Section: New Results
Averaging in control and application to space mechanics
Participants : Bernard Bonnard, HelenClare Henninger, Jana Němcová [Instritute of Chemical Tech, Prague, CZ] , JeanBaptiste Pomet, Jeremy Rouot.
As explained in sections 3.5 and 4.1 , control problems where the non controlled system is conservative and the control effect is small compared to the free dynamics lead to computing an average system. This computation may be explicit or numerical.
Even though it will not be always the case that an explicit expression is available, it is interesting to study that case thoroughly.

In [23] , [24] , a smooth Riemannian metric was introduced to describe the energy minimizing orbital transfer with low propulsion. We have pursued a study of its deformation due to the standard perturbations in space mechanics, e.g. oblate spheroid shape of the Earth and lunar attraction. In [12] , using Hamiltonian formalism, we describe the effects of the perturbations on the orbital transfers and the deformation of the conjugate and cut loci of the original metric. This is done using averaging with respect to both the proper frequency of the space vehicle and the moon frequency.

The average system has the advantage of being more controllable (it has new virtual controls), but often displays singularities that were not present in the original system. It is the case when minimum time is considered instead of the quadratic energy criterium. We are conducted an analysis of this average minimum time Hamiltonian flow.
In [6] , we compare the two problems for planar transfers. While the energy case leads to analyze a 2D Riemannian metric using the standard tools of Riemannian geometry (curvature computations, geodesic convexity), the time minimal case is associated to a Finsler metric which is not smooth. Nevertheless a qualitative analysis of the geodesic flow is given in this article to describe the optimal transfers. In particular we prove geodesic convexity of the elliptic domain.