## Section: New Results

### Averaging in control

Participants : Bernard Bonnard, Jean-Baptiste Pomet, Jana Nemcova.

A reference paper on the construction and properties of an “average control system”, has been submitted [27] ; it is based on Alex Bombrun's doctoral work [54] (defended in 2007). It connects properties of convergence of solutions of highly oscillating control systems to those of an average control system, when the frequency of oscillation goes high. Likewise, it details (on a time-interval that goes to infinity) the properties of solutions of a conservative system with small controls in relation to those of an average system as the magnitude of control goes to zero. It also gives many properties of this average control system that has “more controls” than the original system, and yields, when this number of new controls is maximal, a Finsler metric on the state manifold. It is however difficult to compute explicitly and is never twice differentiable.

In [33] , we study into details this average system arising from low-thrust orbital transfer, in the restricted “meridian” co-planar case, and prove that its trajectories for minimum time never leave the “elliptic domain” where averaging is valid. This gives some ground to using it it as a limit to describe transfer from an elliptic orbit to another.

More exploration on this average system and the corresponding Finsler metric is planned.