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 Rⁿ , with constraints ξ(t)∈U, where U is a given subset of R 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