Dynamic systems and optimal control

On optimal control problems with active infinite horizon

D. V. Khlopin^a, A. O. Belyakov<sup>bc

a</sup> Krasovskii Institute of Mathematics and Mechanics UB RAS, Yekaterinburg, Russian Federation
^b Lomonosov Moscow State University, Russian Federation
^c National University of Science and Technology “MISIS”, Russian Federation

Abstract: The article proposes a new formulation of the optimal control problem on an infinite horizon. Usually in such problems, if a Bolza type problem is considered, then its terminal cost depends only on the initial state, additionally, one or another asymptotic requirement can be presented to the right end of the system. A feature of the proposed formulation is the ability to set control not only as a function of time, but also to choose an action-control at the completion of the process itself. This is primarily interesting from the point of view of economic applications, since it is the endless postponement of a generally unprofitable action (for example, "debt repayment") that often leads to a lack of optimal control. In addition to this new formulation, some necessary optimality conditions for the case of the simplest dynamics are presented. With these conditions the example of optimizing consumption under various borrowing restrictions is investigated.

Keywords: infinite horizon control problem, active infinite horizon, Pontryagin maximum principle, weakly overtaking optimality.

UDC: 517.977

MSC: 49K15, 91B62

Received: 18.09.2025
Revised: 29.10.2025
Accepted: 31.10.2025

Language: English

DOI: 10.26516/1997-7670.2025.54.64