Discrete Approximations and Refined Euler–Lagrange Conditions for Nonconvex Differential Inclusions

Abstract

This paper deals with the Bolza problem $(P)$ for differential inclusions subject to general endpoint constraints. We pursue a twofold goal. First, we develop a finite difference method for studying $(P)$ and construct a discrete approximation to $(P)$ that ensures a strong convergence of optimal solutions. Second, we use this direct method to obtain necessary optimality conditions in a refined Euler--Lagrange form without standard convexity assumptions. In general, we prove necessary conditions for the so-called intermediate relaxed local minimum that takes an intermediate place between the classical concepts of strong and weak minima. In the case of a Mayer cost functional or boundary solutions to differential inclusions, this Euler--Lagrange form holds without any relaxation. The results obtained are expressed in terms of nonconvex-valued generalized differentiation constructions for nonsmooth mappings and sets.