In this work the second-order sufficient optimality conditions (SSC) were verified numerically for several optimal control problems taken from recent papers. For a parabolic problem with analytically known solution a second-order finite-difference discretization was shown to have good accuracy already for moderate discretizations. Computational results on coarse grids from [19] were confirmed. They show a very consistent trend: the smallest eigenvalue of the projected Hessian of the Lagrangian suitably scaled by the discretization parameter behaves nearly constant indicating that the computed stationary solution appears to be a strict local minimizer. Subsequently, the procedure is applied to three other parabolic problems including one with pointwise state constraints and to one elliptic control problem, confirming the SSC in each case.
Open questions that should be addressed in future work include the following: can a formal proof be given of the satisfaction of the SSC which were numerically verified above? Can the results be generalized to the singular and bang-bang controls observed in [16]-[18]? Which numerical results can be obtained for state-constrained parabolic control problems? It is to be expected that such PDE-constrained optimization problems as considered in this paper will be subject of intense research efforts in the near future. We refer again to [24]; see also the short survey [10].