TY - JOUR
T1 - Controllability of the discrete-spectrum Schrodinger equation driven by an external field
JF - Ann. Inst. H. Poincare Anal. Non Lineaire 26 (2009) 329-349
Y1 - 2009
A1 - Thomas Chambrion
A1 - Paolo Mason
A1 - Mario Sigalotti
A1 - Ugo Boscain
AB - 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.
UR - http://hdl.handle.net/1963/2547
U1 - 1572
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Limit Time Optimal Syntheses for a control-affine system on S²
JF - SIAM J. Control Optim. 47 (2008) 111-143
Y1 - 2008
A1 - Paolo Mason
A1 - Rebecca Salmoni
A1 - Ugo Boscain
A1 - Yacine Chitour
AB - 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 $U_\\\\alpha$.
UR - http://hdl.handle.net/1963/1862
U1 - 2360
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Time optimal swing-up of the planar pendulum
JF - 46th IEEE Conference on Decision and Control (2007) 5389 - 5394
Y1 - 2007
A1 - Mireille E. Broucke
A1 - Paolo Mason
A1 - Benedetto Piccoli
AB - This paper presents qualitative and numerical results on the global structure of the time optimal trajectories of the planar pendulum on a cart.
UR - http://hdl.handle.net/1963/1867
U1 - 2355
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Common Polynomial Lyapunov Functions for Linear Switched Systems
JF - SIAM J. Control Optim. 45 (2006) 226-245
Y1 - 2006
A1 - Paolo Mason
A1 - Ugo Boscain
A1 - Yacine Chitour
AB - 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.
UR - http://hdl.handle.net/1963/2181
U1 - 2063
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - RPRT
T1 - Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic field
Y1 - 2006
A1 - Ugo Boscain
A1 - Paolo Mason
AB - 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.
JF - Journal of Mathematical Physics 47, 062101 (2006)
UR - http://hdl.handle.net/1963/1734
U1 - 2418
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - RPRT
T1 - Time minimal trajectories for two-level quantum systems with drift
Y1 - 2005
A1 - Ugo Boscain
A1 - Paolo Mason
AB - 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.
JF - Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC \\\'05. 44th IEEE Conference on
UR - http://hdl.handle.net/1963/1688
U1 - 2445
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - Generic
T1 - On the minimal degree of a common Lyapunov function for planar switched systems
T2 - 43rd IEEE Conference on Decision and Control, 2004, 2786 - 2791 Vol.3
Y1 - 2004
A1 - Paolo Mason
A1 - Ugo Boscain
A1 - Yacine Chitour
AB - 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.
JF - 43rd IEEE Conference on Decision and Control, 2004, 2786 - 2791 Vol.3
PB - IEEE
UR - http://hdl.handle.net/1963/4834
U1 - 4611
U2 - Mathematics
U3 - Functional Analysis and Applications
U4 - -1
ER -