The Kurdyka-{\L}ojasiewicz-Simon inequality and stabilisation in nonsmooth infinite-dimensional gradient systems

TL;DR

This paper establishes a stabilisation result for solutions of abstract gradient systems with nonsmooth energy functions in infinite-dimensional Hilbert spaces, relaxing previous assumptions and broadening applicability.

Contribution

It introduces a new stabilisation theorem for nonsmooth infinite-dimensional gradient systems, simplifying conditions and extending applicability to a wider class of energies.

Findings

Stabilisation results hold under relaxed range assumptions.

Applicable to both smooth and nonsmooth energy functions.

Enhances understanding of infinite-dimensional gradient dynamics.

Abstract

We state and prove a stabilisation result for solutions of abstract gradient systems associated with nonsmooth energy functions on infinite dimensional Hilbert spaces. One feature is that in this general setting the assumption on the range of the solution can be considerably relaxed, which considerably simplifies the applicability of the stabilisation result even in the case of smooth energies.