8. Existence of Solutions for Variational Problems
Society for Industrial and Applied Mathematics eBooks · 1999
Previous chapter Next chapter Classics in Applied Mathematics Convex Analysis and Variational Problems8. Existence of Solutions for Variational Problemspp.231 - 262Chapter DOI:https://doi.org/10.1137/1.9781611971088.ch8PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutExcerpt Orientation In this chapter, we shall study non-convex problems. In Section 1, we shall introduce normal (not necessarily convex) integrands, an important class of functions of two variables without any convexity; we shall establish their main properties, including a measurable selection theorem, and we shall recall the characterization of weakly relatively compact subsets of L1 . In Section 2, these results will be applied to the study of a non-convex optimization problem, and a sufficient condition for the existence of solutions will be given. The final sections will show that a number of problems in the calculus of variations (Section 3) and in optimal control (Section 4) can be put into the above form and from this we deduce theorems on the existence of solutions. Previous chapter Next chapter RelatedDetails Published:1999ISBN:978-0-89871-450-0eISBN:978-1-61197-108-8 https://doi.org/10.1137/1.9781611971088Book Series Name:Classics in Applied MathematicsBook Code:CL28Book Pages:xiv + 394Key words:convex analysis, relaxation, non-convex, variational problems, duality, minimax theorem