Section: New Results

New results: neurophysiology

• In [17] we study the global properties of an optimal control model of geometry of vision due to Petitot, Citti and Sarti. In particular, we consider the problem of minimizing ${\int }_0^L\sqrt{{\xi }^2+K^2\left(s\right)}ds$ for a planar curve having fixed initial and final positions and directions. The total length $L$ is free. Here $s$ is the variable of arclength parametrization, $K\left(s\right)$ is the curvature of the curve and $\xi >0$ a parameter. The main feature of the problem is that, if for a certain choice of boundary conditions there exists a minimizer, then this minimizer is smooth and has no cusp. However, not for all choices of boundary conditions there is a global minimizer. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there is no global minimizer, then there is neither a local minimizer nor a stationary curve (geodesic). We give properties of the set of boundary conditions for which there exists a solution to the problem. Finally, we present numerical computations of this set.

• In [2] we studied the general problem of reconstructing the cost from the observation of trajectories, in a problem of optimal control. It is motivated by the problem of determining what is the cost minimized in human locomotion. This applied question is very similar to the following applied problem, concerning HALE drones: one would like them to decide by themselves for their trajectories, and to behave at least as a good human pilot. These starting points are the reasons for the particular classes of control systems and of costs under consideration. To summarize, our conclusion is that in general, inside these classes, three experiments visiting the same values of the control are needed to reconstruct the cost, and two experiments are in general not enough. The method is constructive. The proof of these results is mostly based upon the Thom's transversality theory.