Upper semicontinuity properties of set valued functions

IN THE present paper we intend to present lower closure theorems in their generality and to discuss their relevant hypotheses, particularly the role of upper semicontinuity properties of the relevant sets, in the light of recent work of the authors, and of Goodman, Olech, Ioffe and Rockafellar. The uppersemicontinuity property (Q) for closed convex set valued functions y Q(y) c B (Gesari [ 1,2]) is usually expressed in terms of union, intersections and closures on subsets Q(y) of a Banach space B depending on an index y ranging on a subset A of a metric space (I: d):

[1]  L. Cesari.  Existence Theorems for Optimal Controls of the Mayer Type , 1968 .

[2]  A necessary and sufficient condition for lower semicontinuity , 1974 .

[3]  Closure theorems without seminormality conditions , 1975 .

[4]  EXISTENCE THEORY IN OPTIMAL CONTROL PROBLEMS - THE UNDERLYING IDEAS , 1975 .

[5]  Lamberto Cesari,et al.  An existence theorem for pareto problems , 1978 .

[6]  L. Cesari.  Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints ; II: Existence theorems. , 1966 .

[7]  L. Cesari.  Geometric and analytic views in existence theorems for optimal control. II. Distributed and boundary controls , 1975 .

[8]  A. Ioffe.  On Lower Semicontinuity of Integral Functionals. II , 1977 .

[9]  L. Cesari,et al.  Nemitsky's operators and lower closure theorems , 1976 .

[10]  L. Cesari.  Closure theorems for orientor fields and weak convergence , 1974 .

[11]  Closure, Lower Closure, and Semicontinuity Theorems in Optimal Control , 1971 .

[12]  L. Cesari.  Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. I , 1966 .

[13]  G. R. Bates.  Lower closure and existence theorems for optimal control problems with infinite horizon , 1978 .

[14]  Lamberto Cesari,et al.  Existence theorems for Pareto optimization; multivalued and Banach space valued functionals , 1978 .

[15]  C. Olech,et al.  Existence theorems for optimal problems with vector-valued cost function , 1969 .

[16]  Seminormality and upper semicontinuity in optimal control , 1971 .

[17]  C. Castaing,et al.  Convex analysis and measurable multifunctions , 1977 .

[18]  G. S. Goodman.  The duality of convex functions and Cesari's property (Q) , 1976 .

[19]  R. Rockafellar,et al.  Integral functionals, normal integrands and measurable selections , 1976 .

[20]  L. Cesari.  Geometric and analytic views in existence theorems for optimal control in Banach spaces. Part 1. Distributed parameters , 1974 .

[21]  Donald E. Kirk,et al.  Optimal Control Theory , 1970 .