## Artigo

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## versão impressa ISSN 1409-2433

### Rev. Mat vol.20 no.2 San José Jul./Dez. 2013

Analysis of optimal control problems for the process of wastewater biological treatment

Análisis de problemas de control óptimo para el proceso de tratamiento biológico de aguas residuales

Elina V. Grigorieva*+, Natalia V. Bondarenko+, Evgenii N. Khailov+, Andrei Korobeinikov§+

*Dirección para correspondencia:

Abstract

We consider a three-dimensional deterministic control model of the process of aerobic wastewater biotreatment. For this model, we formulate and solve two optimal control problems, each of which has a corresponding minimizing functional. For the first problem, the functional is a weighted sum of the pollutant concentration at the end of a fixed time interval and the cumulative biomass  concentration over the interval. For the second problem, the functional is a weighted sum of the pollutant concentration at the end of the time interval and the cumulative oxygen and biomass concentrations over the interval. In order to solve these problems, we apply the Pontryagin Maximum Principle. The switching functions are analytically investigated and uniquely determine the type of the optimal controls for the considered problems. Their properties allow the simplification of the optimal control problems to that of finitedimensional constrained minimization. Numerical solutions of the optimal control problems are also provided.

Keywords: wastewater treatment, nonlinear model, optimal control.

Resumen

Consideramos un modelo de control determinístico tridimensional del proceso de biotratamiento aeróbico de aguas residuales. Para este modelo, formulamos y resolvemos dos problemas de control óptimo, cada uno de los cuales tiene un funcional a minimizar. Para el primer problema, el funcional es una suma ponderada de la concentración del contaminante al final de un intervalo de tiempo fijo y la concentración acumulada de la biomasa sobre el intervalo. Para el segundo problema, el funcional es una suma ponderada de la concentración del contaminante al final del intervalo de tiempo y las concentraciones acumuladas de oxígeno y biomasa sobre el intervalo. Para resolver estos problemas, aplicamos el Principio del Máximo de Pontryagin. Las funciones de conmutación son investigadas analíticamente y determinan unívocamente el tipo de controles óptimos para los problemas considerados. Sus propiedades permiten la simplificación de los problemas de control óptimo para una minimización finitodimensional con restricciones. Se brindan las soluciones numéricas de los problemas de control óptimo.

Palabras clave: tratamiento de aguas residuales, modelo no lineal, control óptimo.

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

Contenido disponible en pdf

References

[1] Bondarenko, N.V.; Grigorieva, E.V.; Khailov, E.N. (2010) “Attainable set of three-dimensional nonlinear system describing the wastewater treatment process”, in: Yu.S. Osipov & A.V. Kryazhimskii (Eds.) Problems of Dynamical Control, 5, MAX Press, Moscow: 28–41.         [ Links ]

[2] Gomez, J.; de Gracia, M.; Ayesa, E.; Garcia-Heras, J.L. (2007) “Mathematical modelling of autothermal thermophilic aerobic digesters”, Water Research 41(5): 959–968.         [ Links ]

[3] Grigorieva, E.; Bondarenko, N.; Khailov, E.; Korobeinikov, A. (2012) “Finite-dimensional methods for optimal control of autothermal thermophilic aerobic digestion”, in: K.Y. Show & X. Guo (Eds.) Industrial Waste, InTech, Croatia: 91–120.         [ Links ]

[4] Grigorieva, E.V.; Bondarenko, N.V.; Khailov, E.N.; Korobeinikov, A. (2012) “Three-dimensional nonlinear control model of wastewater biotreatment”, Neural, Parallel, and Scientific Computations 20: 23–36.         [ Links ]

[5] Grigorieva, E.V.; Khailov, E.N.; Korobeinikov, A. (2012) “Reduction of the operation cost via optimal control of an industrial wastewater biotreatment process”, in: http://jointmathematicsmeetings.org/amsmtgs/2138 abstracts/1077-g5-1378.pdf.         [ Links ]

[6] Krasnov, K.S.; Vorob’ev, N.K.; Godnev, I.N.; et al. (1995) Physical Chemestry 2. Vysshaya Shkola, Moscow.         [ Links ]

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

[8] Pontryagin, L.S.; Boltyanskii, V.G.; Gamkrelidze, R.V.; Mishchenko, E.F. (1962) Mathematical Theory of Optimal Processes. John Wiley & Sons, New York.         [ Links ]

[9] Rojas J.; Burke, M.; Chapwanya, M.; Doherty, K.; Hewitt, I.; Korobeinikov, A.; Meere, M.; McCarthy, S.; O’Brien, M.; Tuoi, V.T.N.; Winstenley, H.; Zhelev, T. (2010) “Modeling of autothermal thermophilic aerobic digestion”, Mathematics-in-Industry Case Studies 2: 34–63.         [ Links ]

[10] Vasil’ev, F.P. (2002) Optimization Methods. Factorial Press, Moscow.         [ Links ]

*Correspondencia a:
Elina V. Grigorieva. Department of Mathematics and Computer Sciences, Texas Woman’s University, Denton, TX 76204, U.S.A. E-Mail: EGrigorieva@mail.twu.edu
Natalia V. Bondarenko. Department of Computational Mathematics and Cybernetics, Moscow State     Lomonosov University, Moscow, 119992, Russia. E-Mail: nataliabonda@mail.ru
Evgenii N. Khailov.
Same address as/Misma dirección que: N.V. Bondarenko. E-Mail: khailov@cs.msu.su
Andrei Korobeinikov. Centre de Recerca Matemática, Campus de Bellaterra, Edifici C - 08193 Bellaterra, Barcelona, Spain. E-Mail: AKorobeinikov@crm.cat
*Department of Mathematics and Computer Sciences, Texas Woman’s University, Denton, TX 76204, U.S.A. E-Mail: EGrigorieva@mail.twu.edu
Department of Computational Mathematics and Cybernetics, Moscow State     Lomonosov University, Moscow, 119992, Russia. E-Mail: nataliabonda@mail.ru
Same address as/Misma dirección que: N.V. Bondarenko. E-Mail: khailov@cs.msu.su
§Centre de Recerca Matemática, Campus de Bellaterra, Edifici C - 08193 Bellaterra, Barcelona, Spain. E-Mail: AKorobeinikov@crm.cat

Received: 27/Feb/2012; Revised: 16/May/2013; Accepted: 24/May/2013

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