Abstract
In this report, the controlled Lotka–Volterra competition model is used to describe the interaction of the concentrations of healthy and cancer cells. For this controlled model, the minimization problem of the terminal functional is considered, which is a weighted difference of the concentrations of cancerous and healthy cells at the final moment of the treatment period. To analyze the optimal solution of this problem, which consists of the optimal control and the corresponding optimal solutions of the differential equations that determine the model, the Pontryagin maximum principle is applied. It allows to highlight the values of the model parameters under which the optimal control corresponding to them is a piecewise-constant function with at most one switching. Also, the values of the model parameters are found, under which the corresponding optimal control is either a bang–bang function with a finite number of switchings, or in addition to the bang–bang-type portions (nonsingular portions), it also contains a singular arc. Further, only numerical investigations of the optimal control are possible. Therefore, the report presents the results of numerical calculations performed using the software BOCOP-2.1.0 that lead us to the conclusions about the possible type of the optimal control and the corresponding optimal solutions.
Research Perspectives CRM Barcelona, Summer 2018, vol. 11, in Trends in Mathematics Springer-Birkhäuser, Basel.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
F. Bonnans, P. Martinon, D. Giorgi, V. Grélard, S. Maindrault, O. Tissot, J. Liu, BOCOP 2.1.0—User Guide (2017), http://www.bocop.org/
E.B. Lee, L. Marcus, Foundations of Optimal Control Theory (Wiley, New York, 1967)
L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, Mathematical Theory of Optimal Processes (Wiley, New York, 1962)
H. Schättler, U. Ledzewicz, Optimal Control For Mathematical Models of Cancer Therapies. An Application of Geometric Methods (Springer, New York-Heidelberg-Dordrecht-London, 2015)
R.V. Solé, T.S. Deisboeck, An error catastrophe in cancer? J. Theor. Biol. 228, 47–54 (2004)
R.V. Solé, I.G. García, J. Costa, Spatial dynamics in cancer, in Complex systems science in biomedicine, ed by T.S. Deisboeck, J.Y. Kresh (Springer, New York, 2006), pp. 557–572
H. Weiss A Mathematical Introduction to Population Dynamics (2010)
M.I. Zelikin, V.F. Borisov, Theory of Chattering Control with Applications to Astronautics, Robotics, Economics and Engineering (Birkhäuser, Boston, 1994)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Khailov, E.N., Klimenkova, A.D., Korobeinikov, A. (2019). Optimal Control for Anticancer Therapy. In: Korobeinikov, A., Caubergh, M., Lázaro, T., Sardanyés, J. (eds) Extended Abstracts Spring 2018. Trends in Mathematics(), vol 11. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-25261-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-25261-8_6
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-25260-1
Online ISBN: 978-3-030-25261-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)