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Optimal control of a multi-level dynamic model for biofuel production. (English) Zbl 1360.49001

Summary: 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.

MSC:

49J15 Existence theories for optimal control problems involving ordinary differential equations
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49N90 Applications of optimal control and differential games
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References:

[1] J. Alford, Bioprocess control: Advances and challenges,, Computers & Chemical Engineering, 30, 1464 (2006) · doi:10.1016/j.compchemeng.2006.05.039
[2] P. T. Benavides, Optimal control of biodiesel production in a batch reactor: Part I: Deterministic control,, Fuel, 94, 211 (2012)
[3] M. S. Branicky, Introduction to hybrid systems,, In Handbook of Networked and Embedded Control Systems, 91 (2005) · doi:10.1007/0-8176-4404-0_5
[4] A. Bressan, <em>Introduction to the Mathematical Theory of Control</em>,, volume 2 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS) (2007) · Zbl 1127.93002
[5] É. Busvelle, On determining unknown functions in differential systems, with an application to biological reactors,, ESAIM Control Optim. Calc. Var., 9, 509 (2003) · Zbl 1063.93011 · doi:10.1051/cocv:2003025
[6] M. Caponigro, Regularization of chattering phenomena via bounded variation control,, preprint (2013)
[7] Y. Chitour, Singular trajectories of control-affine systems,, SIAM J. Control Optim., 47, 1078 (2008) · Zbl 1157.49041 · doi:10.1137/060663003
[8] M. W. Covert, Regulation of gene expression in flux balance models of metabolism,, J Theor Biol., 213, 73 (2001)
[9] M. W. Covert, Integrating metabolic, transcriptional regulatory and signal transduction models in <em>Escherichia coli</em>,, Bioinformatics, 24, 2044 (2008)
[10] M. D. Di Benedetto, <em>Hybrid Systems: Computation and Control</em>,, Lecture Notes in Comput. Sci. 2034. Springer-Verlag (2034)
[11] T. Eevera, Biodiesel production process optimization and characterization to assess the suitability of the product for varied environmental conditions,, Renewable Energy, 34, 762 (2009) · 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, 63 (1963) · doi:10.1080/00207216308937555
[13] M. Garavello, Hybrid necessary principle,, SIAM J. Control Optim., 43, 1867 (2005) · Zbl 1084.49021 · doi:10.1137/S0363012903416219
[14] J.-P. Gauthier, A simple observer for nonlinear systems applications to bioreactors,, IEEE Trans. Automat. Control, 37, 875 (1992) · Zbl 0775.93020 · doi:10.1109/9.256352
[15] J. L. Hjersted, Optimization of fed-batch <em>Saccharomyces cerevisiae</em> fermentation using dynamic flux balance models,, Biotechnol. Prog., 22, 1239 (2006)
[16] J. L. Hjersted, Steady-state and dynamic flux balance analysis of ethanol production by <em>Saccharomyces cerevisiae</em>,, IET Systems Biology, 3, 167 (2009)
[17] J. L. Hjersted, Genome-Scale Analysis of <em>Saccharomyces cerevisiae</em> Metabolism and Ethanol Production in Fed-Batch Culture,, Biotechnology and Bioengineering, 97, 1190 (2007)
[18] E. Jung, Optimal control of treatments in a two-strain tubercolosis model,, Discrete and Continuous Dynamical Systems-Series B, 2, 473 (2002) · Zbl 1005.92018 · doi:10.3934/dcdsb.2002.2.473
[19] D. Kirschner, Optimal control of the chemotherapy of HIV,, J. Math. Biol., 35, 775 (1997) · Zbl 0876.92016 · doi:10.1007/s002850050076
[20] A. Kremling, Analysis of global control of Escherichia coli carbohydrate uptake,, BMC Systems Biology, 1 (2007) · Zbl 1337.92093 · doi:10.1186/1752-0509-1-42
[21] R. Mahadevan, Dynamic flux balance analysis of diauxic growth in <em>Escherichia coli</em>,, Biophys J., 83, 1331 (2002)
[22] J. Moreno, Optimal time control of bioreactors for the wastewater treatment,, Optimal Control Applications Methods, 20, 145 (1999) · doi:10.1002/(SICI)1099-1514(199905/06)20:3<145::AID-OCA651>3.0.CO;2-J
[23] B. O. Palsson, <em>Systems Biology - Property of Reconstructed Networks</em>,, Cambridge University Press (2006)
[24] L. S. Pontryagin, <em>The Mathematical Theory of Optimal Processes</em>,, 1983. (1983) · Zbl 0516.49001
[25] A. Rapaport, Minimal time control of fed-batch processes with growth functions having several maxima,, IEEE Trans. Automat. Contr., 56, 2671 (2011) · Zbl 1368.93190 · doi:10.1109/TAC.2011.2159424
[26] H. J. Sussmann, A nonsmooth hybrid maximum principle,, In Stability and stabilization of nonlinear systems (Ghent, 325 (1999) · Zbl 0967.49016 · doi:10.1007/1-84628-577-1_17
[27] S. Tiwari, Plants as bioreactors for the production of vaccine antigens,, Biotechnology Advances, 27, 449 (2009) · doi:10.1016/j.biotechadv.2009.03.006
[28] K. Yamuna Rani, Control of fermenters - a review,, Bioprocess and Biosystems Engineering, 21, 77 (1999) · doi:10.1007/PL00009066
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