On some control problems for Kolmogorov type systems

Alexandru Hofman; Radu Precup; Alexandru Hofman; Radu Precup

doi:10.3934/mmc.2022011

Mathematical Modelling and Control

2022, Volume 2, Issue 3: 90-99. doi: 10.3934/mmc.2022011

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Research article

On some control problems for Kolmogorov type systems

Alexandru Hofman ¹,
Radu Precup ^{2
,
,}

1.
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania
2.
Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania & Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania

Received: 16 January 2022 Revised: 27 March 2022 Accepted: 13 May 2022 Published: 26 September 2022

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.
- Kolmogorov system,
- control problem,
- fixed point
Citation: Alexandru Hofman, Radu Precup. On some control problems for Kolmogorov type systems[J]. Mathematical Modelling and Control, 2022, 2(3): 90-99. doi: 10.3934/mmc.2022011

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Abstract

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.

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