Section: New Results

New results: neurophysiology

• In recent papers models of the human locomotion by means of an optimal control problem have been proposed. In this paradigm, the trajectories are assumed to be solutions of an optimal control problem whose cost has to be determined. The purpose of [3] is to analyze the class of optimal control problems defined in this way. We prove strong convergence result for their solutions on the one hand for perturbations of the initial and final points (stability), and on the other hand for perturbations of the cost (robustness).

• [8] analyses a class of optimal control problems on geometric paths of the euclidean space, that is, curves parametrized by arc length. In the first part we deal with existence and robustness issues for such problems and we define the associated inverse optimal control problem. In the second part we discuss the inverse optimal control problem in the special case of planar trajectories and under additional assumptions. More precisely we define a criterion to restrict the study to a convenient class of costs based on the analysis of experimentally recorded trajectories. This method applies in particular to the case of human locomotion trajectories.

• The article [17] presents an algorithm implementing the theory of neurogeometry of vision, described by Jean Petitot in his book. We propose a new ingredient, namely working on the group of translations and discrete rotations $\text{SE}(2,N)$. We focus on the theoretical and numerical aspects of integration of an hypoelliptic diffusion equation on this group. Our main tool is the generalized Fourier transform. We provide a complete numerical algorithm, fully parallellizable.