Relaxation of optimal control problems driven by nonlinear evolution equations

Leszek Gasiński; Nikolaos S. Papageorgiou

doi:10.3934/eect.2020050

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2020, Volume 9, Issue 4: 1027-1040. Doi: 10.3934/eect.2020050

This issue Previous Article On dynamic contact problem with generalized Coulomb friction, normal compliance and damage Next Article Measurable solutions for elliptic and evolution inclusions

Relaxation of optimal control problems driven by nonlinear evolution equations

Leszek Gasiński^1, , and
Nikolaos S. Papageorgiou^2,

1.
Pedagogical University of Krakow, Department of Mathematics, 2, 30-084 Kraków, Poland
2.
National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece

^* Corresponding author
^* Corresponding author

Dedicated to Professor Meir Shillor on the occasion of his 70th birthday

Received: October 2019

Early access: March 2020

Published: December 2020

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Abstract

We consider a nonlinear optimal control problem with dynamics described by a nonlinear evolution equation defined on an evolution triple of spaces. Both the dynamics and the cost functional are not convex and so an optimal pair need not exist. For this reason using tools from multivalued analysis and from convex analysis, we introduce a relaxed version of the problem. No Young measures are involved in our relaxation method. We show that the relaxed problem is admissible.

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Mathematics Subject Classification: 49J20, 49J35, 34G20.

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