On a stability of a solution of the loaded heat equation

Abstract

Steadily growing interest in study of loaded differential equations is explained by the range of their applications and a circumstance that loaded equations make a special class of functional - differential equations with specific problems. These equations have applications in study of inverse problems of differential equations with important applied interests. In this paper solvability questions of stabilization problems with a boundary for the loaded heat equation are studied in the given bounded domain Ω≡(-π/2,π/2). The task is to choose boundary conditions (controls), that the solution of the obtained mixed boundary value problem tends to a given stationary solution with the prescribed speed exp(-σ₀t ) as t →∞. At this the control is required to be a feedback control, i.e. that it reacted to the unintended fluctuations of the system, suppressing the results of their impact on the stabilized solution. Stabilization problems have a direct connection with controllability problems. The paper proposes a mathematical formalization of the concept of feedback, and with its help it solves the problem of stabilizability of a loaded heat equation by dint of feedback control given on the part of the boundary is solved.