JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 6, No, 2, 1970 Seminormality and Upper Semicontinuity in Optimal Control I

Abstract. This paper concerns the concept of upper semicominuity of variable sets, precisely the variant of Kuratowski's definition of upper semicontinuity that Cesari has denoted as property (Q). This concept has been used by Cesari n most of his papers on existence theorems for optimal solutions, and later used by Olech, Lasota and Olech, Brunovsky, Baum, Suryanarayana, nd Angetl. First, criteria are given for property (Q) in addition to those which had been already given previously. Then, it is shown that a slight restriction in the concept can be expressed in a form which is similar to Tonelli's concept of seminormality for free problems of the calculus of variations. Thus, the property (Q) appears to be a generalization to Lagrange problems of control of the well-known concept of seminormality for free problems. 1