We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term Elog E. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalized Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator.

%B Communications in Partial Differential Equations %I Taylor & Francis %V 41 %P 32-50 %G eng %U https://doi.org/10.1080/03605302.2015.1095766 %R 10.1080/03605302.2015.1095766 %0 Journal Article %J Journal of Geometric Analysis %D 2013 %T Lipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces %A Ugo Boscain %A Grégoire Charlot %A Roberta Ghezzi %A Mario Sigalotti %XTwo-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–Carathéodory 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 analyze 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 labeled graph associated with it.

%B Journal of Geometric Analysis %V 23 %P 438–455 %8 Jan %G eng %U https://doi.org/10.1007/s12220-011-9262-4 %R 10.1007/s12220-011-9262-4 %0 Journal Article %D 2013 %T Self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces %A Ugo Boscain %A Dario Prandi %G eng %R 10.1016/j.jde.2015.10.011 %0 Journal Article %J SIAM J. Control Optim., 50 (2012) 559–582 %D 2012 %T On 2-step, corank 2 nilpotent sub-Riemannian metrics %A Davide Barilari %A Ugo Boscain %A Jean-Paul Gauthier %X In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics\\r\\nthat are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a byproduct of this study we get some smoothness properties of the spherical Hausdorff measure in the case of a generic 6 dimensional, 2-step corank 2 sub-Riemannian metric. %B SIAM J. Control Optim., 50 (2012) 559–582 %I Society for Industrial and Applied Mathematics %G en %U http://hdl.handle.net/1963/6065 %1 5950 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2012-08-01T12:38:52Z\\nNo. of bitstreams: 1\\n1105.5766v2.pdf: 271835 bytes, checksum: 0836f63f262f14fbd4d44422b6c85686 (MD5) %R 10.1137/110835700 %0 Journal Article %J Calculus of Variations and Partial Differential Equations. Volume 43, Issue 3-4, March 2012, Pages 355-388 %D 2012 %T On the Hausdorff volume in sub-Riemannian geometry %A Andrei A. Agrachev %A Davide Barilari %A Ugo Boscain %X For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative\r\nof the spherical Hausdorff measure with respect to a smooth volume. We prove\r\nthat this is the volume of the unit ball in the nilpotent approximation and it\r\nis always a continuous function. We then prove that up to dimension 4 it is\r\nsmooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4\r\non every smooth curve) but in general not C^5. These results answer to a\r\nquestion addressed by Montgomery about the relation between two intrinsic\r\nvolumes that can be defined in a sub-Riemannian manifold, namely the Popp and\r\nthe Hausdorff volume. If the nilpotent approximation depends on the point (that\r\nmay happen starting from dimension 5), then they are not proportional, in\r\ngeneral. %B Calculus of Variations and Partial Differential Equations. Volume 43, Issue 3-4, March 2012, Pages 355-388 %I SISSA %G en %U http://hdl.handle.net/1963/6454 %1 6399 %2 Mathematics %4 1 %# MAT/05 ANALISI MATEMATICA %$ Submitted by Andrei Agrachev (agrachev@sissa.it) on 2013-02-05T13:55:36Z\r\nNo. of bitstreams: 1\r\n1005.0540v3.pdf: 352986 bytes, checksum: 7d3e71cad3c7ff917dc8769ba4cd96c5 (MD5) %R 10.1007/s00526-011-0414-y %0 Report %D 2012 %T Introduction to Riemannian and sub-Riemannian geometry %A Andrei A. Agrachev %A Davide Barilari %A Ugo Boscain %I SISSA %G en %U http://hdl.handle.net/1963/5877 %1 5747 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2012-05-30T13:23:26Z\r\nNo. of bitstreams: 1\r\n09M_2012_Barilari.pdf: 1146702 bytes, checksum: 28492a78df0fdf2435dd0606fdfc78f5 (MD5) %0 Journal Article %J Proc. Steklov Inst. Math. 270 (2010) 43-56 %D 2010 %T Existence of planar curves minimizing length and curvature %A Ugo Boscain %A Grégoire Charlot %A Francesco Rossi %X 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.\\nFor 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.\\nWe instead prove existence of minimizers for the \\\"time-reparameterized\\\" functional $$\\\\int \\\\| \\\\dot\\\\gamma(t) \\\\|\\\\sqrt{1+K_\\\\ga^2} dt$$ 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. %B Proc. Steklov Inst. Math. 270 (2010) 43-56 %I Springer %G en_US %U http://hdl.handle.net/1963/4107 %1 297 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-12-06T13:30:01Z\\nNo. of bitstreams: 1\\n0906.5290v2.pdf: 305594 bytes, checksum: 3966f5a798b69743ce88a29bf73edc47 (MD5) %R 10.1134/S0081543810030041 %0 Journal Article %J arXiv preprint arXiv:1008.5036 %D 2010 %T A normal form for generic 2-dimensional almost-Riemannian structures at a tangency point %A Ugo Boscain %A Grégoire Charlot %A Roberta Ghezzi %B arXiv preprint arXiv:1008.5036 %G eng %0 Journal Article %J ESAIM COCV 16 (2010) 275-297 %D 2010 %T Projective Reeds-Shepp car on $S^2$ with quadratic cost %A Ugo Boscain %A Francesco Rossi %X Fix two points $x,\\\\bar{x}\\\\in S^2$ and two directions (without orientation) $\\\\eta,\\\\bar\\\\eta$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $$ J[\\\\gamma]=\\\\int_0^T g_{\\\\gamma(t)}(\\\\dot\\\\gamma(t),\\\\dot\\\\gamma(t))+\\nK^2_{\\\\gamma(t)}g_{\\\\gamma(t)}(\\\\dot\\\\gamma(t),\\\\dot\\\\gamma(t)) ~dt$$ along all smooth curves starting from $x$ with direction $\\\\eta$ and ending in $\\\\bar{x}$ with direction $\\\\bar\\\\eta$. Here $g$ is the standard Riemannian metric on $S^2$ and $K_\\\\gamma$ is the corresponding geodesic curvature.\\nThe 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).\\nWe 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. %B ESAIM COCV 16 (2010) 275-297 %G en_US %U http://hdl.handle.net/1963/2668 %1 1429 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-06-11T11:22:54Z\\nNo. of bitstreams: 1\\n0805.4800v1.pdf: 610220 bytes, checksum: b0fa81a60fc43e6da6a4682e91b4d21e (MD5) %R 10.1051/cocv:2008075 %0 Journal Article %J Ann. Inst. H. Poincare Anal. Non Lineaire %D 2010 %T Two-dimensional almost-Riemannian structures with tangency points %A Andrei A. Agrachev %A Ugo Boscain %A Grégoire Charlot %A Roberta Ghezzi %A Mario Sigalotti %XTwo-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.

