Buttazzo, G. ; Casas, E. ; de Teresa, L. ; Glowinski, R. ; Leugering, G. ; Trélat, E. ; Zhang, X.(éd.)
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.
Accepté le :
Première publication :
Publié le :
Keywords: Elliptic PDEs, Navier-Stokes equations, optimal control, bi-objective problems, Nash equilibria, Dubovitskii-Milyutin formalism
@article{COCV_2021__27_1_A52_0,
author = {Fern\'andez-Cara, E. and Mar{\'\i}n-Gayte, I.},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {Bi-objective optimal control of some {PDEs:} {Nash} equilibria and quasi-equilibria},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021050},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021050/}
}
TY - JOUR AU - Fernández-Cara, E. AU - Marín-Gayte, I. ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - Bi-objective optimal control of some PDEs: Nash equilibria and quasi-equilibria JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021050/ DO - 10.1051/cocv/2021050 LA - en ID - COCV_2021__27_1_A52_0 ER -
%0 Journal Article %A Fernández-Cara, E. %A Marín-Gayte, I. %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T Bi-objective optimal control of some PDEs: Nash equilibria and quasi-equilibria %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021050/ %R 10.1051/cocv/2021050 %G en %F COCV_2021__27_1_A52_0
Fernández-Cara, E.; Marín-Gayte, I. Bi-objective optimal control of some PDEs: Nash equilibria and quasi-equilibria. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 50. doi: 10.1051/cocv/2021050
[1] and , On some control problems in fluid mechanics. Theoret. Comput. Fluid Dyn. 1 (1990) 303–325.
[2] , , and , Multi-objective Pareto-optimal control: an application to wastewater management. Comput. Optim. Appl. Berlin 46 (2010) 135–157.
[3] , and , Some optimal control problems for a two-phase field model of solidification. Rev. Mat. Complut. 23 (2010) 49–75.
[4] , and , An optimal control problem for a generalized Boussinesq Model: the time dependent case. Rev. Mat. Complut. 20 (2007) 339–366.
[5] , Functional Analysis, Sobolev Sapces and Partial Differential Equations. Springer, New York, Dordrecht, Heidelberg, London (2011).
[6] and , On the computation of Nash and pareto equilibria for some bi-objective control problems. J. Sci. Comput. 78 (2019) 246–273.
[7] , Introduction to numerical linear algebra and optimisation. Cambridge University Press, Cambridge (1989).
[8] , Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems. Control Optim. Calc. Variat. 1 (1996) 267–302.
[9] , Differential analysis: Differentiation, differential equations and differential inequalities. Cambridge University Press, Cambridge (1980).
[10] , Optimal Control of Distributed Systems: Theory and Applications. American Mathematical Society Boston, MA, USA (2000).
[11] , Lectures on mathematical theory of extremum problems. Vol. 67 of Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin (1972).
[12] , Finite element methods for incompressible viscous flow. Vol. 9 of Handbook of Numerical Analysis. North-Holland, Amsterdam, (2003) 3–1176.
[13] and , Finite element methods for Navier-Stokes equations. Annu. Rev. Fluid Mech. 24 (1992) 167–204.
[14] , http://www.freefem.org.
[15] , Contrôle de Pareto de systèmes distribués. Le cas d’évolution. C.R. Acad. Sci. Paris, Sr. I 302 (1986) 413–417.
[16] , Optimal control of systems governed by partial differential equations. Springer-Verlag, New York (1971).
[17] , Some remarks on Stackelberg optimization. Math. Models Methods Appl. Sci. 4 (1994) 477–487.
[18] , Noncooperative games. Ann. Math. 54 (1951) 286–295.
[19] , Cours d’économie politique. Rouge, Laussane, Switzerland (1896).
[20] , Optimization. Algorithms and consistent approximation. Springer-Verlag, New York (1997).
[21] , and , Nash equilibria for the multiobjective control of linear partial differential equations. J. Optim. Theory Appl. 112 (2002) 457–498.
[22] , and , Pointwise control of the Burgers equation and related Nash equilibrium problems. Comput. Approach: J. Optim. Theory Appl. 112 (2002) 499–516.
[23] , Navier-Stokes equations. Theory and numerical analysis. AMS Chelsea Pub., Providence (2001).
[24] , Marktform und gleichgewicht. Springer, Berlin, Germany (1934).
Cité par Sources :
Partially financed by MINECO, Grant MTM2016-76990-P.
