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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

Térence Bayen 1, , Marc Mazade 1, and  Francis Mairet 2,

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.
Keywords: Pontryagin Maximum Principle, Optimal control, singular control, chemostat model, fed-batch bioreactor..
Mathematics Subject Classification: Primary: 49J15, 49K15, 92B05; Secondary: 93C9.
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), 521-553. 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), 1485-1496. 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), 207-236. doi: 10.1051/cocv:2007012.  Google Scholar

[4]

B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Vol. 40, Springer-Verlag, Berlin, 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), 2598-2610. 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, B. Bride, J. P. Gauthier and I. Kupka), Birkhäuser, 1991, 69-87. 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), 1279-1311. doi: 10.1137/S0363012992241338.  Google Scholar

[8]

U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Vol. 43, Springer-Verlag, Berlin, 2004.  Google Scholar

[9]

U. Boscain and B. Piccoli, Extremal synthesis for generic planar systems, Journal of Dynamical and Control Systems, 7 (2001), 209-258. 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), 12-32. 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., Washington, D. C., 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), 2671-2676. doi: 10.1109/TAC.2011.2159424.  Google Scholar

[13]

D. Dochain and P. Vanrolleghem, Dynamical Modelling and Estimation in Wastewater Treatment Processes, IWA Publishing, U.K., 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), 2827-2856. 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), 1052-1079. 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), 29-48. 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, Garden City, New York, 1961. Google Scholar

[18]

J. Monod, Recherches sur la Croissance des Cultures Bactériennes, Hermann, Paris, 1942. Google Scholar

[19]

J. A. Moreno, Optimal time control of bioreactors for the wastewater treatment, Optim. Control Appl. Meth., 20 (1999), 145-164. 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), 1914-1946. 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), 359-410. doi: 10.1137/S0363012999322031.  Google Scholar

[22]

L. Pontryagin, V. Boltyanski, R. Gamkrelidze and E. Michtchenko, The Mathematical Theory of Optimal Processes, Wiley Interscience, New York, 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), 203-241. doi: 10.1515/form.1993.5.203.  Google Scholar

[24]

H. Schattler and U. Ledzewicz, Geometric Optimal Control, Springer, New York, 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), 2488-2499. 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, Cambridge, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[27]

P. Spinelli and G. Solay Rakotonirayni, Minimum time problem synthesis, Systems and Control Letters, 10 (1988), 281-290. 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), 433-465. 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), 868-904. 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), 1145-1162. 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), 521-553. 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), 1485-1496. 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), 207-236. doi: 10.1051/cocv:2007012.  Google Scholar

[4]

B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Vol. 40, Springer-Verlag, Berlin, 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), 2598-2610. 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, B. Bride, J. P. Gauthier and I. Kupka), Birkhäuser, 1991, 69-87. 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), 1279-1311. doi: 10.1137/S0363012992241338.  Google Scholar

[8]

U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Vol. 43, Springer-Verlag, Berlin, 2004.  Google Scholar

[9]

U. Boscain and B. Piccoli, Extremal synthesis for generic planar systems, Journal of Dynamical and Control Systems, 7 (2001), 209-258. 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), 12-32. 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., Washington, D. C., 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), 2671-2676. doi: 10.1109/TAC.2011.2159424.  Google Scholar

[13]

D. Dochain and P. Vanrolleghem, Dynamical Modelling and Estimation in Wastewater Treatment Processes, IWA Publishing, U.K., 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), 2827-2856. 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), 1052-1079. 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), 29-48. 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, Garden City, New York, 1961. Google Scholar

[18]

J. Monod, Recherches sur la Croissance des Cultures Bactériennes, Hermann, Paris, 1942. Google Scholar

[19]

J. A. Moreno, Optimal time control of bioreactors for the wastewater treatment, Optim. Control Appl. Meth., 20 (1999), 145-164. 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), 1914-1946. 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), 359-410. doi: 10.1137/S0363012999322031.  Google Scholar

[22]

L. Pontryagin, V. Boltyanski, R. Gamkrelidze and E. Michtchenko, The Mathematical Theory of Optimal Processes, Wiley Interscience, New York, 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), 203-241. doi: 10.1515/form.1993.5.203.  Google Scholar

[24]

H. Schattler and U. Ledzewicz, Geometric Optimal Control, Springer, New York, 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), 2488-2499. 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, Cambridge, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[27]

P. Spinelli and G. Solay Rakotonirayni, Minimum time problem synthesis, Systems and Control Letters, 10 (1988), 281-290. 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), 433-465. 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), 868-904. 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), 1145-1162. doi: 10.1137/0325062.  Google Scholar

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