# Jean-Baptiste Caillau - LJAD, Univ. Côte d'Azur & CNRS/Inria, Parc Valrose, F-06108 Nice - Keywords: slow-fast dynamical systems, minimum time control

We consider the minimum time control of dynamical systems with slow and fast state variables. Such models are ubiquitous for complex mechanical systems that exhibit behaviours driven by different time scales. With applications to perturbations of integrable systems in mind, we focus on the case of problems with one or more fast angles, together with a small drift on the slow part modelling a so-called secular evolution of the slow variables. According to Pontrjagin maximum principle, time minimizing trajectories are projections on the state space of Hamiltonian curves. In the case of a single fast angle, the slow and fast parts of these curves are identified thanks to an appropriate symplectic reduction. Then, an approximation of the Hamiltonian flow is obtained using an averaging procedure. It turns out that, provided the drift on the slow part of the original system is small enough, this approximation is of metric nature: Time minimizing trajectories can be approximated by geodesics of a suitable Finsler metric. Moreover, because of the secular evolution of the slow variables, this metric is asymmetric. We report results on asymptotic controllability, existence and convergence for the original control system. As an

application to space mechanics, the effect of the J2 term in the Earth potential on the control of a spacecraft is considered. The J2 perturbation accounts for the oblateness of the Earth, and we provide a qualitative analysis of its influence when minimizing time. In an ongoing work, we address the more involved question of systems having two or more fast angles. In such situations, resonances come into play and complicate significantly the analysis.

Joint work with Lamberto Dell'Elce, Jean-Baptiste Pomet (both Inria Sophia) and Jérémy Rouot (EPF)