no. 1
Nash equilibrium for a multiobjective control problem related to wastewater management
García-Chan, Néstor ; Muñoz-Sola, Rafael ; Vázquez-Méndez, Miguel Ernesto1
1 Departamento de Matemática Aplicada, Universidad de Santiago de Compostela, Escola Politécnica Superior, 27002 Lugo, Spain.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 117-138.

This paper is concerned with mathematical modelling in the management of a wastewater treatment system. The problem is formulated as looking for a Nash equilibrium of a multiobjective pointwise control problem of a parabolic equation. Existence of solution is proved and a first order optimality system is obtained. Moreover, a numerical method to solve this system is detailed and numerical results are shown in a realistic situation posed in the estuary of Vigo (Spain).

DOI : 10.1051/cocv:2008019
Classification : 49J20, 49K20, 90C29, 91B76
Mots clés : optimal control, pointwise control, Nash equilibrium, existence, optimality conditions, numerical simulation, wastewater management
García-Chan, Néstor  ; Muñoz-Sola, Rafael  ; Vázquez-Méndez, Miguel Ernesto 1

1 Departamento de Matemática Aplicada, Universidad de Santiago de Compostela, Escola Politécnica Superior, 27002 Lugo, Spain. 
@article{COCV_2009__15_1_117_0,
     author = {Garc{\'\i}a-Chan, N\'estor and Mu\~noz-Sola, Rafael and V\'azquez-M\'endez, Miguel Ernesto},
     title = {Nash equilibrium for a multiobjective control problem related to wastewater management},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {117--138},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {1},
     year = {2009},
     doi = {10.1051/cocv:2008019},
     mrnumber = {2488571},
     zbl = {1155.49002},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2008019/}
}
TY  - JOUR
AU  - García-Chan, Néstor
AU  - Muñoz-Sola, Rafael
AU  - Vázquez-Méndez, Miguel Ernesto
TI  - Nash equilibrium for a multiobjective control problem related to wastewater management
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2009
SP  - 117
EP  - 138
VL  - 15
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2008019/
DO  - 10.1051/cocv:2008019
LA  - en
ID  - COCV_2009__15_1_117_0
ER  - 
%0 Journal Article
%A García-Chan, Néstor
%A Muñoz-Sola, Rafael
%A Vázquez-Méndez, Miguel Ernesto
%T Nash equilibrium for a multiobjective control problem related to wastewater management
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2009
%P 117-138
%V 15
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2008019/
%R 10.1051/cocv:2008019
%G en
%F COCV_2009__15_1_117_0
García-Chan, Néstor; Muñoz-Sola, Rafael; Vázquez-Méndez, Miguel Ernesto. Nash equilibrium for a multiobjective control problem related to wastewater management. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 117-138. doi : 10.1051/cocv:2008019. http://archive.numdam.org/articles/10.1051/cocv:2008019/

[1] L.J. Álvarez-Vázquez, A. Martínez, C. Rodríguez and M.E. Vázquez-Méndez, Numerical convergence for a sewage disposal problem. Appl. Math. Model. 25 (2001) 1015-1024. | Zbl

[2] L.J. Álvarez-Vázquez, A. Martínez, C. Rodríguez and M.E. Vázquez-Méndez, Numerical optimization for the location of wastewater outfalls. Comput. Optim. Appl. 22 (2002) 399-417. | MR | Zbl

[3] L.J. Álvarez-Vázquez, A. Martínez, C. Rodríguez and M.E. Vázquez-Méndez, Mathematical model for optimal control in wastewater discharges: the global performance. C. R. Biologies 328 (2005) 327-336.

[4] L.J. Álvarez-Vázquez, A. Martínez, R. Muñoz-Sola, C. Rodríguez and M.E. Vázquez-Méndez, The water conveyance problem: Optimal purification of polluted waters. Math. Models Meth. Appl. Sci. 15 (2005) 1393-1416. | MR | Zbl

[5] A. Bermúdez, Numerical modelling of water pollution problems, in Environment, Economics and their Mathematical Models, J.I. Diaz and J.L. Lions Eds., Masson, Paris (1994). | Zbl

[6] A. Bermúdez, C. Rodríguez and M.A. Vilar, Solving shallow water equations by a mixed implicit finite element method. IMA J. Num. Anal. 11 (1991) 79-97. | MR | Zbl

[7] E. Casas, Pontryagin's principle for state constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35 (1997) 1297-1327. | MR | Zbl

[8] R. Gibbons, A Primer in Game Theory. Pearson Higher Education (1992). | Zbl

[9] O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural'Ceva, Linear and quasilinear equations of parabolic type, in Translations of Mathematical Monographs 23, Amer. Math. Soc., Providence (1968). | MR | Zbl

[10] J.L. Lions, Contrôle optimal des systèmes gouvernés par des équations aux derivées partielles. Dunod, Paris (1968). | MR | Zbl

[11] J.L. Lions and E. Magenes, Problèmes aux limites non homogenes et applications. Dunod, Paris (1968). | Zbl

[12] A. Martínez, C. Rodríguez and M.E. Vázquez-Méndez, Theoretical and numerical analysis of an optimal control problem related to wastewater treatment. SIAM J. Control Optim. 38 (2000) 1534-1553. | Zbl

[13] D. Parra-Guevara and Yn. Skiba, Elements of the mathematical modeling in the control of pollutants emissions. Ecol. Model. 167 (2003) 263-275.

[14] O. Pironneau, Finite Element Methods for Fluids. J. Wiley & Sons, Chichester (1989). | MR | Zbl

[15] A.M. Ramos, R. Glowinski and J. Periaux, Nash equilibria for the multiobjetive control of linear partial differential equations. J. Optim. Theory Appl. 112 (2002) 457-498. | MR | Zbl

[16] E. Zeidler, Nonlinear Functional Analysis and its Applications. Springer-Verlag (1993).

Cité par Sources :