Contraction Theory for Dynamical Systems on Hilbert Spaces

TL;DR

This paper extends contraction theory to infinite-dimensional Hilbert spaces, providing new integral conditions for stability and applying the results to reaction-diffusion systems.

Contribution

It introduces contraction concepts for Hilbert space systems, offering novel integral criteria and stability results, expanding the applicability of contraction theory.

Findings

Established a new integral condition for contractivity in Hilbert spaces.

Proved existence of a unique globally exponentially stable equilibrium for time-invariant systems.

Applied the theory to a reaction-diffusion system demonstrating practical relevance.

Abstract

Contraction theory for dynamical systems on Euclidean spaces is well-established. For contractive (resp. semi-contractive) systems, the distance (resp. semi-distance) between any two trajectories decreases exponentially fast. For partially contractive systems, each trajectory converges exponentially fast to an invariant subspace. In this note, we develop contraction theory on Hilbert spaces. First, we provide a novel integral condition for contractivity, and for time-invariant systems, we establish the existence of a unique globally exponentially stable equilibrium. Second, we introduce the notions of partial and semi-contraction and we provide various sufficient conditions for time-varying and time-invariant systems. Finally, we apply the theory on a classic reaction-diffusion system.