On linear-quadratic elliptic control problems of semi-infinite type

Abstract

We derive a priori error estimates for linear-quadratic elliptic optimal control problems with pointwise state constraints in a compact subdomain of the spatial domain Ω for a class of problems with finite-dimensional control space. The problem formulation leads to a class of semi-infinite programming problems, whose constraints are implicitly given by the FE-discretization of the underlying PDEs. We prove an order of for the error |ū − ū ^h | in the controls, and show that it can be improved to an order of h ²|log h| under certain assumptions on the structure of the active set. Numerical experiments underline the proven theoretical results.

Keywords:

• elliptic optimal control problem
• state constraints
• error estimates
• finite element discretization
• semi-infinite optimization

AMS Subject Classifications::

• 49J20
• 80M10
• 49N05
• 41A25
• 90C34