Nonsmoothness and nonconvexity in calculus of variations and optimal control

We discuss a number of questions relating to the modern theory of necessary conditions in optimal control: Is the Euler-Lagrange inclusion necessary for a weak minimum? In case of nonconvex dependence, does there exist an adjoint arc satisfying jointly the Euler-Lagrange inclusion and the Weierstrass-type condition? Is the maximum principle, jointly with either Euler-Lagrange or Hamiltonian adjoint inclusions, necessary for a strong local minimum? What kind of relationship exists between solutions of Hamiltonian and Euler-Lagrange adjoint inclusions? and, Find an alternative proof of Clarke´s Hamiltonian maximum principle for the Mayer problem, with convex-valued inclusion