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

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  • 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.
    Mathematics Subject Classification: Primary: 49J15, 49K15, 92B05; Secondary: 93C95.


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  • [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.


    T. Bayen and F. Mairet, Minimal time control of fed-batch bioreactor with product inhibition, Bioprocess and Biosystems Engineering, 36 (2013), 1485-1496.


    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.


    B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Vol. 40, Springer-Verlag, Berlin, 2003.


    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.


    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.


    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.


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


    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.


    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.


    Jr. A. E. Bryson and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation and Control, Hemisphere Publishing Corp., Washington, D. C., 1975.


    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.


    D. Dochain and P. Vanrolleghem, Dynamical Modelling and Estimation in Wastewater Treatment Processes, IWA Publishing, U.K., 2001.


    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.


    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.


    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.


    A. Miele, Application of Green's theorem to the extremization of linear integrals, in Symp. on Vehicle Systems Optimization, Garden City, New York, 1961.


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


    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}.


    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.


    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.


    L. Pontryagin, V. Boltyanski, R. Gamkrelidze and E. Michtchenko, The Mathematical Theory of Optimal Processes, Wiley Interscience, New York, 1962.


    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.


    H. Schattler and U. Ledzewicz, Geometric Optimal Control, Springer, New York, 2012.doi: 10.1007/978-1-4614-3834-2.


    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.


    H. L. Smith and P. Waltman, The Theory of the Chemostat, Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.doi: 10.1017/CBO9780511530043.


    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.


    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.


    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.


    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.

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