On some parabolic boundary control problems with constraints on the control and functional-constraints on the state

Abstract

In this paper parabolic boundary control problems in a $n$-dimensional bounded domain are investigated, where in addition to a pointwise constraint on the control finitely many linear functional constraints on the state are given. As performance functional the $L_{\infty }$- or $L_2$-norm of the difference between the final state and a given (desired) state is used.

A generalized bang-bang-principle is proved which expresses that under natural assumptions almost everywhere (with respect to the time) either the optimal control or the corresponding state is acting on the boundary of its constraints. The assumptions are discussed by examples.