Second order analysis of optimal control problems with singular arcs. Optimality conditions and shooting algorithm.

Abstract : This thesis deals with optimal control problems for systems that are affine in one part of the control variable. First, we state necessary and sufficient second order conditions when all control variables enter linearly. We have bound control constraints and a bang-singular solution. The sufficient condition is restricted to the scalar control case. We propose a shooting algorithm and provide a sufficient condition for its local quadratic convergence. This condition guarantees the stability of the optimal solution and the local quadratic convergence of the algorithm for the perturbed problem in some cases. We present numerical tests that validate our method. Afterwards, we investigate an optimal control problems with systems that are affine in one part of the control variable. We obtain second order necessary and sufficient conditions for optimality. We propose a shooting algorithm, and we show that the sufficient condition just mentioned is also sufficient for the local quadratic convergence. Finally, we study a model of optimal hydrothermal scheduling. We investigate, by means of necessary conditions due to Goh, the possible occurrence of a singular arc.
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Maria Soledad Aronna. Second order analysis of optimal control problems with singular arcs. Optimality conditions and shooting algorithm.. Optimization and Control [math.OC]. Ecole Polytechnique X, 2011. English. ⟨tel-00682276⟩

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