Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 51, pp. 1-11.

Title: Optimal control for solutions to Sobolev stochastic equations
       
Authors: Evgeniy Bychkov  (South Ural State Univ., Chelyabinsk, Russia)
         Georgy Sviridyuk (South Ural State Univ., Chelyabinsk, Russia)
         Alexey Bogomolov (Russian Academy of Sciences, St. Petersburg, Russia)

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 09, 2021.
Math Subject Classifications: 60H30, 49J20, 47A62, 47D03.
Key Words: Stochastic Sobolev type equations; white noise; space of noises;
           Wiener process; additive white noise.