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## Archive of Issues

Russia Yekaterinburg
Year
2016
Volume
26
Issue
1
Pages
3-14
 Section Mathematics Title Properties of the value function in optimal control problems with infinite horizon Author(-s) Bagno A.L.a, Tarasyev A.M.b Affiliations Ural Federal Universitya, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesb Abstract The article investigates properties of the value function of the optimal control problem on infinite horizon with an unlimited integrand index appearing in the quality functional with a discount factor. The estimate is derived for approximating the value function in a problem with the infinite horizon by levels of value functions in problems with lengthening finite horizons. The structure of the value function is identified basing on stationary value functions which depend only on phase variables. The description is given for the asymptotic growth of the value function generated by various types of the quality functional applied in economic and financial modeling: logarithmic, power, exponential, linear functions. The property of continuity is specified for the value function and estimates are deduced for the Hölder parameters of continuity. These estimates are needed for the development of grid algorithms designed for construction of the value function in optimal control problems with infinite horizon. Keywords optimal control, infinite horizon, value function, estimation of continuity modulus, asymptotic properties UDC 517.977 MSC 49K15 DOI 10.20537/vm160101 Received 27 October 2015 Language Russian Citation Bagno A.L., Tarasyev A.M. Properties of the value function in optimal control problems with infinite horizon, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 1, pp. 3-14. References Capuzzo Dolcetta I.C., Ishii H. Approximate solution of the Bellman equation of deterministic control theory, Appl. Math. Optimiz., 1984, vol. 11, no. 1, pp. 161-181. Nikol'skii M.S. Continuity and the Lipschitz property of the Bellman function in some optimization problems on the semi-infinite interval $[0,+\infty)$, Differential Equations, 2002, vol. 38, no. 11, pp. 1599-1604. Adiatulina R.A., Tarasyev A.M. A differential game of unlimited duration, J. Appl. Math. Mech., 1987, vol. 51, no. 4, pp. 415-420. Intriligator M. Matematicheskie metody optimizatsii i ekonomicheskaya teoriya (Mathematical optimization methods and economic theory), Moscow: Airis press, 2002, 576 p. Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial’nye igry (Positional differential games), Moscow: Nauka, 1974, 456 p. Krushvits L. Finansirovanie i investitsii (Financing and investments), St. Petersburg: Piter, 2000, 381 p. Subbotin A.I. Minimaksnye neravenstva i uravneniya Gamil'tona-Yakobi (Minimax inequalities and Hamilton-Jacobi equations), Moscow: Nauka, 1991, 216 p. Full text