Necessary optimality conditions for parabolic equations with Venttsel boundary control

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. The first and second order necessary conditions are derived for the optimal solution in the case of both unconstrained and constrained problems.