On the shooting algorithm for optimal control problems with state constraints
Audrey Hermant · HAL (Le Centre pour la Communication Scientifique Directe) · 2008
This thesis deals with (deterministic) optimal control problems of an ordinary differential equation subject to one or several state constraints, of arbitrary orders, in the case when the strengthened Legendre-Clebsch condition is satisfied. Pontryagin's minimum principle provides us with a well-known first-order optimality condition. In this thesis we first obtain a second-order sufficient optimality condition which is the weakest possible, i.e. which is as close as possible to the second-order necessary condition and characterizes quadratic growth. This condition allows us to obtain a characterization of the well-posedness of the shooting algorithm in presence of state constraints. Then stability and sensitivity analysis of solutions under perturbation of the data is investigated. We obtain for the first time stability results for state constraints of order greater than or equal to two that make no assumption on the structure of the trajectory. Moreover, results on structural stability of Pontryagin's extremals are given. Finally, the above results on the well-posedness of the shooting algorithm and on stability analysis allow us to design a new continuation method, for state constraints of first and second-order, whose novelty is to automatically detect the structure of the trajectory and initialize the associated shooting parameters.