Skip to main content
SpringerLink
Log in

Optimal Control of Solutions to Showalter–Sidorov Problem for a High Order Sobolev Type Equation with Additive “Noise”

  • Conference paper
  • First Online:
Semigroups of Operators – Theory and Applications (SOTA 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 325))

Included in the following conference series:

Abstract

In this paper, the problem of optimal control of solutions to the Showalter–Sidorov problem for a high-order Sobolev type equation with additive “noise” is investigated. The existence and uniqueness of a strong solution to the Showalter–Sidorov problem for this equation are proved. Sufficient conditions for the existence and uniqueness of an optimal control of such solutions are obtained. For this, we built the space of “noises”. For the differentiation of additive “noise”, we use the derivative of a stochastic process in the sense of Nelson–Gliklikh.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Al’shin, A.B., Korpusov, M.O., Sveshnikov, A.G.: Blow-up in Nonlinear Sobolev Type Equations. Walter de Gruyter and Co., Berlin (2011)

    Google Scholar 

  2. Demidenko, G.V., Uspenskii, S.V.: Partial Differential Equations and Systems Not Solvable with Respect to the Highest Order Derivative. Basel, Hong Kong, Marcel Dekker, Inc, N.Y. (2003)

    Google Scholar 

  3. Favini, A., Yagi, A.: Degenerate Differential Equations in Banach Spaces. Basel, Hong Kong, Marcel Dekker, Inc, N.Y. (1999)

    Google Scholar 

  4. Showalter, R.E.: Hilbert Space Methods for Partial Differential Equations. Pitman, London, San Francisco, Melbourne (1977)

    Google Scholar 

  5. Sviridyuk, G.A., Fedorov, V.E.: Linear Sobolev Type Equations and Degenerate Semigroups of Operators. VSP, Utrecht, Boston, Koln, Tokyo (2003)

    Google Scholar 

  6. Zamyshlyaeva, A.A.: The higher-order Sobolev-type models. Bull. South Ural State Univ. Ser.: Math. Model. Program. Comput. Softw. 7(2), 5–28 (2014). https://doi.org/10.14529/mmp140201 (in Russian)

  7. Gliklikh, YuE: Global and Stochastic Analysis with Applications to Mathematical Physics. Dordrecht, Heidelberg, N.Y., Springer, London (2011)

    Google Scholar 

  8. Zagrebina, S.A., Soldatova, E.A.: The linear Sobolev-type equations with relatively p-bounded operators and additive white noise. Bull. South Ural State Univ. Ser.: Math. Model. Program. Comput. Softw. 6(1), 20–34 (2013)

    Google Scholar 

  9. Kovács, M., Larsson, S.: Introduction to stochastic partial differential equations. In: Proceedings of “New Directions in the Mathematical and Computer Sciences”. National Universities Commission, Abuja, Nigeria, October 8–12, 2007. Publications of the ICMCS, vol. 4, pp. 159–232 (2008)

    Google Scholar 

  10. Favini, A., Sviridyuk, G.A., Zamyshlyaeva, A.A.: One class of Sobolev type equations of higher order with additive "White Noise". Commun. Pure Appl. Anal. 15(1), 185–196 (2016)

    MathSciNet  MATH  Google Scholar 

  11. Shestakov, A.L., Sviridyuk, G.A.: On a new conception of white noise. Obozrenie Prikladnoy i Promyshlennoy Matematiki, Moscow, 19(2) , 287–288 (2012)

    Google Scholar 

  12. Shestakov, A.L., Sviridyuk, G.A.: On the measurement of the “white noise”. Bull. South Ural State Univ. Ser.: Math. Model. Program. Comput. Softw. 27(286), 99–108 (2012)

    Google Scholar 

  13. Gliklikh, Yu.E.: Investigation of Leontieff type equations with white noise protect by the methods of mean derivatives of stochastic processes. Bull. South Ural State Univ. Ser.: Math. Model. Program. Comput. Softw. 27(286), 24–34 (2012)

    Google Scholar 

  14. Gliklikh, YuE: Mean derivatives of stochastic processes and their applications. SMI VSC RAS, Vladikavkaz (2016). (in Russian)

    Google Scholar 

  15. Sviridyuk, G.A., Manakova, N.A.: Dynamic models of Sobolev type with the Showalter–Sidorov condition and additive “Noises”. Bull. South Ural State Univ. Ser.: Math. Model. Program. Comput. Softw. 7(1), 90–103 (2014). (in Russian)

    Google Scholar 

  16. Favini, A., Sviridyuk, G.A., Manakova, N.A.: Linear Sobolev type equations with relatively p-sectorial operators in space of “ Noises”. Abstract Appl. Analy. Article ID: 69741, 8 (2015). https://doi.org/10.1155/2015/697410

  17. Zamyshlyaeva, A.A.: Stochastic incomplete linear Sobolev type high-ordered equations with additive white noise. Bull. South Ural State Univ. Bull. South Ural State Univ. Ser.: Math. Model. Program. Comput. Softw. 40(299), 73–82 (2012). (in Russian)

    Google Scholar 

  18. Lions, Zh-L: Optimal’noe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi. Mir, Optimal Control of Systems Described by Equations with Partial Derivatives. Moscow (1972). (in Russian)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Programming, South Ural State University, 76, Lenin ave, Chelyabinsk, 454080, Russian Federation

    Alyona A. Zamyshlyaeva

  2. Department of Mathematical Physics Equations, South Ural State University, 76, Lenin ave, Chelyabinsk, 454080, Russian Federation

    Olga N. Tsyplenkova

Authors
  1. Alyona A. Zamyshlyaeva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Olga N. Tsyplenkova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alyona A. Zamyshlyaeva .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, South Africa

    Jacek Banasiak

  2. Department of Mathematics, Lublin University of Technology, Lublin, Poland

    Adam Bobrowski

  3. Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland

    Mirosław Lachowicz

  4. Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

    Yuri Tomilov

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Zamyshlyaeva, A.A., Tsyplenkova, O.N. (2020). Optimal Control of Solutions to Showalter–Sidorov Problem for a High Order Sobolev Type Equation with Additive “Noise”. In: Banasiak, J., Bobrowski, A., Lachowicz, M., Tomilov, Y. (eds) Semigroups of Operators – Theory and Applications. SOTA 2018. Springer Proceedings in Mathematics & Statistics, vol 325. Springer, Cham. https://doi.org/10.1007/978-3-030-46079-2_24

Download citation

Publish with us

Policies and ethics

Search

Navigation