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.

1 aBoscain, Ugo1 aPrandi, Dario1 aSeri, M. uhttps://doi.org/10.1080/03605302.2015.109576601360nas a2200181 4500008004100000022001400041245008900055210006900144260000800213300001400221490000700235520080900242100001701051700002301068700002001091700002101111856004601132 2013 eng d a1559-002X00aLipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces0 aLipschitz Classification of AlmostRiemannian Distances on Compac cJan a438–4550 v233 aTwo-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.

1 aBoscain, Ugo1 aCharlot, Grégoire1 aGhezzi, Roberta1 aSigalotti, Mario uhttps://doi.org/10.1007/s12220-011-9262-400494nas a2200097 4500008004100000245012100041210006900162100001700231700001800248856013000266 2013 eng d00aSelf-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces0 aSelfadjoint extensions and stochastic completeness of the Laplac1 aBoscain, Ugo1 aPrandi, Dario uhttps://www.math.sissa.it/publication/self-adjoint-extensions-and-stochastic-completeness-laplace-beltrami-operator-conic-and00934nas a2200133 4500008004100000245005700041210005100098260005100149520050200200100002100702700001700723700002400740856003600764 2012 en d00aOn 2-step, corank 2 nilpotent sub-Riemannian metrics0 a2step corank 2 nilpotent subRiemannian metrics bSociety for Industrial and Applied Mathematics3 aIn 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.1 aBarilari, Davide1 aBoscain, Ugo1 aGauthier, Jean-Paul uhttp://hdl.handle.net/1963/606501194nas a2200133 4500008004100000245005500041210004700096260001000143520080800153100002500961700002100986700001701007856003601024 2012 en d00aOn the Hausdorff volume in sub-Riemannian geometry0 aHausdorff volume in subRiemannian geometry bSISSA3 aFor 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.1 aAgrachev, Andrei, A.1 aBarilari, Davide1 aBoscain, Ugo uhttp://hdl.handle.net/1963/645400389nas a2200121 4500008004100000245005900041210005800100260001000158100002500168700002100193700001700214856003600231 2012 en d00aIntroduction to Riemannian and sub-Riemannian geometry0 aIntroduction to Riemannian and subRiemannian geometry bSISSA1 aAgrachev, Andrei, A.1 aBarilari, Davide1 aBoscain, Ugo uhttp://hdl.handle.net/1963/587701347nas a2200133 4500008004300000245006300043210006300106260001300169520093400182100001701116700002301133700002101156856003601177 2010 en_Ud 00aExistence of planar curves minimizing length and curvature0 aExistence of planar curves minimizing length and curvature bSpringer3 aIn 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.1 aBoscain, Ugo1 aCharlot, Grégoire1 aRossi, Francesco uhttp://hdl.handle.net/1963/410700493nas a2200109 4500008004100000245009300041210006900134100001700203700002300220700002000243856012000263 2010 eng d00aA normal form for generic 2-dimensional almost-Riemannian structures at a tangency point0 anormal form for generic 2dimensional almostRiemannian structures1 aBoscain, Ugo1 aCharlot, Grégoire1 aGhezzi, Roberta uhttps://www.math.sissa.it/publication/normal-form-generic-2-dimensional-almost-riemannian-structures-tangency-point01299nas a2200109 4500008004300000245006000043210005600103520095600159100001701115700002101132856003601153 2010 en_Ud 00aProjective Reeds-Shepp car on $S^2$ with quadratic cost0 aProjective ReedsShepp car on S2 with quadratic cost3 aFix 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.1 aBoscain, Ugo1 aRossi, Francesco uhttp://hdl.handle.net/1963/266801439nas a2200181 4500008004300000245007000043210006800113260001300181300001200194490000700206520090200213100002501115700001701140700002301157700002001180700002101200856003601221 2010 en_Ud 00aTwo-dimensional almost-Riemannian structures with tangency points0 aTwodimensional almostRiemannian structures with tangency points bElsevier a793-8070 v273 aTwo-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.

