# Control theory

In the absence of decoherence, the dynamics of a controlled quantum
system is given by a Schroedinger equation, *x'=Ax+u(t)Bx*, where *x*
lies in some infinite dimensional Hilbert space, *A* is a skew-adjoint
operator, *B* is a skew-symmetric linear operator accounting for the
interaction of the environment with the system (e.g., trough a laser)
and *u* is the time variable scalar intensity of the control. We will
restrict ourselves to the case where *A* has a purely discrete spectrum.
The energy of the system is the
*A1/2*
norm of *x*.

A bilinear system is weakly-coupled if the commutator of *A* and *B* is
relatively bounded with respect to *A*. Most of the physical examples
encountered in the literature have this feature. For weakly-coupled
bilinear systems, there exists an a priori bound of the energy of the
system in terms of the
*L1*
norm of the control *u*. In particular,
such systems can be approached with arbitrary precision by their finite
dimensional Galerkin approximations. This gives a theoretical
justification of the approximations usually done in practice by quantum
physicists and provides constructive control algorithms.

These results have been recently obtained in collaboration with Nabile Boussaid (Besancon, France) and Marco Caponigro (Nancy, France).

An important question in the study of human motor control is to determine which law governs a particular body movement, such as arm pointing motions, human locomotion or eye's saccadic motions. A nowadays widely accepted paradigm in neurophysiology is that, among all possible movements, the accomplished ones satisfy suitable optimality criteria. Once a dynamical model of the movements under consideration is given, one is then led to solve an "inverse optimal control problem": given recorded experimental data, infer a cost function such that the recorded movements are solutions of the associated optimal control problem.

We will present in this talk the approach we have develop to solve this problem this problem in the case of pointing movements of the arm and of goal-oriented human locomotion. In that approach, based on geometric control theory, the cost structure is deduced from qualitative properties highlighted by the experimental data. These works presented here result from a collaboration between physiologists and applied mathematicians.

## Un théorème de Kupka-Smale à la Mañé pour les système hamitoniens : une approche "contrôle"

Nous montrerons comment des méthodes de contrôle peuvent être utilisées efficacement pour obtenir des résultats sur la structure générique de systèmes hamiltoniens. Ceci est un travail en collaboration avec Rafael Ruggiero.