Abstract.
Since the work of Fliess in the 70s, Lie algebraic techniques have become standard tools for investigating (analytic, finite dimensional) nonlinear control systems. On the other hand, it has become apparent in recent years that the geometric foundations of such systems may be described by an even more basic algebraic structure, namely chronological, or Leibniz, algebras.
The name "chronolgical algebra" in a control context is due to the fundational work of the late 70s by Agrachev and Gamkrelidze. More recently, the same algebraic structures have been investigated in the algebraic community, where the term "Leibniz algebra" is used.

In this presentation we exhibit what is gained by working with chronological, rather than Lie algebras. We also make an effort to relate some of the recent work by algebraists to the context of nonlinear control.
Principal applications in control are normal forms for nilpotent (approximating) systems, and associated computationally effective algorithms for path-planning and related problems. On a more theoretical level, this work has consequences for both nonlinear controllability and high order conditions for optimality.

This talk is an outgrowth of joint work with Sussmann and Melancon. The purely algebraic work discussed is primarily due to Loday.

For further information please contact: Matthias Kawski.