# American Institute of Mathematical Sciences

January  2015, 20(1): 39-58. doi: 10.3934/dcdsb.2015.20.39

## Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc

 1 Université Montpellier 2, CC 051, 34095 Montpellier cedex 5, France, France 2 Inria 'BIOCORE' team, Inria Sophia-Antipolis, route des Lucioles, 06902 Sophia-Antipolis, France

Received  November 2013 Revised  July 2014 Published  November 2014

We study a minimal time control problem under the presence of a saturation point on the singular locus. The system describes a fed-batch reactor with one species and one substrate. Our aim is to find an optimal feedback control steering the system to a given target in minimal time. The growth function is of Haldane type implying the existence of a singular arc which is non-necessary admissible everywhere (i.e. the singular control can take values outside the admissible control set). Thanks to Pontrygin's Principle, we provide an optimal synthesis of the problem that exhibits a frame point at the intersection of the singular arc and a switching curve. Numerical simulations allow to compute this curve and the frame point.
Citation: 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
##### References:
 [1] T. Bayen, P. Gajardo and F. Mairet, Optimal synthesis for the minimal time control problems of fed-batch processes for growth functions with two maxima,, J. Optim. Theory and Applications, 158 (2013), 521. doi: 10.1007/s10957-012-0225-0. Google Scholar [2] T. Bayen and F. Mairet, Minimal time control of fed-batch bioreactor with product inhibition,, Bioprocess and Biosystems Engineering, 36 (2013), 1485. Google Scholar [3] B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control,, ESAIM Control Optim. Calc. Var., 13 (2007), 207. doi: 10.1051/cocv:2007012. Google Scholar [4] B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory,, Vol. 40, (2003). Google Scholar [5] B. Bonnard, M. Chyba and D. Sugny, Time-minimal control of dissipative two-level quantum systems: The generic case,, IEEE Trans. Automat. Contr., 54 (2009), 2598. doi: 10.1109/TAC.2009.2031212. Google Scholar [6] B. Bonnard, J.-P. Gauthier and J. de Morant, Geometric time-optimal control for batch reactors,, in Analysis of Controlled Dynamical Systems (eds. B. Bonnard, (1991), 69. doi: 10.1109/CDC.1991.261646. Google Scholar [7] B. Bonnard and J. de Morant, Towards a geometric theory in the time minimal control of chemical batch reactors,, SIAM J. on Control and Opt., 33 (1995), 1279. doi: 10.1137/S0363012992241338. Google Scholar [8] U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds,, Vol. 43, (2004). Google Scholar [9] U. Boscain and B. Piccoli, Extremal synthesis for generic planar systems,, Journal of Dynamical and Control Systems, 7 (2001), 209. doi: 10.1023/A:1013003204923. Google Scholar [10] A. Bressan and B. Piccoli, A generic classification of time optimal planar stabilizing feedbacks,, SIAM J. on Control and Optimization, 36 (1998), 12. doi: 10.1137/S0363012995291117. Google Scholar [11] Jr. A. E. Bryson and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation and Control,, Hemisphere Publishing Corp., (1975). Google Scholar [12] D. Dochain and A. Rapaport, Minimal time control of fed-batch processes with growth functions having