The Properties of Continuity and Differentiation of Solution Sets of Lipschetzean Differential Inclusions

Abstract

For a long time the detailed investigation of optimization problems for differential inclusions encountered great principal difficulties for lack of suitable properties of solutions of differential inclusions. F. Clark avoided those difficulties mainly by using his approximated method and Ekeland’s Theorem (see F. Clark [5]). But we have to remark that for general optimization problems Clark’s necessary conditions of optimality contained restrictions on choosing a rather narrow part of the tangent cone, i.e. Clark’s tangent cone. On the other hand, several attempts were made to develop Pon-tryagin’s direct method of variations from optimal control theory (see [10]) to optimization problems with differential inclusions (see [1], [11]). The attempts were not completely successful because to improve Pontryagin’s method the above authors imposed superfluous restrictions on differential inclusions. But nevertheless, it seemed that there has to exist a direct method that could lead to more general results.