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

Summary In this thesis we focus on optimal control problems for affine systems in one part of the control. First, we give a second order necessary condition for the case where the system is affine in all controls. We have bounds on the controls and a bang-singular solution. A sufficient condition is given for the case of a scalar control. We then propose a shooting algorithm and a sufficient condition for its local quadratic convergence. This condition guarantees the stability of the optimal solution and implies that the algorithm converges locally quadratically for the perturbed problem, in some cases. We present numerical tests that validate our method. Then, we study an affine system in a part of the orders. We obtain necessary and sufficient conditions of the second order. Then, we propose a shooting algorithm and we show that the mentioned sufficient condition guarantees that this algorithm converges locally quadratically. Finally, we study a planning problem for a hydro-thermal power plant. We analyze, by means of the necessary conditions obtained by Goh, the possible appearance of singular arcs.