Contracting dynamical systems in Banach spaces

TL;DR

This paper extends contraction analysis from finite-dimensional inner-product spaces to Banach spaces using weighted semi-inner products, enabling stability analysis of infinite-dimensional systems and applications in PDEs and machine learning.

Contribution

It introduces a unified approach to contraction in Banach spaces via weighted semi-inner products, generalizing stability concepts beyond traditional Riemannian metrics.

Findings

Generalized contraction to Banach spaces using semi-inner products

Applied contraction analysis to PDEs and continuum mechanics

Derived stability conditions for functional gradient descent in Banach spaces

Abstract

Contraction rates of time-varying maps induced by dynamical systems illuminate a wide range of asymptotic properties with applications in stability analysis and control theory. In finite-dimensional smoothly varying inner-product spaces such as $\mathbb{R}^n$ and $\mathbb{C}^n$ with Riemannian metrics, contraction rates can be estimated by upper-bounding the real numerical range of the vector field's Jacobian. However, vector spaces with norms other than $L^2$ commonly arise in the stability analysis of infinite-dimensional systems such as those arising from partial differential equations and continuum mechanics. To this end, we present a unified approach to contraction analysis in Banach spaces using the theory of weighted semi-inner products. We generalize contraction in a geodesic distance to asymptotic stability of perturbations in smoothly varying semi-inner products, and show that the latter is a dynamical invariant similar to the coordinate-invariance of Lyapunov exponents. We show that contraction in particular weighted spaces verifies asymptotic convergence to subspaces and submanifolds, present applications to limit-cycle analysis and phase-locking phenomena, and pose general conditions for inheritance of contraction properties within coupled systems. We discuss contraction rates in Sobolev spaces for retention of regularity in partial differential equations, and suggest a type of weak solution defined by a vanishing contractive term. Lastly, we present an application to machine learning, using weighted semi-inner products to derive stability conditions for functional gradient descent in a Banach space.