Dichotomy theorem for control sets

doi:10.1016/j.sysconle.2019.05.001

Systems & Control Letters

Volume 129, July 2019, Pages 10-16

https://doi.org/10.1016/j.sysconle.2019.05.001 Get rights and content

Abstract

We introduce two notions of equi-invariant sets and unstable sets for control systems and show that a control set with dense interior is either equi-invariant or unstable. This is the analogy to the well-known dichotomy theorem for topological transitive dynamical systems. An illustrative example is given to show that both of the two situations can occur.

Introduction

The key control theoretic notionsdiscussed in the present paper are control sets, which were studied by Colonius and Kliemann [1], [2]. Control sets, as a means for expressing controllability, are the maximal subsets of the state space on which complete approximate controllability holds. The link between control and topological dynamical systems theory is established by the control flow, a dynamical system defined on the product space of admissible control functions and state space. It should be mentioned that Colonius and Kliemann [1], [2] established an elegant connection that control sets correspond to maximal topologically transitive sets of the control flow in the continuous-time case. We refer to [3] for the discrete-time case.

The well-known Auslander–Yorke dichotomy theorem in topological dynamical systems reveals an extreme situation in a minimal (transitive) system. It states that a minimal system is either sensitive or equicontinuous and that a transitive system is either sensitive or almost equicontinuous (see [4], [5], [6]). Motivated by the elegant connection between control sets and maximal transitive sets, we want to explore some interesting evolution for points in control sets in terms of admissible control functions.

In this paper, we first introduce notions, named as equi-invariant sets and unstable sets (see Definition 3.1, Definition 3.9). Then some properties and equivalent characterizations for these notions are given. In particular, the relations between equi-invariant sets and (outer) invariance entropy are investigated. For more systematic study of invariance entropy (pressure) we refer the readers to the references [7], [8], [9], [10], [11], [12], [13], [14], [15]. More importantly, we find an interesting phenomenon in a control set with dense interior (see Theorem 3.13). We call this phenomenon the dichotomy theorem, which is closely linked to the properties of control sets. An illustrative example is given to show that both of the two situations can occur (see Example 3.17).

Section snippets

Control systems

Let $Z$ , $R$ and $N$ denote the sets of all integers, real numbers, and positive integers, respectively. Let $T \in {Z, R}$ and $T_{+} = {t \in T : t \geq 0}$ . When the time set is understood from the text, all intervals are assumed to be restricted to $T$ . For example, $[σ, τ) = {t \in T : σ \leq t < τ}$ . Let $U$ be a nonempty set and $U$ be a collection of maps from $T$ to $U$ . For $σ \leq τ$ , let $U_{[σ, τ)}$ denote the set ${ω |_{[σ, τ)} : ω \in U}$ . Suppose $σ, τ, t \in T$ with $σ \leq τ \leq t$ , $ω_{1} \in U_{[σ, τ)}$ , $ω_{2} \in U_{[τ, t)}$ , the concatenation of $ω_{1}$ and $ω_{2}$ is defined as $ω_{1} ω_{2} (s) = \{\begin{aligned} ω_{1} (s), if s \in [σ, τ), \\ ω_{2} (s) \end{aligned}$

Dichotomy theorem for control sets

In this section, we establish the dichotomy theorem for control sets with dense interior. Firstly, we introduce the notions of equi-invariant sets and unstable sets, which are pivotal for our study in this paper.

Several examples

In this section, three examples are given to illustrate several assertions stated in Section 3.1. Each of these examples is constructed by considering a control system which is given by difference equation $x_{k + 1} = F (x_{k}, u_{k}) = u_{k} (x_{k}), u_{k} \in U, k \in Z_{+},$ where $X$ is a metric space, $U$ is a subset in the space $C (X, X)$ of all continuous maps from $X$ to $X$ . According to [14, Section 2.4], this system can be written as $Σ = (Z, X, U, U^{Z}, φ)$ with $φ (i, x, ω) = \{\begin{aligned} x, & if i = 0, \\ ω_{i - 1} \circ \dots \circ ω_{1} \circ ω_{0} (x), & if i \geq 1, \end{aligned}$ where $U^{Z} = {ω = (\dots ω_{0} ω_{1} \dots) : ω_{i} \in U, i \in Z}$ .

Recall

Acknowledgment

The authors would like to thank the anonymous referees and the associate editor for their critical comments and suggestions that led to the improvement of this manuscript.

Declaration of competing interest

No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.sysconle.2019.05.001.

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^☆: This work was partly supported by National Nature Science Funds of China (11771459, 11701584).

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Dichotomy theorem for control sets☆

Abstract

Introduction

Section snippets

Control systems

Dichotomy theorem for control sets

Several examples

Acknowledgment

Declaration of competing interest

Systems Control Lett.

Systems Control Lett.

Some aspects of control systems as dynamical systems

J. Dynam. Differential Equations

The Dynamics of Control

Some connections between chaotic dynamical systems and control systems

When is a transitive map chaotic?

Interval maps, factors of maps, and chaos

Toˆhoku Math. J.

Sensitive dependence on initial conditions

Nonlinearity

Metric invariance entropy and conditionally invariant measures

Ergodic Theory Dynam. Syst.

T $\hat{o}$ hoku Math. J.