ON STABILIZATION PROBLEM FOR A LOADED HEAT EQUATION: THE TWO-DIMENSIONAL CASE
Ayazbayeva A.M. Imanberdiyev K.B. Kassymbekova A.S.
30 September 2021al-Farabi Kazakh State National University
KazNU Bulletin. Mathematics, Mechanics, Computer Science Series
2021#111Issue 33 - 15 pp.
One of the important properties that characterize the behavior of solutions of boundary value problems for differential equations is stabilization, which has a direct relationship with the problems of controllability. The problems of solvability of stabilization problems of two-dimensional loaded equations of parabolic type with the help of feedback control given on the boundary of the region are investigated in the article. 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 → ∞. 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.
boundary stabilization , eigenfunction , heat equation , loaded Laplace operator , spectrum
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Al-Farabi Kazakh National University, Almaty, Kazakhstan
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Al-Farabi Kazakh National University
Institute of Mathematics and Mathematical Modeling
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