Lie Groups and Lie Algebra for Nonlinear System Dynamics

Question

I have a background in machine learning and control theory. Lately, I shifted my focus towards nonlinear systems. I am interested in rigorous mathematical analysis, stability, and control of nonlinear dynamical systems.

I have a solid mathematical background in measure theory, functional analysis, and ergodic theory.

To be more specific, I am looking for some reference that:

1. Introduces Lie Groups and Lie Algebra.

2. Gives a special treatment for their applications in nonlinear systems.

I am citing the following article as a motivation

$[1]$ Chein-Shan Liu, A Lie-group DSO(n) method for nonlinear dynamical systems, Applied Mathematics Letters, Volume 26, Issue 7, 2013, Pages 710-717, ISSN 0893-9659, https://doi.org/10.1016/j.aml.2013.01.012. (https://www.sciencedirect.com/science/article/pii/S0893965913000530)

Answer 1

If you're interested in control aspects of nonlinear systems you could have a look into the second edition of Sontag's "Mathematical Control Theory" (1998): in Chapter 4 Sontag treats non-linear control systems and how to study them via Lie theory.

Admittedly, this book is not an introduction to Lie groups & Lie algebras but I think it very much satisfies your second criterion which is why I felt inclined to mention it here.

Answer 2

One possibility is Peter Olver's Applications of Lie Groups to Differential Equations (Graduate Texts in Mathematics 107, Springer-Verlag, 1986).

Answer 3

The book "Nonlinear Dynamical Control Systems" by Henk Nijmeijer and Arjan van der Schaft use these concepts in the context of control. This might be a useful reference for you as well, next to the given ones.

link: https://link.springer.com/book/10.1007/978-1-4757-2101-0

Answer 4

Another good reference is

Francesco Bullo and Andrew Lewis, Geometric Control of Mechanical Systems (2005).

• For the first part of your question, see in particular Chapter 5 (Lie groups, systems on groups, and symmetries) including Section 5.2 (Lie groups and Lie algebras).
• For the second part of your question, see Chapter 6 (Stability) and Chapter 7 (Controllability).