**abstract:**
When one’s interested in the minimum time control of mechanical systems, and more generally of dynamics that are affine in the control, 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. 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, which consist in building a Lagrangian submanifold on which the canonical projection of the extremal flow is invertible. Then one can compare final times of neighboring 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.

Tue 21 May, 11:20 - 12:10, Aula Dini

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