Extremal arcs and extended Hamiltonian systems

Abstract

A general variational problem is considered; it involves the minimization of an integrand L of a very general nature. The Lagrangian L is allowed to assume the value + ∞ + \infty , and need satisfy no differentiability or convexity conditions. A Hamiltonian corresponding to the problem is defined via the conjugate function of convex analysis, and it is shown how one obtains necessary conditions in the form of an extended Hamiltonian system. This system is expressed in terms of certain “generalized gradients” previously developed by the author. A further result is given which has the feature that the principal hypotheses required, as well as the ensuing conclusions, are entirely in terms of H. This allows the treatment of classes of problems in which H is more amenable to direct analysis than L. The approach also sheds light on the relation between existence theory and the theory of necessary conditions, since the results may easily be compared with R. T. Rockafellar’s recent work on existence theory, in which H also plays a central role. As an example of its application the main result is specialized to a differential inclusion problem. A specific example of its use is also given, an unorthodox optimal control problem with a discontinuous cost functional.