Abstract:
In this presentation we address the problem of smoothness of the minimum time function for some classes of differential inclusions under some mild smoothness requirement on the Hamiltonian function associated to the system. Generalized differentiability results are proved, together with propagation of non-superdifferentiability along the trajectories of the generalized Hamiltonian system. We derive from it the classical result of constancy of the Hamiltonian along optimal trajectories also in this non-smooth setting. Partial sufficient conditions for optimality are also established. Finally, we use the result to give a sufficient condition to upper bound the Hausdorff measure of the set of points around which the minimum time function does not exhibit a Lipschitz behaviour, yielding special bounded variation regularity.