Further results on stabilization of periodic trajectories for a chemostat with two species

IEEE Transactions on Automatic Control

Volumn 53, Issue SPECIAL ISSUE, 2008, Pages 66-74

  Mazenc, Frédéric ^a   Malisoff, Michael ^b   Harmand, Jérôme ^c  

^a INRIA   (France)

^b LOUISIANA STATE UNIVERSITY   (United States)

^c LABORATOIRE DE BIOTECHNOLOGIE DE L ENVIRONNEMENT   (France)

Author keywords

Chemostats; Lyapunov functions; Stabilization; Systems biology; Tracking

Indexed keywords

CLOSED LOOPS; DILUTION RATE; EXPONENTIALLY STABLE; GLOBAL ERRORS; LINEAR FEEDBACK CONTROL; LYAPUNOV FUNCTION METHOD; OSCILLATORY BEHAVIORS; PERIODIC TRAJECTORIES; REFERENCE TRAJECTORIES; SPECIES CONCENTRATION; STABILIZATION SYSTEMS; TIME VARYING; OSCILLATORY REFERENCE TRAJECTORY; PERIODIC TRAJECTORY;

CONCENTRATION (PROCESS); DIFFERENTIAL EQUATIONS; FEEDBACK CONTROL; LYAPUNOV FUNCTIONS; NAVIGATION; NONLINEAR FEEDBACK; STABILIZATION; TRAJECTORIES; ASYMPTOTIC STABILITY; CHEMOSTATS; CLOSED LOOP CONTROL SYSTEMS; COMPUTER SIMULATION; DILUTION; LINEAR CONTROL SYSTEMS; SYSTEM STABILITY;

CHEMOSTATS; TIME VARYING SYSTEMS;

EID: 39849094915     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2007.911315     Document Type: Article

References (36)

