Skip to main content

Advertisement

SpringerLink
Log in

Optimal Buffer Zone for the Control of Groundwater Pollution from Agricultural Activities

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

Abstract

We consider an optimal control model of groundwater pollution due to agricultural activities, the objective of the optimal manager being the optimization of the trade-off between the fertilizer use and the cleaning costs. The size of a buffer zone defined around the water production sites for limiting pollution effects may also be chosen by the manager. The spread of the pollution from crops to captation wells is modeled using a convection–diffusion–reaction equation. The main hydrogeological features of the dynamics are taken into account: the process is convection-dominated, dispersion effects are included, generic nonlinear reaction terms are considered. The existence and the uniqueness of the optimal solution is proven. Using asymptotic analysis, we then rigorously prove that a one-dimensional static model can be substituted to the full dynamic 3D model for the long time study of the optimal solution. We prove that this new optimal control problem is also well-posed. All these theoretical results are used for characterizing the optimal buffer zone.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. The continuity can be relaxed by assuming that f is a bounded upper semi-continuous function.

  2. Here \(u\otimes v\) denotes the tensor product, \((u\otimes v)_{ij}=u_{i}v_{j}\).

  3. The work [25] only deals with the case \({\underline{p}} =0\) in (1). Nevertheless, a slight modification of the proof of the maximum principle for the concentration c gives Proposition 1.

  4. We use the superscript diamond for the dimensional form of the equations.

  5. The better regularity of \(c^*\), compared to the one stated in Proposition 3, comes obviously from the additional assumptions of the present subsection.

  6. The terminology ‘dual problem’ is the one used in Benosman et al. [30] and Barbu and Iannelli [31].

  7. All along the paper, we denote the subsequences the same for the sake of the simplicity.

  8. Bear in mind that the elliptic equation completed by a Neumann boundary condition defining \(\phi _\epsilon \) may be completed by a prescription of the mean value of \(\phi _\epsilon \) for ensuring the uniqueness of its solution.

References

  1. Commissariat Général au Développement durable. L’eau et les milieux aquatiques - Chiffres clés - Edition 2016. Repères (2016)

  2. CNRS, dossiers scientifiques. L’eau ; Dégradations ; La pollution par les nitrates. http://www.cnrs.fr/cw/dossiers/doseau/-decouv/degradation/07_pollution.htm

  3. Lankoski, J., Ollikainen, M.: Innovations in nonpoint source pollution policy—European perspectives. Mag. Food Farm Resour. 28(3), 1–5 (2013)

    Google Scholar 

  4. Miquel, G., de Marsily, G.: Rapport de l’Office parlementaire d’évaluation des choix scientifiques et technologiques (2003)

  5. Correll, D.L.: Buffer zones and water quality protection: general principles. Buffer zones: their processes and potential in water protection, Quest Environmental, Harpenden, pp. 7–20 (1996)

  6. Leandri, M.: The shadow price of assimilative capacity in optimal flow pollution control. Ecol. Econ. 68(4), 1020–1031 (2009)

    Article  Google Scholar 

  7. Keeler, E., Spence, M., Zeckhauser, R.: The optimal control of pollution. J. Econ. Theory 4, 19–34 (1972). https://doi.org/10.1016/0022-0531(72)90159-7

    Article  MATH  Google Scholar 

  8. Kossioris, G., Plexousakis, M., Xepapadeas, A., De Zeeuw, A.J., Maler, G.: Feedback Nash equilibria for non-linear differential games in pollution control. J. Econ. Dyn. Control 32(4), 1312–1331 (2008)

    Article  MathSciNet  Google Scholar 

  9. Kim, C., Hostetler, J., Amacher, G.: The regulation of groundwater quality with delayed responses. Water Resour. Res. 29(5), 1369–1377 (1993). https://doi.org/10.1029/93WR00287

    Article  Google Scholar 

  10. Lalbat, F., Blavoux, B., Banton, O.: Description of a simple hydrochemical indicator to estimate groundwater residence time in carbonate aquifers. Geophys. Res. Lett. (2007). https://doi.org/10.1029/2007GL031390

  11. Winkler, R.: Optimal control of pollutants with delayed stock accumulation, CER-ETH Working Paper 08/91 (2008)

  12. Winkler, R.: A note on the optimal control of stocks accumulating with a delay. Macroecon. Dyn. 15, 565–578 (2011). https://doi.org/10.1017/S1365100510000234

    Article  MATH  Google Scholar 

  13. Augeraud-Véron, E., Leandri, M.: Optimal pollution control with distributed delays. J. Math. Econ. 55, 24–32 (2014). https://doi.org/10.1016/j.jmateco.2014.09.010

    Article  MathSciNet  MATH  Google Scholar 

  14. Brock, W., Xepapadeas, A.: Diffusion induced instability and pattern formation in infinite horizon recursive optimal control. J. Econ. Dyn. Control 32, 2745–2787 (2008). https://doi.org/10.2139/ssrn.895682

    Article  MathSciNet  MATH  Google Scholar 

  15. de Frutos, J., Martin-Herran, G.: Spatial effects and strategic behavior in a multiregional transboundary pollution dynamic game. J. Environ. Econ. Manag. (2017). https://doi.org/10.1016/j.jeem.2017.08.001

  16. Lions, J.L.: Asymptotic problems in distributed systems. Riviere Meml. Lect. IMA Preprint Ser. 147, 241–258 (1985). https://doi.org/10.1007/978-1-4613-8704-6_14

    Article  Google Scholar 

  17. Fonseca, I., Francfort, G.: 3D–2D asymptotic analysis of an optimal design problem for thin films. Journal für die reine und angewandte Mathematik (Crelles Journal) 505, 173–202 (1998)

