Time-optimal control problem for a fourth order parabolic equation with involution in a two-dimensional domain

This paper considers the optimal time control problem for a fourth-order parabolic equation with involution in a square domain. The solution with the control function on the border of the considered domain 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 initial-boundary problem is solved by the Fourier method, and the control problem under consideration is analyzed with the Volterra integral equation of the first kind. The existence of the control function was proved by the Laplace transform method, and an optimal estimate of the minimum time required for a thin film to reach a given average height is found.