On the firing algorithm for optimal control problems with state constraints.

Summary This thesis is concerned with the optimal (deterministic) control problem of an ordinary differential equation subject to one or more constraints on the state, of any order, in the case where the strong Legendre-Clebsch condition is satisfied. The Pontryaguine minimum principle provides a well-known necessary optimality condition. In this thesis, we first obtain a second order sufficient optimality condition that is as close as possible to the second order necessary condition and characterizes the quadratic growth. This condition allows us to obtain a characterization of the well-posedness of the shooting algorithm in the presence of constraints on the state. Then we perform a stability and sensitivity analysis of the solutions when the problem data is perturbed. For constraints of order greater than or equal to two, we obtain for the first time a stability result for the solutions making no assumption on the structure of the trajectory. Moreover, results on the structural stability of Pontryaguine extremals are given. Finally, these results on the one hand on the shooting algorithm and on the other hand on the stability analysis allow us to propose, for constraints on the state of order one and two, a homotopy algorithm whose novelty is to automatically determine the structure of the trajectory and to initialize the associated shooting parameters.