Boundary control for a pseudo-parabolic equation

Previously, a mathematical model for the following problem was considered. On a part of the border of the region $\Omega\subset\mathbb{R}^3$ there is a heater with controlled temperature. It is required to find such a mode of its operation that the average temperature in some subregion $D$ of $\Omega$ reaches some given value.
In this paper, we consider a similar boundary control problem associated with a pseudo-parabolic equation on a segment. On the part of the border of the considered segment, the value of the solution with control parameter is given. Restrictions on the control are given in such a way that the average value of the solution in some part of the considered segment gets a given value. The auxiliary problem is solved by the method of separation of variables, while the problem in consideration is reduced to the Volterra integral equation of the second kind. By Laplace transform method, the existence and uniqueness theorems for admissible control are proved.