THE CONTROL PROBLEM ASSOCIATED WITH HEATING PROCESS OF A ROD (International Journal of Mathematics and Physics, 2025)
In this paper, the control problem for a heat conduction equation with periodic boundary condition is considered. The solution with the control function on the border of the rod is given. The constraints on the control are determined to ensure that the average value of the solution within the considered domain attains a given value. The considered control problem is reduced to the Volterra integral equation, which is the first type, using the Fourier method. The proof of the existence of admissible control is related to the existence of a solution of the integral equation. The existence of the solution of the integral equation is shown using the Laplace transform method and the admissibility of the solution is proved. In the last section, the control function needed to achieve the average temperature given certain values of the parameters is found.Keywords: heat conduction equation, average temperature, periodic boundary condition, control problem, Volterra integral equation, Laplace transform.