%B Ann. Inst. H. Poincare Anal. Non Lineaire %I Elsevier %V 27 %P 793-807 %G en_US %U http://hdl.handle.net/1963/3870 %1 839 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-05-31T14:15:39Z\\nNo. of bitstreams: 1\\n0908.2564v1.pdf: 302590 bytes, checksum: 15369151ee10bb886dc6678350dee7f5 (MD5) %R 10.1016/j.anihpc.2009.11.011 %0 Journal Article %J Ann. Inst. H. Poincare Anal. Non Lineaire 26 (2009) 329-349 %D 2009 %T Controllability of the discrete-spectrum Schrodinger equation driven by an external field %A Thomas Chambrion %A Paolo Mason %A Mario Sigalotti %A Ugo Boscain %X We prove approximate controllability of the bilinear Schrodinger equation in the case in which the uncontrolled Hamiltonian has discrete nonresonant\\nspectrum. 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\\nGalerkin 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 controlled by suitable potentials. %B Ann. Inst. H. Poincare Anal. Non Lineaire 26 (2009) 329-349 %G en_US %U http://hdl.handle.net/1963/2547 %1 1572 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-01-10T13:32:19Z\\nNo. of bitstreams: 1\\n2008-ttinger.pdf: 270196 bytes, checksum: e706dbae08d996576cefe55f53d7284e (MD5) %R 10.1016/j.anihpc.2008.05.001 %0 Journal Article %J J. Funct. Anal. 256 (2009) 2621-2655 %D 2009 %T The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups %A Andrei A. Agrachev %A Ugo Boscain %A Jean-Paul Gauthier %A Francesco Rossi %X 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.\\nWe 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.\\nOur 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. %B J. Funct. Anal. 256 (2009) 2621-2655 %G en_US %U http://hdl.handle.net/1963/2669 %1 1428 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-06-11T11:38:01Z\\nNo. of bitstreams: 1\\n0806.0734v1.pdf: 494960 bytes, checksum: 640ace795ac663f09426814440b15432 (MD5) %R 10.1016/j.jfa.2009.01.006 %0 Journal Article %J Discrete Contin. Dyn. Syst. 20 (2008) 801-822 %D 2008 %T A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds %A Andrei A. Agrachev %A Ugo Boscain %A Mario Sigalotti %X 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. %B Discrete Contin. Dyn. Syst. 20 (2008) 801-822 %G en_US %U http://hdl.handle.net/1963/1869 %1 2353 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-10-11T07:14:06Z\\nNo. of bitstreams: 1\\nmath.oc0609566.pdf: 920147 bytes, checksum: c6f020bc9676ee3966b64f4135e4ce52 (MD5) %R 10.3934/dcds.2008.20.801 %0 Journal Article %J SIAM J. Control Optim. 47 (2008) 1851-1878 %D 2008 %T Invariant Carnot-Caratheodory metrics on S3, SO(3), SL(2) and Lens Spaces %A Ugo Boscain %A Francesco Rossi %X In this paper we study the invariant Carnot-Caratheodory metrics on SU(2) \\\' S3,\\nSO(3) and SL(2) induced by their Cartan decomposition. 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\\ngiven on SU(2) projects on the so called Lens Spaces L(p; q). Also for Lens Spaces, we compute\\nthe cut loci (globally). %B SIAM J. Control Optim. 47 (2008) 1851-1878 %G en_US %U http://hdl.handle.net/1963/2144 %1 2099 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-09-27T08:46:25Z\\nNo. of bitstreams: 1\\nBoscain-Rossi-2007.pdf: 587872 bytes, checksum: 6b4998700cd692ddbd99e14289405bdf (MD5) %R 10.1137/070703727 %0 Journal Article %J SIAM J. Control Optim. 