Mario Sigalotti

Controllability of the discrete-spectrum Schrödinger equation driven by an external field

We prove approximate controllability of the bilinear Schrodinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant 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.

A Weak Spectral Condition for the Controllability of the Bilinear Schrödinger Equation with Application to the Control of a Rotating Planar Molecule

In this paper we prove an approximate controllability result for the bilinear Schrödinger equation. This result requires less restrictive non-resonance hypotheses on the spectrum of the uncontrolled Schrödinger operator than those present in the literature. The control operator is not required to be bounded and we are able to extend the controllability result to the density matrices. The proof is based on fine controllability properties of the finite dimensional Galerkin approximations and allows to get estimates for the L1 norm of the control. The general controllability result is applied to the problem of controlling the rotation of a bipolar rigid molecule confined on a plane by means of two orthogonal external fields.

On the Local Structure of Optimal Trajectories in R 3

We analyze the structure of a control function u(t) corresponding to an optimal trajectory for the system

Generic Controllability Properties for the Bilinear Schrödinger Equation

\dot q =f(q)+u\, g(q)

Brief paper: On the algebraic characterization of invariant sets of switched linear systems

in a three-dimensional manifold, near a point where some nondegeneracy conditions are satisfied. The kind of optimality which is studied includes time-optimality. The control turns out to be the concatenation of some bang and some singular arcs. Studying the index of the second variation of the switching times, the number of such arcs is bounded by four.

Converse Lyapunov Theorems for Switched Systems in Banach and Hilbert Spaces

In [16] we proposed a set of sufficient conditions for the approximate controllability of a discrete-spectrum bilinear Schrödinger equation. These conditions are expressed in terms of the controlled potential and of the eigenpairs of the uncontrolled Schrödinger operator. The aim of this paper is to show that these conditions are generic with respect to the uncontrolled and the controlled potential, denoted respectively by V and W. More precisely, we prove that the Schrödinger equation is approximately controllable generically with respect to W when V is fixed and also generically with respect to V when W is fixed and non-constant. The results are obtained by analytic perturbation arguments and through the study of asymptotic properties of eigenfunctions.

On the marginal instability of linear switched systems

In this paper, a suitable LaSalle principle for continuous-time linear switched systems is used to characterize invariant sets and their associated switching laws. An algorithm to determine algebraically these invariants is proposed. The main novelty of our approach is that we require no dwell time conditions on the switching laws. By not focusing on restricted control classes we are able to describe the asymptotic properties of the considered switched systems. Observability analysis of a flying capacitor converter is proposed as an illustration.

Stability Analysis of Singularly Perturbed Switched Linear Systems

We consider switched systems on Banach and Hilbert spaces governed by strongly continuous one-parameter semigroups of linear evolution operators. We provide necessary and sufficient conditions for their global exponential stability, uniform with respect to the switching signal, in terms of the existence of a Lyapunov function common to all modes.

Adiabatic Control of the Schrödinger Equation via Conical Intersections of the Eigenvalues

Abstract Stability properties for continuous-time linear switched systems are at first determined by the (largest) Lyapunov exponent associated with the system, which is the analogue of the joint spectral radius for the discrete-time case. The purpose of this paper is to provide a characterization of marginally unstable systems, i.e., systems for which the Lyapunov exponent is equal to zero and there exists an unbounded trajectory, and to analyze the asymptotic behavior of their trajectories. Our main contribution consists in pointing out a resonance phenomenon associated with marginal instability. In the course of our study, we derive an upper bound of the state at time t , which is polynomial in t and whose degree is computed from the resonance structure of the system. We also derive analogous results for discrete-time linear switched systems.

On the Stabilization of Persistently Excited Linear Systems

This note is concerned with the stability of planar linear singularly perturbed switched systems. We propose a characterization of the stability properties of such multiple time scale switched systems as the perturbation parameter goes to zero. We also study transitions as this parameter varies and we restrict their number and nature. Finally, we compare the results obtained in this way with the Tikhonov-type results for differential inclusions available in the literature.