An Extended Mathematical Framework for Barrier Methods in Function Space

Abstract

In this note, we extend the mathematical framework in [7] of barrier methods for state constrained optimal control problems with PDEs to a more general setting. In [7] we modelled the state equation by Ly = u with L a closed, densely defined, surjective operator. This restricts the applicability of our theory mainly to certain distributed control problems. Motivated by the discussion in [6], we consider in this work operator equations of the more general form Ay-Bu = 0, where A is closed, densely defined and with closed range and B is continuous. While this change in framework only neccessitates minor modificatios in the theory, it extends its applicability to large additional classes of control problems, such as boundary control and finite dimensional control. To make this paper as self contained as possible, assumptions and results of [7] are recapitulated, but for brevity proofs and more detailed information are only given when there are differences to [7]. This is possible, because our extension has only a very local effect.