Some Generalizations of the Showalter - Sidorov Problem for Sobolev-Type Models

At present, investigations of Sobolev-type models are actively developing. In the solution of applied problems the results allowing to get their numerical solutions are very
significant. The initial Showalter - Sidorov condition is not simply a generalization of the Cauchy condition for Sobolev-type models. It allows to find an approximate solution without checking the coordination of initial data. This article presents an overview of some results of the Chelyabinsk mathematical school on Sobolev type equations obtained using either directly Showalter - Sidorov condition or its generalizations.
The article consists of seven sections. The first one includes results on investigation of solvability of an optimal measurement problem for the Shestakov - Sviridyuk model. The second section provides an overview of the currently existing approaches to the concept of white noise. The third section contains results on solvability of a weakened Showalter - Sidorov problem for the Leontief type system with additive 'white noise'. In the fourth section we present results on the unique solvability of multipoint initial-final value problem for the Sobolev type equation of the first order. A study of optimal control of solutions to this problem is discussed in the fifth section. The sixth and the seventh sections contain results related to research of optimal control of solutions to the Showalter - Sidorov problem and initial-final value problem for the Sobolev-type equation of the second order, respectively.

Keywords

Sobolev type equations; Leontief type sistems; optimal control; Showalter - Sidorov problem; the (multipoint) initial-finale value condition; optimal measurement.

References

1. Keller A.V. The Algorithm for Solution of the Showalter - Sidorov Problem for Leontief Type Models. tBulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2011, no. 4 (241), issue 7, pp. 40-46. (in Russian)
2. Sviridyuk G.A., Zagrebina S.A. The Showalter - Sidorov Problem as a Phenomena of the Sobolev-Type Equations. The Bulletin of Irkutsk State University. Series: Mathematics, 2010, vol. 3, no. 1, pp. 51-72. (in Russian)
3. Zagrebina S.A. The Initial-Finite Problems for Nonclassical Models of Mathematical Physics. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2013, vol. 6, issue 2, pp. 5-24. (in Russian)
4. Shestakov A.L., Sviridyuk G.A. Optimal Measurement of Dynamically Distorted Signals. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2011, no. 17 (234), issue 8, pp. 70-75.
5. Shestakov A.L., Keller A.V., Nazarova E.I. The Numerical Solution of the Optimal Demension Problem. Automation and Remote Control, 2011, vol. 73, no. 1, pp. 97-104. DOI: 10.1134/S0005117912010079
6. Khudyakov Yu.V. The Numerical Algorithm to Investigate Shestakov - Sviridyuk's Model of Measuring Device with Inertia and Resonances. Yakutian Mathematical Journal, 2013, vol. 20, no. 2, pp. 211-221. (in Russian)
7. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Koln, Tokyo, VSP, 2003. DOI: 10.1515/9783110915501
8. Zagrebina S.A., Soldatova E.A., Sviridyuk G.A. The Stochastic Linear Oskolkov Model of the Oil Transportation by the Pipeline. Semigroups of Operators - Theory and Applications. [International Conference, Bedlewo, Poland, Oktober 2013]. Heidelberg, N.-Y., Dordrecht, London, Springer Int. Publ. Switzerland, 2015, pp. 317-325.
9. Keller A.V., Shestakov A.L., Sviridyuk G.A., Khudyakov Yu.V. The Numerical Algorithms for the Measurement of the Deterministic and Stochastic Signals. Semigroups of Operators - Theory and Applications. [International Conference, Bedlewo, Poland, Oktober 2013]. Heidelberg, N.-Y., Dordrecht, London, Springer Int. Publ. Switzerland, 2015, pp. 183-195.
10. Sagadeeva M.A., Sviridyuk G.A. The Nonautonomous Linear Oskolkov Model on a Geometrical Graph: The Stability of Solutions and the Optimal Control. Semigroups of Operators - Theory and Applications. [International Conference, Bedlewo, Poland, Oktober 2013]. Heidelberg, N.-Y., Dordrecht, London, Springer Int. Publ. Switzerland, 2015, pp. 257-271.
11. Shestakov A.L., Sviridyuk G.A., Sagadeeva M.A. Reconstruction of a Dynamically Distorted Signal with Respect to the Measuring Transducer Degradation. Applied Mathematical Sciences, 2014, vol. 8, no. 43. (41-44), pp. 2125-2130. DOI: 10.12988/ams.2014.312718
12. Nelson E. Dynamical Theories of Brownian Motion. Princeton, Princeton University Press, 1967.
13. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Dordrecht, Heidelberg, N.-Y., Springer, 2011.
14. Shestakov A.L., Sviridyuk G.A. On the Measurement of the 'White Noise'. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 27 (286), issue 13, pp. 99-108.
15. Manakova N.A., Dylkov A. G. Optimal Control of the Solutions of the Initial-Finish Problem for the Linear Hoff Model. tMathematical Notes, 2013, vol. 94, issue 2, pp. 220-230. DOI: 10.1134/S0001434613070225
16. Zagrebina S.A., Sagadeeva M.A. The Generalized Splitting Theorem for Linear Sobolev type Equations in Relatively Radial Case. The Bulletin of Irkutsk State University. Series: Mathematics, 2014, vol. 7, pp. 19-33.
17. Zamyshlyaeva A.A., Tsyplenkova O.N. Optimal Control of Solutions of the Showalter - Sidorov - Dirichlet Problem for the Boussinesq - Love Equation. Differential Equations, 2013, vol. 49, no. 11, pp. 1356-1365. DOI: 10.1134/S0012266113110049
18. Zamyshlyaeva A.A., Yuzeeva A.V. Initial-Final Problem for the Boussinesq - Love Equation on a Graph. The Bulletin of Irkutsk State University. Series: Mathematics, 2010, vol. 3, no. 2, pp. 18-29. (in Russian)
19. Zamyshlyaeva A.A., Tsyplenkova O.N. The Optimal Control over Solutions of the Initial-Finish Value Problem for the Boussinesque-Love Equation. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, vol. 5 (264), issue 11, pp. 13-24. (in Russian)
20. Gantmacher F.R. The Theory of Matrices. AMS Chelsea Publishing: Reprinted by American Mathematical Society, 2000. 660 p.
21. Sagadeeva M.A. Dichotomy of Solutions of Linear Sobolev Type Equations. Chelyabinsk, Publ. Center of the South Ural State University, 2012. 107 p. (in Russian)
22. Manakova N.A. Optimal Control Problem for the Semilinear Sobolev Type Equations. Chelyabinsk, Publ. Center of the South Ural State University, 2012. 88 p. (in Russian)
23. Manakova N.A., Dylkov A.G. On One Optimal Control Problem with a Penalty Functional in General Form. Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya: Fiziko-Matematicheskie Nauki, 2011, no. 4 (25), pp. 18-24. (in Russian)
24. Manakova N.A., Dyl'kov A.G. Optimal Control of Solutions of the Initial-Finish Value Problem for a Evolutionary Models. Yakutian Mathematical Journal, 2012, vol. 19, no. 2, pp. 111-127. (in Russian)
25. Zamyshlyaeva A.A. Linear Sobolev Type Equations of High Order. Chelyabinsk, Publ. Center of the South Ural State University, 2012. 107 p. (in Russian)