# American Institute of Mathematical Sciences

June  2015, 4(2): 143-158. doi: 10.3934/eect.2015.4.143

## Input/state invariant LQ-optimal control: Application to competitive coexistence in a chemostat

 1 CENAERO, Bâtiment Eole, 29 rue des Frères Wright, 6041 Gosselies, Belgium 2 University of Namur, naXys and Department of Mathematics, 8 Rempart de la Vierge, B-5000 Namur, Belgium 3 Université Catholique de Louvain, INMA-ICTEAM, CESAME, 4-6 avenue G. Lemaitre, 1348 Louvain-la-Neuve, Belgium

Received  July 2014 Revised  December 2014 Published  May 2015

This paper is devoted to the application of the input/state-invariant linear quadratic (LQ) problem in order to solve the problem of coexistence of species in competition in a chemostat. The methodology that is used has for objective to guarantee the local positive input/state-invariance of the nonlinear system which describes the chemostat model by ensuring the input/state-invariance of its linear approximation around an equilibrium. This is achieved by applying an appropriate LQ-optimal control to the system, following two different approaches, namely a receding horizon method and an inverse problem approach.
Citation: Charlotte Beauthier, Joseph J. Winkin, Denis Dochain. Input/state invariant LQ-optimal control: Application to competitive coexistence in a chemostat. Evolution Equations and Control Theory, 2015, 4 (2) : 143-158. doi: 10.3934/eect.2015.4.143
##### References:
 [1] E. Barany, M. Ballyk and H. Noussi, Stabilization of chemostats using feedback linearization, in Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, USA, December 2007. [2] Ch. Beauthier, The LQ-Optimal Control Problem for Invariant Linear Systems, PhD thesis, University of Namur (FUNDP, Namur, Belgium), 2011. Available from: http://hdl.handle.net/2078.2/76840. [3] Ch. Beauthier and J. J. Winkin, Input/state-invariant LQ-optimal control, in preparation, 2014. [4] R. E. Bixby, Implementation of the simplex method: The initial basis, ORSA Journal on Computing, 4 (1992), 267-284. doi: 10.1287/ijoc.4.3.267. [5] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics, Philadelphia, 1994. doi: 10.1137/1.9781611970777. [6] G. J. Butler, S. B. Hsu and P. Waltman, A mathematical model of the chemostat with periodic washout rate, SIAM Journal on Applied Mathematics, 45 (1985), 435-449. doi: 10.1137/0145025. [7] G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM Journal on applied mathematics, 45 (1985), 138-151. doi: 10.1137/0145006. [8] V. Chellaboina, S. Bhat, W. M. Haddad and D. S. Bernstein, Modeling and analysis of mass-action kinetics, IEEE Control Systems Magazine, 29 (2009), 60-78. doi: 10.1109/MCS.2009.932926. [9] P. De Leenheer, D. Angeli and E. D. Sontag, Crowding effects promote coexistence in the chemostat, J. Math. Anal. Appl., 319 (2006), 48-60. doi: 10.1016/j.jmaa.2006.02.036. [10] P. De Leenheer and S. S. Pilyugin, Feedback-mediated oscillatory