Seminormality and upper semicontinuity in optimal control

This paper concerns the concept of upper semicontinuity 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 in most of his papers on existence theorems for optimal solutions, and later used by Olech, Lasota and Olech, Brunovsky, Baum, Suryanarayana, and Angell. 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.This research was partially supported by AFOSR Research Project No. 69-1662.