Conclusion

In this work the second-order sufficient optimality conditions (SSC) were verified numerically for a number of different optimal control problems taken from five 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. Since dense numerical linear algebra was used for the verification of the SSC they were not checked for the finest discretization used but still 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 another parabolic and ten elliptic control problems, 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]-[19]? 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 [21]; see also the short survey [10].

2001-06-17