several maxima,, IEEE Trans. Automat. Contr., 56 (2011), 2671. doi: 10.1109/TAC.2011.2159424. Google Scholar [13] D. Dochain and P. Vanrolleghem, Dynamical Modelling and Estimation in Wastewater Treatment Processes,, IWA Publishing, (2001). Google Scholar [14] P. Gajardo, H. Ramirez and A. Rapaport, Minimal time sequential batch reactors with bounded and impulse controls for one or more species,, SIAM J. Control Optim., 47 (2008), 2827. doi: 10.1137/070695204. Google Scholar [15] U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem,, SIAM J. on Control and Optimization, 46 (2007), 1052. doi: 10.1137/060665294. Google Scholar [16] J. Lee, S. Y. Lee, S. Park and A. P. J. Middelberg, Control of fed-batch fermentations,, Biotechnology Advances, 17 (1999), 29. doi: 10.1016/S0734-9750(98)00015-9. Google Scholar [17] A. Miele, Application of Green's theorem to the extremization of linear integrals,, in Symp. on Vehicle Systems Optimization, (1961). Google Scholar [18] J. Monod, Recherches sur la Croissance des Cultures Bactériennes,, Hermann, (1942). Google Scholar [19] J. A. Moreno, Optimal time control of bioreactors for the wastewater treatment,, Optim. Control Appl. Meth., 20 (1999), 145. doi: {10.1002/(SICI)1099-1514(199905/06)20:3<145::AID-OCA651>3.0.CO;2-J}. Google Scholar [20] B. Piccoli, Classification of generic singularities for the planar time-optimal synthesis,, SIAM J. on Control and Optimization, 34 (1996), 1914. doi: 10.1137/S0363012993256149. Google Scholar [21] B. Piccoli and H. J. Sussmann, Regular synthesis and sufficiency conditions for optimality,, SIAM J. on Control and Optimization, 39 (2000), 359. doi: 10.1137/S0363012999322031. Google Scholar [22] L. Pontryagin, V. Boltyanski, R. Gamkrelidze and E. Michtchenko, The Mathematical Theory of Optimal Processes,, Wiley Interscience, (1962). Google Scholar [23] H. Schattler and M. Jankovic, A synthesis of time-optimal controls in the presence of saturated singular arcs,, Forum Mathematicum, 5 (1993), 203. doi: 10.1515/form.1993.5.203. Google Scholar [24] H. Schattler and U. Ledzewicz, Geometric Optimal Control,, Springer, (2012). doi: 10.1007/978-1-4614-3834-2. Google Scholar [25] C. J. Silva and E. Trélat, Smooth regularization of bang-bang optimal control problems,, IEEE Trans. Automat. Control, 55 (2010), 2488. doi: 10.1109/TAC.2010.2047742. Google Scholar [26] H. L. Smith and P. Waltman, The Theory of the Chemostat, Dynamics of Microbial Competition,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043. Google Scholar [27] P. Spinelli and G. Solay Rakotonirayni, Minimum time problem synthesis,, Systems and Control Letters, 10 (1988), 281. doi: 10.1016/0167-6911(88)90018-7. Google Scholar [28] H. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: The $C^{\infty}$ nonsingular case,, SIAM J. on Control and Optimization, 25 (1987), 433. doi: 10.1137/0325025. Google Scholar [29] H. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: The general real analytic case,, SIAM J. on Control and Optimization, 25 (1987), 868. doi: 10.1137/0325048. Google Scholar [30] H. Sussmann, Regular synthesis for time-optimal control of single-input real analytic systems in the plane,, SIAM J. on Control and Optimization, 25 (1987), 1145. doi: 10.1137/0325062. Google Scholar

