Control semiflows, chain controllability, and the Selgrade decomposition for linear delay systems

TL;DR

This paper introduces a continuous semiflow framework for linear delay control systems, establishing the existence of a unique chain control set and a decomposition into exponentially separated subbundles, extending Selgrade's theorem.

Contribution

It develops a novel semiflow approach for delay systems and generalizes Selgrade's decomposition to infinite-dimensional settings.

Findings

Existence of a unique chain control set for the semiflow.

Lifting of the semiflow to an infinite-dimensional vector bundle.

Decomposition into exponentially separated subbundles.

Abstract

A continuous semiflow is introduced for linear control systems with delays in the states and controls and bounded control range. The state includes the control functions. It is proved that there exists a unique chain control set which corresponds to the chain recurrent set of the semiflow. The semiflow can be lifted to a linear semiflow on an infinite dimensional vector bundle with chain transitive base flow. A decomposition into exponentially separated subbundles is provided by a recent generalization of Selgrade's theorem.