| Issue |
ESAIM: COCV
Volume 27, 2021
Special issue in honor of Enrique Zuazua's 60th birthday
|
|
|---|---|---|
| Article Number | 50 | |
| Number of page(s) | 30 | |
| DOI | https://doi.org/10.1051/cocv/2021050 | |
| Published online | 04 June 2021 | |
Bi-objective optimal control of some PDEs: Nash equilibria and quasi-equilibria*
1
Dpto. EDAN e IMUS, Universidad de Sevilla, Aptdo. 1160,
41080
Sevilla, Spain.
2
Departamento EDAN, Universidad de Sevilla, Campus Reina Mercedes,
41012
Sevilla, Spain.
** Corresponding author: cara@us.es
Received:
24
April
2020
Accepted:
5
May
2021
This paper deals with the solution of some multi-objective optimal control problems for several PDEs: linear and semilinear elliptic equations and stationary Navier-Stokes systems. Specifically, we look for Nash equilibria associated with standard cost functionals. For linear and semilinear elliptic equations, we prove the existence of equilibria and we deduce related optimality systems. For stationary Navier-Stokes equations, we prove the existence of Nash quasi-equilibria, i.e. solutions to the optimality system. In all cases, we present some iterative algorithms and, in some of them, we establish convergence results. For the existence and characterization of Nash quasi-equilibria in the Navier-Stokes case, we use the formalism of Dubovitskii and Milyutin. In this context, we also present a finite element approximation and we illustrate the techniques with numerical experiments.
Mathematics Subject Classification: 35Q30 / 76D05 / 76N10
Key words: Elliptic PDEs / Navier-Stokes equations / optimal control / bi-objective problems / Nash equilibria / Dubovitskii-Milyutin formalism
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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