On an optimal scanning control problem in a one-dimensional space

In this note we consider an optimal scanning control problem for parabolic systems in a one-dimensional space. It is shown that for the optimal regulator problem involving the minimization of a given final-time quadratic cost functional there exist optimal locations for the scanning control. Conditions for optimality are given. Also, it is shown that for the heat equation the "best" location of the control cannot be a fixed point in the spatial domain of the system.