Control theory and splitting methods

TL;DR

This paper explores the deep connections between numerical splitting methods and control theory, revealing new insights into the order of methods and controllability using Lie algebra tools.

Contribution

It introduces a control-theoretic interpretation of splitting methods, proving the existence of high-order schemes with complex coefficients and linking order restrictions to Lie brackets.

Findings

Existence of arbitrary order schemes with complex coefficients for splitting methods.

Lie algebra rank condition is equivalent to small-time local controllability with complex controls.

Order restrictions are linked to 'bad' Lie brackets obstructing controllability.

Abstract

Our goal is to highlight some deep connections between numerical splitting methods and control theory. We consider evolution equations of the form $\dot{x} = f_0(x) + f_1(x)$, where $f_0$ encodes non-reversible dynamics, motivating schemes that involve only forward flows of $f_0$. In this context, a splitting method can be interpreted as a trajectory of the control-affine system $\dot{x}(t)=f_0(x(t))+u(t)f_1(x(t))$, associated with a control $u$ that is a finite sum of Dirac masses. The goal is then to find a control such that the flow generated by $f_0 + u(t)f_1$ is as close as possible to the flow of $f_0+f_1$. Using this interpretation and classical tools from control theory, we revisit well-known results on numerical splitting methods and prove several new ones. First, we show that there exist numerical schemes of arbitrary order involving only forward flows of $f_0$, provided one allows complex coefficients for $f_1$. Equivalently, for complex-valued controls, we prove that the Lie algebra rank condition is equivalent to small-time local controllability. Second, for real-valued coefficients, we show that the well-known order restrictions are linked to so-called "bad" Lie brackets from control theory, which are known to obstruct small-time local controllability. We investigate the conditions under which high-order methods exist, thanks to a basis of the free Lie algebra that we recently constructed.