@article {2009,
	title = {Controllability of the discrete-spectrum  Schrodinger equation driven by an external field},
	journal = {Ann. Inst. H. Poincare Anal. Non Lineaire 26 (2009) 329-349},
	number = {SISSA;01/2008/M},
	year = {2009},
	abstract = {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.},
	doi = {10.1016/j.anihpc.2008.05.001},
	url = {http://hdl.handle.net/1963/2547},
	author = {Thomas Chambrion and Paolo Mason and Mario Sigalotti and Ugo Boscain}
}
@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 {2007,
	title = {Time optimal swing-up of the planar pendulum},
	journal = {46th IEEE Conference on Decision and Control (2007) 5389 - 5394},
	number = {SISSA;50/2006/M},
	year = {2007},
	abstract = {This paper presents qualitative and numerical results on the global structure of the time optimal trajectories of the planar pendulum on a cart.},
	doi = {10.1109/CDC.2007.4434688},
	url = {http://hdl.handle.net/1963/1867},
	author = {Mireille E. Broucke and Paolo Mason and Benedetto Piccoli}
}
@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 {2006,
	title = {Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic field},
	number = {SISSA;82/2005/M},
	year = {2006},
	abstract = {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 {\textrightarrow} $\infty$ 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 {\textrightarrow} 0, giving a partial proof of a conjecture formulated in a previous paper.},
	doi = {10.1063/1.2203236},
	url = {http://hdl.handle.net/1963/1734},
	author = {Ugo Boscain and Paolo Mason}
}
@article {2005,
	title = {Time minimal trajectories for two-level quantum systems with drift},
	number = {SISSA;5/2005/M},
	year = {2005},
	abstract = {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.},
	url = {http://hdl.handle.net/1963/1688},
	author = {Ugo Boscain and Paolo Mason}
}
@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}
}
