SciELO - Scientific Electronic Library Online

vol.18 issue1Creation of a model of pollutans diffusion in soil-water system using a cellular automataA non-standard generating function for continuous dual q-hahn polynomials author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand




Related links

  • Have no similar articlesSimilars in SciELO


Revista de Matemática Teoría y Aplicaciones

Print version ISSN 1409-2433

Rev. Mat vol.18 n.1 San José Jun. 2011


Optimal control of pollution stock through ecological interaction of the manufacturer and the state

Control óptimo de contaminación almacenada a través de interacción ecológica entre el fabricante y el estado

Ellina V. Grigorieva*
Evgenii N. Khailov
E.I. Kharitonova

*Department of Mathematics and Computer Sciences, Texas Woman’s University, Denton, TX 76204, U.S.A. E-Mail:
†Department of Computer Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992, Russia. E-Mail:
‡Misma direccin que/same address as E.N. Khailov.

Dirección para correspondencia


A model of an interaction between a manufacturer and the state where the  manufacturer produces a single product and the state controls the level of pollution is  created and investigated. A local economy with a stock pollution problem that must  choose between productive and environmental investments (control functions) is  considered. The model is described by a nonlinear system of two differential equations  with two bounded controls. The best optimal strategy is found analytically with the use  of the Pontryagin Maximum Principle and Green’s Theorem.

Keywords: optimal control, nonlinear model, environmental problem.


Se ha creado e investigado un modelo de interaccin entre un fabricante y el estado  donde el fabricante produce un solo producto y el estado controla el nivel de  contaminación. Se considera una economía local con un problema de contaminación  almacenada, que debe escoger entre inversiones en producción y medio ambiente  (funciones de control). El modelo es descrito por un sistema de dos ecuaciones  diferenciales con dos controles acotados. La mejor estrategia de control se encuentra  analíticamente usando el Principio del Máximo de Pontryagin y el Teorema de Green.

Palabras clave: control óptimo, modelo no lineal, problema ambiental.

Mathematics Subject Classification: 49J15, 49N90, 93C10, 93C95.

Ver contenido disponible en pdf


[1] Brock, W.; Taylor, M.S. (2005) “Economic growth and the environment: a review of  theory and empirics”, in: S. Durlauf & P. Aghion (Eds.) Handbook of Economic Growth, Elsevier, Amsterdam: 1749–1821.         [ Links ]

[2] World Bank (1992) World Development Report. Oxford University, New York.         [ Links ]

[3] Grossman, G.; Krueger, A. (1995) “Economic growth and the environment”,  Quarterly Journal of Economics 110: 353–377.         [ Links ]

[4] Cabo, F.; Escudero, E.: Martin-Herran, G. (2006) “Time consistent agreement in an  interregional differential game on pollution and trade”, International Game Theory Review 8(3): 369–393.         [ Links ]

[5] Jorgensen, S.; Zaccour, G. (2001) “Time consistent side payments in a dynamic game of downstream pollution”, Journal of Economic Dynamics and Control 25(2): 1973–1987.         [ Links ]

[6] Jorgensen, S.; Zaccour, G. (2003) “Agreeability and time-consistency in linear-state differential games”, Journal of Optimization Theory and Applications 119(1): 49–63.         [ Links ]

[7] Carraro, C. (1999) Envinonmental Conflict, Bargaining and Cooperation, Handbook  of Environment and Resource Economics. Edward Elgar, Cheltenham.         [ Links ]

[8] Chimeli, A.; Braden, J.B. (2001) “Economic growth and the dynamics of  environmental quality”, Encontro Brasileiro de Econometria 23: 379–398.         [ Links ]

[9] Holmaker, K.; Sterner, T. (1999) “Growth or environmental concern: which comes first? Optimal control with pure stock pollutants”, Environmental Economics and Policy  Studies 2: 167–185.         [ Links ]

[10] Keeler, E.; Spence, M.: Zeckhauser, R. (1971) “The optimal control of pollution”, Journal of Economic Theory 4: 19–34.         [ Links ]

[11] del Brio, A.; Fernandez, E. (2007) “Customer interaction in environmental  innovation: the case of cloth diaper laundering”, Service Business 1(2): 141–158.         [ Links ]

[12] Sethi, S.; Thompson, G. (2003) Optimal Control Theory: Application to Management Science and Economics. Kluwer Academic Publishers, Boston-Dordrecht-London.         [ Links ]

[13] Dockner, E.; Jorgensen, S. (2006) Differential Games in Economics and  Management Science. Cambridge University Press, Cambridge.         [ Links ]

[14] Filippov, A.F. (1962) “On certain questions in the theory of optimal control, SIAM Journal on Control 1: 76–84.         [ Links ]

[15] Lee, E.B.; Marcus, L. (1967) Foundations of Optimal Control Theory. John Wiley & Sons, New York.         [ Links ]

[16] Bonnard, B.; Chyba, M. (2003) Singular Trajectories and their Role in Conrol Theory. Springer-Verlag, Berlin-Heidelberg-New York.         [ Links ]

[17] Hajek, O. (1991) Control Theory in the Plane, Lecture Notes in Control and  Information Science 153. Springer-Verlag, Berlin-Heidelberg-New York.         [ Links ]

[18] Krabs, W. (1979) Optimization and Approximation. John Wiley & Sons, New York.         [ Links ]

[19] Mangasarian, O.L. (1994) Nonlinear Programming. SIAM, Philadelphia.         [ Links ]

Correspondencia a: Ellina V. Grigorieva. Department of Mathematics and Computer Sciences, Texas Woman’s University, Denton, TX 76204, U.S.A. E-Mail:
Evgenii N. Khailov. Department of Computer Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992, Russia. E-Mail:
E.I. Kharitonova. Misma direccin que/same address as E.N. Khailov

Received: 24 Sep 2010; Revised: 19 Nov 2010; Accepted: 26 Nov 2010

Creative Commons License All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License