# On the time-optimal control problem for a fourth order parabolic equation in a two-dimensional domain

## DOI:

https://doi.org/10.31489/2024m2/57-70## Keywords:

initial-boundary problem, fourth-order parabolic equation, minimal time, admissible control, Volterra integral equation, Laplace transform method## Abstract

Previously, boundary control problems for the second order parabolic type equation in the bounded domain were studied. In this paper, a boundary control problem associated with a fourth-order parabolic equation in a bounded two-dimensional domain was considered. On the part of the considered domain’s boundary, the value of the solution with control function is given. Restrictions on the control are given in such a way that the average value of the solution in the considered domain gets a given value. By the method of separation of variables the given problem is reduced to a Volterra integral equation of the first kind. The existence of the control function was proved by the Laplace transform method and an estimate was found for the minimal time at which the given average temperature in the domain is reached.