01061nas 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/1963/254701845nas 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/1963/186200512nas a2200121   4500008004300000245004900043210004800092520014900140100002500289700001700314700002300331856003600354       2007                        en_Ud  00aTime optimal swing-up of the planar pendulum0 aTime optimal swingup of the planar pendulum3 aThis paper presents qualitative and numerical results on the global structure of the time optimal trajectories of the planar pendulum on a cart.1 aBroucke, Mireille E.1 aMason, Paolo1 aPiccoli, Benedetto  uhttp://hdl.handle.net/1963/186701052nas 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/1963/218102083nas 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/1963/173400714nas 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/1963/168800956nas 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/1963/4834