The averaging method for the optimal control problem of a parabolic inclusion with fast-oscillating coefficients on a finite time interval

Authors

DOI:

https://doi.org/10.17721/1812-5409.2024/2.6

Keywords:

optimal control problem, parabolic differential inclusion, averaging method, rapidly oscillating variables

Abstract

In this paper we investigate the optimal control problem for a parabolic differential inclusion with rapidly oscillating variables in the finite interval. There are many approaches intended for the investigation of control problems for differential equations and inclusions. Thus, in particular, the asymptotic methods are used fairly extensively. Among these methods, we can especially mention the averaging method, which was mathematically rigorously substantiated by Krylov M.M. and Bogolyubov M.M. The well-known Krasnoselski–Krein theorem and its multi-valued analogue play an essential role for the investigation of the above-mentioned problems. The averaging method was substantiated, in particular, for ordinary differential inclusions, inclusions with partial derivatives, and inclusions with the Hukuhara derivative. When dealing with multi-valued mappings one faces specific problems, such as closedness, convexity of the family of solutions, existence of limit solutions, selection of solutions with given properties, etc. However, the well-developed apparatus of mathematical analysis applied to the study of multi-valued functions makes it possible to apply the averaging method to the optimal control problem described above. Thus, using the averaging method  the convergence of optimal controls and optimal trajectories of solutions of the exact problem to optimal control and the trajectory of the averaged problem is proved in the paper.

Pages of the article in the issue: 33 - 40

Language of the article: English

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Published

2025-01-29

How to Cite

Kapustyan, O., Kasimova, N., Sobchuk, V., & Stanzhytskyi, O. (2025). The averaging method for the optimal control problem of a parabolic inclusion with fast-oscillating coefficients on a finite time interval. Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, 79(2), 33–40. https://doi.org/10.17721/1812-5409.2024/2.6

Issue

Section

Differential equations, mathematical physics and mechanics

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