BSU bulletin. Mathematics, Informatics
, PARAMETERIZATION OF OPTIMAL CONTROL PROBLEMS APPLIED TO A SINGLE MODEL OF BIOLOGICAL WATER TREATMENT // BSU bulletin. Mathematics, Informatics. - 2021. №1. . - С. 3-12.
PARAMETERIZATION OF OPTIMAL CONTROL PROBLEMS APPLIED TO A SINGLE MODEL OF BIOLOGICAL WATER TREATMENT
A mathematical model related to the process of biological wastewater treatment by eliminating pathogenic microorganisms and reducing the concentration of organic substances is considered. The process is described using a controlled three- dimensional system of differential equations. The adequacy of the phase trajectories to the meaningful meaning of the variables under consideration is investigated. Two op- timal control problems are set for the minimum of terminal and integral functionals that make sense of the concentration of wastewater pollution. In modern conditions, such tasks are quite relevant. The study of problems is carried out based on the maximum principle. The analysis of the control switching function leads to the conclusion that there are no special modes and allows us to specify the structure of optimal controls by the number of switching points. As a result, optimal control problems are reduced to minimizing the functions of one or two variables with the possibility of using deriva- tives.
biological water purification; the system of differential equations; opti- mal control problems; maximum principle; parametrization of the problem.
List of references:
Bondarenko N. V., Grigorieva E. V., Khaylov E. N. Problems of Pollution Minimization in the Mathematical Model of Biological Wastewater Treatment. Compu- tational Mathematics and Mathematical Physics. 2012. Vol. 52, no. 4. Pp. 614–627.
Gabasov R., Kirillova F. M. The Maximum Principle in the Theory of Optimal Control. Moscow: Librocom Publ., 2011. 272 p.
Gorbunov V. K. On Reducing Optimal Control Problems to Finitedimensional Ones. Computational Mathematics and Mathematical Physics. 1978. Vol. 18, no. 5. Pp. 1083–1095.
Srochko V. A. Iterative Methods for Solving Optimal Control Problems. Mos- cow: Fizmatlit Publ., 2000. 160 p.
Rojas J., Burke M., Chapwanya M. Modeling of Autothermal Thermophilic Aerobic Digestion. Math. Industry Case Studies J. 2010. V. 2. Pp. 34–63.