Fine metrizable convex relaxations of parabolic optimal control problems

TL;DR

This paper introduces a new convex relaxation method for parabolic optimal control problems in infinite-dimensional spaces, enabling better analysis and solution existence by balancing the coarseness and fineness of the relaxation.

Contribution

It develops a novel metrizable convex compactification combining Young measures and Choquet theory for improved control problem analysis.

Findings

Ensures existence of solutions for parabolic optimal control problems.

Provides optimality conditions within the new relaxation framework.

Balances nonlinearity handling with topological properties.

Abstract

Nonconvex optimal-control problems governed by evolution problems in infinite-dimensional spaces (as e.g. parabolic boundary-value problems) needs a continuous (and possibly also smooth) extension on some (preferably convex) compactification, called relaxation, to guarantee existence of their solutions and to facilitate analysis by relatively conventional tools. When the control is valued in some subsets of Lebesgue spaces, the usual extensions are either too coarse (allowing in fact only very restricted nonlinearities) or too fine (being nonmetrizable). To overcome these drawbacks, a compromising convex compactification is here devised, combining classical techniques for Young measures with Choquet theory. This is applied to parabolic optimal control problems as far as existence and optimality conditions concerns.