Hierarchical Stability Notions and Lyapunov Functions for PDEs

TL;DR

This paper introduces a hierarchical framework for classifying stability notions and Lyapunov functions for PDEs, simplifying their interpretation by relating various stability concepts to a fundamental state.

Contribution

It proposes a novel hierarchical classification of stability notions for PDEs based on a fundamental state, unifying diverse Lyapunov stability concepts.

Findings

Hierarchy established for Lyapunov, exponential, and finite-energy stability

Sufficient Lyapunov conditions formulated via operator inequalities

Framework includes illustrative examples and computational tools

Abstract

Unlike linear ordinary differential equations (ODEs), linear partial differential equations (PDEs) admit a multitude of non-equivalent notions of stability. This variety makes interpretation of Lyapunov stability results challenging. To simplify this interpretation, we propose a framework for hierarchical classification of notions of stability and Lyapunov conditions. To do this, for every well-posed PDE and set of boundary conditions, we define a fundamental state on $L_2$ corresponding to the minimal information needed to uniquely forward propagate the solution. Stability notions and Lyapunov functions are then defined in terms of this fundamental state. This gives rise to a hierarchy of stability notions, the weakest being fundamental state to PDE state stability. Other stability notions and Lyapunov conditions may then be interpreted relative to this weakest notion. Hierarchies are established for: Lyapunov, exponential and finite-energy stability. Sufficient Lyapunov conditions are defined in terms of operator inequalities. Illustrative examples and computational tools are provided.