Seminormality properties of convex sets

In discussing Lagrange problems of optimal control for simple as well as for multiple integrals Cesari has introduced an upper semicontinuity property of variable sets called property (Q) which plays a role analogous to that of Tonelli's and McShane's concept, of seminormality for free problems of the calculus of variations. This paper deals with analytical criteria for property (Q) which is the unifying idea in the study of lower semicontinuity and lower closure with unbounded controls. In section 1 we state the concepts of seminormality, normality, and property (Q). In section 2 we establish new criteria for property (Q) in the particular situation whenf ₀(t,x,u) is continuous and seminormal (or normal) andf(t,x,u) is linear in the variableu. In section 3 we consider the role of property (Q) for restricted sets. In section 4 we discuss the intermediate properties (Q _p).