Constructive Proof of Global Lyapunov Function as Potential Function

TL;DR

This paper establishes a constructive proof linking global Lyapunov functions in engineering with potential functions in physics, opening new avenues for constructing Lyapunov functions for nonlinear systems.

Contribution

It provides a formal proof of the equivalence between Lyapunov and potential functions and connects the Lyapunov equation to the Einstein relation for linear systems.

Findings

Proves the equivalence of Lyapunov and potential functions.

Suggests new methods for constructing Lyapunov functions for nonlinear systems.

Connects Lyapunov equation to Einstein relation in linear systems.

Abstract

We provide a constructive proof on the equivalence of two fundamental concepts: the global Lyapunov function in engineering and the potential function in physics, establishing a bridge between these distinct fields. This result suggests new approaches on the significant unsolved problem namely to construct Lyapunov functions for general nonlinear systems through the analogy with existing methods on potential functions. In addition, we show another connection that the Lyapunov equation is a reduced form of the generalized Einstein relation for linear systems.