coexistence in the chemostat, Lecture Notes in Control and Information Sciences, 341 (2006), 97-104. doi: 10.1007/3-540-34774-7_13. [11] P. De Leenheer and H. Smith, Feedback control for a chemostat with two organisms, in Cédérom Proceedings of the Symposium on Mathematical Theory of Networks and Systems (MTNS), University of Notre-Dame, 2002. [12] P. De Leenheer and H. Smith, Feedback control for chemostat models, Journal of Mathematical Biology, 46 (2003), 48-70. doi: 10.1007/s00285-002-0170-x. [13] S. B. Hsu, A competition model for a seasonally fluctuating nutrient, J. Math. Biol., 9 (1980), 115-132. doi: 10.1007/BF00275917. [14] G. E. Hutchinson, The paradox of the plankton, The American Naturalist, 95 (1961), 137-145. doi: 10.1086/282171. [15] M. Ikeda and D. D. Siljak, Optimality and robustness of linear quadratic control for nonlinear systems, Automatica, 26 (1990), 499-511. doi: 10.1016/0005-1098(90)90021-9. [16] T. Kačzorek, Locally positive nonlinear systems, Int. J. Appl. Math. Comput. Sci., 13 (2003), 505-509. [17] W. S. Keeran, P. De Leenheer and S. S. Pilyugin, Feedback-mediated coexistence and oscillations in the chemostat, DCDS - B, 9 (2008), 321-351. [18] H. K. Khalil, Nonlinear Systems, Third edition, Prentice Hall, 2002. [19] C. Lobry and J. Harmand, A new hypothesis to explain the coexistence of $n$ species in the presence of a single resource, C. R. Biologies, 329 (2006), 40-46. doi: 10.1016/j.crvi.2005.10.004. [20] J. Löfberg, http://users.isy.liu.se/johanl/yalmip/, Last updated on February 08, 2011. [21] S. Losseau, Coexistence Dans un Modèle de Chemostat Par Feedback Linéarisant, Master's thesis, Department of Mathematics, University of Namur, Belgium, 2012. [22] F. Mazenc and M. Malisoff, Remarks on output feedback stabilization of two-species chemostat models, Automatica, 46 (2010), 1739-1742. doi: 10.1016/j.automatica.2010.06.035. [23] N. S. Rao and E. O. Roxin, Controlled growth of competing species, SIAM J. Appl. Math., 50 (1990), 853-864. doi: 10.1137/0150049. [24] A. Rapaport, D. Dochain and J. Harmand, Long run coexistence in the chemostat with multiple species, Journal of Theoritical Biology, 257 (2009), 252-259. doi: 10.1016/j.jtbi.2008.11.015. [25] A. Rapaport, J. Harmand and F. Mazenc, Coexistence in the design of a series of two chemostats, Nonlinear Analysis: Real World Applications, 9 (2008), 1052-1067. doi: 10.1016/j.nonrwa.2007.02.003. [26] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, 1995. [27] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043. [28] G. Stephanopoulos, A. G. Fredrickson and R. Asis, The growth of competing microbial populations in CSTR with periodically varying inputs, Amer. Instit. Chem. Eng. J., 25 (1979), 863-872. doi: 10.1002/aic.690250515. [29] J. G. VanAntwerp and R. D. Braatz, A tutorial on linear and bilinear matrix inequalities, Journal of Process Control, 10 (2000), 363-385. doi: 10.1016/S0959-1524(99)00056-6.

