American Institute of Mathematical Sciences

Advanced
June  2017, 7(2): 235-257. doi: 10.3934/mcrf.2017008

Optimal control of a multi-level dynamic model for biofuel production

Roberta Ghezzi 1,, and  Benedetto Piccoli 2,

1. 

Institut de Mathématiques de Bourgogne, COMUE Université Bourgogne-Franche Comté, 9 Avenue Alain Savary, 21078 Dijon, France
2. 

Department of Mathematical Sciences and Center, for Computational and Integrative Biology, Rutgers University 311 N 5th St, 08102 Camden NJ, USA

* Corresponding author

Received  March 2016 Revised  October 2016 Published  April 2017

Dynamic flux balance analysis of a bioreactor is based on the coupling between a dynamic problem, which models the evolution of biomass, feeding substrates and metabolites, and a linear program, which encodes the metabolic activity inside cells. We cast the problem in the language of optimal control and propose a hybrid formulation to model the full coupling between macroscopic and microscopic level. On a given location of the hybrid system we analyze necessary conditions given by the Pontryagin Maximum Principle and discuss the presence of singular arcs. For the multi-input case, under suitable assumptions, we prove that generically with respect to initial conditions optimal controls are bang-bang. For the single-input case the result is even stronger as we show that optimal controls are bang-bang.

Keywords: Optimal control, bioprocesses, Pontryagin Maximum Principle.
Mathematics Subject Classification: Primary: 49J15, 49J30, 49N90.
Citation: Roberta Ghezzi, Benedetto Piccoli. Optimal control of a multi-level dynamic model for biofuel production. Mathematical Control & Related Fields, 2017, 7 (2) : 235-257. doi: 10.3934/mcrf.2017008
References:
[1]

J. Alford, Bioprocess control: Advances and challenges, Computers & Chemical Engineering, 30 (2006), 1464-1475.  doi: 10.1016/j.compchemeng.2006.05.039.  Google Scholar

[2]

P. T. Benavides and U. Diwekar, Optimal control of biodiesel production in a batch reactor: Part Ⅰ: Deterministic control, Fuel, 94 (2012), 211-217.   Google Scholar

[3]

M. S. Branicky, Introduction to hybrid systems, In Handbook of Networked and Embedded Control Systems, Control Eng. , pages 91-116. Birkhäuser Boston, Boston, MA, 2005. doi: 10.1007/0-8176-4404-0_5.  Google Scholar

[4]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, volume 2 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.  Google Scholar

[5]

É. Busvelle and J.-P. Gauthier, On determining unknown functions in differential systems, with an application to biological reactors, ESAIM Control Optim. Calc. Var., 9 (2003), 509-551.  doi: 10.1051/cocv:2003025.  Google Scholar

[6]

M. Caponigro, R. Ghezzi, B. Piccoli and E. Trélat, Regularization of chattering phenomena via bounded variation control, preprint, 2013, arXiv: 1303.5796. Google Scholar

[7]

Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems, SIAM J. Control Optim., 47 (2008), 1078-1095.  doi: 10.1137/060663003.  Google Scholar

[8]

M. W. Covert, C. Schilling and B. Palsson, Regulation of gene expression in flux balance models of metabolism, J Theor Biol., 213 (2001), 73-88.   Google Scholar

[9]

M. W. Covert, N. Xiao, T. J. Chen and J. R. Karr, Integrating metabolic, transcriptional regulatory and signal transduction models in Escherichia coli, Bioinformatics, 24 (2008), 2044-2050.   Google Scholar

[10]

M. D. Di Benedetto and A. Sangiovanni-Vincentelli, Hybrid Systems: Computation and Control Lecture Notes in Comput. Sci. 2034. Springer-Verlag, Berlin, Heidelberg, 2001. Google Scholar

[11]

T. Eevera, K. Rajendran and S. Saradha, Biodiesel production process optimization and characterization to assess the suitability of the product for varied environmental conditions, Renewable Energy, 34 (2009), 762-765.  doi: 10.1016/j.renene.2008.04.006.  Google Scholar

[12]

A. T. Fuller, Study of an optimum non-linear control system, J. Electronics Control (1), 15 (1963), 63-71.  doi: 10.1080/00207216308937555.  Google Scholar

[13]

M. Garavello and B. Piccoli, Hybrid necessary principle, SIAM J. Control Optim., 43 (2005), 1867-1887 (electronic).  doi: 10.1137/S0363012903416219.  Google Scholar

[14]

J.-P. Gauthier, H. Hammouri and S. Othman, A simple observer for nonlinear systems applications to bioreactors, IEEE Trans. Automat. Control, 37 (1992), 875-880.  doi: 10.1109/9.256352.  Google Scholar

[15]

J. L. Hjersted and M. A. Henson, Optimization of fed-batch Saccharomyces cerevisiae fermentation using dynamic flux balance models, Biotechnol. Prog., 22 (2006), 1239-1248.   Google Scholar

[16]

J. L. Hjersted and M. A. Henson, Steady-state and dynamic flux balance analysis of ethanol production by Saccharomyces cerevisiae, IET Systems Biology, 3 (2009), 167-179.   Google Scholar

