The paper deals with some control problems related to the Kolmogorov system for two interacting populations. For the first problem, the control acts in time over the per capita growth rates of the two populations in order for the ratio between their sizes to follow a prescribed evolution. For the second problem, the control is a constant which adjusts the per capita growth rate of a single population so that it reaches the desired size at a certain time. For the third problem the control acts on the growth rate of one of the populations in order that the total population to reach a prescribed level. The solution of the three problems is done within an abstract scheme, by using operator-based techniques. Some examples come to illustrate the results obtained. One refers to a system that models leukemia, and another to the SIR model with vaccination.
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania
Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babeş-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca
Kolmogorov system, control problem, fixed point.
Al. Hofman, R. Precup, On some control problems for Kolmogorov type systems, Mathematical Modelling and Control, 2 (2022) no. 3, pp. 90-99, http://doi.org/10.3934/mmc.2022011
Mathematical Modelling and Control
google scholar link
 V. Barbu, Mathematical methods in optimization of differential systems, Dordrecht: Springer Science+Business Media, 1994.
 I.-Ş. Haplea, L.-G. Parajdi, R. Precup, On the controllability of a system modeling cell dynamics related to leukemia, Symmetry, 13 (2021), 1867. https://doi.org/10.3390/sym13101867 doi: 10.3390/sym13101867
 R. Precup, Fixed point theorems for decomposable multivalued maps and applications, Zeitschrift fu¨r Analysis und ihre Anwendungen, 22 (2003), 843–861. https://doi.org/10.4171/ZAA/1176
 R. Precup, On some applications of the controllability principle for fixed point equations, Results in Applied Mathematics, 13 (2022), 100236. https://doi.org/10.1016/j.rinam.2021.100236 doi: 10.1016/j.rinam.2021.100236
 M. E. M. Meza, A. Bhaya, E. Kaszkurewicz, Controller design techniques for the Lotka-Volterra nonlinear system, Sba: Controle and Automaça~, 16 (2005), 124–135. https://doi.org/10.1590/S0103-17592005000200002
 K. Balachandran, J. P. Dauer, Controllability of nonlinear systems via fixed-point theorems, J. Optimiz. Theory Appl., 53 (1987), 345–352. https://doi.org/10.1007/BF00938943 doi: 10.1007/BF00938943
 N. Carmichael, M. D. Quinn, Fixed point methods in nonlinear control, In: F. Kappel, K. Kunisch, W. Schappacher (eds) Distributed parameter systems, Lecture Notes in Control and Information Sciences, vol 75, Berlin: Springer, 1985.
 L. Górniewicz, S. K. Ntouyas, D. O’Regan, Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces, Reports on Mathematical Physics, 56 (2005), 437–470. https://doi.org/10.1016/S0034-4877(05)80096-5 doi: 10.1016/S0034-4877(05)80096-5
 J. Klamka, Schauder’s fixed-point theorem in nonlinear controllability problems, Control Cybern., 29 (2000), 153–165.
 J. Klamka, A. Babiarz, M. Niezabitowski, Banach fixed-point theorem in semilinear controllability problems – a survey, B. Pol. Acad. Sci.-Tech., 64 (2016), 21–35. https://doi.org/10.1515/bpasts-2016-0004 doi: 10.1515/bpasts-2016-0004
 H. Leiva, Rothe’s fixed point theorem and controllability of semilinear nonautonomous systems, Syst. Control Lett., 67 (2014), 14–18. https://doi.org/10.1016/j.sysconle.2014.01.008 doi: 10.1016/j.sysconle.2014.01.008
 J.-M. Coron, Control and nonlinearity, Mathematical Surveys and Monographs Vol. 136, Providence: Amer. Math. Soc., 2007.
 A. N. Kolmogorov, Sulla teoria di Volterra della lotta per l’esistenza, Giornale dell’Istituto Italiano degli Attuari, 7 (1936), 74–80.
 K. Sigmund, Kolmogorov and population dynamics, In: É. Charpentier, A. Lesne, N. K. Nikolski (eds) Kolmogorov’s heritage in mathematics, Berlin: Springer, 2007.
 A. Granas, J. Dugundji, Fixed point theory, New York: Springer, 2003.
 R. Precup, Methods in nonlinear integral equations, Dordrecht: Springer Science+Business Media, 2002.
 B. Neiman, A mathematical model of chronic myelogenous leukemia, Oxford: Oxford University, 2000.