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April  2020, 19(4): 2101-2126. doi: 10.3934/cpaa.2020093

## Theoretical and numerical results for some bi-objective optimal control problems

 1 Departamento EDAN and IMUS, Universidad de Sevilla, Campus Reina Mercedes, Sevilla, SPAIN, 41012 2 Departamento EDAN, Universidad de Sevilla, Campus Reina Mercedes Sevilla, SPAIN, 41012

Decicated to Prof. T. Caraballo for his 60th birthday

Received  December 2018 Revised  April 2019 Published  January 2020

Fund Project: The first author is supported by MINECO (Spain), grant MTM2016-76990-P. The second author is supported by CEI (Junta de Andalucía, Spain), Grant FQM-131, FPU 2017 (Ministerio de Educación, Spain).

This article deals with the solution of some multi-objective optimal control problems for several PDEs: linear and semilinear elliptic equations and stationary Navier-Stokes systems. More precisely, we look for Pareto equilibria associated to standard cost functionals. First, we study the linear and semilinear cases. We prove the existence of equilibria, we deduce appropriate optimality systems, we present some iterative algorithms and we establish convergence results. Then, we analyze the existence and characterization of Pareto equilibria for the Navier-Stokes equations. Here, we use the formalism of Dubovitskii and Milyutin. In this framework, we also present a finite element approximation of the bi-objective problem and we illustrate the techniques with several numerical experiments.

Citation: Enrique Fernández-Cara, Irene Marín-Gayte. Theoretical and numerical results for some bi-objective optimal control problems. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2101-2126. doi: 10.3934/cpaa.2020093
##### References:

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##### References:
The domain and the "rough" mesh; $\Omega$ is composed of the band $\omega$, the large rectangle $\mathcal{O}_1$ and the small rectangle $\mathcal{O}_2$. Number of nodes: 1519. Number of triangles: 2876
The function $u_{1d}$
The final velocity fields computed with ALG 8 for various $\alpha$ in the case $a = 1.5$ and $\nu = 0.06$. Number of nodes: 3449. Number of triangles: 6658
The final velocity field, the adjoint and the control computed with ALG 9 for $\alpha = 0.5$ in the case $a = 0.8$ and $\nu = 0.1$. Number of nodes: 6003. Number of triangles: 11684
The final velocity field, the adjoint and the control computed with ALG 10 for $\alpha = 0.5$ in the case $a = 1.8$ and $\nu = 0.00204$. Number of nodes: 1519. Number of triangles: 2876
The logarithms of the functionals $J_1$, $J_2$ for $\alpha = 0.5$ and $J_{(\alpha)}$ for $\alpha = 0.1,$ $0.5$ and $0.9$, with $a = 1.5$ and $\nu = 0.06$. Number of nodes: 1519. Number of triangles: 2876
Number of iterates with $\alpha = 0.5, 1519$ nodes and $\varepsilon = 10^{-6}$ for various values of $a$ and $\nu$
Precision in iteration 50 for $\alpha = 0.5$ and $a = 1.5$ (NP is the number of nodes)
Precision in iteration 50 for $\alpha = 0.5$ and $\nu = 0.06$
Computation times (in seconds) and numbers of iterates to reach an error less than $\varepsilon = 10^{-6},$ for $\alpha = 0.5$, $a = 1.5$ and $\nu = 0.06$
The cost $J_{(\alpha)}$ for various methods and parameters
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