“Good Lie Brackets” for Control-affine Systems

*This event has been postponed*
On (date TBA), 2023:

Organized by: FAU DCN-AvH, Chair for Dynamics, Control, Machine Learning and Numerics – Alexander von Humboldt Professorship at FAU, Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany)
Title: “Good Lie Brackets” for Control-affine Systems
Speaker: Prof. Dr. Andrei Agrachev
Affiliation: SISSA

Abstract. We consider smooth systems of the form \dot x=f_0(x)+\sum\limits_{i=1}^ku_if_i(x),\ x\in M, and study controllability issues on the group \mathrm{Diff}M. There exists a satisfactory theory for control-linear systems \dot{x}=\sum\limits_{i=0}^ku_if_i(x). Namely, the system can arbitrarily well approximate the movement in the direction of any Lie bracket polynomial of f_0,\ldots,f_k. Moreover, the controls that approximate the movement in the desired direction depend only on the structure of the Lie bracket polynomial and not on f_i. Any Lie bracket polynomial is “good” in this sense.

In the case of a control-affine system, we are not able to go (approximately) in the direction of any Lie bracket polynomial but some Lie bracket polynomials are still fine. I am going to characterize Lie bracket polynomials that are good for any control-affine system, this leads to interesting objects in the free Lie algebras, which. I am grateful to Alexander Zuev and Khazhgali Kozhasov for very useful and stimulating conversations.

WHERE?

On-site / Online

[On-site] Room TBA.
Department Mathematik. FAU, Friedrich-Alexander-Universität Erlangen-Nürnberg
Cauestrasse 11, 91058 Erlangen(Germany)

[Online] Zoom meeting link
Meeting ID: 614 4658 159 | PIN code: 914397