[17]

J. L. Hjersted, M. A. Henson and R. Mahadevan, Genome-Scale Analysis of Saccharomyces cerevisiae Metabolism and {E}thanol Production in Fed-Batch Culture, Biotechnology and Bioengineering, 97 (2007), 1190-1204.   Google Scholar

[18]

E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tubercolosis model, Discrete and Continuous Dynamical Systems-Series B, 2 (2002), 473-482.  doi: 10.3934/dcdsb.2002.2.473.  Google Scholar

[19]

D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, J. Math. Biol., 35 (1997), 775-792.  doi: 10.1007/s002850050076.  Google Scholar

[20]

A. Kremling, K. Bettenbrock and E. Gilles, Analysis of global control of Escherichia coli carbohydrate uptake BMC Systems Biology, 1 (2007), p42. doi: 10.1186/1752-0509-1-42.  Google Scholar

[21]

R. Mahadevan, J. Edwards and F. Doyle, Dynamic flux balance analysis of diauxic growth in Escherichia coli, Biophys J., 83 (2002), 1331-1340.   Google Scholar

[22]

J. Moreno, Optimal time control of bioreactors for the wastewater treatment, Optimal Control Applications Methods, 20 (1999), 145-164.  doi: 10.1002/(SICI)1099-1514(199905/06)20:3<145::AID-OCA651>3.0.CO;2-J.  Google Scholar

[23] B. O. Palsson, Systems Biology -Property of Reconstructed Networks, Cambridge University Press, 2006.   Google Scholar
[24]

L. S. Pontryagin, V. G. Boltyanskiǐ, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes ,"Nauka", Moscow, fourth edition, 1983.  Google Scholar

[25]

A. Rapaport and D. Dochain, Minimal time control of fed-batch processes with growth functions having several maxima, IEEE Trans. Automat. Contr., 56 (2011), 2671-2676.  doi: 10.1109/TAC.2011.2159424.  Google Scholar

[26]

H. J. Sussmann, A nonsmooth hybrid maximum principle, In Stability and stabilization of nonlinear systems (Ghent, 1999), volume 246 of Lecture Notes in Control and Inform. Sci. , pages 325-354. Springer, London, 1999. doi: 10.1007/1-84628-577-1_17.  Google Scholar

[27]

S. Tiwari, P. Verma, P. Singh and R. Tuli, Plants as bioreactors for the production of vaccine antigens, Biotechnology Advances, 27 (2009), 449-467.  doi: 10.1016/j.biotechadv.2009.03.006.  Google Scholar

[28]

K. Yamuna Rani and V. S. Ramachandra Rao, Control of fermenters -a review, Bioprocess and Biosystems Engineering, 21 (1999), 77-88.  doi: 10.1007/PL00009066.  Google Scholar

show all references

References:
[1]

J. Alford, Bioprocess control: Advances and challenges, Computers & Chemical Engineering, 30 (2006), 1464-1475.  doi: 10.1016/j.compchemeng.2006.05.039.  Google Scholar

[2]

P. T. Benavides and U. Diwekar, Optimal control of biodiesel production in a batch reactor: Part Ⅰ: Deterministic control, Fuel, 94 (2012), 211-217.   Google Scholar

[3]

M. S. Branicky, Introduction to hybrid systems, In Handbook of Networked and Embedded Control Systems, Control Eng. , pages 91-116. Birkhäuser Boston, Boston, MA, 2005. doi: 10.1007/0-8176-4404-0_5.  Google Scholar

[4]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, volume 2 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.  Google Scholar

[5]

É. Busvelle and J.-P. Gauthier, On determining unknown functions in differential systems, with an application to biological reactors, ESAIM Control Optim. Calc. Var., 9 (2003), 509-551.  doi: 10.1051/cocv:2003025.  Google Scholar

[6]

M. Caponigro, R. Ghezzi, B. Piccoli and E. Trélat, Regularization of chattering phenomena via bounded variation control, preprint, 2013, arXiv: 1303.5796. Google Scholar

[7]

Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems, SIAM J. Control Optim., 47 (2008), 1078-1095.  doi: 10.1137/060663003.  Google Scholar

[8]

M. W. Covert, C. Schilling and B. Palsson, Regulation of gene expression in flux balance models of metabolism, J Theor Biol., 213 (2001), 73-88.   Google Scholar

[9]

M. W. Covert, N. Xiao, T. J. Chen and J. R. Karr, Integrating metabolic, transcriptional regulatory and signal transduction models in Escherichia coli, Bioinformatics, 24 (2008), 2044-2050.   Google Scholar

[10]

M. D. Di Benedetto and A. Sangiovanni-Vincentelli, Hybrid Systems: Computation and Control Lecture Notes in Comput. Sci. 2034. Springer-Verlag, Berlin, Heidelberg, 2001. Google Scholar

[11]

T. Eevera, K. Rajendran and S. Saradha, Biodiesel production process optimization and characterization to assess the suitability of the product for varied environmental conditions, Renewable Energy, 34 (2009), 762-765.  doi: 10.1016/j.renene.2008.04.006.  Google Scholar

[12]

