Optimal control in linear Sobolev type mathematical models

Optimal control in linear Sobolev type mathematical models

Автор: Zamyshlyaeva A.A., Manakova N.A., Tsyplenkova O.N.

Журнал: Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование @vestnik-susu-mmp

Рубрика: Обзорные статьи

Статья в выпуске: 1 т.13, 2020 года.

Бесплатный доступ

The article presents a review of the work of the Chelyabinsk mathematical school on Sobolev type equations in studying the optimal control problems for linear Sobolev type models with initial Cauchy (Showalter-Sidorov) conditions or initial-final conditions. To identify the nonemptiness of the set of feasible solutions to the control problem we use the phase space method, which has already proved itself in solving Sobolev type equations. The method reduces the singular equation to a regular one defined on some subspace of the original space and applies the theory of degenerate (semi)groups of operators to the case of relatively bounded, sectorial and radial operators. Here mathematical models are reduced to initial (initial-final) problems for an abstract Sobolev type equation. Abstract results are applied to the study of control problems for the Barenblatt-Zheltov-Kochina mathematical model, which describes fluid filtration in a fractured-porous medium, the Hoff model on a graph simulating the dynamics of I-beam bulging in a construction, and the Boussinesq-Löve model describing longitudinal vibrations in a thin elastic rod, taking into account inertia and under external load, or the propagation of waves in shallow water.

Еще

Sobolev type equations, strong solutions, optimal control, phase space, barenblatt-zheltov-kochina model, model of an i-beam bulging, boussinesq-löve model, dzektzer model, chen-gurtin model

