|Michael Orieux (SISSA)|
Résumé. Optimal control of systems whose dynamics are affine in the control have a wide range of applications, from energy minimisation in orbit transfer problems to quantum control. The necessary conditions give the optimal trajectory as the projection of the integral curves of an Hamiltonian system defined on the cotangent bundle of the phase space. In that regard, minimum time control plays a singular role with respect to other criteria because the integral curves of Hamiltonian do not depend on the cost, but only on the initial dynamics. Those curves are called extremal, and the time minimisation induces a lack of regularity: the Hamiltonian is not smooth, and has codimension 2 singularities. In this talk we will prove sufficient conditions for optimality of these singular extremals. Our method uses techniques from symplectic geometry, and consist in building a Lagrangian submanifold on which the canonical projection of the extremal flow is invertible. Then one can compare final times of neighbouring trajectories by lifting them to the cotangent bundle and evaluate the Poincaré-Cartan form along their lifts. The main difficulty is the definition of these objects without the required regularity, and an extended study of the extremal flow is necessary.