Oberseminar "Dynamische Systeme und Kontrolltheorie" - Dr. Gunther Dirr

Given a parameter dependent fixed point equation x = F (x, u), we derive an abstract
compactness principle for the fixed point map u 7→ x ∗ (u) under the assumptions that
(i) the fixed point equation can be solved by the contraction principle and (ii) the map
u 7→ F (x, u) is compact for fixed x.
This result is applied to infinite-dimensional, semi-linear control systems and their
reachable sets. More precisely, we extend a non-controllability result of Ball, Marsden,
and Slemrod to semi-linear systems. First we consider L p -controls, p > 1. Subsequently
we analyze the case p = 1.