In this section numerical results will be reported for the application of the method outlined in the previous section to three parabolic control problems from [1,9] and an exemplary elliptic control problems from [19]. The first parabolic problem is particularly interesting because for it in [1] an analytical exact solution is given and the continuous SSC conditions are verified. For the sake of completeness, the specification will be given for each problem. The discretizations used are those defined in section 2 for the parabolic and in [16][18] for the elliptic problem.
Problem (P) from section 2 is solved without state constraints
and with the following data

In Table 1 the maximum errors over the grid points are listed for the computed control and state functions while in Table 2 the smallest eigenvalue is listed as well as the value scaled by . The errors exhibit quadratic convergence while the scaled eigenvalue stays nearly constant with a slightly increasing tendency. These facts justify the application of the technique to other control problems for which no exact solution is known and it also shows which scaling of is appropriate for problem . This scaling is also suggested by known quadratic convergence estimates for fixed control in since the differential equation as well as all boundary conditions were discretized with secondorder consistency. An argument could be made that the smallness of the eigenvalue may indicate that it is, in fact, negative and the relatively coarse discretization prevents it from exhibiting this fact. This seems highly unlikely in the light of the very uniform and even slightly growing behavior of the scaled value combined with the fact that all the eigenvalues were computed and the smallest one is not considerably smaller than the next ones but they grow only gradually.
As a second parabolic case one from [9] was chosen. The data for
(P) are
We include a plot of the optimal control and state for this problem.
Due to the consistent results obtained the computations for
were not done in these cases.
Finally, we solve a boundary control problem for the instationary
Burgers equation, see also [30]. For simplicity we used the
same data as for case (II) above with and the control bounds
both of which become active in the
solution. This case is denoted by 5.2IV. It clearly shows that for
this problem the suitable normalization of the eigenvalue is .
This will have to be confirmed by a careful comparison of the
continuous and discrete SSC as had been done for the first order
necessary conditions in [16,17]. A similar situation
arose in the elliptic boundary control problem in [18] which
is addressed next below. For it in [19] the scaling by
seemed appropriate in contrast to the other such problems considered.
Next, we present the data for the elliptic boundary control problem from [18] and the eigenvalues obtained. The domain is the unit square. The problem (EB) together with (2.1) and (2.2) is considered and .
We also include a plot of the optimal control and state for this problem. For plots of the adjoint variables and further details, we refer to [18].