On the approximability of Koopman-based operator Lyapunov equations

TL;DR

This paper investigates the approximation of Lyapunov functions for nonlinear systems using Koopman operator theory, establishing a connection between operator Lyapunov equations and finite-dimensional approximations with promising numerical results.

Contribution

It introduces a novel approach to approximate Lyapunov functions via operator Lyapunov equations in Koopman framework, enabling efficient finite-rank numerical methods.

Findings

Operator Lyapunov solutions exhibit rapid eigenvalue decay.

Finite rank approximations are justified and effective.

Numerical examples demonstrate practical benefits.

Abstract

Lyapunov functions play a vital role in the context of control theory for nonlinear dynamical systems. Besides its classical use for stability analysis, Lyapunov functions also arise in iterative schemes for computing optimal feedback laws such as the well-known policy iteration. In this manuscript, the focus is on the Lyapunov function of a nonlinear autonomous finite-dimensional dynamical system which will be rewritten as an infinite-dimensional linear system using the Koopman or composition operator. Since this infinite-dimensional system has the structure of a weak-* continuous semigroup, in a specially weighted $\mathrm{L}^p$-space one can establish a connection between the solution of an operator Lyapunov equation and the desired Lyapunov function. It will be shown that the solution to this operator equation attains a rapid eigenvalue decay which justifies finite rank approximations with numerical methods. The potential benefit for numerical computations will be demonstrated with two short examples.