A Lanczos-like method for non-autonomous linear ordinary differential equations

TL;DR

This paper introduces a *-Lanczos algorithm based on generalized Krylov subspaces for efficiently approximating the time-ordered exponential of non-autonomous linear differential equations, a key challenge in system dynamics and control.

Contribution

It presents a novel *-Lanczos algorithm derived from generalized Krylov subspaces, with proven properties and a proposed numerical implementation strategy.

Findings

Proves the matching moment property of the *-Lanczos algorithm.

Provides a framework for numerical implementation of the algorithm.

Addresses a longstanding challenge in evaluating the time-ordered exponential.

Abstract

The time-ordered exponential is defined as the function that solves a system of coupled first-order linear differential equations with generally non-constant coefficients. In spite of being at the heart of much system dynamics, control theory, and model reduction problems, the time-ordered exponential function remains elusively difficult to evaluate. The *-Lanczos algorithm is a (symbolic) algorithm capable of evaluating it by producing a tridiagonalization of the original differential system. In this paper, we explain how the *-Lanczos algorithm is built from a generalization of Krylov subspaces, and we prove crucial properties, such as the matching moment property. A strategy for its numerical implementation is also outlined and will be subject of future investigation.