47 (2008) 111-143 %D 2008 %T Limit Time Optimal Syntheses for a control-affine system on S² %A Paolo Mason %A Rebecca Salmoni %A Ugo Boscain %A Yacine Chitour %X For $\\\\alpha \\\\in ]0,\\\\pi/2[$, let $(\\\\Sigma)_\\\\alpha$ 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 and of respective norms $\\\\cos(\\\\alpha)$ and $\\\\sin(\\\\alpha)$. In this paper, we study the time optimal synthesis (TOS) from the north pole $(0,0,1)^T$ associated to $(\\\\Sigma)_\\\\alpha$, as the parameter $\\\\alpha$ tends to zero; this problem is motivated by specific issues in the control of quantum systems. We first prove that the TOS is characterized by a \\\"two-snakes\\\" configuration on the whole $S^2$, except for a neighborhood $U_\\\\alpha$ of the south pole $(0,0,-1)^T$ of diameter at most ${\\\\cal O}(\\\\alpha)$. We next show that, inside $U_\\\\alpha$, the TOS depends on the relationship between $r(\\\\alpha):=\\\\pi/2\\\\alpha-[\\\\pi/2\\\\alpha]$ and $\\\\alpha$. More precisely, we characterize three main relationships by considering sequences $(\\\\alpha_k)_{k\\\\geq 0}$ satisfying (a) $r(\\\\alpha_k)=\\\\bar{r}$, (b) $r(\\\\alpha_k)=C\\\\alpha_k$, and (c) $r(\\\\alpha_k)=0$, 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 $\\\\alpha$ tends to zero, of the corresponding TOS inside $U_\\\\alpha$. %B SIAM J. Control Optim. 47 (2008) 111-143 %G en_US %U http://hdl.handle.net/1963/1862 %1 2360 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-09-28T11:24:18Z\\nNo. of bitstreams: 1\\n48-M.pdf: 5823285 bytes, checksum: 7f66cf9d280da78b56a423c415d9078c (MD5) %R 10.1137/060675988 %0 Journal Article %J Commun. Pure Appl. Anal. 7 (2008) 1-21 %D 2008 %T Stability of planar switched systems: the nondiagonalizable case %A Ugo Boscain %A Moussa Balde %B Commun. Pure Appl. Anal. 7 (2008) 1-21 %G en_US %U http://hdl.handle.net/1963/1857 %1 2361 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-09-28T09:15:19Z\\nNo. of bitstreams: 1\\n44-M.pdf: 3158552 bytes, checksum: c4fee14d81a9fcf026efd4448da88c75 (MD5) %R 10.3934/cpaa.2008.7.1 %0 Journal Article %J Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007) 333-342 %D 2007 %T Gaussian estimates for hypoelliptic operators via optimal control %A Ugo Boscain %A Sergio Polidoro %X We obtain Gaussian lower bounds for the fundamental solution of a class of hypoelliptic equations, by using repeatedly an invariant Harnack inequality. Our main result is given in terms of the value function of a suitable optimal control problem. %B Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007) 333-342 %G en_US %U http://hdl.handle.net/1963/1994 %1 2202 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-08-09T13:14:42Z\\nNo. of bitstreams: 1\\n45-2007M.pdf: 181670 bytes, checksum: 6f262dccd2b3004cfd0b0b711bd011b2 (MD5) %R 10.4171/RLM/499 %0 Report %D 2007 %T High-order angles in almost-Riemannian geometry %A Ugo Boscain %A Mario Sigalotti %X 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 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. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula for domains with piecewise-C2 boundary. The main feature of such formula is the presence of terms that play the role of high-order angles at the intersection points with the set of singularities. %G en_US %U http://hdl.handle.net/1963/1995 %1 2201 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-08-24T12:00:46Z\\nNo. of bitstreams: 1\\nhigh-order.pdf: 168685 bytes, checksum: 11d6f55bf9b07b4da01dae3f3deb969a (MD5) %0 Journal Article %J J. Math. Sci. 135 (2006) 3109-3124 %D 2006 %T Classification of stable time-optimal controls on 2-manifolds %A Ugo Boscain %A Igor Nikolaev %A Benedetto Piccoli %B J. Math. Sci. 135 (2006) 3109-3124 %G en_US %U http://hdl.handle.net/1963/2196 %1 2048 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-11T07:48:50Z\\nNo. of bitstreams: 1\\nboscain-paper.pdf: 322031 bytes, checksum: 560c6fdce3f93d196d1cc9d9c4115edb (MD5) %R 10.1007/s10958-006-0148-0 %0 Journal Article %J SIAM J. Control Optim. 