Volume 14, no. 4Pages 36 - 45
Research of the Optimal Control Problem for One Mathematical Model of the Sobolev TypeK.V. Perevozchikova, N.A. Manakova
At the moment, optimal control problems for various linear and nonlinear Sobolev-type equations have been widely studied. This article is devoted to the study of optimal control for one mathematical model of the Sobolev type, which is based on the model equation, which describes various processes (for example, deformation processes, processes occurring in semiconductors, wave processes, etc.) depending on the parameters and can belong either to the class of degenerate (for lambda> 0) equations or to the class of nondegenerate (for lambda <0) equations. This work is the first attempt to study the control problem for mathematical semilinear models of the Sobolev type in the absence of the property of non-negative definiteness of the operator at the time derivative, i.e. the construction of a singular optimality system in accordance with the singular situation caused by the instability of the model. Conditions for the existence of a control-state pair are presented, and conditions for the existence of an optimal control are found.Full text
- Sobolev type equations; phase space method; optimal control problem.
- 1. Korpusov M.O., Sveshnikov A.G. Three-Dimensional Nonlinear Evolution Equations of Pseudoparabolic Type in Problems of Mathematical Physics. II. Computational Mathematics and Mathematical Physics, 2003, vol. 43, no. 12, pp. 1835-1869.
2. Sviridyuk G.A. The Manifold of Solutions of an Operator Singular Pseudoparabolic Equation. Doklady Akademii Nauk SSSR, 1986, vol. 289, no. 6, pp. 1-31.
3. Manakova N.A., Sviridyuk G.A. Non-Classical Equations of Mathematical Physics. Phase Spaces of Semilinear Sobolev Equations. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2016, vol. 8, no. 3, pp. 31-51. DOI: 10.14529/mmph160304 (in Russian)
4. Sviridyuk G.A. Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev Type. Izvestiya RAN. Seriya Matematicheskaya, 1994, vol. 42, no. 3, pp. 601-614.
5. Sviridyuk G.A., Klimentev M.V. Phase Spaces of Sobolev-Type Equations with s-Monotone or Strongly Coercive Operators. Russian Mathematics, 1994, vol. 38, no 11, pp. 72-79.
6. Lions J.-L. Quelques maerthodes de resolution des problemes aux limites non lineaires. Paris, Dunod, 1968. (in French)
7. 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.
8. Manakova N.A. Mathematical Models and Optimal Control of The Filtration and Deformation Processes. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 3, pp. 5-24. (in Russian) DOI: 10.14529/mmp150301
9. Gavrilova O.V. A Numerical Study of the Optimal Control Problem for Degenerate Multicomponent Mathematical Model of the Propagation of a Nerve Impulse in the System of Nerves. Journal of Computational and Engineering Mathematics, 2020, vol. 7, no. 1, pp. 47-61.
10. Kotlovanov K.Yu., Bychkov E.V., Bogomolov A.V. Optimal Control in the Mathematical Model of Internal Waves. Journal of Computational and Engineering Mathematics, 2020, vol. 7, no. 1, pp. 62-71.
11. 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
12. Shestakov A.L., Sviridyuk G.A. A New Approach to Measurement of Dynamically Perturbed Signals. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2010, no. 16 (192), issue 5, pp. 116-120.
13. Shestakov A.L., Keller A.V., Zamyshlyaeva A.A., Manakova N.A., Zagrebina S.A., Sviridyuk G.A. The Optimal Measurements Theory as a New Paradigm in the Metrology. Journal of Computational and Engineering Mathematics, 2020, vol. 7, no. 1, pp. 3-23. DOI: 10.14529/jcem200101
14. Lions J.-L. Upravlenie singuljarnymi raspredelennymi sistemami [Singular Distributed Systems Management]. Moscow, Nauka, 1987.
15. Sviridyuk G.A., Sukacheva T.G. Cauchy problem for a class of semilinear equations of Sobolev type. Siberian Mathematical Journal, 1990, vol. 31, pp. 794-802.
16. Leng S. Introduction to Differentiable Manifolds. N.Y., Springer, 2002.
17. Manakova N.A., Vasiuchkova K.V. Research of one Mathematical Model of the Distribution of Potentials in a Crystalline Semiconductor. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2019, vol. 12, no. 2, pp. 150-157. DOI: 10.14529/mmp190213