Section: New Results

Averaging and filtering for optimal control in Space mechanics

Participants : Jean-Baptiste Caillau, Thierry Dargent, Florentina Nicolau, Jean-Baptiste Pomet, Jérémy Rouot.

Investigating averaging in optimal control for space mechanics with low thrust, or more generally with conservative systems with “small” controls is an ongoing subject in the team. It is also central in the research contract with CNES mentioned in Section 7.1.

Convergence properties of the Maximum principle

Part of Jérémy Rouot's PhD [2] was devoted to convergence properties in the Hamiltonian system resulting from Pontryagin's Maximum principle when the small parameter representing the ratio between slow and fast velocities tends to zero. The difference with previous work is that we give a clear method to sort fast and slow variables in the adjoint variables, and we provide convergence of these under some conditions. A more complete publication is under preparation.

Approximation by filtering in optimal control and applications

Minimum time control of slow-fast systems is considered. In the case of only one fast angle, averaging techniques are available for such systems. The approach introduced in [54] and [43] is recalled, then extended to time dependent systems by means of a suitable filtering operator. The process relies upon approximating the dynamics by means of sliding windows. The size of these windows is an additional parameter that provides intermediate approximations between averaging over the whole fast angle period and the original dynamics. The method was applied to problems coming from space mechanics, and is exposed in [31].

Averaging with reconstruction of the fast variable

We have been studying a way to modify the initial condition of the average equation in order to approach better (but in the mean) the slow variable while reconstructing asymptotically the fast variable. This follows an idea that was shown to work numerically in [54].

In [32], we give a construction for Cauchy problems. It is lighter than second order averaging, in that oscillating signals and ODEs are not used, and still provides a second order error in the mean, together with convergence of the fast variable. This remains to be developed for two-point boundary value problems like in optimal control.