  We are concerned with the optimal control problem of the well known nonlocal
thermistor problem, i.e., in studying the heat transfer in the resistor device
whose electrical conductivity is strongly dependent on the temperature.
Existence of an optimal control is proved. The optimality system consisting of
the state system coupled with adjoint equations is derived, together with a
characterization of the optimal control. Uniqueness of solution to the
optimality system, and therefore the uniqueness of the optimal control, is
established. The last part is devoted to numerical simulations.
