Analysis of a time optimal control problem related to the management of a bioreactor

Lino J. Alvarez-Vázquez; Francisco J. Fernández; Aurea Martínez

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 3, page 722-748
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Alvarez-Vázquez, Lino J., Fernández, Francisco J., and Martínez, Aurea. "Analysis of a time optimal control problem related to the management of a bioreactor." ESAIM: Control, Optimisation and Calculus of Variations 17.3 (2011): 722-748. <http://eudml.org/doc/272765>.

@article{Alvarez2011,
abstract = {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.},
author = {Alvarez-Vázquez, Lino J., Fernández, Francisco J., Martínez, Aurea},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {time optimal control; partial differential equations; optimality conditions; existence; bioreactor},
language = {eng},
number = {3},
pages = {722-748},
publisher = {EDP-Sciences},
title = {Analysis of a time optimal control problem related to the management of a bioreactor},
url = {http://eudml.org/doc/272765},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Alvarez-Vázquez, Lino J.
AU - Fernández, Francisco J.
AU - Martínez, Aurea
TI - Analysis of a time optimal control problem related to the management of a bioreactor
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2011
PB - EDP-Sciences
VL - 17
IS - 3
SP - 722
EP - 748
AB - 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.
LA - eng
KW - time optimal control; partial differential equations; optimality conditions; existence; bioreactor
UR - http://eudml.org/doc/272765
ER -

References

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  1. [1] W. Allegretto, C. Mocenni and A. Vicino, Periodic solutions in modelling lagoon ecological interactions. J. Math. Biol.51 (2005) 367–388. Zbl1087.92056MR2213040
  2. [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. Zbl1147.49003MR2422713
  3. [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. Zbl1147.92038MR2455737
  4. [4] N. Arada and J.-P. Raymond, Time optimal problems with Dirichlet boundary controls. Discrete Contin. Dyn. Syst.9 (2003) 1549–1570. Zbl1076.49012MR2017681
  5. [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. Zbl0984.92032MR1794939
  6. [6] R.P. Canale, Modeling biochemical processes in aquatic ecosystems. Ann Arbor Science Publishers, Ann Arbor (1976). 
  7. [7] P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems. ESAIM: COCV 12 (2006) 350–370. Zbl1105.93007MR2209357
  8. [8] E. Casas, Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim.31 (1993) 993–1006. Zbl0798.49020MR1227543
  9. [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. Zbl0989.92023
  10. [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. [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. Zbl1133.49006MR2394510
  12. [12] S. Li and G. Wang, The time optimal control of the Boussinesq equations. Numer. Funct. Anal. Optim.24 (2003) 163–180. Zbl1062.49018MR1978959
  13. [13] F. Lunardini and G. Di Cola, Oxygen dynamics in coastal and lagoon ecosystems. Math. Comput. Model.31 (2000) 135–141. Zbl1042.86501MR1756750
  14. [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. [15] J.P. Raymond and H. Zidani, Pontryagin's principle for time-optimal problems. J. Optim. Theory Appl.101 (1999) 375–402. Zbl0952.49020MR1684676
  16. [16] J.P. Raymond and H. Zidani, Time optimal problems with boundary controls. Differ. Integr. Equat.13 (2000) 1039–1072. Zbl0983.49016MR1775245
  17. [17] T. Roubíček, Nonlinear partial differential equations with applications. Birkhäuser-Verlag, Basel (2005). Zbl1270.35005
  18. [18] G. Wang, The existence of time optimal control of semilinear parabolic equations. Syst. Control Lett.53 (2004) 171–175. Zbl1157.49301MR2092507
  19. [19] L. Wang and G. Wang, The optimal time control of a phase-field system. SIAM J. Control Optim.42 (2003) 1483–1508. Zbl1048.93034MR2044806
  20. [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. [21] E. Zeidler, Nonlinear Functional Analysis and Its Applications – Part 3: Variational Methods and Optimization. Springer-Verlag, Berlin (1985). Zbl0583.47051MR768749

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