## Section: New Results

### Advances in optimal control

#### Algebraic and geometric techniques in medical resonance imaging

Participants : Bernard Bonnard, Jean-Charles Faugère [EPI PolSys] , Alain Jacquemard [Univ. de Bourgogne] , Mohab Safey El Din [EPI PolSys] , Thibaut Verron [EPI PolSys] .

In the framework of the ANR-DFG project Explosys (see Section 8.3) we use computer algebra methods to analyze the controlled Bloch equations, modeling the contrast problem in MRI. The problem boils down to analyzing the so called singular extremals associated to the problem. Thanks to the linear dependance of the problem with respect to the state variables and the relaxation parameters the problem is algebraic and is equivalent to determining equilibrium points and eigenvalues of the linearized system at such points together with the algebraic classification of the surface associated to the switches between bang and singular arcs. Preliminary results are described in ISSAC paper [12] using Grobner basis and stratifications of singularities of determinantal varieties. This work was a part of T. Verron's PhD and is continuing in particular with him (Post doc APO-ENSEEIHT).

#### Local minima, second order conditions

Participants : Jean-Baptiste Caillau, Zheng Chen, Yacine Chitour [Univ. Paris-Sud] , Ariadna Farrés [Univ. Barcelona] .

It is well known that the PMP gives necessary conditions for optimality, but curves satisfying this condition may be local minima or critical sadle points. Roughly speaking, the PMP is a first order condition. Higher order conditions give finer necessary conditions (and sufficient in some special cases), but they require differentiability that is not always satisfied when commutations occur. Furthermore, these local conditions cannot distinguish local from global minima. In [4] and [19], we make contributions respectively to extending higher order conditions to non-smooth cases and to exploring local and global minima on an example of interest.

Second order systems whose drift is defined by the gradient of a given potential are considered, and minimization of the L${}^{1}$-norm of the control is addressed in [4]. An analysis of the extremal flow emphasizes the role of singular trajectories of order two [78], [81]; the case of the two-body potential is treated in detail. In L${}^{1}$-minimization, regular extremals are associated with bang-bang controls (saturated ocnstraint on the norm); in order to assess their optimality properties, sufficient conditions are given for broken extremals and related to the no-fold conditions of [75]. Two examples of numerical verification of these conditions are proposed on a problem coming from space mechanics.

In another direction, we have been studying the structure of local minima for time minimization in the controlled three-body problem. In [19], several homotopies are systematically used to unfold the structure of these local minimizers, and the resulting singularity of the path associated with the value function is analyzed numerically.

#### Solving chance-constrained optimal control problems in aerospace engineering via Kernel Density Estimation

Participants : Jean-Baptiste Caillau, Max Cerf [Airbus Industries] , Achille Sassi, Emmanuel Trélat [Univ. P. & M. Curie] , Hasna Zidani [ENSTA ParisTech] .

The goal of [30] is to show how non-parametric statistics can be used to solve chance-constrained optimization and optimal control problems by reformulating them into deterministic ones, focusing on the details of the algorithmic approach. We use the Kernel Density Estimation method to approximate the probability density function of a random variable with unknown distribution, from a relatively small sample. In the paper it is shown how this technique can be applied to a class of chance-constrained optimization problem, focusing on the implementation of the method. In particular, in our examples we analyze a chance-constrained version of the well known problem in aerospace optimal control: the Goddard problem.