Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Linear Test for Global Nonlinear Controllability

Karthik Elamvazhuthi

TL;DR

The paper investigates whether invertibility of the horizontal sub-Laplacian $\Delta_H$ implies approximate controllability for nonlinear control-affine systems without drift. By linking the invertibility of $\Delta_H$ to the controllability of the associated continuity equation with potentially non-smooth controls through the Ambrosio-Gigli-Savare superposition principle, it presents two complementary viewpoints, Perron-Frobenius and Koopman, to establish a controllability relation. It proves that if $\Delta_H$ is invertible on $L^2_{\perp}(\Omega)$, then the backward reachable set $\mathrm{Reach}^{-1}(B_R(y))$ has full Lebesgue measure for every ball $B_R(y)$, yielding a spectral-gap based degree of controllability via a Poincaré inequality with constant $\lambda$. The dual Koopman argument shows that injectivity leads to full-measure forward reachability, and the results extend to weighted Laplacians and manifolds, providing a computable, infinite-dimensional criterion for approximate nonlinear controllability.

Abstract

It is known that if a nonlinear control affine system without drift is bracket generating, then its associated sub-Laplacian is invertible under some conditions on the domain. In this note, we investigate the converse. We show how invertibility of the sub-Laplacian operator implies a weaker form of controllability, where the reachable sets of a neighborhood of a point have full measure. From a computational point of view, one can then use the spectral gap of the (infinite-dimensional) self-adjoint operator to define a notion of degree of controllability. An essential tool to establish the converse result is to use the relation between invertibility of the sub-Laplacian to the the controllability of the corresponding continuity equation using possibly non-smooth controls. Then using Ambrosio-Gigli-Savare's superposition principle from optimal transport theory we relate it to controllability properties of the control system. While the proof can be considered of the Perron-Frobenius type, we also provide a second dual Koopman point of view.

A Linear Test for Global Nonlinear Controllability

TL;DR

The paper investigates whether invertibility of the horizontal sub-Laplacian implies approximate controllability for nonlinear control-affine systems without drift. By linking the invertibility of to the controllability of the associated continuity equation with potentially non-smooth controls through the Ambrosio-Gigli-Savare superposition principle, it presents two complementary viewpoints, Perron-Frobenius and Koopman, to establish a controllability relation. It proves that if is invertible on , then the backward reachable set has full Lebesgue measure for every ball , yielding a spectral-gap based degree of controllability via a Poincaré inequality with constant . The dual Koopman argument shows that injectivity leads to full-measure forward reachability, and the results extend to weighted Laplacians and manifolds, providing a computable, infinite-dimensional criterion for approximate nonlinear controllability.

Abstract

It is known that if a nonlinear control affine system without drift is bracket generating, then its associated sub-Laplacian is invertible under some conditions on the domain. In this note, we investigate the converse. We show how invertibility of the sub-Laplacian operator implies a weaker form of controllability, where the reachable sets of a neighborhood of a point have full measure. From a computational point of view, one can then use the spectral gap of the (infinite-dimensional) self-adjoint operator to define a notion of degree of controllability. An essential tool to establish the converse result is to use the relation between invertibility of the sub-Laplacian to the the controllability of the corresponding continuity equation using possibly non-smooth controls. Then using Ambrosio-Gigli-Savare's superposition principle from optimal transport theory we relate it to controllability properties of the control system. While the proof can be considered of the Perron-Frobenius type, we also provide a second dual Koopman point of view.
Paper Structure (8 sections, 9 theorems, 46 equations)

This paper contains 8 sections, 9 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Perron-Frobenius view
  3. Koopman View
  4. Notation and Background
  5. Proof of the main result
  6. Perron-Frobenius Approach
  7. Koopman Approach
  8. Acknowldegements

Key Result

Theorem 1.1

(Bracket Generation implies invertibility of the sub-Laplacian) Suppose the system eq:ctrsys is bracket generating of order $r$. Let $\Omega$ be a $\epsilon-\delta$ subdomain of $\mathbb{R}^d$. then $\Delta_H$ is invertible on $L^{\perp}_2(\Omega)$, the subset of functions in $L_2(\Omega)$ that inte

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1
  • Lemma 2.1
  • Theorem 2.2
  • Corollary 2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 7 more