45 (2006) 226-245 %D 2006 %T Common Polynomial Lyapunov Functions for Linear Switched Systems %A Paolo Mason %A Ugo Boscain %A Yacine Chitour %X In this paper, we consider linear switched systems $\\\\dot x(t)=A_{u(t)} x(t)$, $x\\\\in\\\\R^n$, $u\\\\in U$, and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching ({\\\\bf UAS} for short). We first prove that, given a {\\\\bf UAS} system, it is always possible to build a common polynomial Lyapunov function. Then our main result is that the degree of that common polynomial Lyapunov function is not uniformly bounded over all the {\\\\bf UAS} systems. This result answers a question raised by Dayawansa and Martin. A generalization to a class of piecewise-polynomial Lyapunov functions is given. %B SIAM J. Control Optim. 45 (2006) 226-245 %G en_US %U http://hdl.handle.net/1963/2181 %1 2063 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-04T14:36:21Z\\nNo. of bitstreams: 1\\n0403209v2.pdf: 244831 bytes, checksum: 1762d79876f9eb915a68dbc25d9a3a21 (MD5) %R 10.1137/040613147 %0 Report %D 2006 %T Stability of planar nonlinear switched systems %A Ugo Boscain %A Grégoire Charlot %A Mario Sigalotti %X We consider the time-dependent nonlinear system ˙ q(t) = u(t)X(q(t)) + (1 − u(t))Y (q(t)), where q ∈ R2, X and Y are two smooth vector fields, globally asymptotically stable at the origin and u : [0,∞) → {0, 1} is an arbitrary measurable function. Analysing the topology of the set where X and Y are parallel, we give some sufficient and some necessary conditions for global asymptotic stability, uniform with respect to u(.). Such conditions can be verified without any integration or construction of a Lyapunov function, and they are robust under small perturbations of the vector fields. %B Discrete Contin. Dyn. Syst. 15 (2006) 415-432 %G en_US %U http://hdl.handle.net/1963/1710 %1 2441 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-01-18T08:35:31Z\\nNo. of bitstreams: 1\\nmath.OC0502361.pdf: 322404 bytes, checksum: e56f0d709d97e2e300e3cb9d4a629a1b (MD5) %0 Report %D 2006 %T Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic field %A Ugo Boscain %A Paolo Mason %X In this paper we consider the minimum time population transfer problem for the z-component\\nof 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 |omega (t)| ≤ 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 wR = 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 → ∞ 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 → 0, giving a partial proof of a conjecture formulated in a previous paper. %B Journal of Mathematical Physics 47, 062101 (2006) %G en_US %U http://hdl.handle.net/1963/1734 %1 2418 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-02-03T08:14:36Z\\nNo. of bitstreams: 1\\nquant-ph0512074.pdf: 550597 bytes, checksum: 588a3483219fa1360d5f71c131e2488e (MD5) %R 10.1063/1.2203236 %0 Journal Article %J Discrete Contin. Dyn. Syst. Ser. B 5 (2005) 957-990 %D 2005 %T Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy %A Ugo Boscain %A Thomas Chambrion %A Grégoire Charlot %X 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),\\n2) 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. %B Discrete Contin. Dyn. Syst. Ser. B 5 (2005) 957-990 %G en_US %U http://hdl.handle.net/1963/2259 %1 1988 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-18T11:55:27Z\\nNo. of bitstreams: 1\\n0409022v2.pdf: 578605 bytes, checksum: db7298996e781c3a8546c3d01ee28384 (MD5) %0 Book Section %B Contrôle non linéaire et applications: Cours donnés à l\\\'école d\\\'été du Cimpa de l\\\'Université de Tlemcen / Sari Tewfit [ed.]. - Paris: Hermann, 2005 %D 2005 %T A short introduction to optimal control %A Ugo Boscain %A Benedetto Piccoli %B Contrôle non linéaire et applications: Cours donnés à l\\\'école d\\\'été du Cimpa de l\\\'Université de Tlemcen / Sari Tewfit [ed.]