In this paper we consider the minimum time population transfer problem for the z-component of 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 |Ω(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 ω R = 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.

We prove approximate controllability of the bilinear Schrödinger equation in the case in which the uncontrolled Hamiltonian has discrete nonresonant spectrum. 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 Galerkin 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, both controlled by suitable potentials.Résumé Nous montrons la contrôlabilité approchée de l'équation de Schrödinger bilinéaire dans le cas où l'hamiltonien non contrôlé a un spectre discret et nonrésonnant. Les résultats obtenus sont valables que le domaine soit borné ou non, et que le potentiel de contrôle soit borné ou non. La preuve repose sur des méthodes de dimension finie appliquées aux approximations de Galerkyn du système. Ces méthodes permettent en plus d'obtenir des résultats de contrôlabilité des matrices de densité. Deux exemples sont présentés, l'oscillateur harmonique et le puit de potentiel en dimension trois, munis de potentiels de contrôle adéquats.

This paper is concerned with the stability problem for the planar linear switched systemẋ(t) = u(t)A1x(t)+(1−u(t))A2x(t), where the real matrices A1, A2 ∈ R 2×2 are Hurwitz and u(·) : [0, ∞[→ {0, 1} is a measurable function. We give coordinate-invariant necessary and sufficient conditions on A1 and A2 under which the system is asymptotically stable for arbitrary switching functions u(·). The new conditions unify those given in previous papers and are simpler to be verified since we are reduced to study 4 cases instead of 20. Most of the cases are analyzed in terms of the function Γ(A1, A2) = 1 2 (tr(A1)tr(A2) − tr(A1A2)).