Compactness of Fixed Point Maps and the Ball-Marsden-Slemrod Conjecture

TL;DR

This paper develops an abstract compactness principle for fixed point maps in parameter-dependent equations and applies it to extend non-controllability results for semi-linear control systems, including cases with different control regularities.

Contribution

It introduces a new compactness principle for fixed point maps and applies it to extend the Ball-Marsden-Slemrod non-controllability result to semi-linear systems.

Findings

Extended non-controllability to semi-linear systems

Analyzed control regularity cases p>1 and p=1

Provided a new abstract compactness criterion

Abstract

Given a parameter dependent fixed point equation $x = F(x,u)$, we derive an abstract compactness principle for the fixed point map $u \mapsto x^*(u)$ under the assumptions that (i) the fixed point equation can be solved by the contraction principle and (ii) the map $u \mapsto F(x,u)$ 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 [1] to semi-linear systems. First we consider $L^p$-controls, $p>1$. Subsequently we analyze the case $p=1$.