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Nonautonomous control systems and skew product flows

Fritz Colonius, Roberta Fabbri

TL;DR

The work develops a framework to analyze controllability of nonautonomous control-affine systems via skew-product/drive-flow representations. It defines and interrelates chain control sets and nonautonomous control sets, proving that chain control sets are fiber-determined and correspond to maximal chain transitive sets of the control flow, while nonautonomous equilibria in the uncontrolled system lie inside control sets and can induce topologically mixing dynamics. The results generalize autonomous and almost periodic cases and provide structural tools for bifurcation analysis and computation, leveraging hull constructions, minimal equicontinuous flows, and transfer between fiberwise and global dynamics. Together, this advances a cohesive theory linking nonautonomous controllability to skew-product dynamics and offers practical criteria for identifying control regions and equilibria in driven systems.

Abstract

For nonautonomous control systems with compact control range, associated control flows are introduced. This leads to several skew product flows with various base spaces. The controllability and chain controllability properties are studied and related to properties of the associated skew product flows.

Nonautonomous control systems and skew product flows

TL;DR

The work develops a framework to analyze controllability of nonautonomous control-affine systems via skew-product/drive-flow representations. It defines and interrelates chain control sets and nonautonomous control sets, proving that chain control sets are fiber-determined and correspond to maximal chain transitive sets of the control flow, while nonautonomous equilibria in the uncontrolled system lie inside control sets and can induce topologically mixing dynamics. The results generalize autonomous and almost periodic cases and provide structural tools for bifurcation analysis and computation, leveraging hull constructions, minimal equicontinuous flows, and transfer between fiberwise and global dynamics. Together, this advances a cohesive theory linking nonautonomous controllability to skew-product dynamics and offers practical criteria for identifying control regions and equilibria in driven systems.

Abstract

For nonautonomous control systems with compact control range, associated control flows are introduced. This leads to several skew product flows with various base spaces. The controllability and chain controllability properties are studied and related to properties of the associated skew product flows.
Paper Structure (5 sections, 17 theorems, 117 equations)

This paper contains 5 sections, 17 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chain Control Sets
  4. Relation to chain transitive sets
  5. Control sets

Key Result

Theorem 2

Let $(X,\phi)$ be a continuous flow on a compact metric space. If it is minimal, then it is either equicontinuous or sensitive with respect to initial conditions, i.e., there is $\delta>0$ such that whenever $U$ is a nonvoid open set there exist $x,y\in U$ such that $d(\phi (T,x),\phi(T,y))>\delta$

Theorems & Definitions (32)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Example 4
  • Definition 5
  • Definition 6
  • Remark 7
  • Lemma 8
  • Proposition 9
  • Proposition 10
  • ...and 22 more