A Ball-Marsden-Slemrod type result for semi-linear equations with analytic semigroups

Abstract

We extend a well-known non-controllability result of Ball, Marsden, and Slemrod on infinite-dimensional bilinear systems (with bounded control term) to control-affine semi-linear systems whose linear part generates an analytic semigroup and whose control term is possibly unbounded as well. Here control inputs are assumed to lie in some suitable Lp-space, p > 1. The result allows an application to PDEs whose control terms include not only the state but also lower oder derivatives compared to the uncontrolled leading linear part. The proof relies on an abstract compactness principle for parameter dependent fixed point maps.