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 - 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 - JOUR T1 - Time Optimal Synthesis for Left-Invariant Control Systems on SO(3) JF - SIAM J. Control Optim. 44 (2005) 111-139 Y1 - 2005 A1 - Ugo Boscain A1 - Yacine Chitour AB - 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. UR - http://hdl.handle.net/1963/2258 U1 - 1989 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 - TY - JOUR T1 - Controllability for discrete systems with a finite control set JF - Math. Control Signals Systems 14 (2001) 173-193 Y1 - 2001 A1 - Yacine Chitour A1 - Benedetto Piccoli PB - Springer UR - http://hdl.handle.net/1963/3114 U1 - 1219 U2 - Mathematics U3 - Functional Analysis and Applications ER -