Existence and relaxation for finite-dimensional optimal control problems driven by maximal monotone operators
type: Item
instance of: scholarly article
title: Existence and relaxation for finite-dimensional optimal control problems driven by maximal monotone operators
zbMATH Open document ID: 1040.49021
DOI: 10.4171/ZAA/1177
author: Nikolaos S. Papageorgiou
author: Francesca Papalini
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2004-06-15T00:00:00Z
review text: Summary: In this paper we study the optimal control of a class of nonlinear finite-dimensional optimal control problems driven by a maximal monotone operator which is not necessarily everywhere defined. So our model problem incorporates systems monitored by variational inequalities. First we prove an existence theorem using the reduction method of Berkovitz and Cesàri. This requires a convexity hypothesis. When this convexity condition is not satisfied, we have to pass to an augmented, convexified problem known as the ``relaxed problem''. We present four relaxation methods. The first uses Young measures, the second uses multi-valued dynamics, the third is based on Carathéodory's theorem for convex sets in \(\mathbb{R}^N\) and the fourth uses lower semicontinuity arguments and \(\Gamma\)-limits. We show that they are equivalent and admissible, which roughly speaking means that the corresponding relaxed problem is in a sense the ``closure'' of the original one.
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 49J24
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 2075234
zbMATH Keywords: maximal monotone operator
zbMATH Keywords: variational inequalities
zbMATH Keywords: reduction method
zbMATH Keywords: Young measure
zbMATH Keywords: Carathéodory theorem
zbMATH Keywords: multi-valued dynamics
zbMATH Keywords: regularization
zbMATH Keywords: admissible relaxation
zbMATH Keywords: Gamma-limit
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2075234