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Singular Arcs in the Generalized Goddard’s Problem

Author

Listed:
  • F. Bonnans

    (Ecole Polytechnique, and INRIA Futurs)

  • P. Martinon

    (Ecole Polytechnique, and INRIA Futurs)

  • E. Trélat

    (Ecole Polytechnique, and INRIA Futurs
    Université d’Orléans)

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.

Suggested Citation

  • F. Bonnans & P. Martinon & E. Trélat, 2008. "Singular Arcs in the Generalized Goddard’s Problem," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 439-461, November.
    • Handle: RePEc:spr:joptap:v:139:y:2008:i:2:d:10.1007_s10957-008-9387-1
    DOI: 10.1007/s10957-008-9387-1
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    Cited by:

    1. E. Trélat, 2012. "Optimal Control and Applications to Aerospace: Some Results and Challenges," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 713-758, September.
    2. E. Cristiani & P. Martinon, 2010. "Initialization of the Shooting Method via the Hamilton-Jacobi-Bellman Approach," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 321-346, August.
    3. M. Soledad Aronna & J. Frédéric Bonnans & Pierre Martinon, 2013. "A Shooting Algorithm for Optimal Control Problems with Singular Arcs," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 419-459, August.

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