1 aAgrachev, Andrei, A.1 aBoscain, Ugo1 aCharlot, Grégoire1 aGhezzi, Roberta1 aSigalotti, Mario uhttp://hdl.handle.net/1963/387001061nas a2200133 4500008004300000245009500043210006900138520060700207100002200814700001700836700002100853700001700874856003600891 2009 en_Ud 00aControllability of the discrete-spectrum Schrodinger equation driven by an external field0 aControllability of the discretespectrum Schrodinger equation dri3 aWe 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.1 aChambrion, Thomas1 aMason, Paolo1 aSigalotti, Mario1 aBoscain, Ugo uhttp://hdl.handle.net/1963/254701520nas a2200133 4500008004300000245008600043210006900129520106500198100002501263700001701288700002401305700002101329856003601350 2009 en_Ud 00aThe intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups0 aintrinsic hypoelliptic Laplacian and its heat kernel on unimodul3 aWe 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.1 aAgrachev, Andrei, A.1 aBoscain, Ugo1 aGauthier, Jean-Paul1 aRossi, Francesco uhttp://hdl.handle.net/1963/266901551nas a2200121 4500008004300000245007900043210006900122520113900191100002501330700001701355700002101372856003601393 2008 en_Ud 00aA Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds0 aGaussBonnetlike formula on twodimensional almostRiemannian manif3 aWe 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.1 aAgrachev, Andrei, A.1 aBoscain, Ugo1 aSigalotti, Mario uhttp://hdl.handle.net/1963/186900870nas a2200109 4500008004300000245007800043210006900121520049600190100001700686700002100703856003600724 2008 en_Ud 00aInvariant Carnot-Caratheodory metrics on S3, SO(3), SL(2) and Lens Spaces0 aInvariant CarnotCaratheodory metrics on S3 SO3 SL2 and Lens Spac3 aIn 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).1 aBoscain, Ugo1 aRossi, Francesco uhttp://hdl.handle.net/1963/214401845nas a2200133 4500008004300000245006800043210006700111520142200178100001701600700002101617700001701638700002001655856003601675 2008 en_Ud 00aLimit Time Optimal Syntheses for a control-affine system on S²0 aLimit Time Optimal Syntheses for a controlaffine system on S²3 aFor $\\\\alpha \\\\in ]0,\\\\pi/2[$, 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 $U_\\\\alpha$.1 aMason, Paolo1 aSalmoni, Rebecca1 aBoscain, Ugo1 aChitour, Yacine uhttp://hdl.handle.net/1963/186200349nas a2200097 4500008004300000245006900043210006800112100001700180700001800197856003600215 2008 en_Ud 00aStability of planar switched systems: the nondiagonalizable case0 aStability of planar switched systems the nondiagonalizable case1 aBoscain, Ugo1 aBalde, Moussa uhttp://hdl.handle.net/1963/185700617nas a2200109 4500008004300000245007000043210006900113520025100182100001700433700002100450856003600471 2007 en_Ud 00aGaussian estimates for hypoelliptic operators via optimal control0 aGaussian estimates for hypoelliptic operators via optimal contro3 aWe 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.1 aBoscain, Ugo1 aPolidoro, Sergio uhttp://hdl.handle.net/1963/199401180nas a2200109 4500008004300000245005200043210005000095520085100145100001700996700002101013856003601034 2007 en_Ud 00aHigh-order angles in almost-Riemannian geometry0 aHighorder angles in almostRiemannian geometry3 aLet 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.1 aBoscain, Ugo1 aSigalotti, Mario uhttp://hdl.handle.net/1963/199500378nas a2200109 4500008004300000245006600043210006400109100001700173700001900190700002300209856003600232 2006 en_Ud 00aClassification of stable time-optimal controls on 2-manifolds0 aClassification of stable timeoptimal controls on 2manifolds1 aBoscain, Ugo1 aNikolaev, Igor1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/219601052nas a2200121 4500008004300000245006900043210006900112520065900181100001700840700001700857700002000874856003600894 2006 en_Ud 00aCommon Polynomial Lyapunov Functions for Linear Switched Systems0 aCommon Polynomial Lyapunov Functions for Linear Switched Systems3 aIn 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.1 aMason, Paolo1 aBoscain, Ugo1 aChitour, Yacine uhttp://hdl.handle.net/1963/218100968nas a2200121 4500008004300000245005100043210005100094520060400145100001700749700002300766700002100789856003600810 2006 en_Ud 00aStability of planar nonlinear switched systems0 aStability of planar nonlinear switched systems3 aWe 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.1 aBoscain, Ugo1 aCharlot, Grégoire1 aSigalotti, Mario uhttp://hdl.handle.net/1963/171002083nas a2200109 4500008004300000245007400043210006900117520171700186100001701903700001701920856003601937 2006 en_Ud 00aTime Minimal Trajectories for a Spin 1/2 Particle in a Magnetic field0 aTime Minimal Trajectories for a Spin 12 Particle in a Magnetic f3 aIn 