## Section: New Results

### Average control systems

Participants : Alex Bombrun [ Univ. of Heidelberg ] , Jean-Baptiste Pomet.

In the terminology of [54] , [55] ,
a *Kepler control system* is a system in dimension n whose drift has
n-1 first integral and compact trajectories
and where the control is “small” in the sense that we are interested in asymptotic properties as the bound on
the control tends to zero.
It is the case in low thrust orbital transfer, see section
4.4 , for negative energy, *i.e.* in the so-called elliptic domain.

For this class of systems, a notion of *average control system* is
introduced in [54] , [55] .
Using averaging techniques in this context is rather natural, since the free
system produces a fast periodic motion and the *small* control a slow
one; averaging is a widespread tool in perturbations of integrable Hamiltonian
systems, and the small control is in some sense a “perturbation”.
In some recent literature, one proceeds as follows: the control is
pre-assigned, for instance to time optimal control via
Pontryagin's Maximum Principle or else to
some feedback designed beforehand. Then, averaging is performed on the
resulting ordinary differential equation,
whose limit behavior is analyzed when the control magnitude tends to zero.

The novelty of [54] (see also [56] )
is to average *before* assigning the control, hence getting a
*control system* that describes the limit behavior better.
For that reason, the average control system is a convenient tool
when comparing different control strategies.

It allowed us to answer an open question stated in [61] on the minimum transfer-time between two elliptic orbit when the thrust magnitude tends to zero, see [57] .

Under some controllability conditions that are trivially satisfied in the case at hand, we proved that the average system is one where the velocity set has nonempty interior, i.e. all velocity directions are allowed at any point, and the constraint is convex; mathematically this yields a Finsler structure (in the same way as a controllable system without drift with a quadratic constraint on the control yields a sub-Riemannian structure). An article is in progress, reporting on these results [55] .