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

Abstract

top
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).


How to cite

top

García-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
  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.  Zbl1197.76080
  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.  Zbl1013.90088
  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. Biologies328 (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.  Zbl1089.49037
  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).  Zbl0843.90026
  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.  Zbl0713.76069
  7. 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
  8. R. Gibbons, A Primer in Game Theory. Pearson Higher Education (1992).  Zbl0759.90106
  9. 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).  
  10. J.L. Lions, Contrôle optimal des systèmes gouvernés par des équations aux derivées partielles. Dunod, Paris (1968).  Zbl0179.41801
  11. J.L. Lions and E. Magenes, Problèmes aux limites non homogenes et applications. Dunod, Paris (1968).  Zbl0165.10801
  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.  Zbl0961.49002
  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).  Zbl0665.73059
  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.  Zbl1012.49020
  16. E. Zeidler, Nonlinear Functional Analysis and its Applications. Springer-Verlag (1993).  Zbl0794.47033

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.