Singular arcs in the generalized Goddard's Problem

J. Frederic Bonnans 1 Pierre Martinon 1 Emmanuel Trélat 2
1 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
CNRS - Centre National de la Recherche Scientifique : UMR7641, Polytechnique - X, UMA - Unité de Mathématiques Appliquées, Inria Saclay - Ile de France, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
Abstract : We investigate variants of Goddard's problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this report, performing an analysis based on the Pontryagin Maximum Principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle with the problem of nonsmoothness of the optimal control.
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[Research Report] RR-6157, INRIA. 2007, pp.25
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J. Frederic Bonnans, Pierre Martinon, Emmanuel Trélat. Singular arcs in the generalized Goddard's Problem. [Research Report] RR-6157, INRIA. 2007, pp.25. 〈inria-00139273v2〉

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