Geodesic
Stabilization of solutions of two-dimensional parabolic equations and related spectral problems
Eurasian mathematical journal, Tome 11 (2020) no. 1, pp. 72-85 
M. Jenaliyev; K. Imanberdiyev; A. Kassymbekova; K. Sharipov. Stabilization of solutions of two-dimensional parabolic equations and related spectral problems. Eurasian mathematical journal, Tome 11 (2020) no. 1, pp. 72-85. http://geodesic.mathdoc.fr/item/EMJ_2020_11_1_a5/
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Voir la notice de l'article provenant de la source Math-Net.Ru

One of the important properties that characterize the behaviour of solutions of boundary value problems for differential equations is stabilization, which has a direct relationship with the problems of controllability. In this paper, the problems of solvability are investigated for stabilization problems of two-dimensional loaded equations of parabolic type with the help of feedback control given on the boundary of the region. These equations have numerous applications in the study of inverse problems for differential equations. The problem consists in the choice of boundary conditions (controls), so that the solution of the boundary value problem tends to a given stationary solution at a certain speed at $t\to\infty$. This requires that the control is feedback, i.e. that it responds to unintended fluctuations in the system, suppressing the results of their impact on the stabilized solution. The spectral properties of the loaded two-dimensional Laplace operator, which are used to solve the initial stabilization problem, are also studied. The paper presents an algorithm for solving the stabilization problem, which consists of constructively implemented stages. The idea of reducing the stabilization problem for a parabolic equation by means of boundary controls to the solution of an auxiliary boundary value problem in the extended domain of independent variables belongs to A.V. Fursikov. At the same time, recently, the so-called loaded differential equations are actively used in problems of mathematical modeling and control of nonlocal dynamical systems.

[1] A. V. Fursikov, “Stabilizability a quasilinear parabolic equation by using the boundary control with feedback”, Mathematicheskii sbornik, 192:4 (2001), 115–160 (in Russian) | DOI | MR | Zbl

[2] A. V. Fursikov, “Stabilization for the 3D Navier-Stokes system by feedback boundary control”, Discrete and Continues Dynamical Systems, 10:1–2 (2004), 289–314 | MR | Zbl

[3] A. V. Fursikov, A. V. Gorshkov, “Certain questions of feedback stabilization for Navier-Stokes equations”, Evolution equations and control theory, 1:1 (2012), 109–140 | DOI | MR | Zbl

[4] A. V. Fursikov, “Stabilization of the simplest normal parabolic equation by starting control”, Communication on Pure and Applied Analysis, 13:5 (2014), 1815–1854 | DOI | MR | Zbl

[5] A. M. Nakhushev, Loaded equations, their applications, Nauka, M., 2012 (in Russian)

[6] A. M. Nakhushev, “Loaded equations, their applications”, Differential Equations, 19:1 (1983), 86–94 (in Russian) | MR | Zbl

[7] M. Amangalieva, D. Akhmanova, M. Dzhenaliev (Jenaliyev), M. Ramazanov, “Boundary value problems for a spectrally loaded heat operator with load line approaching the time axis at zero or infinity”, Differential Equations, 47 (2011), 231–243 | DOI | MR | Zbl

[8] D. Akhmanova, M. Dzhenaliev (Jenaliyev), M. Ramazanov, “On a particular second kind Volterra integral equation with a spectral parameter”, Siberian Mathematical Journal, 52 (2011), 1–10 | DOI | MR | Zbl

[9] M. Dzhenaliev (Jenaliyev), M. Ramazanov, “On a boundary value problem for a spectrally loaded heat operator: I”, Differential Equations, 43 (2007), 513–524 | DOI | MR | Zbl

[10] M. Dzhenaliev (Jenaliyev), M. Ramazanov, “On a boundary value problem for a spectrally loaded heat operator: II”, Differential Equations, 43 (2007), 806–812 | DOI | MR | Zbl

[11] Anna Sh. Lyubanova, “On nonlocal problems for systems of parabolic equations”, Journal of Mathematical Analisys and Applications, 421 (2015), 1767–1778 | DOI | MR | Zbl

[12] M. Amangaliyeva, M. Jenaliyev, K. Imanberdiyev, M. Ramazanov, “On spectral problems for loaded two-dimension Laplace operator”, AIP Conference Proceedings, 1759 (2016), 020049 | DOI

[13] M. T. Jenaliyev, M. M. Amangaliyeva, K. B. Imanberdiyev, M. I. Ramazanov, “On a stability of a solution of the loaded heat equation”, Bulletin of the Karaganda University. «Mathematics» series, 90:2 (2018), 56–71 | DOI

[14] M. T. Jenaliyev, M. I. Ramazanov, “Stabilization of solutions of loaded on zero-dimensional manifolds heat equation with using boundary controls”, Mathematical journal, 15:4 (2015), 33–53 (in Russian)

[15] M. T. Jenaliyev, K. B. Imanberdiyev, A. S. Kassymbekova, K. S. Sharipov, “Spectral problem arizaing in the stabilization problem for the loaded heat equation: two-dimensional and multi-points cases”, Eurasian Journal of Mathematical and Computer Applications, 7:1 (2019), 23–37 | DOI

[16] F. Riesz, B. Sz. Nagy, Lecons D'Analyse Fonctionnelle, Akademiai Kiado, Budapest, 1968 | MR

[17] F. R. Gantmakher, Theory of matricies, Fizmatlit, M., 2004 (in Russian) | MR