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Trudy Inst. Mat. i Mekh. UrO RAN, 2017, Volume 23, Number 4, Pages 265–280 (Mi timm1486)  

This article is cited in 3 scientific papers (total in 3 papers)

Optimal result in a control problem with piecewise monotone dynamics

N. N. Subbotinaab, N. G. Novoselovaab

a Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia
b Ural Federal University, Yekaterinburg, 620002 Russia

Abstract: We consider an optimal control problem for a deterministic nonlinear system with piecewise monotone dynamics. The mathematical model under consideration describes the process of a chemotherapy treatment of a malignant tumor. The research makes it possible to analyze the influence of the type of nonmonotonicity on the structure of the optimal control. We consider the case when the therapy function, which describes the effect of the drug on the cell growth rate, has two maxima. Comparisons are made with the results for the previously studied case of a single maximum of the therapy function in this model. This paper is devoted to the construction of the value function for the optimal control problem under consideration. As is known, the value function is the basis for constructing an optimal synthesis, i.e., an optimal feedback strategy in the therapy. We use the fact that the value function is the unique minimax (viscosity) solution of the Cauchy problem for the basic Hamilton–Jacobi–Bellman (HJB) equation. By means of the continuous gluing of a finite number of smooth functions obtained by the Cauchy method of characteristics for auxiliary HJB equations, a continuous function $\varphi$ is constructed. A new element of the construction is the line of nonsmooth gluing with the use of the Rankin–Hugoniot conditions. This line plays a key role for the optimal feedback strategy, because it determines its discontinuity line. We prove that the constructed function $\varphi$ coincides with the minimax solution of the Cauchy problem for the basic HJB equation.

Keywords: optimal control, Rankine–Hugoniot line, Hamilton–Jacobi–Bellman equation, Cauchy method of characteristics.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-01-00074
Russian Academy of Sciences - Federal Agency for Scientific Organizations PRAS-18-01


DOI: https://doi.org/10.21538/0134-4889-2017-23-4-265-280

Full text: PDF file (240 kB)
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Bibliographic databases:

UDC: 517.977
MSC: 47N05, 37N25, 37N40
Received: 02.09.2017

Citation: N. N. Subbotina, N. G. Novoselova, “Optimal result in a control problem with piecewise monotone dynamics”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 4, 2017, 265–280

Citation in format AMSBIB
\Bibitem{SubNov17}
\by N.~N.~Subbotina, N.~G.~Novoselova
\paper Optimal result in a control problem with piecewise monotone dynamics
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2017
\vol 23
\issue 4
\pages 265--280
\mathnet{http://mi.mathnet.ru/timm1486}
\crossref{https://doi.org/10.21538/0134-4889-2017-23-4-265-280}
\elib{https://elibrary.ru/item.asp?id=30713980}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. N. N. Subbotina, N. G. Novoselova, “The value function in a problem of chemotherapy of a malignant tumor growing according to the Gompertz law”, IFAC-PapersOnLine, 51:32 (2018), 855–860  crossref  isi  scopus
    2. N. N. Subbotina, N. G. Novoselova, “On Applications of the Hamilton–Jacobi Equations and Optimal Control Theory to Problems of Chemotherapy of Malignant Tumors”, Proc. Steklov Inst. Math., 304 (2019), 257–267  mathnet  crossref  crossref  mathscinet  isi  elib
    3. N. G. Novoselova, “Numerical constructions of optimal feedback in models of chemotherapy of a malignant tumor”, J. Bioinform. Comput. Biol., 17:1, SI (2019), 1940004  crossref  isi  scopus
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