The averaging method for the optimal control problem of a parabolic inclusion with fast-oscillating coefficients on a finite time interval
DOI:
https://doi.org/10.17721/1812-5409.2024/2.6Keywords:
optimal control problem, parabolic differential inclusion, averaging method, rapidly oscillating variablesAbstract
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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Copyright (c) 2024 Oleksiy Kapustyan, Nina Kasimova, Valentyn Sobchuk, Oleksandr Stanzhytskyi

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