A Shooting Algorithm for Optimal Control Problems with Singular Arcs

M. Soledad Aronna (), J. Frédéric Bonnans () and Pierre Martinon ()
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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

Journal of Optimization Theory and Applications, 2013, vol. 158, issue 2, No 8, 419-459

Abstract: 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.

Keywords: Optimal control; Singular arc; Bang-singular control; Shooting algorithm; Second order optimality condition; Gauss–Newton method; Stability analysis (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10957-012-0254-8

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