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Investigating a Nonlinear Optimal Control Problem with Discounting

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In this article, we consider one-dimensional optimization models that are relevant for mining applications. The maximand functional is interpreted as discounted profit. Several alternative formulations of the optimization problem are considered on finite and infinite horizons. Optimal solutions are constructed in analytical form. The optimal control is determined in the form of a time function (a program) and in the form of a function of the phase coordinate (feedback — the hyperbolic tangent law). The theoretical basis for our results is provided by Pontryagin’s maximum principle and Bellman’s dynamic programming method. The theoretical results provide a foundation for computer experiments with prototype and real-life data. The article is an expanded version of [5]. The discussion is accompanied by graphs.

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References

  1. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Moscow (1961).

  2. R. Bellman et al., Optimization Techniques with Application to Space Flight Mechanics [Russian translation], G. Leitman, ed., Nauka, Moscow (1965).

  3. S. N. Avvakumov, Yu. N. Kiselev, and M. V. Orlov, “Investigation of a one-dimensional optimization model MD1. Infinite horizon. Finite horizon,” Problemy Dinamicheskogo Upravleniya, MAKS Press, Moscow, Vyp. 1 (2005), pp. 111–136.

  4. N. L. Grigorenko, D. V. Kamzolkin, and L. N. Luk’yanova, “Solution of an optimal control problem,” Problemy Dinamicheskogo Upravleniya, MAKS Press, Moscow, Vyp. 1 (2005), pp. 137–144.

  5. Yu. N. Kiselev, S. N. Avvakumov, and M. V. Orlov, “The hyperbolic tangent law for optimal control synthesis in a nonlinear model with discounting,” Diff. Uravn., 42, No. 11, 1490–1506 (2006).

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  6. Yu. N. Kiselev, “Sufficient optimality conditions in terms of Pontryagin’s maximum principle constructs,” in: Mathematical Models in Economics and Biology. Proc. of a Scientific Seminar, Planernoe, Moscow Province, MAKS Press, Moscow (2003), pp. 57–67.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Faculty of Computation Mathematics and Cybernetics, Moscow, Russia

    S. N. Avvakumov, Yu. N. Kiselev & M. V. Orlov

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  1. S. N. Avvakumov
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  2. Yu. N. Kiselev
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  3. M. V. Orlov
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Translated from Problemy Dinamicheskogo Upravleniya, Issue 2, 2007, pp. 4–38.

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Avvakumov, S.N., Kiselev, Y.N. & Orlov, M.V. Investigating a Nonlinear Optimal Control Problem with Discounting. Comput Math Model 25, 239–269 (2014). https://doi.org/10.1007/s10598-014-9223-4

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  • DOI: https://doi.org/10.1007/s10598-014-9223-4

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