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Singular Arcs in the Generalized Goddard’s Problem

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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 article, 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 that 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 the problem of nonsmoothness of the optimal control.

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Authors and Affiliations

  1. CMAP UMR CNRS 7641, Ecole Polytechnique, and INRIA Futurs, 91128, Palaiseau, France

    F. Bonnans, P. Martinon & E. Trélat

  2. Math. Labo. MAPMO, UMR CNRS 6628, Université d’Orléans, Route de Chartres, BP 6759, 45067, Orléans Cedex 2, France

    E. Trélat

Authors
  1. F. Bonnans
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  2. P. Martinon
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  3. E. Trélat
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Correspondence to P. Martinon.

Additional information

Support from the French Space Agency CNES (Centre National d’Etudes Spatial) is gratefully acknowledged.

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Bonnans, F., Martinon, P. & Trélat, E. Singular Arcs in the Generalized Goddard’s Problem. J Optim Theory Appl 139, 439–461 (2008). https://doi.org/10.1007/s10957-008-9387-1

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