Nonlinear Control and Combinatorics of Words
This article attempts to bring together some problems in nonlinear control systems and in algebraic combinatorics of words. One the control side recent results in controllability, nilpotent approximating systems, and applications to path planning are surveyed. On the other side, after a brief survey of various bases for free Lie algebras recent developments in the algebraic combinatorics of words are discussed. These two areas are linked together by the applications to and of the Chen series, Fliess series, and product expansions of these series, culminating in the introduction of chronological products. 0 Introduction This is a survey article about an area of overlap of nonlinear control theory and algebraic combinatorics. The unifying theme is the increasingly sophisticated treatment, analysis and usage of the Chen-Fliess series. On the control side we restrict our attention to the problems of local controllability, nilpotent approximations and very brieey, the path planning problem. On the combinatorial side we survey the development of the theory of bases for free Lie algebras, and then concentrate on the problem of resolving the identity map on some free algebra, which leads to the problem of nding explicit formulas for the dual bases of the Poincar e-Birkhoo-Witt basis built on bases of the free Lie algebra. In trying to be accessible to audiences of both control theorists and the combina-torists, most topics are motivated at an elementary level, and complete proofs are given only where they are particularly illuminating (mostly in the last sections which unite