The convergence problem for dissipative autonomous systems: classical methods and recent advances

TL;DR

This paper reviews classical and recent methods addressing the convergence problem in dissipative autonomous systems, emphasizing developments based on the Łojasiewicz gradient inequality and connecting foundational theories with recent advances.

Contribution

It provides an updated synthesis of classical and recent approaches to the convergence problem, including new developments in infinite-dimensional dynamics and the Łojasiewicz gradient inequality.

Findings

Enhanced understanding of convergence in dissipative systems

Integration of recent advances with classical methods

Clarification of the connection between background theory and recent research

Abstract

The initial motivation of this text was to provide an up to date translation of the monograph [45] written in french by the first author, taking account of more recent developments of infinite dimensional dynamics based on the {\L}ojasiewicz gradient inequality. In order to keep the present work within modest size bounds and to make it available to the readers without too much delay, we decided to make a first volume entirely dedicated to the so-called convergence problem for autonomous systems of dissipative type. We hope that this volume will help the interested reader to make the connection between the rather simple background developed in the french monograph and the rather technical specialized literature on the convergence problem which grew up rather fast in the recent years.