The Erdmann Condition and Hamiltonian Inclusions in Optimal Control and the Calculus of Variations

Abstract

Consider the basic problem in the calculus of variations, that of minimizing1.1over a class of functions x satisfying certain boundary conditions at 0 and 1. One of the classical first order necessary conditions for optimality is the second Erdmann condition, which asserts, in the case in which L is independent mof t, that1.2along any local solution x. This formula is the customary basis for solving many of the classical problems, such as the brachistochrone. When it is possible to define via the Legendre transform a Hamiltonian H(t, x, p) corresponding to L, the second Erdmann condition, again in the autonomous case, is the assertion that1.3a relation which always evokes classical Hamiltonian mechanics and conservation laws.