Короткий адрес: https://sciup.org/147232983

IDR: 147232983   |   DOI: 10.14529/mmp200101

Список литературы Optimal control in linear Sobolev type mathematical models

  • Barenblatt G.I., Zheltov Yu.P., Kochina I.N. Basic Concepts in the Theory of Seepage of Homogeneous Fluids in Fissurized Rocks. Journal of Applied Mathematics and Mechanics, 1960, vol. 24, no. 4, pp. 1268-1303. DOI: 10.1016/0021-8928(60)90107-6
  • Sviridyuk G.A., Efremov A.A. Optimal Control Problem for One Class of Linear Sobolev Type Equations. Russian Mathematics, 1996, vol. 40, no. 12, pp. 60-71.
  • Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht; Boston; Köln; Tokyo, VSP, 2003. DOI: 10.1515/9783110915501
  • 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, no. 2, pp. 5-24. (in Russian)
  • Hoff N.J. Creep Buckling. Journal of the Aeronautical Sciences, 1956, no. 7, pp. 1-20. DOI: 10.1017/S0001925900010106
  • Sviridyuk G.A., Shemetova V.V. Hoff Equations on Graphs. Differential Equations, 2006, vol. 42, no. 1, pp. 139-145.
  • DOI: 10.1134/S0012266106010125
  • Sviridyuk G.A., Kazak V.O. The Phase Space of an Initial-Boundary Value Problem for the Hoff Equation. Mathematical Notes, 2002, vol. 71, no. 1-2, pp. 262-266.
  • DOI: 10.4213/mzm347
  • Manakova N.A., Dyl'kov A.G. Optimal Control of the Solutions of the Initial-Finish Problem for the Linear Hoff Model. Mathematical Notes, 2013, vol. 94, no. 2, pp. 220-230.
  • DOI: 10.1134/S0001434613070225
  • Sviridyuk G.A., Manakova N.A. An Optimal Control Problem for the Hoff Equation. Journal of Applied and Industrial Mathematics, 2007, no. 2, pp. 247-253.
  • DOI: 10.1134/S1990478907020147
  • 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
  • Zamyshlyaeva A.A., Bychkov E.V., Tsyplenkova O.N. Mathematical Models Based on Boussinesq-Love Equation. Applied Mathematical Sciences, 2014, vol. 8, pp. 5477-5483.
  • DOI: 10.12988/ams.2014.47546
  • Dzektser E.S. [The Generalization of the Equations of Motion of Groundwater]. Dokl. Akad. Nauk SSSR, 1972, no. 5, pp. 1031-1033. (in Russian)
  • Sviridyuk G.A., Efremov A.A. Optimal Control of Sobolev Type Linear Equations with Relativity p-Sectorial Operators. Differential Equations, 1995, vol. 31, no. 11, pp. 1882-1890.
  • 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)
  • Chen P.J., Gurtin M.E. On a Theory of Heat Conduction Involving Two Temperatures. Zeitschrift für angewandte mathematik und physik, 1968, vol. 19, pp. 614-627.
  • DOI: 10.1007/BF01594969
  • Manakova N.A., Sviridyuk G.A. An Optimal Control of the Solutions of the Initial-Final Problem for Linear Sobolev Type Equations with Strongly Relatively p-Radial Operator. Springer Proceedings in Mathematics and Statistics, 2015, vol. 13, pp. 213-224.
  • DOI: 10.1007/978-3-319-12145-1_13
  • Sagadeeva M.A., Zagrebina S.A., Manakova N.A. Optimal Control of Solutions of a Multipoint Initial-Final Problem for Non-Autonomous Evolutionary Sobolev Type Equation. Evolution Equations and Control Theory, 2019, vol. 8, no. 3, pp. 473-488.
  • DOI: 10.3934/eect.2019023
  • Demidenko G.V., Uspenskii S.V. Partial Differential Equations and Systems Not Solvable with Respect to the Highest-Order Derivative. N.Y., Basel, Hong Kong, Marcel Dekker, 2003.
  • DOI: 10.1201/9780203911433
  • Favini A., Yagi A. Degenerate Differential Equations in Banach Spaces. N.Y., Basel, Hong Kong, Marcel Dekker, 1999.
  • DOI: 10.1201/9781482276022
  • Sviridyuk G.A. [The Manifold of Solutions of an Operator Singular Pseudoparabolic Equation]. Dokldy Akademii Nauk SSSR, 1986, vol. 289, no. 6, pp. 1315-1318. (in Russian)
  • Al'shin A.B., Korpusov M.O., Sveshnikov A.G. Blow-up in Nonlinear Sobolev Type Equations. De Gruyter, 2011.
  • DOI: 10.1515/9783110255294
  • Showalter R.E. The Sobolev Equation. Applicable Analysis, 1975, vol. 5, no. 1, pp. 15-22; vol. 5, no. 2, pp. 81-89.
  • DOI: 10.1080/00036817508839103
  • 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. 104-125. (in Russian)
  • Keller A.V., Zagrebina S.A. Some Generalizations of the Showalter-Sidorov Problem for Sobolev-Type Models. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 2, pp. 5-23. (in Russian)
  • DOI: 10.14529/mmp150201
  • Sviridyuk G.A., Zagrebina S.A. Verigin's Problem for Linear Equations of the Sobolev Type with Relatively p-Sectorial Operators. Differential Equations, 2002, vol. 38, no. 12, pp. 1745-1752. DOI: 10.1023/A:1023812213901
  • Sviridyuk G.A., Efremov A.A. Optimal Control for a Class of Degenerate Linear Equations. Doklady Akademii Nauk, 1999, vol. 364, no. 3, pp. 323-325.
  • Fedorov V.E., Plekhanova M.V. Optimal Control of Sobolev Type Linear Equations. Differential Equations, 2004, vol. 40, no. 11, pp. 1627-1637.
  • DOI: 10.1007/s10625-004-0013-1
  • Fedorov V.E., Ruzakova O.A. Controllability in Dimensions One and Two of Sobolev-Type Equations in Banach Spaces. Mathematical Notes, 2003, vol. 74, no. 3-4, pp. 583-592. DOI: 10.1023/A:1026160314352
  • Zamyshlyaeva A.A. The Higher-Order Sobolev-Type Models. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 2, pp. 5-28. (in Russian)
  • Zamyshlyaeva A.A., Tsyplenkova O.N. Optimal Control of Solutions to Cauchy Problem for Sobolev Type Equation of Higher Order. Journal of Computational and Engineering Mathematics, 2014, vol. 1, no. 2, pp. 62-67.
  • Zamyshlyaeva A.A., Tsyplenkova O.N., Bychkov E.V. Optimal Control of Solutions to the Initial-Final Problem for Sobolev Type Equation of Higher Order. Journal of Computational and Engineering Mathematics, 2016, vol. 3, no. 2, pp. 57-67.
  • DOI: 10.14529/jcem1602007
  • Bayazitova A.A. The Sturm-Liouville Problem on Geometric Graph. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2010, no. 16 (192), issue 5, pp. 4-10. (in Russian)
  • Fedorov V.E [About Some Relations in the Theory of Degenerate Operator Semigroups]. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2008, no. 15 (115), issue 7, pp. 89-99. (in Russian)
  • Shestakov A.L., Keller A.V., Nazarova E.I. Numerical Solution of the Optimal Measurement Problem. Automation and Remote Control, 2012, vol. 73, no. 1, pp. 97-104.
  • DOI: 10.1134/S0005117912010079
  • Keller A.V., Shestakov A.L., Sviridyuk G.A., Khudyakov Y.V. The Numerical Algorithms for the Measurement of the Deterministic and Stochastic Signals. Springer Proceedings in Mathematics and Statistics, 2015, vol. 113, pp. 183-195.
  • DOI: 10.1007/978-3-319-12145-1_11
  • Shestakov A.L., Sagadeeva M.A. Stochastic Leontieff-Type Equations with Multiplicative Effect in Spaces of Complex-Valued Noises. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 4, pp. 132-139.
  • DOI: 10.14529/mmp140412
  • Zagrebina S.A., Soldatova E.A., Sviridyuk G.A. The Stochastic Linear Oskolkov Model of the Oil Transportation by the Pipeline. Springer Proceedings in Mathematics and Statistics, 2015, vol. 113, pp. 317-325.
  • DOI: 10.1007/978-3-319-12145-1_20
  • Sviridyuk G.A., Zagrebina S.A., Konkina A.S. The Oskolkov Equations on the Geometric Graphs as a Mathematical Model of the Traffic Flow. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 3, pp. 148-154.
  • DOI: 10.14529/mmp1503010
  • Zagrebina S.A., Konkina A.S. The Multipoint Initial-Final Value Condition for the Navier-Stokes Linear Model. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 1, pp. 132-136.
  • DOI: 10.14529/mmp150111
  • 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
  • Favini A., Zagrebina S.A., Sviridyuk G.A. Multipoint Initial-Final Value Problems for Dynamical Sobolev-Type Equations in the Space of ''Noises''. Electronic Journal of Differential Equations, 2018, vol. 2018, no. 128, pp. 1-10.
Еще
Статья научная
QR-код для установки приложения SciUp Наведите камеру и скачайте бесплатное приложение SciUp