Control Sets for Affine Systems, Spectral Properties and Projective Spaces

TL;DR

This paper investigates the structure of control sets in affine control systems, revealing how spectral properties determine boundedness and uniqueness, with distinctions between hyperbolic and nonhyperbolic systems.

Contribution

It characterizes control sets for affine systems using spectral analysis and explores their properties in compactified state spaces, highlighting differences based on hyperbolicity.

Findings

Hyperbolic systems have a unique bounded control set with nonvoid interior.

Nonhyperbolic systems have unbounded control sets.

In compactified spaces, a unique chain control set relates to the system's homogeneous part.

Abstract

For affine control systems with bounded control range the control sets, i.e., the maximal subsets of complete approximate controllability, are studied using spectral properties. For hyperbolic systems there is a unique control set with nonvoid interior and it is bounded. For nonhyperbolic systems, these control sets are unbounded. In an appropriate compactification of the state space there is a unique chain control set and the relations to the homogenous part of the control system are worked out.