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A Shooting Algorithm for Optimal Control Problems with Singular Arcs

Author

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  • M. Soledad Aronna

    (ITN Marie Curie Network SADCO at Università degli Studi di Padova)

  • J. Frédéric Bonnans

    (INRIA Saclay and CMAP Ecole Polytechnique)

  • Pierre Martinon

    (INRIA Saclay and CMAP Ecole Polytechnique)

Abstract

In this article, we propose a shooting algorithm for a class of optimal control problems for which all control variables appear linearly. The shooting system has, in the general case, more equations than unknowns and the Gauss–Newton method is used to compute a zero of the shooting function. This shooting algorithm is locally quadratically convergent, if the derivative of the shooting function is one-to-one at the solution. The main result of this paper is to show that the latter holds whenever a sufficient condition for weak optimality is satisfied. We note that this condition is very close to a second order necessary condition. For the case when the shooting system can be reduced to one having the same number of unknowns and equations (square system), we prove that the mentioned sufficient condition guarantees the stability of the optimal solution under small perturbations and the invertibility of the Jacobian matrix of the shooting function associated with the perturbed problem. We present numerical tests that validate our method.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:158:y:2013:i:2:d:10.1007_s10957-012-0254-8
    DOI: 10.1007/s10957-012-0254-8
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    References listed on IDEAS

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    1. Pierre Martinon & Joseph Gergaud, 2006. "An Application of PL Continuation Methods to Singular Arcs Problems," Lecture Notes in Economics and Mathematical Systems, in: Alberto Seeger (ed.), Recent Advances in Optimization, pages 163-186, Springer.
    2. U. Felgenhauer, 2012. "Structural Stability Investigation of Bang-Singular-Bang Optimal Controls," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 605-631, March.
    3. Eligius M. T. Hendrix & Boglárka G.-Tóth, 2010. "Nonlinear Programming algorithms," Springer Optimization and Its Applications, in: Introduction to Nonlinear and Global Optimization, chapter 5, pages 91-136, Springer.
    4. G. Vossen, 2010. "Switching Time Optimization for Bang-Bang and Singular Controls," Journal of Optimization Theory and Applications, Springer, vol. 144(2), pages 409-429, February.
    5. 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.
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    Cited by:

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    2. Ursula Felgenhauer, 2016. "Discretization of semilinear bang-singular-bang control problems," Computational Optimization and Applications, Springer, vol. 64(1), pages 295-326, May.

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