Convex Integral Functionals and Duality

Abstract

This chapter discusses convex integral functionals and duality. Recent work on convex integral functionals has been motivated by broader applications to the extremum problems and variational principles. Many problems in the calculus of variations or optimal control involve extended-real-valued functionals where z: [a, b]→ Z is a curve in a linear space Z. The notion of continuous addition of convex functions provides motivation for a general theory of convex integral functional. Duality has always played a fundamental role in the analysis of convex integral functionals. General convex integral functionals conjugate to each other were first investigated for X = Rn. The chapter also presents some of the basic theorems about convex integral functionals in a more general form.