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 Mat. Zametki, 2018, Volume 103, Issue 2, Pages 163–171 (Mi mz11657)

On Balder's Existence Theorem for Infinite-Horizon Optimal Control Problems

K. O. Besov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Balder's well-known existence theorem (1983) for infinite-horizon optimal control problems is extended to the case in which the integral functional is understood as an improper integral. Simultaneously, the condition of strong uniform integrability (over all admissible controls and trajectories) of the positive part $\max\{f_0,0\}$ of the utility function (integrand) $f_0$ is relaxed to the requirement that the integrals of $f_0$ over intervals $[T,T']$ be uniformly bounded above by a function $\omega(T,T')$ such that $\omega(T,T')\to 0$ as $T,T'\to\infty$. This requirement was proposed by A.V. Dmitruk and N.V. Kuz'kina (2005); however, the proof in the present paper does not follow their scheme, but is instead derived in a rather simple way from the auxiliary results of Balder himself. An illustrative example is also given.

Keywords: optimal control, existence theorem, infinite horizon.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.4213/mzm11657

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English version:
Mathematical Notes, 2018, 103:2, 167–174

Bibliographic databases:

UDC: 517.977.57

Citation: K. O. Besov, “On Balder's Existence Theorem for Infinite-Horizon Optimal Control Problems”, Mat. Zametki, 103:2 (2018), 163–171; Math. Notes, 103:2 (2018), 167–174

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz11657
• https://doi.org/10.4213/mzm11657
• http://mi.mathnet.ru/eng/mz/v103/i2/p163

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This publication is cited in the following articles:
1. S. M. Aseev, “An existence result for infinite-horizon optimal control problem with unbounded set of control constraints”, IFAC Proceedings Volumes (IFAC-PapersOnline), 51:32 (2018), 281–285
2. S. M. Aseev, K. O. Besov, S. Yu. Kaniovski, “Optimal Policies in the Dasgupta–Heal–Solow–Stiglitz Model under Nonconstant Returns to Scale”, Proc. Steklov Inst. Math., 304 (2019), 74–109
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