Normal curves in sub-Finsler Lie groups

In this talk, I will present recent advances in the study of sub-Finsler geometry on Lie groups. I will begin by reformulating the Pontryagin Maximum Principle in this setting using tools from subdifferential calculus and the adjoint representation. After recalling key properties of normal curves, I will analyze chattering in sub-Finsler Lie groups equipped with polyhedral norms, showing that the control of a normal curve must locally take values on a face of the unit ball. This yields a proof of the local optimality of normal curves in Carnot groups. I will then discuss branching phenomena, providing sufficient conditions for branching of geodesics in the presence of strongly convex norms. Finally, I will turn to the global dynamics of normal curves in Carnot groups, demonstrating that all normal trajectories escape every compact set as time tends to infinity.