Boundary control problem for a heat equation in a multidimensional parallelepiped

In this paper, we consider a boundary control problem associated with the heat equation in a bounded $ n $-dimensional domain. The value of the solution is prescribed on a part of the boundary, and the objective is to determine a control function that ensures a prescribed average temperature is attained within the domain. The initial-boundary problem is solved by the Fourier method, and the control problem under consideration is that of the Volterra integral equation. The existence of admissible control is proved by the Laplace transform method.