We study the minimum time control problem of a series of two interconnected chemostats under the input constraint $u_{2}\leq u_{1}$, where $u_{i}$ are the respective dilution rates in the tanks. This constraint brings controllability issues in the study of the optimal strategies. We overcome this difficulty by splitting the state domain into two subdomains, one with no lack of controllability of the target, and its complement where any optimal trajectory satisfies $u_{1}=u_{2}$. We explicitly compute the complete optimal synthesis that depends on the position of the target with respect to a semipermeable curve that passes through a steady-state singular point.


  1. optimal control
  2. minimal time problem
  3. Pontryagin's maximum principle
  4. optimal synthesis
  5. chemostat model
  6. gradostat model
  7. nonlinear controllability
  8. semipermeability

MSC codes

  1. 49J15
  2. 49K15
  3. 49N25

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D. Ami, P. Cartigny, and A. Rapaport, Can marine protected areas enhance both economic and biological situations?, C. R. Biologies, 328, 4 (2005), pp. 357--366.
W. Bakker, H. Beeftink, C. de Gooijer, and J. Tramper, Bioreactors in series: An overview of design procedures and practical applications, Enzyme Microbial Technol., 18 (1996), pp. 202--219.
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Visosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhauser, Basel, 1997.
T. Bayen, F. Mairet, and P. Gajardo, Optimal synthesis for the minimum time control problems of fed-batch bioprocesses for growth functions with two maxima, J. Optim. Theory Appl., 158 (2013), pp. 521--553.
B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Math. Appl. 40, Springer, Berlin, 2002.
U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Math. Appl. 43, Springer, Berlin, 2004.
P. Cadarliaguet, On the regularity of semipermeable surfaces in control theory with application to the optimal exit-time problem (part I), SIAM J. Control Optim., 35 (1997), pp. 1638--1652.
P. Cadarliaguet, On the regularity of semipermeable surfaces in control theory with application to the optimal exit-time problem (part II), SIAM J. Control Optim., 35 (1997), pp. 1653--1671.
C. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd ed., John Wiley & Sons, New York, 1990.
I. Cacciari, S. Grego, and E. Di Mattia, Eco-physiological characterization of soil bacterial populations in different states of growth, Microb. Ecol., 43 (2002), pp. 34--43.
C. Camarasa, T. Clement, M. Perez, J. R. Mouret, and J. M. Sablayrolles, Use of a continuous multistage bioreactor to mimic winemaking fermentation, Internat. J. Food Microbiol., 150 (2011), pp. 42--49.
D. Dochain and A. Rapaport, Minimal time control of fed-batch processes for growth functions with several maxima, IEEE Trans. Automat. Control, 56 (2011), pp. 2671--2676.
A. Dramé, J. Harmand, C. Lobry, and A. Rapaport, Multiple steady state profiles in interconnected biological systems, Math. Comput. Model. Dyn. Syst., 12 (2006), pp. 379--393.
B. Dubey, P. Chandra, and P. Sinha, A model for fishery resource with reserve area, Nonlinear Anal.: Real World Appl., 4 (2003), pp. 625--637.
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., 13 (2007), pp. 207--236.
D. Dochain and P. Vanrolleghem, Dynamical Modelling and Estimation in Wastewater Treatment Processes, IWA Publishing, London, 2001.
H. El-Owaidy and O. El-Leithy, Theoretical studies on extinction in the gradostat, Math. Biosci., 101 (1990), pp. 1--26.
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), pp. 2827--2856.
F. Gérard, I. Haidar, and A. Rapaport, Effects of spatial structure and diffusion on the performances of the chemostat, Math. Biosci. Eng., 8 (2011), pp. 953--971.
R. Hannesson, Marine reserves: What would they accomplish?, Marine Resource Economics, 13 (1998), pp. 159--170.
J. Harmand, A. Rapaport, and A. Trofino, Optimal design of two interconnected bioreactors--some new results, Amer. Inst. Chemical Engineering J., 49 (1999), pp. 1433--1450.
Y. Higashi, N. Ytow, H. Saida, and H. Seki, In situ gradostat for the study of natural phytoplankton community with an experimental nutrient gradient, Environ. Pollution, 99 (1998), pp. 395--404.
G. Hill and C. Robinson, Minimum tank volumes for CFST bioreactors in series, Canadian J. Chemical Engineering, 67 (1989), pp. 818--824.
E. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley $\&$ Sons, New York, 1967.
U. Ledzewicz and H. Schattler, Geometric Optimal Control, Springer, New York, 2012.
R. Lovitt and J. Wimpenny, The gradostat: A bidirectional compound chemostat and its applications in microbial research, J. General Microbiol., 127 (1981), pp. 261--268.
A. Miele, Application of Green's Theorem to the extremization of linear integrals, in Proceedings of the Symposium on Vehicle Systems Optimization, Garden City, NY, (1961), pp. 26--35.
K. Mischaikow, H. L. Smith, and H. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Liapunov functions, Trans. Amer. Math. Soc., 347 (1995), pp. 1669--1685.
J. A. Moreno, Optimal time control of bioreactors for the wastewater treatment, Optim. Control Appl. Meth., 20 (1999), pp. 145--164.
M. Nelson and H. Sidhu, Evaluating the performance of a cascade of two bioreactors, Chemical Engineering Sci., 61 (2006), pp. 3159--3166.
M. Quincampoix, Differential inclusions and target problems, SIAM J. Control Optim., 30 (1992), pp. 324--335.
J. A. Sanchirico and J. E. Wilen, Bioeconomics of spacial exploitation in a patchy environment, J. Environmental Economics Management, 37 (1999), pp. 129--15.
H. L. Smith and P. Waltman, The gradostat: A model of competition along a nutrient gradient, J. Microb. Ecol., 22 (1991), pp. 207--226.
H. L. Smith and P. Waltman, The theory of the chemostat, in Dynamics of Microbial Competition, Cambridge University Press, Cambridge, UK, 1995.
B. Tang, Mathematical investigations of growth of microorganisms in the gradostat, J. Math. Biol., 23 (1986), pp. 319--339.
H. Veldcamp, Ecological studies with the chemostat, Adv. Microbial Ecology, 1 (1977), pp. 59--95.

Information & Authors


Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 2568 - 2594
ISSN (online): 1095-7138


Submitted: 23 December 2013
Accepted: 28 May 2014
Published online: 28 August 2014


  1. optimal control
  2. minimal time problem
  3. Pontryagin's maximum principle
  4. optimal synthesis
  5. chemostat model
  6. gradostat model
  7. nonlinear controllability
  8. semipermeability

MSC codes

  1. 49J15
  2. 49K15
  3. 49N25



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