Geodesic
Numerical methods for construction of value functions in optimal control problems on an infinite horizon
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 53 (2019), pp. 15-26.

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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. 
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A. L. Bagno; A. M. Tarasyev. Numerical methods for construction of value functions in optimal control problems on an infinite horizon. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 53 (2019), pp. 15-26. http://geodesic.mathdoc.fr/item/IIMI_2019_53_a1/

[1] Aseev S. M., Veliov V. M., “Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions”, Proceedings of the Steklov Institute of Mathematics, 291, Suppl. 1 (2015), 22–39 | DOI | MR

[2] Crandall M. G., Lions P.-L., “Viscosity solutions of Hamilton–Jacobi equations”, Trans. Amer. Math. Soc., 277:1 (1983), 1–42 | DOI | MR | Zbl

[3] Capuzzo Dolcetta I., “On a discrete approximation of the Hamilton–Jacobi equation of dynamic programming”, Applied Mathematics and Optimization, 10:1 (1983), 367–377 | DOI | MR

[4] Capuzzo Dolcetta I., Ishii H., “Approximate solution of the Bellman equation of deterministic control theory”, Applied Mathematics and Optimization, 11:1 (1984), 161–181 | DOI | MR | Zbl

[5] Falcone M., “A numerical approach to the infinite horizon problem of deterministic control theory”, Applied Mathematics and Optimization, 15:1 (1987), 1–13 | DOI | MR | Zbl

[6] Skritek B., Veliov V. M., “On the infinite-horizon optimal control of age-structured systems”, Journal of Optimization Theory and Applications, 167:1 (2015), 243–271 | DOI | MR | Zbl

[7] Souganidis P. E., “Approximation schemes for viscosity solutions of Hamilton–Jacobi equations”, Journal of Differential Equations, 59:1 (1985), 1–43 | DOI | MR | Zbl

[8] Adiatulina R. A., Taras'ev A. M., “A differential game of unlimited duration”, Journal of Applied Mathematics and Mechanics, 51:4 (1987), 415–420 | DOI | MR | Zbl

[9] Aseev S. M., Kryazhimskii A. V., “The Pontryagin maximum principle and optimal economic growth problems”, Proceedings of the Steklov Institute of Mathematics, 257:1 (2007), 1–255 | DOI | MR | Zbl

[10] Bagno A. L., Taras'ev A. M., “Discrete approximation of the Hamilton–Jacobi equation for the value function in an optimal control problem with infinitesimal horizon”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 24, no. 1, 2018, 27–39 (in Russian) | DOI

[11] Bagno A. L., Tarasyev A. M., “Estimate for the accuracy of a backward procedure for the Hamilton–Jacobi equation in an infinite-horizon optimal control problem”, Proceedings of the Steklov Institute of Mathematics, 304:1 (2019), 110–123 | DOI | DOI | MR | Zbl

[12] 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, 26:1 (2016), 3–14 (in Russian) | DOI | MR | Zbl

[13] Bagno A. L., Taras'ev A. M., “Stability properties of the value function in an infinite horizon optimal control problem”, Proceedings of the Steklov Institute of Mathematics, 301, suppl. 1 (2018), 1–14 | DOI | MR

[14] Bellman R., Dynamic programming, Princeton University Press, New York, 1957 | MR | MR | Zbl

[15] Besov K. O., “On necessary optimality conditions for infinite-horizon economic growth problems with locally unbounded instantaneous utility function”, Proceedings of the Steklov Institute of Mathematics, 284:1 (2014), 50–80 | DOI | DOI | MR | Zbl

[16] Krasovskii N. N., Subbotin A. I., Positional differential games, Nauka, M., 1974 | MR

[17] Taras'yev A. M., Ushakov V. N., Khripunov A. P., “On a computational algorithm for solving game control problems”, Journal of Applied Mathematics and Mechanics, 51:2 (1987), 167–172 | DOI | MR | Zbl

[18] Subbotin A. I., Minimax inequalities and Hamilton–Jacobi equations, Nauka, M., 1991

[19] Subbotin A. I., Taras'ev A. M., “Conjugate derivatives of the value function of a differential game”, Soviet Math. Dokl., 32 (1985), 162–166 | MR | Zbl