Evgeniy Bychkov, Georgy Sviridyuk, Alexey Bogomolov
Abstract:
This article concerns the optimal control problem for internal gravitational waves in
a model with additive "white noise". This mathematical models based on the stochastic
Sobolev equation, Dirichlet boundary conditions, and a Cauchy initial condition.
The inhomogeneity describes random heterogeneities of the medium and fluctuations.
By white noise we realize the Nelson-Gliklikh derivative of the Wiener process.
The study was carried out within the framework of the theory of relatively bounded
operators and the theory of Sobolev-type stochastic equations of higher order and
the theory of (semi) groups of operators.
We show the existence and uniqueness of a strong solutions, and obtain
sufficient conditions for the existence of an optimal control of such solutions.
The theorem about the existence and uniqueness of the optimal control is based on the
works of J.-L. Lyons.
Submitted March 21, 2021. Published June 9, 2021.
Math Subject Classifications: 60H30, 49J20, 47A62, 47D03.
Key Words: Stochastic Sobolev type equations; white noise; space of noises;
Wiener process; additive white noise.
DOI: https://doi.org/10.58997/ejde.2021.51
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Evgeniy Bychkov South Ural State University Department of Equations of Mathematical Physics Lenin avenue, 76 454080 Chelyabinsk, Russia email: bychkovev@susu.ru |
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Georgy Sviridyuk South Ural State University Department of Equations of Mathematical Physics Lenin avenue, 76 454080 Chelyabinsk, Russia email: sviridiukga@susu.ru |
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Alexey Bogomolov St. Petersburg Federal Research Center of the Russian Academy of Sciences Intelligent Systems Laboratory 14th lin. Vasilievskii Ostrov, 39, 199178 St. Petersburg, Russia email: a.v.bogomolov@gmail.com |
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