    Article  MathSciNet  Google Scholar 

  18. Kogut, P.I ., Leugering, G.R.: Optimal control problems for partial differential equations on Reticulated domains, approximation and asymptotic analysis, systems and control: foundations and applications, chap. 9 and 10, Birkhaüser, (2011). https://doi.org/10.1007/978-0-8176-8149-4

  19. Casado-Díaz, J., Couce-Calvo, J., Martin-Gómez, J.D.: A rigorous asymptotic expansion for a control problem in a thin domain. Asympt. Anal. 50(1–2), 69–91 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Porretta, A., Zuazua, E.: Long time versus steady state optimal control. SIAM J. Control Optim. 51(6), 4242–4273 (2013). https://doi.org/10.1137/130907239

    Article  MathSciNet  MATH  Google Scholar 

  21. Abbott, M.B., Bathurst, J.C., Cunge, J.A., O’Connell, P.E., Rasmussen, J.: An introduction to the European hydrological system “she” 2: structure of a physically-based, distributed modelling system. J. Hydrol. 87(1), 61–77 (1986). https://doi.org/10.1016/0022-1694(86)90114-9

    Article  Google Scholar 

  22. Willey, R.W.: Resource use in intercropping systems. Agric. Water Manag. 17, 215–231 (1990)

    Article  Google Scholar 

  23. GEVES, French Variety and Seed Study and Control Group, Service plants. https://www.geves.fr/variety-seed-expertise/service-plants/service-plants/

  24. Louison, L.: Modeling plant nutrient uptake: mathematical analysis and optimal control. Evol. Equ. Control Theory 4(2), 193–203 (2015). https://doi.org/10.3934/eect.2015.4.193

    Article  MathSciNet  MATH  Google Scholar 

  25. Augeraud-Véron, E., Choquet, C., Comte, É.: Optimal control for a groundwater pollution ruled by a convection-diffusion-reaction problem. J. Optim. Theory Appl. 173(3), 941–966 (2017). https://doi.org/10.1007/s10957-016-1017-8

    Article  MathSciNet  MATH  Google Scholar 

  26. Scheidegger, A.E.: The Physics of Flow through Porous Media. Univ. Toronto Press, Toronto (1974)

    MATH  Google Scholar 

  27. Bear, J., Verruijt, A.: Modeling Groundwater Flow and Pollution, Theory and Applications of Transport in Porous Media. Springer, New York (1987)

    Book  Google Scholar 

  28. The economics of shallow lakes. The Economics of Non-Convex Ecosystems, pp. 105–126. Springer, Amsterdam (2004)

  29. Comte, É.: Pollution agricole des ressources en eau : approches couplées hydrogéologique et économique, PhD Thesis, La Rochelle University (2017)

  30. Benosman, C., Aïnseba, B.E., Ducrot, A.: Optimization of cytostatic leukemia therapy in an advection-reaction-diffusion model. J. Optim. Theory Appl. 167(1), 296–325 (2015)

    Article  MathSciNet  Google Scholar 

  31. Barbu, V., Iannelli, M.: Optimal control of population dynamics. J. Optim. Theory Appl. 102, 1–14 (1999)

    Article  MathSciNet  Google Scholar 

  32. Kennedy, J.O.: Dynamic programming: applications to agriculture and natural resources. Springer, Berlin (2012)

  33. Godart, C., Jayet, P.A., Niang, B., Bamiere, L., De Cara, S., Debove, E., Baranger, E.: Rapport Final—APR GICC 2002, Convention de recherche MEDD num 02-00019 Institut National de la Recherche Agronomique, UMR Economie Publique INRA-INAPG (2002)

  34. Ledoux, E., Gomez, E., Monget, J.M., Viavattene, C., Viennot, P., Ducharne, A., Benoit, M., Mignolet, C., Schott, C., Mary, B.: Agriculture and groundwater nitrate contamination in the Seine basin. The STICS-MODCOU modelling chain. Sci. Total Environ. 375, 33–47 (2007). https://doi.org/10.1016/j.scitotenv.2006.12.002

    Article  Google Scholar 

  35. Ann. Mat. Pura Appl. Compact sets in the space \(L^p(0,T;B)\). CXLVI, 65–96 (1987)

  36. Tartar, L.C.: Compensated compactness and applications to partial differential equations, In: R. J. Knops, Ed., Research Notes in Mathematics, Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, Vol. 4, Pitman Press, London (1979)

  37. Rockafellar, R.T.: Integrals which are convex functionals. Pac. J. Math. 24(3), 525–539 (1968)

    Article  MathSciNet  Google Scholar 

  38. Stefanelli, U.: The Brezis-Ekeland principle for doubly nonlinear equations. SIAM J. Control Optim. 47(3), 1615–1642 (2008). https://doi.org/10.1137/070684574

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Bordeaux, GREThA, Avenue Léon Duguit, 33608, Pessac Cedex, France

    Emmanuelle Augeraud-Véron

  2. La Rochelle Université, MIA, 23, Avenue A. Einstein, BP 33060, 17031, La Rochelle, France

    Catherine Choquet

  3. Université de Lorraine, CNRS, Inria, IECL, 54000, Nancy, France

    Éloïse Comte

Authors
  1. Emmanuelle Augeraud-Véron
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Catherine Choquet
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Éloïse Comte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Catherine Choquet.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Augeraud-Véron, E., Choquet, C. & Comte, É. Optimal Buffer Zone for the Control of Groundwater Pollution from Agricultural Activities. Appl Math Optim 84, 51–83 (2021). https://doi.org/10.1007/s00245-019-09638-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-019-09638-2

Keywords

Mathematics Subject Classification

Search

Navigation