A. T. Fuller, Study of an optimum non-linear control system, J. Electronics Control (1), 15 (1963), 63-71.  doi: 10.1080/00207216308937555.  Google Scholar

[13]

M. Garavello and B. Piccoli, Hybrid necessary principle, SIAM J. Control Optim., 43 (2005), 1867-1887 (electronic).  doi: 10.1137/S0363012903416219.  Google Scholar

[14]

J.-P. Gauthier, H. Hammouri and S. Othman, A simple observer for nonlinear systems applications to bioreactors, IEEE Trans. Automat. Control, 37 (1992), 875-880.  doi: 10.1109/9.256352.  Google Scholar

[15]

J. L. Hjersted and M. A. Henson, Optimization of fed-batch Saccharomyces cerevisiae fermentation using dynamic flux balance models, Biotechnol. Prog., 22 (2006), 1239-1248.   Google Scholar

[16]

J. L. Hjersted and M. A. Henson, Steady-state and dynamic flux balance analysis of ethanol production by Saccharomyces cerevisiae, IET Systems Biology, 3 (2009), 167-179.   Google Scholar

[17]

J. L. Hjersted, M. A. Henson and R. Mahadevan, Genome-Scale Analysis of Saccharomyces cerevisiae Metabolism and {E}thanol Production in Fed-Batch Culture, Biotechnology and Bioengineering, 97 (2007), 1190-1204.   Google Scholar

[18]

E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tubercolosis model, Discrete and Continuous Dynamical Systems-Series B, 2 (2002), 473-482.  doi: 10.3934/dcdsb.2002.2.473.  Google Scholar

[19]

D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, J. Math. Biol., 35 (1997), 775-792.  doi: 10.1007/s002850050076.  Google Scholar

[20]

A. Kremling, K. Bettenbrock and E. Gilles, Analysis of global control of Escherichia coli carbohydrate uptake BMC Systems Biology, 1 (2007), p42. doi: 10.1186/1752-0509-1-42.  Google Scholar

[21]

R. Mahadevan, J. Edwards and F. Doyle, Dynamic flux balance analysis of diauxic growth in Escherichia coli, Biophys J., 83 (2002), 1331-1340.   Google Scholar

[22]

J. Moreno, Optimal time control of bioreactors for the wastewater treatment, Optimal Control Applications Methods, 20 (1999), 145-164.  doi: 10.1002/(SICI)1099-1514(199905/06)20:3<145::AID-OCA651>3.0.CO;2-J.  Google Scholar

[23] B. O. Palsson, Systems Biology -Property of Reconstructed Networks, Cambridge University Press, 2006.   Google Scholar
[24]

L. S. Pontryagin, V. G. Boltyanskiǐ, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes ,"Nauka", Moscow, fourth edition, 1983.  Google Scholar

[25]

A. Rapaport and D. Dochain, Minimal time control of fed-batch processes with growth functions having several maxima, IEEE Trans. Automat. Contr., 56 (2011), 2671-2676.  doi: 10.1109/TAC.2011.2159424.  Google Scholar

[26]

H. J. Sussmann, A nonsmooth hybrid maximum principle, In Stability and stabilization of nonlinear systems (Ghent, 1999), volume 246 of Lecture Notes in Control and Inform. Sci. , pages 325-354. Springer, London, 1999. doi: 10.1007/1-84628-577-1_17.  Google Scholar

[27]

S. Tiwari, P. Verma, P. Singh and R. Tuli, Plants as bioreactors for the production of vaccine antigens, Biotechnology Advances, 27 (2009), 449-467.  doi: 10.1016/j.biotechadv.2009.03.006.  Google Scholar

[28]

K. Yamuna Rani and V. S. Ramachandra Rao, Control of fermenters -a review, Bioprocess and Biosystems Engineering, 21 (1999), 77-88.  doi: 10.1007/PL00009066.  Google Scholar

Figure 1.  Bioprocess scheme exhibiting full coupling between metabolic activity and external dynamics
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110
[2]

Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control & Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021
[3]

Xiao-Li Ding, Iván Area, Juan J. Nieto. Controlled singular evolution equations and Pontryagin type maximum principle with applications. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021059
[4]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[5]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61
[6]

Térence Bayen, Marc Mazade, Francis Mairet. Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 39-58. doi: 10.3934/dcdsb.2015.20.39
[7]

Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174
[8]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161
[9]

Zhen Wu, Feng Zhang. Maximum principle for discrete-time stochastic optimal control problem and stochastic game. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021031
[10]

Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004
[11]

H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77
[12]

Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022
[13]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067
[14]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581
[15]

H. O. Fattorini. The maximum principle in infinite dimension. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557
[16]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018
[17]

Miaomiao Chen, Rong Yuan. Maximum principle for the optimal harvesting problem of a size-stage-structured population model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021245
[18]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499
[19]

Shi'an Wang, N. U. Ahmed. Optimal control and stabilization of building maintenance units based on minimum principle. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1713-1727. doi: 10.3934/jimo.2020041
[20]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations & Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035

2020 Impact Factor: 1.284

Tools

Metrics

  • PDF downloads (181)
  • HTML views (48)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy