Butcher series and control theory

TL;DR

This paper demonstrates how solutions to nonlinear differential equations can be expressed using Butcher series indexed by planar trees, and applies this to establish controllability of such systems when the nonlinear term is small.

Contribution

It introduces a novel approach to control nonlinear differential equations using Butcher series and provides explicit control expressions based on minimization of functionals.

Findings

Controllability of nonlinear systems is guaranteed if the linearized system is controllable and the nonlinear term is sufficiently small.

Explicit control functions are derived as sums over planar trees, facilitating practical control design.

The method bridges series expansion techniques with control theory for nonlinear systems.

Abstract

We show how solutions of a non--linear differential equation can be written as sum indexed by planar trees: the Butcher series. Then we use that property in order to control non--linear differential equation. We show that if the linearized system is controllable then the system itself is controllable if the nonlinear term is small enough and we express explicitly the control as a sum indexed by planar tree which each terms is obtained by minimization of a functional.