Analysis of a time optimal control problem related to the management of a bioreactor
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, p. 722-748

We consider a time optimal control problem arisen from the optimal management of a bioreactor devoted to the treatment of eutrophicated water. We formulate this realistic problem as a state-control constrained time optimal control problem. After analyzing the state system (a complex system of coupled partial differential equations with non-smooth coefficients for advection-diffusion-reaction with Michaelis-Menten kinetics, modelling the eutrophication processes) we demonstrate the existence of, at least, an optimal solution. Then we present a detailed derivation of a first order optimality condition (involving the corresponding adjoint systems) characterizing these optimal solutions. Finally, a numerical example is shown.

DOI : https://doi.org/10.1051/cocv/2010020
Classification:  35D05,  49J20,  93C20
Keywords: time optimal control, partial differential equations, optimality conditions, existence, bioreactor
@article{COCV_2011__17_3_722_0,
author = {Alvarez-V\'azquez, Lino J. and Fern\'andez, Francisco J. and Mart\'\i nez, Aurea},
title = {Analysis of a time optimal control problem related to the management of a bioreactor},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {17},
number = {3},
year = {2011},
pages = {722-748},
doi = {10.1051/cocv/2010020},
zbl = {1230.49002},
mrnumber = {2826977},
language = {en},
url = {http://www.numdam.org/item/COCV_2011__17_3_722_0}
}

Alvarez-Vázquez, Lino J.; Fernández, Francisco J.; Martínez, Aurea. Analysis of a time optimal control problem related to the management of a bioreactor. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, pp. 722-748. doi : 10.1051/cocv/2010020. http://www.numdam.org/item/COCV_2011__17_3_722_0/

[1] W. Allegretto, C. Mocenni and A. Vicino, Periodic solutions in modelling lagoon ecological interactions. J. Math. Biol. 51 (2005) 367-388. | MR 2213040 | Zbl 1087.92056

[2] L.J. Alvarez-Vázquez, F.J. Fernández and R. Muñoz-Sola, Analysis of a multistate control problem related to food technology. J. Differ. Equ. 245 (2008) 130-153. | MR 2422713 | Zbl 1147.49003

[3] L.J. Alvarez-Vázquez, F.J. Fernández and R. Muñoz-Sola, Mathematical analysis of a three-dimensional eutrophication model. J. Math. Anal. Appl. 349 (2009) 135-155. | MR 2455737 | Zbl 1147.92038

[4] N. Arada and J.-P. Raymond, Time optimal problems with Dirichlet boundary controls. Discrete Contin. Dyn. Syst. 9 (2003) 1549-1570. | MR 2017681 | Zbl 1076.49012

[5] O. Arino, K. Boushaba and A. Boussouar, A mathematical model of the dynamics of the phytoplankton-nutrient system. Nonlinear Anal. Real World Appl. 1 (2000) 69-87. | MR 1794939 | Zbl 0984.92032

[6] R.P. Canale, Modeling biochemical processes in aquatic ecosystems. Ann Arbor Science Publishers, Ann Arbor (1976).

[7] P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems. ESAIM: COCV 12 (2006) 350-370. | Numdam | MR 2209357 | Zbl 1105.93007

[8] E. Casas, Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31 (1993) 993-1006. | MR 1227543 | Zbl 0798.49020

[9] F. Cioffi and F. Gallerano, Management strategies for the control of eutrophication processes in Fogliano lagoon (Italy): a long-term analysis using a mathematical model. Appl. Math. Model. 25 (2001) 385-426. | Zbl 0989.92023

[10] M. Drago, B. Cescon and L. Iovenitti, A three-dimensional numerical model for eutrophication and pollutant transport. Ecol. Model. 145 (2001) 17-34.

[11] M. Gugat and G. Leugering, L∞-norm minimal control of the wave equation: on the weakness of the bang-bang principle. ESAIM: COCV 14 (2008) 254-283. | Numdam | MR 2394510 | Zbl 1133.49006

[12] S. Li and G. Wang, The time optimal control of the Boussinesq equations. Numer. Funct. Anal. Optim. 24 (2003) 163-180. | MR 1978959 | Zbl 1062.49018

[13] F. Lunardini and G. Di Cola, Oxygen dynamics in coastal and lagoon ecosystems. Math. Comput. Model. 31 (2000) 135-141. | MR 1756750 | Zbl 1042.86501

[14] K. Park, H.-S. Jung, H.-S. Kim and S.-M. Ahn, Three-dimensional hydrodynamic-eutrophication model (HEM-3D): application to Kwang-Yang Bay, Korea. Mar. Environ. Res. 60 (2005) 171-193.

[15] J.P. Raymond and H. Zidani, Pontryagin's principle for time-optimal problems. J. Optim. Theory Appl. 101 (1999) 375-402. | MR 1684676 | Zbl 0952.49020

[16] J.P. Raymond and H. Zidani, Time optimal problems with boundary controls. Differ. Integr. Equat. 13 (2000) 1039-1072. | MR 1775245 | Zbl 0983.49016

[17] T. Roubíček, Nonlinear partial differential equations with applications. Birkhäuser-Verlag, Basel (2005). | Zbl 1270.35005

[18] G. Wang, The existence of time optimal control of semilinear parabolic equations. Syst. Control Lett. 53 (2004) 171-175. | MR 2092507 | Zbl 1157.49301

[19] L. Wang and G. Wang, The optimal time control of a phase-field system. SIAM J. Control Optim. 42 (2003) 1483-1508. | MR 2044806 | Zbl 1048.93034

[20] Y. Yamashiki, M. Matsumoto, T. Tezuka, S. Matsui and M. Kumagai, Three-dimensional eutrophication model for Lake Biwa and its application to the framework design of transferable discharge permits. Hydrol. Proc. 17 (2003) 2957-2973.

[21] E. Zeidler, Nonlinear Functional Analysis and Its Applications - Part 3: Variational Methods and Optimization. Springer-Verlag, Berlin (1985). | MR 768749 | Zbl 0583.47051