%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 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 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 Math. Control Signals Systems 14 (2001) 173-193
%D 2001
%T Controllability for discrete systems with a finite control set
%A Yacine Chitour
%A Benedetto Piccoli
%B Math. Control Signals Systems 14 (2001) 173-193
%I Springer
%G en_US
%U http://hdl.handle.net/1963/3114
%1 1219
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-15T10:33:44Z\\nNo. of bitstreams: 1\\ncontrollability.pdf: 247896 bytes, checksum: 6f94f9b27348f59bb506c1e79beba6e1 (MD5)
%R 10.1007/PL00009881