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The chain control set of discrete-time linear system on the affine two-dimensional Lie group

Thiago Cavalheiro, Alexandre Santana, João Cossich

Abstract

In this paper, we present conditions for the existence and uniqueness of chain control sets of discrete-time linear systems on the affine two-dimensional Lie group. More specifically, we prove that these chain control sets are given by the union of an infinite number of control sets with empty interiors.

The chain control set of discrete-time linear system on the affine two-dimensional Lie group

Abstract

In this paper, we present conditions for the existence and uniqueness of chain control sets of discrete-time linear systems on the affine two-dimensional Lie group. More specifically, we prove that these chain control sets are given by the union of an infinite number of control sets with empty interiors.
Paper Structure (14 sections, 10 theorems, 103 equations, 3 figures)

This paper contains 14 sections, 10 theorems, 103 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Control sets of $\Sigma$
  4. Chain control sets of $\Sigma$
  5. Controllability sets and chain controllability
  6. Declarations
  7. Funding
  8. Conflict of interest/Competing interests
  9. Ethics approval
  10. Consent to participate
  11. Consent for publication
  12. Availability of data and materials
  13. Code availability
  14. Authors' contributions

Key Result

Proposition 5

Consider a discrete-time linear control system on a Lie group $G$ defined by $x_{k+1} = f(u_k,x_k)$, $u_k \in U$. Then for all $g \in G$ and $u = (u_i)_{i \in \mathbb{Z}} \in \mathcal{U}$ we have

Figures (3)

  • Figure 1: A representation of the control sets in case of $g(u) = u$, $a=0$ and $U = [-1,1]$. In the general case, the point $(x,0)$ does not need necessarily to be in the relative interior (by the induced topology) of $D_x$.
  • Figure 2: A representation of the chain control set of the Proposition \ref{['chaincontrolsetd>0']}, when $a < 0$.
  • Figure 3: Region $G_0$ which contains every control set in the form $D_x = \{x\} \times \mathbb{R}$, with $d = 1$ and $(a+g(u_m))(a+g(u_M)) > 0$.

Theorems & Definitions (16)

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