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Necessary conditions for local controllability of a particular class of systems with two scalar controls

Laetitia Giraldi, Pierre Lissy, Clément Moreau, Jean-Baptiste Pomet

Abstract

We consider affine control systems with two scalar controls, such that one control vector field vanishes at an equilibrium state. We state two necessary conditions of local controllability around this equilibrium, involving the iterated Lie brackets of the system vector fields, with controls that are either bounded, small in L $\infty$ or small in W 1,$\infty$. These results are illustrated with several examples.

Necessary conditions for local controllability of a particular class of systems with two scalar controls

Abstract

We consider affine control systems with two scalar controls, such that one control vector field vanishes at an equilibrium state. We state two necessary conditions of local controllability around this equilibrium, involving the iterated Lie brackets of the system vector fields, with controls that are either bounded, small in L or small in W 1,. These results are illustrated with several examples.

Paper Structure

This paper contains 19 sections, 6 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Definitions of local controllability
  4. Known results for single-input systems
  5. Main results
  6. Obstruction coming from the first bad bracket
  7. Obstruction coming from the second bad bracket
  8. Illustrating examples and applications
  9. Examples for the first bracket obstruction
  10. Case where the second control cannot help to neutralize $B_1$
  11. Case where STLC is retrieved thanks to the second control
  12. Application to micro-swimmer robots
  13. Examples for the second bracket obstruction
  14. Existence of several neutralizing brackets
  15. Role of the bracket $[f_1,[f_2,f_1]]$
  16. ...and 4 more sections

Key Result

Proposition 1

If and, for all $k$ in $\mathbb{N}$, then system eq:1 is STLC.

Theorems & Definitions (28)

  • Definition 1: STLC
  • Definition 2: $\alpha$-STLC
  • Remark 1
  • Remark 2
  • Definition 3: $\mathrm{W}^{k,\infty}$-STLC
  • Remark 3
  • Remark 4
  • Proposition 1: sussmann1983lie, Theorem 2.1, p.688
  • Proposition 2
  • Proposition 3
  • ...and 18 more