Existence of complete Lyapunov functions with prescribed orbital derivative

TL;DR

This paper proves the existence of complete Lyapunov functions with prescribed orbital derivatives outside the chain-recurrent set, providing a theoretical basis for their numerical construction and further analysis.

Contribution

It establishes that complete Lyapunov functions can have their orbital derivatives prescribed in compact sets outside the chain-recurrent set, with smoothness matching the vector field.

Findings

Existence of Lyapunov functions with prescribed derivatives

Smoothness of Lyapunov functions matches the vector field

Foundation for numerical construction of Lyapunov functions

Abstract

Complete Lyapunov functions for a dynamical system, given by an autonomous ordinary differential equation, are scalar-valued functions that are strictly decreasing along orbits outside the chain-recurrent set. In this paper we show that we can prescribe the (negative) values of the derivative along orbits in any compact set, which is contained in the complement of the chain-recurrent set. Further, the complete Lyapunov function is as smooth as the vector field defining the dynamics. This delivers a theoretical foundation for numerical methods to construct complete Lyapunov functions and renders them accessible for further theoretical analysis and development.