## Section: Application Domains

### Spatial mechanics

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

Generally speaking, aerospace engineering requires sophisticated control techniques for which optimization is often crucial, due to the extreme functioning conditions. The use of satellites in telecommunication networks motivates a lot of research in the area of signal and image processing; see for instance section 4.3 for an illustration. Of course, this requires that satellites be adequately controlled, both in position and orientation (attitude). This problem and similar ones continue to motivate research in control. The team has been working for six years on control problems in orbital transfer with low-thrust engines, including four years under contract with Thales Alenia Space (formerly Alcatel Space) in Cannes.

Technically, the reason for using these (ionic) low thrust engines, rather than chemical engines that deliver a much higher thrust, is that they require much less “fuel”; this is decisive because the total mass is limited by the capacity of the launchers: less fuel means more payload, while fuel represents today an impressive part of the total mass.

From the control point of view, the low thrust makes the transfer problem
delicate. In principle of course, the control law leading to the right orbit
in minimum time exists, but it is quite heavy to obtain numerically and
the computation is non-robust against many unmodelled phenomena. Considerable progress on
the approximation of such a law by a feedback has been carried out
using *ad hoc* Lyapunov functions.These
approximate surprisingly well time-optimal trajectories. The
easy implementation of such control laws makes them attractive as compared to
genuine optimal control.
Here the n-1 first integrals are an easy means to build control
Lyapunov functions
since any function of these first integrals can be made
monotone decreasing by a suitable control. See [54] and
the references therein.