Closure, Lower Closure, and Semicontinuity Theorems in Optimal Control

First we discuss in detail the concept of lower closure for Lagrange problems, and we extend in various ways previous closure and lower closure theorems. In particular, we show that the present lower closure theorems and related concepts are extensions to Lagrange problems of optimal control of well-known semicontinuity theorems for free problems of the calculus of variations and the related concept of seminormality. Finally, we prove by a new approach that the convexity condition usually requested in lower closure theorems is, in a suitable sense, both a necessary and sufficient condition for lower closure.