Abstract:
Properties of the value function are examined in an infinite horizon optimal control problem with an integrand index appearing in the quality functional with a discount factor. The properties are analyzed in the case when the payoff functional of the control system includes aquality index represented by an unbounded function. An upper estimate is given for the growth rate of the value function. Necessary and sufficient conditions are obtained to ensure that the value function satisfies the infinitesimal stability properties. The question of coincidence of the value function with the generalized minimax solution of the Hamilton-Jacobi-Bellman-Isaacs equation is discussed. The uniqueness of the corresponding minimax solution is shown. The growth asymptotic behavior of the value function is described for the logarithmic, power, and exponential quality functionals, which arise in economic and financial modeling. The obtained results can be used toconstruct grid approximation methods for the value function as the generalized minimax solution of the Hamilton-Jacobi-Bellman-Isaacs equation. These methods are effective tools in the modeling of economic growth processes.
Citation:
A. L. Bagno, A. M. Tarasyev, “Stability properties of the value function in an infinite horizon optimal control problem”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 1, 2017, 43–56; Proc. Steklov Inst. Math. (Suppl.), 301, suppl. 1 (2018), 1–14
\Bibitem{BagTar17}
\by A.~L.~Bagno, A.~M.~Tarasyev
\paper Stability properties of the value function in an infinite horizon optimal control problem
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2017
\vol 23
\issue 1
\pages 43--56
\mathnet{http://mi.mathnet.ru/timm1383}
\crossref{https://doi.org/10.21538/0134-4889-2017-23-1-43-56}
\elib{https://elibrary.ru/item.asp?id=28409367}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2018
\vol 301
\issue , suppl. 1
\pages 1--14
\crossref{https://doi.org/10.1134/S0081543818050012}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000453520500006}
Linking options:
https://www.mathnet.ru/eng/timm1383
https://www.mathnet.ru/eng/timm/v23/i1/p43
This publication is cited in the following 3 articles:
A. L. Bagno, A. M. Tarasyev, “Estimate for the Accuracy of a Backward Procedure for the Hamilton–Jacobi Equation in an Infinite-Horizon Optimal Control Problem”, Proc. Steklov Inst. Math., 304 (2019), 110–123
A. L. Bagno, A. M. Tarasev, “Chislennye metody postroeniya funktsii tseny v zadachakh optimalnogo upravleniya na beskonechnom gorizonte”, Izv. IMI UdGU, 53 (2019), 15–26
A. L. Bagno, A. M. Tarasev, “Diskretnaya approksimatsiya uravneniya Gamiltona - Yakobi dlya funktsii tseny v zadache optimalnogo upravleniya s beskonechnym gorizontom”, Vypusk posvyaschen 70-letnemu yubileyu Aleksandra Georgievicha Chentsova, Tr. IMM UrO RAN, 24, no. 1, 2018, 27–39
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