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

top
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

top

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/221924>.

@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; time optimal control; optimality conditions},
language = {eng},
month = {8},
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/221924},
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
DA - 2011/8//
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; time optimal control; optimality conditions
UR - http://eudml.org/doc/221924
ER -

References

top
  1. W. Allegretto, C. Mocenni and A. Vicino, Periodic solutions in modelling lagoon ecological interactions. J. Math. Biol.51 (2005) 367–388.  
  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.  
  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.  
  4. N. Arada and J.-P. Raymond, Time optimal problems with Dirichlet boundary controls. Discrete Contin. Dyn. Syst.9 (2003) 1549–1570.  
  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.  
  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: COCV12 (2006) 350–370.  
  8. E. Casas, Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim.31 (1993) 993–1006.  
  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.  
  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: COCV14 (2008) 254–283.  
  12. S. Li and G. Wang, The time optimal control of the Boussinesq equations. Numer. Funct. Anal. Optim.24 (2003) 163–180.  
  13. F. Lunardini and G. Di Cola, Oxygen dynamics in coastal and lagoon ecosystems. Math. Comput. Model.31 (2000) 135–141.  
  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.  
  16. J.P. Raymond and H. Zidani, Time optimal problems with boundary controls. Differ. Integr. Equat.13 (2000) 1039–1072.  
  17. T. Roubíček, Nonlinear partial differential equations with applications. Birkhäuser-Verlag, Basel (2005).  
  18. G. Wang, The existence of time optimal control of semilinear parabolic equations. Syst. Control Lett.53 (2004) 171–175.  
  19. L. Wang and G. Wang, The optimal time control of a phase-field system. SIAM J. Control Optim.42 (2003) 1483–1508.  
  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).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.