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.1 aBoscain, Ugo1 aMason, Paolo uhttp://hdl.handle.net/1963/173401293nas a2200121 4500008004300000245009700043210006900140520086400209100001701073700002201090700002301112856003601135 2005 en_Ud 00aNonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy0 aNonisotropic 3level quantum systems complete solutions for minim3 aWe 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.1 aBoscain, Ugo1 aChambrion, Thomas1 aCharlot, Grégoire uhttp://hdl.handle.net/1963/225900333nas a2200109 4500008004300000020001800043245004400061210004200105100001700147700002300164856003600187 2005 en_Ud a2 7056 6511 000aA short introduction to optimal control0 ashort introduction to optimal control1 aBoscain, Ugo1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/225700714nas a2200109 4500008004300000245007100043210006900114520035100183100001700534700001700551856003600568 2005 en_Ud 00aTime minimal trajectories for two-level quantum systems with drift0 aTime minimal trajectories for twolevel quantum systems with drif3 aOn 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.1 aBoscain, Ugo1 aMason, Paolo uhttp://hdl.handle.net/1963/168801372nas a2200109 4500008004300000245007100043210006800114520100700182100001701189700002001206856003601226 2005 en_Ud 00aTime Optimal Synthesis for Left-Invariant Control Systems on SO(3)0 aTime Optimal Synthesis for LeftInvariant Control Systems on SO33 aConsider 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.1 aBoscain, Ugo1 aChitour, Yacine uhttp://hdl.handle.net/1963/225800956nas a2200133 4500008004100000245008400041210006900125260000900194520052900203100001700732700001700749700002000766856003600786 2004 en d00aOn the minimal degree of a common Lyapunov function for planar switched systems0 aminimal degree of a common Lyapunov function for planar switched bIEEE3 aIn 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.1 aMason, Paolo1 aBoscain, Ugo1 aChitour, Yacine uhttp://hdl.handle.net/1963/483401180nas a2200121 4500008004300000245007900043210006900122260001700191520077400208100001700982700002300999856003601022 2004 en_Ud 00aResonance of minimizers for n-level quantum systems with an arbitrary cost0 aResonance of minimizers for nlevel quantum systems with an arbit bEDP Sciences3 aWe 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.1 aBoscain, Ugo1 aCharlot, Grégoire uhttp://hdl.handle.net/1963/291000435nas a2200121 4500008004100000245008600041210006900127260001800196100001700214700002200231700002400253856003600277 2002 en d00aOn the K+P problem for a three-level quantum system: optimality implies resonance0 aKP problem for a threelevel quantum system optimality implies re bSISSA Library1 aBoscain, Ugo1 aChambrion, Thomas1 aGauthier, Jean-Paul uhttp://hdl.handle.net/1963/160101393nas a2200109 4500008004100000245007100041210006900112260000900181520104000190100001701230856003601247 2002 en d00aStability of planar switched systems: the linear single input case0 aStability of planar switched systems the linear single input cas bSIAM3 aWe 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.1 aBoscain, Ugo uhttp://hdl.handle.net/1963/152900345nas a2200109 4500008004100000245005000041210005000091260001800141100001700159700002300176856003600199 2001 en d00aExtremal synthesis for generic planar systems0 aExtremal synthesis for generic planar systems bSISSA Library1 aBoscain, Ugo1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/150300380nas a2200109 4500008004100000245006800041210006700109260001800176100001700194700002300211856003600234 2001 en d00aMorse properties for the minimum time function on 2-D manifolds0 aMorse properties for the minimum time function on 2D manifolds bSISSA Library1 aBoscain, Ugo1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/154100351nas a2200109 4500008004100000245005300041210005300094260001800147100001700165700002300182856003600205 2000 en d00aAbnormal extremals for minimum time on the plane0 aAbnormal extremals for minimum time on the plane bSISSA Library1 aBoscain, Ugo1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/150800367nas a2200109 4500008004100000245006100041210006100102260001800163100001700181700002300198856003600221 1999 en d00aProjection singularities of extremals for planar systems0 aProjection singularities of extremals for planar systems bSISSA Library1 aBoscain, Ugo1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/130400347nas a2200109 4500008004100000245005100041210005100092260001800143100001700161700002300178856003600201 1998 en d00aGeometric control approach to synthesis theory0 aGeometric control approach to synthesis theory bSISSA Library1 aBoscain, Ugo1 aPiccoli, Benedetto uhttp://hdl.handle.net/1963/1277