Sufficient Optimality in a Parabolic Control Problem

We define a class of parabolic problems with control and state constraints and identify a problem within this class which possesses a locally unique critical point satisfying the second order sufficient optimality conditions. In this solution inequality constraints on the control are strongly active. The second derivative of the Lagrangian is not globally coercive. This is both shown analytically as well as verified numerically for a finite difference discretization.