Time-optimal control problem associated with a fourth-order parabolic equation

We consider a boundary control problem for a fourth-order parabolic equation in a bounded one-dimensional domain. At a part of the boundary, a value of the solution is given and it is required to find control to get the average value of solution. By the method of separation of variables, the problem is reduced to the Volterra integral equation of the first kind. The existence of the control function was proven by the Laplace transform method and an estimate on the minimum time to reach the given average temperature in the rod was found.

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