Contracting differential equations in weighted Banach spaces

TL;DR

This paper generalizes geodesic contraction analysis to infinite-dimensional Banach spaces using weighted semi-inner products, enabling fixed point existence and applications to nonlinear PDEs and control systems.

Contribution

It introduces a novel framework for contraction analysis in weighted Banach spaces, extending finite-dimensional concepts to infinite-dimensional PDEs and control applications.

Findings

Negative contraction rates imply asymptotic norm-contraction.

Contraction in weighted spaces leads to convergence to subspaces, submanifolds, or limit cycles.

Method for constructing weak solutions via vanishing Lipschitz approximations.

Abstract

Geodesic contraction in vector-valued differential equations is readily verified by linearized operators which are uniformly negative-definite in the Riemannian metric. In the infinite-dimensional setting, however, such analysis is generally restricted to norm-contracting systems. We develop a generalization of geodesic contraction rates to Banach spaces using a smoothly-weighted semi-inner product structure on tangent spaces. We show that negative contraction rates in bijectively weighted spaces imply asymptotic norm-contraction, and apply recent results on asymptotic contractions in Banach spaces to establish the existence of fixed points. We show that contraction in surjectively weighted spaces verify non-equilibrium asymptotic properties, such as convergence to finite- and infinite-dimensional subspaces, submanifolds, limit cycles, and phase-locking phenomena. We use contraction rates in weighted Sobolev spaces to establish existence and continuous data dependence in nonlinear PDEs, and pose a method for constructing weak solutions using vanishing one-sided Lipschitz approximations. We discuss applications to control and order reduction of PDEs.