Lower Closure Problems with Weak Convergence Conditions in a New Perspective

Abstract

A transparent approach is taken towards the class of (lower) closure problems with weak convergence conditions for the “derivatives”. A deparametrization procedure is formulated for the abstract lower closure problem of this class, as well as for its variant in a control problem of Lagrange type. It is shown that, if one follows this procedure, the solution of the lower closure problem merely lies in proving that a certain “modified Lagrangian” is a normal integrand. A lower closure result is obtained that generalizes, in itself, all comparable results of [2a], [6e–f], [7a], [21]. For control problems of Lagrange type with uniform boundedness conditions on the “controls,” a special approximate Lagrangian can be formulated by which the deparametrization procedure yields results that are superior to those obtained previously [2b], [7b–c]. An important novelty in deriving the measurability properties of Lagrangians is also introduced here: it consists of the employment of the Kunugui–Novikov–Stchegolkov projection theorem [4].