. - Paris: Hermann, 2005 %@ 2 7056 6511 0 %G en_US %U http://hdl.handle.net/1963/2257 %1 1990 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-18T11:30:24Z\\nNo. of bitstreams: 1\\nNotes-OptCont.pdf: 442775 bytes, checksum: 196e4d9cc52950dd18cf17feb0e89808 (MD5) %0 Report %D 2005 %T Time minimal trajectories for two-level quantum systems with drift %A Ugo Boscain %A Paolo Mason %X On a two-level quantum system driven by an external field, we consider the population transfer problem from the first to the second level, minimizing the time of transfer, with bounded field 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. %B Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC \\\'05. 44th IEEE Conference on %G en_US %U http://hdl.handle.net/1963/1688 %1 2445 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2005-06-20T12:25:55Z\\nNo. of bitstreams: 1\\nquant-ph0502151.pdf: 136766 bytes, checksum: d23bf76326e4a9a09555f4ead6edc808 (MD5) %0 Journal Article %J SIAM J. Control Optim. 44 (2005) 111-139 %D 2005 %T Time Optimal Synthesis for Left-Invariant Control Systems on SO(3) %A Ugo Boscain %A Yacine Chitour %X 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 $\\\\|g\\\\|=\\\\sin(\\\\al)$, $\\\\al\\\\in ]0,\\\\pi/4[$. In this paper, we provide an upper bound and a lower bound for $N(\\\\alpha)$, 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 $\\\\pi/(4\\\\alpha).$ 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. %B SIAM J. Control Optim. 44 (2005) 111-139 %G en_US %U http://hdl.handle.net/1963/2258 %1 1989 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-18T11:43:01Z\\nNo. of bitstreams: 1\\n0502483v1.pdf: 429552 bytes, checksum: 9f72f53d7031cdc7ccb2aca8b8ec16de (MD5) %R 10.1137/S0363012904441532 %0 Conference Proceedings %B 43rd IEEE Conference on Decision and Control, 2004, 2786 - 2791 Vol.3 %D 2004 %T On the minimal degree of a common Lyapunov function for planar switched systems %A Paolo Mason %A Ugo Boscain %A Yacine Chitour %X In this paper, we consider linear switched systems x(t) = Au(t)x(t), x ε Rn, u ε U, and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching (UAS for short). We first prove that, given a UAS system, it is always possible to build a polynomial common Lyapunov function. Then our main result is that the degree of that the common polynomial Lyapunov function is not uniformly bounded over all the UAS systems. This result answers a question raised by Dayawansa and Martin. %B 43rd IEEE Conference on Decision and Control, 2004, 2786 - 2791 Vol.3 %I IEEE %G en %U http://hdl.handle.net/1963/4834 %1 4611 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-19T07:47:35Z\\nNo. of bitstreams: 0 %R 10.1109/CDC.2004.1428884 %0 Journal Article %J ESAIM COCV 10 (2004) 593-614 %D 2004 %T Resonance of minimizers for n-level quantum systems with an arbitrary cost %A Ugo Boscain %A Grégoire Charlot %X We consider an optimal control problem describing a laser-induced population transfer on a $n$-level quantum system.\\nFor a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for $n=2$ and $n=3$): instead of looking for minimizers on the sphere $S^{2n-1}\\\\subset\\\\C^n$ one is reduced to look just for minimizers on the sphere $S^{n-1}\\\\subset \\\\R^n$. Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer. %B ESAIM COCV 10 (2004) 593-614 %I EDP Sciences %G en_US %U http://hdl.handle.net/1963/2910 %1 1790 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-11T12:22:45Z\\nNo. of bitstreams: 1\\n0308103v2.pdf: 290972 bytes, checksum: 2195a0a0002da9f91cbc9fff24262981 (MD5) %R 10.1051/cocv:2004022 %0 Journal Article %J J.Dynam. Control Systems 8 (2002),no.4, 547 %D 2002 %T On the K+P problem for a three-level quantum system: optimality implies resonance %A Ugo Boscain %A Thomas Chambrion %A Jean-Paul Gauthier %B J.Dynam. Control Systems 8 (2002),no.4, 547 %I SISSA Library %G en %U http://hdl.handle.net/1963/1601 %1 2517 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T13:05:08Z (GMT). No. of bitstreams: 1\\nmath.OC0204233.pdf: 252630 bytes, checksum: 14283368a7848e46caf6447b4bad85d4 (MD5)\\n Previous issue date: 2002 %R 10.1023/A:1020767419671 %0 Journal Article %J SIAM J. Control Optim. 