Singularities of the Hamiltonian Vectorfield in Optimal Control Problems

Abstract

Variational problems with n degrees of freedom give rise (by the Pontriaguine maximum principle) to a hamiltonian vectorfield in T*Rn, that presents singularities (non-smoothness points) when the lagrangean is not convex. For the problems of the calculus of variations, the singularities that occur are points where the hamiltonian vectorfield is not C0. For optimal control problems, we show that besides these singularities there appear other ones: points where the hamiltonian vectorfield is C0 but not C1, and we classify them.