satisfying $(a)$$r(α_k)=\bar{r}$; $(b)$ $r(α_k)=Cα_k$ and $(c)$ $r(α_k)=0$, where $\bar{r}\in (0,1)$ and $C>0$. In each case, we describe the TOS and provide, after a suitable rescaling, the limiting behavior, as $α$ tends to zero, of the corresponding TOS inside $U_α$.

math.OC↗

arXiv2006

A Gauss-Bonnet-like Formula on Two-Dimensional Almost-Riemannian Manifolds

We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and $Y$ are everywhere linearly independent, then they define a classical Riemannian metric on $M$ (the metric for which they are orthonormal) and they give to $M$ the structure of metric space. If $X$ and $Y$ become linearly dependent somewhere on $M$, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way are called almost-Riemannian structures. They are special cases of rank-varying sub-Riemannian structures, which are naturally defined in terms of submodules of the space of smooth vector fields on $M$. Almost-Riemannian structures show interesting phenomena, in particular for what concerns the relation between curvature, presence of conjugate points, and topology of the manifold. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula.

math.OC↗

arXiv2006

Stability of Planar Switched Systems: the Nondiagonalizable Case

Consider the planar linear switched system $\dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t),$ where $A$ and $B$ are two $2\times2$ real matrices, $x \in \R^2$, and $u(.):[0,\infty[\to\{0,1\}$ is a measurable function. In this paper we consider the problem of finding a (coordinate-invariant) necessary and sufficient condition on $A$ and $B$ under which the system is asymptotically stable for arbitrary switching functions $u(.)$. This problem was solved in previous works under the assumption that both $A$ and $B$ are diagonalizable. In this paper we conclude this study, by providing a necessary and sufficient condition for asymptotic stability in the case in which $A$ and/or $B$ are not diagonalizable. To this purpose we build suitable normal forms for $A$ and $B$ containing coordinate invariant parameters. A necessary and sufficient condition is then found without looking for a common Lyapunov function but using "worst-trajectory'' type arguments.

math.OC↗

arXiv2005

Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic Field

In this paper we consider the minimum time population transfer problem for the $z$-component of the spin of a (spin 1/2) particle driven by a magnetic field, controlled along the x axis, with bounded amplitude. On the Bloch sphere (i.e. after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on 2-D manifolds. Let $(-E,E)$ be the two energy levels, and $|Ω(t)|\leq M$ the bound on the field amplitude. For each couple of values $E$ and $M$, we determine the time optimal synthesis starting from the level $-E$ and we provide the explicit expression of the time optimal trajectories steering the state one to the state two, in terms of a parameter that can be computed solving numerically a suitable equation. For $M/E<<1$, every time optimal trajectory is bang-bang and in particular the corresponding control is periodic with frequency of the order of the resonance frequency $ω_R=2E$. On the other side, for $M/E>1$, the time optimal trajectory steering the state one to the state two is bang-bang with exactly one switching. Fixed $E$ we also prove that for $M\to\infty$ the time needed to reach the state two tends to zero. In the case $M/E>1$ there are time optimal trajectories containing a singular arc. Finally we compare these results with some known results of Khaneja, Brockett and Glaser and with those obtained by controlling the magnetic field both on the $x$ and $y$ directions (or with one external field, but in the rotating wave approximation). As byproduct we prove that the qualitative shape of the time optimal synthesis presents different patterns, that cyclically alternate as $M/E\to0$, giving a partial proof of a conjecture formulated in a previous paper.

quant-ph↗

arXiv2005

Nonisotropic 3-level Quantum Systems: Complete Solutions for Minimum Time and Minimum Energy

We apply techniques of subriemannian geometry on Lie groups and of optimal synthesis on 2-D manifolds to the population transfer problem in a three-level quantum system driven by two laser pulses, of arbitrary shape and frequency. In the rotating wave approximation, we consider a nonisotropic model i.e. a model in which the two coupling constants of the lasers are different. The aim is to induce transitions from the first to the third level, minimizing 1) the time of the transition (with bounded laser amplitudes), 2) the energy of lasers (with fixed final time). After reducing the problem to real variables, for the purpose 1) we develop a theory of time optimal syntheses for distributional problem on 2-D-manifolds, while for the purpose 2) we use techniques of subriemannian geometry on 3-D Lie groups. The complete optimal syntheses are computed.

quant-ph↗

arXiv2005

Time Optimal Synthesis for Left--Invariant Control Systems on SO(3)

Consider the control system given by $\dot x=x(f+ug)$, where $x\in SO(3)$, $|u|\leq 1$ and $f,g\in so(3)$ define two perpendicular left-invariant vector fields normalized so that $\|f\|=\cos(\al)$ and $\|g\|=\sin(\al)$, $\al\in ]0,π/4[$. In this paper, we provide an upper bound and a lower bound for $N(α)$, the maximum number of switchings for time-optimal trajectories. More precisely, we show that $N_S(\al)\leq N(\al)\leq N_S(\al)+4$, where $N_S(\al)$ is a suitable integer function of $\al$ which for $\al\to 0$ is of order $π/(4α).$ The result is obtained by studying the time optimal synthesis of a projected control problem on $R P^2$, where the projection is defined by an appropriate Hopf fibration. Finally, we study the projected control problem on the unit sphere $S^2$. It exhibits interesting features which will be partly rigorously derived and partially described by numerical simulations.

math.OC↗