  In this paper we study the optimality condition for the Venttsel boundary
control of a parabolic equation, that is, the state of the dynamic system is
governed by a parabolic equation together with an initial condition while the
control is applied to the system via the Venttsel boundary condition. A first
order necessary condition is derived for the optimal solution in the case of
both unconstrained and constrained problems. The condition is also sufficient
for the unconstrained problem.
