A posteriori error estimation for nonlinear parabolic boundary control

We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: Let an arbitrary control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second order sufficient optimality condition for the (unknown) locally optimal control, we are able to estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian. We apply this technique to a model reduced optimal control problem obtained by proper orthogonal decomposition (POD). The distance between a (suboptimal) local solution of the reduced problem to a local solution of the original problem is estimated.