Spectral problems arising in the stabilization problem for the loaded heat equation: a two-dimensional and multi-point cases

Spectral properties of a loaded two-dimensional Laplace operator, studied in this work are the application with the stabilization of solutions of problems for the heat equation. The stabilization problem (of forming a cylinder) of a solution of boundary value problem for heat equation with the loaded two-dimensional Laplace operator is considered. An algorithm is proposed for approximate construction of boundary controls providing the required stabilization of the solution. The work continues the research of the authors carried out earlier for the loaded one-dimensional heat equation. 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.