41 (2002), no. 1, 89-112 %D 2002 %T Stability of planar switched systems: the linear single input case %A Ugo Boscain %X We study the stability of the origin for the dynamical system $\\\\dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t),$ where A and B are two 2 × 2 real matrices with eigenvalues having strictly negative real part, $x\\\\in {\\\\mbox{{\\\\bf R}}}^2$, and $u(.):[0,\\\\infty[\\\\to[0,1]$ is a completely random measurable function. More precisely, we find a (coordinates invariant) necessary and sufficient condition on A and B for the origin to be asymptotically stable for each function u(.). The result is obtained without looking for a common Lyapunov function but studying the locus in which the two vector fields Ax and Bx are collinear. There are only three relevant parameters: the first depends only on the eigenvalues of A, the second depends only on the eigenvalues of B, and the third contains the interrelation among the two systems, and it is the cross ratio of the four eigenvectors of A and B in the projective line CP1. In the space of these parameters, the shape and the convexity of the region in which there is stability are studied. %B SIAM J. Control Optim. 41 (2002), no. 1, 89-112 %I SIAM %G en %U http://hdl.handle.net/1963/1529 %1 2634 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T13:03:40Z (GMT). No. of bitstreams: 0\\n Previous issue date: 2000 %R 10.1137/S0363012900382837 %0 Journal Article %J J. Dynam. Control Systems, 2001, 7, 209 %D 2001 %T Extremal synthesis for generic planar systems %A Ugo Boscain %A Benedetto Piccoli %B J. Dynam. Control Systems, 2001, 7, 209 %I SISSA Library %G en %U http://hdl.handle.net/1963/1503 %1 2660 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T13:03:19Z (GMT). No. of bitstreams: 0\\n Previous issue date: 2000 %R 10.1023/A:1013003204923 %0 Journal Article %J J. Dynam. Control Systems 7 (2001), no. 3, 385--423 %D 2001 %T Morse properties for the minimum time function on 2-D manifolds %A Ugo Boscain %A Benedetto Piccoli %B J. Dynam. Control Systems 7 (2001), no. 3, 385--423 %I SISSA Library %G en %U http://hdl.handle.net/1963/1541 %1 2622 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T13:03:51Z (GMT). No. of bitstreams: 0\\n Previous issue date: 2000 %R 10.1023/A:1013190914234 %0 Journal Article %D 2000 %T Abnormal extremals for minimum time on the plane %A Ugo Boscain %A Benedetto Piccoli %I SISSA Library %G en %U http://hdl.handle.net/1963/1508 %1 2655 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T13:03:23Z (GMT). No. of bitstreams: 0\\n Previous issue date: 2000 %0 Journal Article %D 1999 %T Projection singularities of extremals for planar systems %A Ugo Boscain %A Benedetto Piccoli %I SISSA Library %G en %U http://hdl.handle.net/1963/1304 %1 3151 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:56:02Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1999 %0 Journal Article %J Rend. Sem. Mat. Univ. Politec. Torino 56 (1998), no. 4, 53-68 (2001) %D 1998 %T Geometric control approach to synthesis theory %A Ugo Boscain %A Benedetto Piccoli %B Rend. Sem. Mat. Univ. Politec. Torino 56 (1998), no. 4, 53-68 (2001) %I SISSA Library %G en %U http://hdl.handle.net/1963/1277 %1 3178 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:55:40Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1999