\documentclass[12pt,a4paper,draft]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsxtra}
\usepackage{latexsym}
\setlength{\topmargin}{20mm}\setlength{\headheight}{12pt}\setlength{\headsep}{18pt}%
\setlength{\textwidth}{16.5cm}\setlength{\textheight}{25cm}\setlength{\marginparwidth}{0mm}%
\parindent0.8cm\parskip0cm\voffset-1.4in\hoffset-0.6in
\newcounter{N}
\begin{document}
\noindent%
\textbf{Combinatorics and algebra of series expansions in nonlinear control}
\vskip2mm \noindent%
\textbf{M.Kawski}%
\vskip5mm

In the absence of closed-form explicit integrals, series expansions
are one of the most useful tools for the analysis of systems that
involve noncommuting flows. They are used in essential ways for
studying controllability and optimality, and more generally for
trajectory analysis and approximation, for designing tracking
controls, but they also are of fundamental importance outside
control theory, for example in numerical analysis for integration
of differential equations with algebraic constraints.
%
We will start with briefly comparing and contrasting a variety
of such series expansions, such as the Magnus series, Volterra
series, Chen-Fliess series and product expansions thereof, as
well as the chronological calculus of Agrachev and Gamkrelidze.
While all are closely related, they serve different purposes
in different settings.

With the advent of modern high performance computing, it also
has become more important to have effective implementations for
calculating higher order terms taking advantage of large numbers
of redundancies among the terms.
This leads to problems of algebraic and combinatorial nature,
involving e.g. the Hopf algebras of rooted trees and of labelled
binary trees, as well as bases for free Lie algebras, chronological
algebras, and Zinbiel algebras.
We will discuss the respective advantages and shortcomings of the
various formulations, how they map to each other, and their
geometric interpretations.

Explicit exponential product expansions of the Chen-Fliess series
relying on Hall bases were introduced into control over twenty
years ago, and have proven extremely useful since.
It is clear from first principles that one alternatively may
write such series as an exponential of a single Lie series,
which appears very attractive from both a computational point
of view as well as for the analysis and design of nonlinear
control systems.
But in spite of numerous efforts by several authors,
variously known under names such as
{\em continuous Campbell Baker Hausdorff formula} or
{\em coordinates of the first kind},
none of the proposed algorithms yield formulas that
are anywhere close to the extreme elegance and simplicity of
the formulas for the coordinates of the second kind, which
using the Zinbiel product may be written as:
$\xi_{HK}=\xi_H\star \xi_K$ for Hall words $H,K,HK$.
We have traced this difficulty to the structure of the commonly
used Hall bases for free Lie algebras which by construction,
based on Lazard elimination, are perfect matches for coordinates
of the second kind only.
We present some alternative strategies and report on preliminary
results which show possible alternatives.

Finally, we will point out how these algebraic and combinatorial
objects map to the geometric and analytic properties of control
systems.
One of the most intriguing observations is that while in control
we naturally consider noncommuting flows to lead to Lie algebras,
algebraically the formulas appear to live more naturally in a
Leibniz algebra -- that is, simply speaking, a Lie algebra without
the anticommutativity requirement.
Geometrically, this is related to connections with not necessarily
zero torsion.
Further specializations allow various simplifications -- special
cases include systems that are affine in the control as a subclass
of fully nonlinear systems, or second order models as they arise
naturally in mechanical systems where certain symmetric products
are a popular choice for working with affine connections.
\vskip 7mm

\noindent%
\footnotesize{%
Department of Mathematics and Statistics, \\
Arizona State University. \\
Tempe, Arizona 85287-1804, USA. \\
\ttfamily http://math.asu.edu/$\tilde{\;}\;$kawski  \\
kawski@asu.edu}
\end{document}