On the time-optimal control problem associated with fourth-order parabolic equation in a ball

In this paper, we consider the optimal time control problem for a fourth-order parabolic equation describing the growth process of a thin film. The control function is given on a certain part of the boundary. The solution of the control problem is reduced to a first-order Volterra integral equation. As a result, it is proved that there is a control function necessary to achieve a given average thickness of the thin film, and an optimal estimate for the minimum time is found.

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