Stability of Solutions to the Stochastic Oskolkov Equation and Stabilization

This paper studies the stability of solutions to the stochastic Oskolkov equation describing a plane-parallel flow of a viscoelastic fluid. This is the equation we consider in the form of a stochastic semilinear Sobolev type equation. First, we consider the solvability of the stochastic Oskolkov equation by the stochastic phase space method. Secondly, we consider the stability of solutions to this equation. The necessary conditions for the existence of stable and unstable invariant manifolds of the stochastic Oskolkov equation are proved. When solving the stabilization problem, this equation is considered as a reduced stochastic system of equations. The stabilization problem is solved on the basis of the feedback principle; graphs of the solution before stabilization and after stabilization are shown.

Keywords

the Oskolkov equation; stochastic Sobolev-type equations; invariant manifolds; the stabilization problem.

References

1. Oskolkov A.P. Nonlocal Problems for Some Class Nonlinear Operator Equations Arising in the Theory Sobolev Type Equations. Journal of Mathematical Sciences, 1993, vol. 64, no. 1, pp. 724-736. DOI: 10.1007/BF02988478
2. Sviridyuk G.A., Yakupov M.M. The Phase Space of the Initial-Boundary Value Problem for the Oskolkov System. Differential Equations, 1996, vol. 32, no. 11, pp. 1535-1540.
3. Kitaeva O.G. Invariant Manifolds of Semilinear Sobolev Type Equations. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2022, vol. 15, no. 1, pp. 101-111. DOI: 10.14529/mmp220106
4. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Springer, 2011. DOI: 10.1007/978-0-85729-163-9
5. Sviridyuk G.A., Manakova N.A. The Dynamical Models of Sobolev Type with Showalter-Sidorov Condition and Additive "Noises". Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 1, pp. 90-103. DOI: 10.14529/mmp140108 (in Russian)
6. Favini A., Sviridyuk G.A., Manakova N.A. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of "Noises". Abstract and Applied Analysis, 2015, article ID: 69741, 8 p. DOI: 10.1155/2015/697410
7. Favini A., Sviridyuk G.A., Sagadeeva M.A. Linear Sobolev Type Equations with Relatively p-Radial Operators in Space of "Noises". Mediterranean Journal of Mathematics, 2016, vol. 13, no. 6, pp. 4607-4621. DOI: 10.1007/s00009-016-0765-x
8. Vasiuchkova K.V., Manakova N.A., Sviridyuk G.A. Some Mathematical Models with a Relatively Bounded Operator and Additive "White Noise" in Spaces of Sequences. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2017, vol. 10, no. 4, pp. 5-14. DOI: 10.14529/mmp170401
9. Vasiuchkova K.V., Manakova N.A., Sviridyuk G.A. Degenerate Nonlinear Semigroups of Operators and their Applications. Semigroups of Operators - Theory and Applications. SOTA 2018. Springer Proceedings in Mathematics and Statistics. Springer, Cham, 2020, no. 325, pp. 363-378. DOI: 10.1007/978-3-030-46079-2_21
10. Kitaeva O.G. Invariant Spaces of Oskolkov Stochastic Linear Equations on the Manifold. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2021, vol. 13, no. 2, pp. 5-10. DOI: 10.14529/mmph210201
11. Kitaeva O.G. Exponential Dichotomies of a Non-Classical Equations of Differential Forms on a Two-Dimensional Torus with "Noises". Journal of Computational and Engineering Mathematics, 2019, vol. 6, no. 3, pp. 26-38. DOI: 10.14529/jcem190303
12. Kitaeva O.G. Dichotomies of Solutions to the Stochastic Ginzburg-Landau Equation on a Torus. Journal of Computational and Engineering Mathematics, 2020, vol. 7, no. 4, pp. 17-25. DOI: 10.14529/jcem200402
13. Kitaeva O.G. Exponential Dichotomies of a Stochastic Non-Classical Equation on a Two-Dimensional Sphere. Journal of Computational and Engineering Mathematics, 2021. vol. 8, no. 1, pp. 60-67. DOI: 10.14529/jcem210105
14. Kitaeva O.G. Invariant Manifolds of the Hoff Model in "Noises" Spaces. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2021, vol. 14, no. 4, pp. 24-35. DOI: 10.14529/mmp210402
15. Kitaeva O.G. Invariant Manifolds of the Stochastic Benjamin-Bona-Mahony Equation. Global and Stochastic Analysis, 2022, vol. 9, no. 3, pp. 9-17.
16. Kitaeva O.G. Stabilization of the Stochastic Barenblatt-Zheltov-Cochina Equation. Journal of Computational and Engineering Mathematics, 2023, vol. 10, no. 1, pp. 21-29. DOI: 10.14529/jcem230103