show all references

##### References:
 [1] T. Bayen, P. Gajardo and F. Mairet, Optimal synthesis for the minimal time control problems of fed-batch processes for growth functions with two maxima,, J. Optim. Theory and Applications, 158 (2013), 521. doi: 10.1007/s10957-012-0225-0. Google Scholar [2] T. Bayen and F. Mairet, Minimal time control of fed-batch bioreactor with product inhibition,, Bioprocess and Biosystems Engineering, 36 (2013), 1485. Google Scholar [3] B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control,, ESAIM Control Optim. Calc. Var., 13 (2007), 207. doi: 10.1051/cocv:2007012. Google Scholar [4] B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory,, Vol. 40, (2003). Google Scholar [5] B. Bonnard, M. Chyba and D. Sugny, Time-minimal control of dissipative two-level quantum systems: The generic case,, IEEE Trans. Automat. Contr., 54 (2009), 2598. doi: 10.1109/TAC.2009.2031212. Google Scholar [6] B. Bonnard, J.-P. Gauthier and J. de Morant, Geometric time-optimal control for batch reactors,, in Analysis of Controlled Dynamical Systems (eds. B. Bonnard, (1991), 69. doi: 10.1109/CDC.1991.261646. Google Scholar [7] B. Bonnard and J. de Morant, Towards a geometric theory in the time minimal control of chemical batch reactors,, SIAM J. on Control and Opt., 33 (1995), 1279. doi: 10.1137/S0363012992241338. Google Scholar [8] U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds,, Vol. 43, (2004). Google Scholar [9] U. Boscain and B. Piccoli, Extremal synthesis for generic planar systems,, Journal of Dynamical and Control Systems, 7 (2001), 209. doi: 10.1023/A:1013003204923. Google Scholar [10] A. Bressan and B. Piccoli, A generic classification of time optimal planar stabilizing feedbacks,, SIAM J. on Control and Optimization, 36 (1998), 12. doi: 10.1137/S0363012995291117. Google Scholar [11] Jr. A. E. Bryson and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation and Control,, Hemisphere Publishing Corp., (1975). Google Scholar [12] D. Dochain and A. Rapaport, Minimal time control of fed-batch processes with growth functions having several maxima,, IEEE Trans. Automat. Contr., 56 (2011), 2671. doi: 10.1109/TAC.2011.2159424. Google Scholar [13] D. Dochain and P. Vanrolleghem, Dynamical Modelling and Estimation in Wastewater Treatment Processes,, IWA Publishing, (2001). Google Scholar [14] P. Gajardo, H. Ramirez and A. Rapaport, Minimal time sequential batch reactors with bounded and impulse controls for one or more species,, SIAM J. Control Optim., 47 (2008), 2827. doi: 10.1137/070695204. Google Scholar [15] U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem,, SIAM J. on Control and Optimization, 46 (2007), 1052. doi: 10.1137/060665294. Google Scholar [16] J. Lee, S. Y. Lee, S. Park and A. P. J. Middelberg, Control of fed-batch fermentations,, Biotechnology Advances, 17 (1999), 29. doi: 10.1016/S0734-9750(98)00015-9. Google Scholar [17] A. Miele, Application of Green's theorem to the extremization of linear integrals,, in Symp. on Vehicle Systems Optimization, (1961). Google Scholar [18] J. Monod, Recherches sur la Croissance des Cultures Bactériennes,, Hermann, (1942). Google Scholar [19] J. A. Moreno, Optimal time control of bioreactors for the wastewater treatment,, Optim. Control Appl. Meth., 20 (1999), 145. doi: {10.1002/(SICI)1099-1514(199905/06)20:3<145::AID-OCA651>3.0.CO;2-J}. Google Scholar [20] B. Piccoli, Classification of generic singularities for the planar time-optimal synthesis,, SIAM J. on Control and Optimization, 34 (1996), 1914. doi: 10.1137/S0363012993256149. Google Scholar [21] B. Piccoli and H. J. Sussmann, Regular synthesis and sufficiency conditions for optimality,, SIAM J. on Control and Optimization, 39 (2000), 359. doi: 10.1137/S0363012999322031. Google Scholar [22] L. Pontryagin, V. Boltyanski, R. Gamkrelidze and E. Michtchenko, The Mathematical Theory of Optimal Processes,, Wiley Interscience, (1962). Google Scholar [23] H. Schattler and M. Jankovic, A synthesis of time-optimal controls in the presence of saturated singular arcs,, Forum Mathematicum, 5 (1993), 203. doi: 10.1515/form.1993.5.203. Google Scholar [24] H. Schattler and U. Ledzewicz, Geometric Optimal Control,, Springer, (2012). doi: 10.1007/978-1-4614-3834-2. Google Scholar [25] C. J. Silva and E. Trélat, Smooth regularization of bang-bang optimal control problems,, IEEE Trans. Automat. Control, 55 (2010), 2488. doi: 10.1109/TAC.2010.2047742. Google Scholar [26] H. L. Smith and P. Waltman, The Theory of the Chemostat, Dynamics of Microbial Competition,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043. Google Scholar [27] P. Spinelli and G. Solay Rakotonirayni, Minimum time problem synthesis,, Systems and Control Letters, 10 (1988), 281. doi: 10.1016/0167-6911(88)90018-7. Google Scholar [28] H. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: The $C^{\infty}$ nonsingular case,, SIAM J. on Control and Optimization, 25 (1987), 433. doi: 10.1137/0325025. Google Scholar [29] H. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: The general real analytic case,, SIAM J. on Control and Optimization, 25 (1987), 868. doi: 10.1137/0325048. Google Scholar [30] H. Sussmann, Regular synthesis for time-optimal control of single-input real analytic systems in the plane,, SIAM J. on Control and Optimization, 25 (1987), 1145. doi: 10.1137/0325062. Google Scholar
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