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 Izv. IMI UdGU, 2019, Volume 53, Pages 15–26 (Mi iimi367)

Numerical methods for construction of value functions in optimal control problems on an infinite horizon

A. L. Bagnoa, A. M. Tarasyevba

a Ural Federal University, pr. Lenina, 51, Yekaterinburg, 620083, Russia
b Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16, Yekaterinburg, 620990, Russia

Abstract: This article deals with the optimal control problem on an infinite horizon, the quality functional of which is contained in the integrand index and the discounting factor. A special feature of this formulation of the problem is the assumption of the possible unboundedness of the integrand index. The problem reduces to an equivalent optimal control problem with a stationary value function as a generalized (minimax, viscosity) solution of the Hamilton–Jacobi equation satisfying the Hölder condition and the condition of linear growth. The article describes the backward procedure on an infinite horizon. It is the method of numerical approximation of the generalized solution of the Hamilton–Jacobi equation. The main result of the article is an estimate of the accuracy of approximation of a backward procedure for solving the original problem. Problems of the analyzed type are related to modeling processes of economic growth and to problems of stabilizing dynamic systems. The results obtained can be used to construct numerical finite-difference schemes for calculating the value function of optimal control problems or differential games.

Keywords: optimal control, generalized solutions of Hamilton–Jacobi equations, value function, approximation schemes, backward procedures.

DOI: https://doi.org/10.20537/2226-3594-2019-53-02

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UDC: 517.977
MSC: 49K15

Citation: A. L. Bagno, A. M. Tarasyev, “Numerical methods for construction of value functions in optimal control problems on an infinite horizon”, Izv. IMI UdGU, 53 (2019), 15–26

Citation in format AMSBIB
\Bibitem{BagTar19} \by A.~L.~Bagno, A.~M.~Tarasyev \paper Numerical methods for construction of value functions in optimal control problems on an infinite horizon \jour Izv. IMI UdGU \yr 2019 \vol 53 \pages 15--26 \mathnet{http://mi.mathnet.ru/iimi367} \crossref{https://doi.org/10.20537/2226-3594-2019-53-02} \elib{http://elibrary.ru/item.asp?id=38503195}