# American Institute of Mathematical Sciences

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

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

 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.

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. [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. [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. [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. [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. [6] M. Caponigro, R. Ghezzi, B. Piccoli and E. Trélat, Regularization of chattering phenomena via bounded variation control, preprint, 2013, arXiv: 1303.5796. [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. [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. [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. [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. [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. [12] A. T. Fuller, Study of an optimum non-linear control system, J. Electronics Control (1), 15 (1963), 63-71. doi: 10.1080/00207216308937555. [13] M. Garavello and B. Piccoli, Hybrid necessary principle, SIAM J. Control Optim., 43 (2005), 1867-1887 (electronic). doi: 10.1137/S0363012903416219. [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. [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. [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. [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. [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. [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. [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. [21] R. Mahadevan, J. Edwards and F. Doyle, Dynamic flux balance analysis of diauxic growth in Escherichia coli, Biophys J., 83 (2002), 1331-1340. [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. [23] B. O. Palsson, Systems Biology -Property of Reconstructed Networks, Cambridge University Press, 2006. [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. [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. [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. [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. [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.

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. [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. [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. [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. [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. [6] M. Caponigro, R. Ghezzi, B. Piccoli and E. Trélat, Regularization of chattering phenomena via bounded variation control, preprint, 2013, arXiv: 1303.5796. [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. [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. [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. [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. [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. [12] A. T. Fuller, Study of an optimum non-linear control system, J. Electronics Control (1), 15 (1963), 63-71. doi: 10.1080/00207216308937555. [13] M. Garavello and B. Piccoli, Hybrid necessary principle, SIAM J. Control Optim., 43 (2005), 1867-1887 (electronic). doi: 10.1137/S0363012903416219. [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. [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. [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. [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. [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. [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. [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. [21] R. Mahadevan, J. Edwards and F. Doyle, Dynamic flux balance analysis of diauxic growth in Escherichia coli, Biophys J., 83 (2002), 1331-1340. [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. [23] B. O. Palsson, Systems Biology -Property of Reconstructed Networks, Cambridge University Press, 2006. [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. [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. [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. [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. [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.
Bioprocess scheme exhibiting full coupling between metabolic activity and external dynamics
 [1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77 [9] 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 [10] 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 [11] H. O. Fattorini. The maximum principle in infinite dimension. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557 [12] 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 [13] Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499 [14] Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571 [15] Dingjun Yao, Kun Fan. Optimal risk control and dividend strategies in the presence of two reinsurers: Variance premium principle. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1055-1083. doi: 10.3934/jimo.2017090 [16] Chiun-Chuan Chen, Li-Chang Hung, Hsiao-Feng Liu. N-barrier maximum principle for degenerate elliptic systems and its application. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 791-821. doi: 10.3934/dcds.2018034 [17] Yunkyong Hyon, Do Young Kwak, Chun Liu. Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1291-1304. doi: 10.3934/dcds.2010.26.1291 [18] Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335 [19] Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395 [20] Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897

2017 Impact Factor: 0.631