The Euler-Lagrange equation under weak regularity conditions

This paper is a continuation of the author's previous work (1994, 1996, 1997, 1998) on a general nonsmooth version of the finite-dimensional Pontryagin maximum principle. The purpose here is to present a further extension of this result for the special case of finite-dimensional classical calculus of variations problems in R/sup n/, with constraints /spl xi/(t)/spl isin/U, where U is a given subset of R/sup n/. For simplicity, we only consider problems with fixed end points. Naturally, the result is a generalization of the classical Euler-Lagrange equations with the Weierstrass's side conditions, stated in the Hamiltonian language of optimal control theory. As in the author's previous work on the maximum principle, the approach used here is classical, in the sense that it involves "needle variations" and uses a fixed-point argument.