A wealth of literature exists on theoretical and computational aspects of
control problems for ordinary differential equations. This includes in
particular results on necessary and sufficient optimality conditions.
Exemplarily, we mention the fundamental works [11,15] in which
it was shown that a two-norm discrepancy cannot be avoided in the
continuous case. Other recent work on such problems includes ,
, and .
Research on control problems governed by partial
differential equations has started more recently as did the investigation of
optimality conditions for these problems. First, nonlinear parabolic control
problems were considered, for example, in , then elliptic
problems . Some of the most recent work is [5,21] in which such problems with control and state constraints are treated.
Second order sufficient optimality conditions (SSC) for elliptic control
problems without such constraints were
considered in [12,13] while SSC were shown and partly also
numerically verified for finite element (semi-) discretizations of the
one dimensional Burgers equation in [28,29].
The establishment of SSC
requires a highly technical machinery including the use of a third norm as
was done before in [6,14].
The verification of the, partly restrictive, continuous optimality condition
is possible only in rare circumstances, such as an analytically known
solution for which all conditions can be evaluated analytically; for a
parabolic example, see .
For the numerical solution of such control problems there are two different approaches, a direct discretization of the entire problem leading to a large finite-dimensional constrained optimization problem. This approach has also been called ``all-at-once'' method, see, for example, , ``one-shot'' method, "full discretization approach" etc. An iterative method can be applied in function space resulting in a series of infinite-dimensional linear-quadratic control problems which then still need to be discretized. The literature on the latter is quite extensive while the former has more recently been considered for the solution of real-life problems, when progress had been made on solution techniques for very large constrained nonlinear programming (NLP) problems. Exemplarily, we cite the special issue .
In - we have considered general semilinear elliptic control problems with control and state constraints. The first-order necessary conditions were formally derived in the continuous case. These were compared to the finite-dimensional conditions for a direct finite-difference discretization of the control problem and the latter was solved by applying state-of-the-art interior point and SQP methods.
The direct discretization will lead to huge NLP's for three-dimensional elliptic or two and three-dimensional parabolic problems. If the discretization is consistent, preferably of higher order in the discretization parameter, the numerical verification of the SSC for these problems, even for relatively coarse discretizations, may be expected to yield clues on the optimality of the approximated continuous solution. It is in this spirit that in  a class of both parabolic and elliptic control problems in one respectively two space dimensions was considered. The problems were discretized and the SSC were verified. However, the use of dense linear algebra techniques limited the size of the problems considerably and an argument could be made that these results need to be confirmed for the same problem sizes for which the control problems themselves were solved and for which, at least in some cases, actual error evaluations could be done showing the quality of approximation of the underlying continuous problem by its discretization.
In the following section both the parabolic and elliptic control problems are stated. Some known SSC results are quoted in the third section. In section 4 the way in which the discrete SSC are verified is described followed in section 5 by results for all parabolic cases from , in addition one with state constraints as well as the instationary Burgers equation, and one exemplary elliptic boundary control problem from [19,18]. Concluding remarks are made in the last section.