# Nash equilibrium for a multiobjective control problem related to wastewater management

Néstor García-Chan; Rafael Muñoz-Sola; Miguel Ernesto Vázquez-Méndez

ESAIM: Control, Optimisation and Calculus of Variations (2009)

- Volume: 15, Issue: 1, page 117-138
- ISSN: 1292-8119

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topGarcía-Chan, Néstor, Muñoz-Sola, Rafael, and Vázquez-Méndez, Miguel Ernesto. "Nash equilibrium for a multiobjective control problem related to wastewater management." ESAIM: Control, Optimisation and Calculus of Variations 15.1 (2009): 117-138. <http://eudml.org/doc/250574>.

@article{García2009,

abstract = {
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).
},

author = {García-Chan, Néstor, Muñoz-Sola, Rafael, Vázquez-Méndez, Miguel Ernesto},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Optimal control; pointwise control; Nash equilibrium; existence; optimality conditions; numerical simulation; wastewater management; optimal control},

language = {eng},

month = {1},

number = {1},

pages = {117-138},

publisher = {EDP Sciences},

title = {Nash equilibrium for a multiobjective control problem related to wastewater management},

url = {http://eudml.org/doc/250574},

volume = {15},

year = {2009},

}

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

DA - 2009/1//

PB - EDP Sciences

VL - 15

IS - 1

SP - 117

EP - 138

AB -
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).

LA - eng

KW - Optimal control; pointwise control; Nash equilibrium; existence; optimality conditions; numerical simulation; wastewater management; optimal control

UR - http://eudml.org/doc/250574

ER -

## References

top- 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. Zbl1197.76080
- 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. Zbl1013.90088
- 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. Biologies328 (2005) 327–336.
- 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. Zbl1089.49037
- 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). Zbl0843.90026
- 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. Zbl0713.76069
- E. Casas, Pontryagin's principle for state constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim.35 (1997) 1297–1327. Zbl0893.49017
- R. Gibbons, A Primer in Game Theory. Pearson Higher Education (1992). Zbl0759.90106
- O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural'ceva, Linear and quasilinear equations of parabolic type, in Translations of Mathematical Monographs23, Amer. Math. Soc., Providence (1968).
- J.L. Lions, Contrôle optimal des systèmes gouvernés par des équations aux derivées partielles. Dunod, Paris (1968). Zbl0179.41801
- J.L. Lions and E. Magenes, Problèmes aux limites non homogenes et applications. Dunod, Paris (1968). Zbl0165.10801
- 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. Zbl0961.49002
- D. Parra-Guevara and YN. Skiba, Elements of the mathematical modeling in the control of pollutants emissions. Ecol. Model.167 (2003) 263–275.
- O. Pironneau, Finite Element Methods for Fluids. J. Wiley & Sons, Chichester (1989). Zbl0665.73059
- 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. Zbl1012.49020
- E. Zeidler, Nonlinear Functional Analysis and its Applications. Springer-Verlag (1993). Zbl0794.47033

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