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Contraction theory: Hausdorff--Riemann Measures as Set-Based Lyapunov Functions

A. Matveev, A. Pogromsky

Abstract

We offer a measure-theoretic extension of the concept and theory of $k$-contraction, including their generalization on fractional dimensions $d$. The respective contraction property is defined through the exponential decay of the $d$-dimensional volume of compact sets transported by a nonlinear flow. For autonomous systems on positively invariant compact sets, we derive comprehensive, i.e., necessary and sufficient, conditions for $d$-contractivity in two complementary forms. The first is expressed in terms of the finite-time Lyapunov characteristic exponents and is akin in spirit to the first Lyapunov method. The second one is consonant with the second Lyapunov method and relies on existence of a Riemannian metric ensuring exponential decay of the metric-induced $d$-dimensional Hausdorff measure. To acquire monotone measure-theoretic-based Lyapunov functions, we introduce a family of \emph{Hausdorff-Riemann measures}, which are elliptic, metric-dependent $d$-measures that strictly decrease along the trajectories and thus may serve as Lyapunov functions. These measures enable an anytime characterization of the rate of contraction and provide constructive tools for stability analysis and feedback design. To illustrate the applicability of the approach, we derive tractable criteria for orbital stability of periodic solutions of autonomous ODE's and employ several prototypical particular examples, including a rigid body with dissipation and constant torque, the Rössler system, and the Langford system.

Contraction theory: Hausdorff--Riemann Measures as Set-Based Lyapunov Functions

Abstract

We offer a measure-theoretic extension of the concept and theory of -contraction, including their generalization on fractional dimensions . The respective contraction property is defined through the exponential decay of the -dimensional volume of compact sets transported by a nonlinear flow. For autonomous systems on positively invariant compact sets, we derive comprehensive, i.e., necessary and sufficient, conditions for -contractivity in two complementary forms. The first is expressed in terms of the finite-time Lyapunov characteristic exponents and is akin in spirit to the first Lyapunov method. The second one is consonant with the second Lyapunov method and relies on existence of a Riemannian metric ensuring exponential decay of the metric-induced -dimensional Hausdorff measure. To acquire monotone measure-theoretic-based Lyapunov functions, we introduce a family of \emph{Hausdorff-Riemann measures}, which are elliptic, metric-dependent -measures that strictly decrease along the trajectories and thus may serve as Lyapunov functions. These measures enable an anytime characterization of the rate of contraction and provide constructive tools for stability analysis and feedback design. To illustrate the applicability of the approach, we derive tractable criteria for orbital stability of periodic solutions of autonomous ODE's and employ several prototypical particular examples, including a rigid body with dissipation and constant torque, the Rössler system, and the Langford system.
Paper Structure (26 sections, 38 theorems, 156 equations, 1 figure)

This paper contains 26 sections, 38 theorems, 156 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some motivation issues
  3. Uniformly $d$-contractive systems
  4. Hausdorff measure as a generalization of the $k$-measured volume
  5. Definition of the uniform $d$-contraction
  6. Compound matrices and a related sufficient criterion for $k$-contraction with a natural $k$
  7. Characterization of the uniform $d$-contraction via the Lyapunov exponents
  8. Characterization of the uniform $d$ contraction in the vein of the second Lyapunov method
  9. Hausdorff-Riemann elliptic $d$-measures
  10. Elliptic $d$-measures
  11. $P$-elliptic $d$-measures
  12. Hausdorff-Riemann elliptic $d$-measure
  13. Area type inequality for Hausdorff-Riemann measures
  14. Proofs of Theorems \ref{['equiv_uniform']}, \ref{['firstmethod']}, and \ref{['second_method']}
  15. Proof of Theorem \ref{['equiv_uniform']}
  16. ...and 11 more sections

Key Result

Proposition 3.1

For any $d \in [0, \infty)$, the following statements hold:

Figures (1)

  • Figure 1: A trajectory of the Langford system \ref{['lanford']} for $a=\tfrac{2}{3}$. The points are colored according to their depth relative to the viewing direction: warmer (red) tones correspond to points closer to the viewer, while colder (blue) tones indicate points farther away. A small coordinate frame is shown at the origin for reference.

Theorems & Definitions (64)

  • Proposition 3.1
  • Definition 3.1
  • Theorem 3.1
  • Corollary 3.1
  • Lemma 4.1
  • proof
  • Theorem 4.1
  • Remark 4.1
  • Theorem 5.1
  • Remark 5.1
  • ...and 54 more