L$^1$-minimization for mechanical systems

Abstract : 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, Birkhäuser, 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. Schättler, 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.
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Zheng Chen, Jean-Baptiste Caillau, Yacine Chitour. L$^1$-minimization for mechanical systems. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2016, 54 (3), pp.1245-1265. ⟨10.1137/15M1013274⟩. ⟨hal-01136676⟩

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