[Paper Review] A Differential Lyapunov Framework for Contraction Analysis

Fulvio Forni, Rodolphe Sepulchre|arXiv (Cornell University)|Aug 14, 2012

Control and Stability of Dynamical Systems48 references15 citations

TL;DR

This paper introduces a differential Lyapunov framework for contraction analysis by lifting classical Lyapunov functions to the tangent bundle, endowing the state space with a Finsler structure. The key contribution is a sufficient pointwise condition—via a Finsler-Lyapunov function—ensuring incremental stability through infinitesimal contraction, with global convergence inferred by integration along solution curves.

ABSTRACT

Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is inferred from infinitesimal contraction of the Finsler metrics through integration along solutions curves.

Motivation & Objective

• To bridge classical Lyapunov stability theory with contraction theory by formulating a differential version of Lyapunov's second theorem.
• To address limitations in existing incremental stability analysis, particularly the ad-hoc construction of error systems and difficulties in defining global distances.
• To unify diverse contraction analysis approaches—Riemannian, matrix measure, and non-Riemannian—under a single Finsler geometric framework.
• To extend classical Lyapunov tools, such as LaSalle’s invariance principle, to incremental stability via the proposed differential framework.
• To introduce and formalize the concept of horizontal contraction for systems with symmetries, where contraction is only required in specific subspaces of the tangent bundle.

Proposed method

• Lift a classical Lyapunov function from the state space to the tangent bundle, defining a Finsler-Lyapunov function that induces a Finsler structure on the manifold.
• Use the differential of the vector field to define a time derivative of the Finsler-Lyapunov function along solution curves, ensuring pointwise decay in the tangent space.
• Establish incremental stability by showing that the integrated distance (defined via the Finsler metric) decays along solution trajectories.
• Introduce the notion of horizontal contraction by restricting the decay condition to a subspace (horizontal subspace) of the tangent space, allowing for symmetry-invariant directions.
• Apply the framework to prove asymptotic attractivity of a set A under boundedness and contraction conditions on the horizontal subspace.
• Utilize class K and KL functions to characterize decay rates and convergence behavior in the presence of asymptotic invariance.

Experimental results

Research questions

• RQ1 How can Lyapunov’s second theorem be generalized to incremental stability using a differential geometric framework?
• RQ2 Can a Finsler structure naturally unify Riemannian and matrix measure-based contraction analyses?
• RQ3 What is the role of horizontal contraction in systems with symmetries, and how can it be formalized within a Lyapunov framework?
• RQ4 How can classical Lyapunov tools like LaSalle’s invariance principle be adapted to incremental stability via the tangent bundle approach?
• RQ5 Under what conditions does contraction in the horizontal subspace imply global asymptotic attractivity of a set A?

Key findings

• A Finsler-Lyapunov function provides a sufficient pointwise condition for incremental stability by ensuring infinitesimal contraction of the Finsler metric along solution curves.
• The global distance between solutions is implicitly constructed via integration of the Finsler metric, and its decay along trajectories proves incremental stability.
• Horizontal contraction allows for the exclusion of symmetry directions from contraction requirements, making the framework suitable for tracking, observer design, and synchronization.
• The proposed framework enables a direct extension of LaSalle’s invariance principle to incremental stability, proving that ω-limit sets are contained in the attractor A under boundedness and horizontal contraction.
• The absence of periodic orbits is guaranteed when the horizontal Finsler-Lyapunov function satisfies a decay condition and the ω-limit set contains no equilibria.
• The framework unifies prior approaches: Riemannian contraction (Lohmiller and Slotine) and matrix measure-based contraction (Russo et al.) are shown to be special cases of the general Finsler-Lyapunov formulation.