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Optimal control of stochastic equations of porous environments
Stanzhytsʹkyy Oleksandr Mykolayovych 1 , Lastivka Ivan 2 , Ryzhov Anton 1
1 Taras Shevchenko National University of Kyiv, Kyiv, 01033, Ukraine
2 Department of Higher Mathematics, National university «Kyiv aviation institute» , Kyiv, 03058, Ukraine
Keywords: Wiener process, optimal control, weak convergence, averaging, compactness, reaction-diffusion, rapidly oscillating coefficients
Abstract

The technique of averaging of differential operators is applied to optimal control problems for stochastic equations describing porous media, which are parabolic equations with rapidly oscillating coefficients. It is proved that the optimal controls, trajectories, and minimum values of the quality criteria for the original problem converge, in the corresponding metrics, to the optimal controls, trajectories, and quality criteria of the averaged problems.

References

[1] Allaire G. Homogenization and two-scale convergence. SIAM J. Math. Anal. 1992, 23 (6), 1482–1518. doi:10.1137/0523084

[2] Bensoussan A. Homogenization of a Class of Stochastic Partial Differential Equations. In: Dal Maso G., Dell’Antonio G.F. (Eds.) Composite Media and Homogenization Theory. Progress in Nonlinear Differential Equations and Their Applications, 5, Birkh¨auser Boston, Boston, 1991.

[3] Chow P.-L. Stochastic Partial Differential Equations. Chapman and Hall/CRC, Boca Raton, 2007.

[4] Ichihara N. Homogenization for stochastic partial differential equations derived from nonlinear filtering with feedback. J. Math. Soc. Japan 2005, 57 (2), 593–603. doi:10.2969/jmsj/1158242072

[5] Jikov V.V., Kozlov S.M., Oleinik O.A. Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin, 1994.

[6] Kapustian O.A., Sobchuk V.V. Approximate homogenized synthesis for distributed optimal control problem with superposition type cost functional. Stat. Optim. Inf. Comput. 2018, 6 (2), 233–239. doi:10.19139/soic.v6i2.305

[7] Kapustyan O.A., Sukretna A.V. Approximate averaged synthesis of the problem of optimal control for a parabolic equation. Ukrainian Math. J. 2004, 56 (10), 1653–1664. doi:10.1007/s11253-005-0141-7

[8] Liu W., R¨ockner M. Stochastic Partial Differential Equations: An Introduction. Springer, Cham, 2015.

[9] Misiats O., Stanzhytskyi O., Yip N.K. Effective macroscopic dynamics of stochastic partial differential equations in perforated domains. Stoch. Dyn. 2019, 19 (2), 1950015. doi:10.1142/S0219493719500151

[10] Da Prato G., Zabczyk J. Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge, 1992.

[11] Spagnolo S. Convergence of parabolic equations. Boll. Un. Mat. Ital. 1977, 14-B, 547–568.

[12] Wang W., Cao D., Duan J. Effective macroscopic dynamics of stochastic partial differential equations in perforated domains. SIAM J. Math. Anal. 2007, 38 (5), 593–603. doi:10.1137/050648766

Paper Received 8/19/2026
Paper Accepted 9/22/2026
Published Online 10/1/2026
Cite
ACS Style
Stanzhytsʹkyy , O.M.; Lastivka, I.; Ryzhov, A. Optimal control of stochastic equations of porous environments. Bukovinian Mathematical Journal. 2026, 14 https://doi.org/https://doi.org/10.31861/bmj2026.02.02
AMA Style
Stanzhytsʹkyy OM, Lastivka I, Ryzhov A. Optimal control of stochastic equations of porous environments. Bukovinian Mathematical Journal. 2026; 14(2). https://doi.org/https://doi.org/10.31861/bmj2026.02.02
Chicago/Turabian Style
Oleksandr Mykolayovych Stanzhytsʹkyy , Ivan Lastivka, Anton Ryzhov. 2026. "Optimal control of stochastic equations of porous environments". Bukovinian Mathematical Journal. 14 no. 2. https://doi.org/https://doi.org/10.31861/bmj2026.02.02
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