Optimal Control and the True Hamiltonian

The natural Hamiltonian function in optimal control is generally not differentiable. However, it is possible to use the theory of generalized gradients (which we discuss as a preliminary) to obtain necessary conditions in the form of a “Hamiltonian inclusion”. The advantages of using the true Hamiltonian include the possibility of considering control problems that do not lie within the usual formulation, or problems with nonsmooth data, and the possibility of bringing to bear upon control problems certain Hamiltonian methods from the classical calculus of variations.