We consider a mathematical model that describes biological treatment of wastewater. The model is a controlled nonlinear three-dimensional system of differential equations. A minimum-time optimal control problem is posed that achieves the required contaminant concentration in the shortest possible time. The problem is investigated by Pontryagin’s maximum principle. The optimal control in this problem is represented by a switching function and its properties are analyzed. The results of this analysis make it possible to reduce the solution of the problem to a successive application of finite-dimensional constrained minimization and the parameter continuation method. The application of these approaches is discussed in detail, and the relevant numerical results are reported.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
J. Rojas, M. Burke, M. Chapwanya, K. Doherty, I. Hewitt, A. Korobeinikov, M. Meere, S. McCarthy, M. O’Brien, V. T. N. Tuoi, H. Winstenley, and T. Zhelev, “Modeling of autothermal thermophilic aerobic direction,” Mathematics-in-Industry Case Studies Journal, 2, 34–63 (2010).
J. Gomes, M. de Gracia, E. Ayesa, and J. L. Garcia-Heras, “Mathematical modelling of autothermal thermophilic aerobic digesters,” Water Research,41, No. 5, 959–968 (2007).
K. S. Krasnov, N. K. Vorob’ev, I. N. Godnev and others, Physical Chemistry. Vol. 2. Electrochemistry. Chemical Kinetics and Catalysis [in Russian], Vysshaya Shkola, Moscow (1995).
N. V. Bondarenko, E. V. Grigor’eva, and E. N. Khailov, “Contaminant minimization problems in a mathematical model of biological wastewater treatment,” Zh. Vychisl. Mat. i Matem. Fiz.,52, No. 4, 614–637 (2012).
W. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control [Russian translation], Mir, Moscow (1978).
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1983).
N. V. Bondarenko, Finite-Dimensional Methods in Applied Optimal Control Problems [in Russian], Dissertation, Moscow (2013).
F. P. Vasil’ev, Optimization Methods [in Russian], Faktorial Press, Moscow (20020.
N. V. Bondarenko, E. V. Grigor’eva, and E. N. Khailov, “Solution of a controllability problem for a nonlinear three-dimensional system,” Vestn. MGU, Ser. 15, Vychisl. Mat. Kibern., No. 1, 9–14 (2012).
V. I. Shalashilin and E. B. Kuznetsov, Parameter Continuation Method and Best Parametrization [in Russian], Editorial URSS, Moscow (1999).
V. A. Il’in and G. M. Pozdnyak, Principles of Mathematical Analysis [in Russian], Vol. 1, Nauka, Moscow (1983).
L. S. Pontryagin, Ordinary Differential Equations [in Russian], Nauka, Moscow (1974).
E. N. Khailov, “Finding the switch times of extremal control in the nonlinear minimum-time problem,” Diff. Uravn.,28, No. 11, 1988–1993 (1992).
About this article
Cite this article
Grigor’eva, E.V., Khailov, E.N. Minimum-Time Optimal Control for a Model of Biological Wastewater Treatment. Comput Math Model 31, 179–189 (2020). https://doi.org/10.1007/s10598-020-09487-7
- biological wastewater treatment
- minimum-time optimal control
- mathematical model
- Pontryagin’s maximum principle