@article {2008,
title = {Limit Time Optimal Syntheses for a control-affine system on S{\texttwosuperior}},
journal = {SIAM J. Control Optim. 47 (2008) 111-143},
number = {SISSA;48/2006/M},
year = {2008},
abstract = {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 $U_\\\\alpha$.},
doi = {10.1137/060675988},
url = {http://hdl.handle.net/1963/1862},
author = {Paolo Mason and Rebecca Salmoni and Ugo Boscain and Yacine Chitour}
}
@article {2006,
title = {Common Polynomial Lyapunov Functions for Linear Switched Systems},
journal = {SIAM J. Control Optim. 45 (2006) 226-245},
number = {arXiv.org;math/0403209v2},
year = {2006},
abstract = {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.},
doi = {10.1137/040613147},
url = {http://hdl.handle.net/1963/2181},
author = {Paolo Mason and Ugo Boscain and Yacine Chitour}
}
@article {2005,
title = {Time Optimal Synthesis for Left-Invariant Control Systems on SO(3)},
journal = {SIAM J. Control Optim. 44 (2005) 111-139},
number = {SISSA;21/2004/M},
year = {2005},
abstract = {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.},
doi = {10.1137/S0363012904441532},
url = {http://hdl.handle.net/1963/2258},
author = {Ugo Boscain and Yacine Chitour}
}
@proceedings {2004,
title = {On the minimal degree of a common Lyapunov function for planar switched systems},
year = {2004},
publisher = {IEEE},
abstract = {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.},
doi = {10.1109/CDC.2004.1428884},
url = {http://hdl.handle.net/1963/4834},
author = {Paolo Mason and Ugo Boscain and Yacine Chitour}
}
@article {2001,
title = {Controllability for discrete systems with a finite control set},
journal = {Math. Control Signals Systems 14 (2001) 173-193},
number = {SISSA;148/97/M},
year = {2001},
publisher = {Springer},
doi = {10.1007/PL00009881},
url = {http://hdl.handle.net/1963/3114},
author = {Yacine Chitour and Benedetto Piccoli}
}