1. J. Arino, J.-L. Gouzé, and A. Sciandra, "A discrete, size-structured model of phytoplankton growth in the chemostat. Introduction of non constant cell division," J. Math. Biol., vol.45, pp. 313-333, 2002.
2. R. A. Armstrong and R. McGehee, "Competitive exclusion," Amer. Naturalist, vol.115, pp. 151-170, 1980.
3. R. A. Armstrong and R. McGehee, "Coexistence of species competing for shared resources," Theor. Pop. Biol., vol.9, pp. 317-328, 1989.
4. O. Bernard and J.-L. Gouzé, "Nonlinear qualitative signal processing for biological systems: Application to the algal growth in bioreactors," Math. Biosci., vol.157, pp. 357-372, 1999.
5. G. J. Butler, S. B. Hsu, and P.Waltman, "A mathematical model of the chemostat with periodic washout rate," SIAM J. Appl. Math., vol.45, pp. 435-449, 1985.
6. P. De Leenheer, B. Li, and H. L. Smith, "Competition in the chemostat: Some remarks," Canad. Appl. Math. Q., vol.11, pp. 229-248, 2003.
7. P. De Leenheer and S. S. Pilyugin, "Feedback-mediated oscillatory coexistence in the chemostat," in Proc. 2nd Multidisciplinary Int. Symp. Positive Systems: Theory and Applications (POSTA 06), Grenoble, France, C. Commault and N. Marchand, Eds., Grenoble, France, 2006, pp. 97-104.
8. P. De Leenheer and H. L. Smith, "Feedback control for chemostat models," J. Math. Biol., vol.46, pp. 48-70, 2003.
9. S. F. Ellermeyer, S. S. Pilyugin, and R. Redheffer, "Persistence criteria for a chemostat with variable nutrient input," J. Diff. Eq., vol.171, pp. 132-147, 2001.
10. T. C. Gard, "A new Liapunov function for the simple chemostat," Nonlinear Analysis: Real World Applications, vol.3, pp. 211-216, 2002.
11. G. F. Gause and A. A. Witt, "Behavior of mixed populations and the problem of natural selection," Amer. Naturalist, vol.69, pp. 596-609, 1935.
12. J.-L. Gouzé and G. Robledo, "Feedback control for nonmonotone competition models in the chemostat," Nonlinear Anal., vol.6, pp. 671-690, 2005.
13. F. Grognard, F. Mazenc, and A. Rapaport, "Polytopic Lyapunov functions for the stability analysis of persistence of competing species," in Proc. 44th IEEE Conf. Decision Contr. Eur. Contr. Conf., Seville, Spain, 2005, pp. 3699-3704.
14. J. K. Hale and A. S. Somolinos, "Competition for fluctuating nutrient," J. Math. Biol., vol.18, pp. 255-280, 1983.
15. G. Hardin, "The competitive exclusion principle," Science, vol.131, pp. 1292-1298, 1960.
16. G. E. Hutchinson, "The paradox of the plankton," American Naturalist, vol.95, pp. 137-145, 1961.
17. A. Karama, O. Bernard, J.-L. Gouzé, A. Benhammou, and D. Dochain, "Hybrid neural modelling of an anaerobic digester with respect to biological constraints," Wat. Sci. Technol., vol.43, pp. 1-8, 2001.
18. H. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2002.
19. P. Lenas and S. Pavlou, "Coexistence of three competing microbial populations in a chemostat with periodically varying dilution rate," Math. Biosci., vol.129, pp. 111-142, 1995.
20. B. Li, "Asymptotic behavior of the chemostat: General responses and different removal rates," SIAM J. Appl. Math., vol.59, pp. 411-422, 1998.
21. C. Lobry, F. Mazenc, and A. Rapaport, "Persistence in ecological models of competition for a single resource," Comp. Rend. Math. Acad. des Sci., vol.340, pp. 199-204, 2005.
22. F. Mazenc, P. De Leenheer, and M. Malisoff, "Stabilizing a periodic solution in the chemostat: A case study in tracking," in Proc. 45th IEEE Conf. Decision Contr., San Diego, CA, 2006, pp. 1794-1799.
23. F. Mazenc, C. Lobry, and A. Rapaport, "Persistence in ratio-dependent models for consumer-resource dynamics," Electron. J. Diff. Eq., vol.15, pp. 211-220, 2007.
24. F. Mazenc, M. Malisoff, and P. De Leenheer, "On the stability of periodic solutions in the perturbed chemostat," Math. Biosci. Eng., vol.4, pp. 319-338, 2007.
25. F. Mazenc and D. Nesic, "Lyapunov functions for time varying systems satisfying generalized conditions of Matrosov theorem," in Proc. 44th IEEE Conf. Decision Contr. Eur. Contr. Conf., Seville, Spain, 2005, pp. 2432-2437.
26. K. S. McCann, "The diversity-stability debate," Nature, vol.405, pp. 228-233, 2000.
27. J. Monod, Recherches sur la Croissance des Cultures Bactériennes. Paris: , 1942, Eds. Hermann et cie.
28. A. Novick and L. Szilard, "Experiments with the chemostat on spontaneous mutations of bacteria," Proc. Nat. Acad. Sci., vol.36, pp. 708-719, 1950.
29. S. Pilyugin and P. Waltman, "Competition in the unstirred chemostat with periodic input and washout," SIAM J. Appl. Math., vol.59, pp. 1157-1177, 1999.
30. N. S. Rao and E. O. Roxin, "Controlled growth of competing species," SIAM J. Appl. Math., vol.50, pp. 853-864, 1990.
31. S. Sastry, Nonlinear Systems, Analysis, Stability and Control. New York: Springer Verlag, 1999.
32. H. L. Smith, "Competitive coexistence in an oscillating chemostat," SIAM J. Appl. Math., vol.40, pp. 498-522, 1981.
33. H. L. Smith, "Microbial growth in periodic gradostats," Rocky Mountain J. Math., vol.20, pp. 1173-1194, 1990.
34. H. L. Smith and P. Waltman, The Theory of the Chemostat. Cambridge: Cambridge University Press, 1995.
35. H. L. Smith and P. Waltman, "Competition in the periodic gradostat," Nonlinear Analysis: Real World Applications, vol.1, pp. 177-188, 2000.
36. G. Wolkowicz and Z. Lu, "Global dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates," SIAM J. Appl. Math., vol.52, pp. 222-233, 1992.