UDC 517.977.56, 517.977.54
MSC: 49L12, 35F20, 49L20, 49L25, 82C21
https://doi.org/10.21538/0134-4889-2026-32-1-fon-01
The research was funded by the Russian Science Foundation (project no. 24-21-00373,
https://rscf.ru/project/24-21-00373/)
This paper studies an optimal control problem for a system governed by a nonlocal balance equation, which models the evolution of a particle distribution. In the examined model, particles move according to a vector field and may disappear. The phase space for this problem is the space of non-negative measures. We prove the existence of an optimal relaxed control, establish a dynamic programming principle, and demonstrate that the value function is a viscosity solution of the corresponding Hamilton–Jacobi equation on the space of non-negative measures.
Keywords: controlled balance equation, optimal control problem, viscosity solution, space of non-negative measures
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Received October 10, 2025
Revised October 31, 2025
Accepted November 3, 2025
Published online November 20, 2025
Funding Agency: The research was funded by the Russian Science Foundation (project no.~24-21-00373, https://rscf.ru/project/24-21-00373/).
Yurii Vladimirovich Averboukh, Dr. Sci. in Math., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: ayv@imm.uran.ru
Cite this article as: Yu. Averboukh. Value function of the optimal control problem for nonlocal balance equation. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 1, pp. 27–43.
Русский
Ю.В. Авербух. Функция цены в задаче оптимального управления уравнением баланса
Данная статья посвящена задаче оптимального управления для системы, описываемой нелокальным уравнением баланса, которое моделирует эволюцию распределения частиц. В рассматриваемой модели частицы движутся в соответствии с векторным полем и могут исчезать. Фазовым пространством для данной задачи является пространство неотрицательных мер. Мы доказываем существование оптимального обобщенного управления, устанавливаем принцип динамического программирования и показываем, что функция значения является вязкостным решением соответствующего уравнения Гамильтона — Якоби в пространстве неотрицательных мер.
Ключевые слова: управляемое уравнение баланса, задача оптимального управления, вязкостное решение, пространство неотрицательных мер
