Verification of second-order sufficient optimality conditions for semilinear elliptic and parabolic control problems

We study optimal control problems for semilinear parabolic equations subjec t to control constraints and for semilinear elliptic equations subject to c ontrol and state constraints. We quote known second-order sufficient optima lity conditions (SSC) from the literature. Both problem classes, the parabo lic one with boundary control and the elliptic one with boundary or distrib uted control, are discretized by a finite difference method. The discrete S SC are stated and numerically verified in all cases providing an indication of optimality where only necessary conditions had been studied before.