Second order systems whose drift is defined by the gradient of a given potential are considered, and minimization of the ${\mathrm{L^1}}$-norm of the control is addressed. An analysis of the extremal flow emphasizes the role of singular trajectories of order two [H. M. Robbins, AIAA J., 3 (1965), pp. 1094--1098; M. I. Zelikin and V. F. Borisov, Theory of Chattering Control, Birkhauser, Basel, 1994]; the case of the two-body potential is treated in detail. In ${\mathrm{L^1}}$-minimization, regular extremals are associated with controls whose norm is bang-bang; in order to assess their optimality properties, sufficient conditions are given for broken extremals and related to the no-fold conditions of [J. Noble and H. Schattler, J. Math. Anal. Appl., 269 (2002), pp. 98--128]. Two examples of numerical verification of these conditions are proposed on a problem coming from space mechanics.
['Zheng Chen', 'Jean-Baptiste Caillau', 'Yacine Chitour']
$L^1$-Minimization for Mechanical Systems