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##### References:
 [1] E. Barany, M. Ballyk and H. Noussi, Stabilization of chemostats using feedback linearization, in Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, USA, December 2007. [2] Ch. Beauthier, The LQ-Optimal Control Problem for Invariant Linear Systems, PhD thesis, University of Namur (FUNDP, Namur, Belgium), 2011. Available from: http://hdl.handle.net/2078.2/76840. [3] Ch. Beauthier and J. J. Winkin, Input/state-invariant LQ-optimal control, in preparation, 2014. [4] R. E. Bixby, Implementation of the simplex method: The initial basis, ORSA Journal on Computing, 4 (1992), 267-284. doi: 10.1287/ijoc.4.3.267. [5] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics, Philadelphia, 1994. doi: 10.1137/1.9781611970777. [6] G. J. Butler, S. B. Hsu and P. Waltman, A mathematical model of the chemostat with periodic washout rate, SIAM Journal on Applied Mathematics, 45 (1985), 435-449. doi: 10.1137/0145025. [7] G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM Journal on applied mathematics, 45 (1985), 138-151. doi: 10.1137/0145006. [8] V. Chellaboina, S. Bhat, W. M. Haddad and D. S. Bernstein, Modeling and analysis of mass-action kinetics, IEEE Control Systems Magazine, 29 (2009), 60-78. doi: 10.1109/MCS.2009.932926. [9] P. De Leenheer, D. Angeli and E. D. Sontag, Crowding effects promote coexistence in the chemostat, J. Math. Anal. Appl., 319 (2006), 48-60. doi: 10.1016/j.jmaa.2006.02.036. [10] P. De Leenheer and S. S. Pilyugin, Feedback-mediated oscillatory coexistence in the chemostat, Lecture Notes in Control and Information Sciences, 341 (2006), 97-104. doi: 10.1007/3-540-34774-7_13. [11] P. De Leenheer and H. Smith, Feedback control for a chemostat with two organisms, in Cédérom Proceedings of the Symposium on Mathematical Theory of Networks and Systems (MTNS), University of Notre-Dame, 2002. [12] P. De Leenheer and H. Smith, Feedback control for chemostat models, Journal of Mathematical Biology, 46 (2003), 48-70. doi: 10.1007/s00285-002-0170-x. [13] S. B. Hsu, A competition model for a seasonally fluctuating nutrient, J. Math. Biol., 9 (1980), 115-132. doi: 10.1007/BF00275917. [14] G. E. Hutchinson, The paradox of the plankton, The American Naturalist, 95 (1961), 137-145. doi: 10.1086/282171. [15] M. Ikeda and D. D. Siljak, Optimality and robustness of linear quadratic control for nonlinear systems, Automatica, 26 (1990), 499-511. doi: 10.1016/0005-1098(90)90021-9. [16] T. Kačzorek, Locally positive nonlinear systems, Int. J. Appl. Math. Comput. Sci., 13 (2003), 505-509. [17] W. S. Keeran, P. De Leenheer and S. S. Pilyugin, Feedback-mediated coexistence and oscillations in the chemostat, DCDS - B, 9 (2008), 321-351. [18] H. K. Khalil, Nonlinear Systems, Third edition, Prentice Hall, 2002. [19] C. Lobry and J. Harmand, A new hypothesis to explain the coexistence of $n$ species in the presence of a single resource, C. R. Biologies, 329 (2006), 40-46. doi: 10.1016/j.crvi.2005.10.004. [20] J. Löfberg, http://users.isy.liu.se/johanl/yalmip/, Last updated on February 08, 2011. [21] S. Losseau, Coexistence Dans un Modèle de Chemostat Par Feedback Linéarisant, Master's thesis, Department of Mathematics, University of Namur, Belgium, 2012. [22] F. Mazenc and M. Malisoff, Remarks on output feedback stabilization of two-species chemostat models, Automatica, 46 (2010), 1739-1742. doi: 10.1016/j.automatica.2010.06.035. [23] N. S. Rao and E. O. Roxin, Controlled growth of competing species, SIAM J. Appl. Math., 50 (1990), 853-864. doi: 10.1137/0150049. [24] A. Rapaport, D. Dochain and J. Harmand, Long run coexistence in the chemostat with multiple species, Journal of Theoritical Biology, 257 (2009), 252-259. doi: 10.1016/j.jtbi.2008.11.015. [25] A. Rapaport, J. Harmand and F. Mazenc, Coexistence in the design of a series of two chemostats, Nonlinear Analysis: Real World Applications, 9 (2008), 1052-1067. doi: 10.1016/j.nonrwa.2007.02.003. [26] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, 1995. [27] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043. [28] G. Stephanopoulos, A. G. Fredrickson and R. Asis, The growth of competing microbial populations in CSTR with periodically varying inputs, Amer. Instit. Chem. Eng. J., 25 (1979), 863-872. doi: 10.1002/aic.690250515. [29] J. G. VanAntwerp and R. D. Braatz, A tutorial on linear and bilinear matrix inequalities, Journal of Process Control, 10 (2000), 363-385. doi: 10.1016/S0959-1524(99)00056-6.
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