A boundary control problem for a one-dimensional P=pseudo-parabolic equation with involution and periodic boundary conditions

In this study, we investigate a boundary control problem for a pseudo-parabolic equation involving involution and periodic boundary conditions. The main objective is to determine a boundary control function that ensures the average temperature of a rod reaches a prescribed function over time. By applying separation of variables, the control problem is reduced to a Volterra integral equation of the second kind. The continuity of the kernel and the solvability of the integral equation are established using Laplace transform method. The admissibility of the control function is also proven.