# What are good sources for studying geometric control theory?

I am studying geometric control theory, and I am focusing on the concepts of manifolds, Lie algebra and Lie brackets, distributions and the Frobenius theorem.

My problem is that I am having a lot of difficulties on undestanding the concepts. I have tried searching different sources, but I still don't have these concepts clear.

Can somebody provide me some material where these concepts are explained clearly?

• The first pages of your book on geometric control should list the prerequisites. Commented Apr 5, 2020 at 11:57

There two-three standard books that everyone refers, which are:

(1) Agrachev, Sachkov, Control theory from geometric viewpoint.

(2) V.Jurdjevic, Geometric control theory.

(3) Jerzy Zabczyk, Mathematical Control theory.

But these are heavily mathematical which is not a surprise because the subject itself is. If you don't have any background on Topology, differential geometry then don't pick up the above books mentioned, as these books starts with compact lie groups and vector fields etc.

I am focusing on the concepts of manifolds, Lie algebra and Lie brackets, distributions and the Frobenius theorem

Keeping this in mind, i would suggest you to go through (1) $$\textit{Introduction to topological manifolds}$$ by John Lee, maybe chapter two and three, then (2)$$\textit{ Introduction to smooth manifolds}$$ by Lee where you'll find necessary background on smooth structures, tangent space, vector bundles, Lie groups and Lie abgebra etc. Another good reference is Lafontaine's differential manifolds. There's two other books which i would also recommend

(4) Abraham, Marsden, Foundations of mechanics

(5) Darryl Holm, Geometric mechanics part I

You can also start with any of these two to build up the background on geometry, topology and lie groups. Compared to Marsden (which is considered as bible of geometric mechanics and control) Holm's book is quite readable and precise specially the lie group and algebra, matrix lie group section, and full of examples but his treatment of abstract manifold is restricted to submanifolds, i.e consider the unit sphere $$\mathbb{S}^2$$ which is itself a manifold irrespective of the ambient space it's embedded on, while Holm's always consider it as something which is embedded in $$\mathbb{R}^3$$ but it won't be a big issue. Anyway, learning all the basics first is a dead machine, so it's better to start somewhere at the middle, move upwards as enlarging your base as you go along. Good luck !

If you are new to manifolds, Lie brackets, and Frobenius, read something more intuitive to start with. Not everyone loves Applied Differential Geometry by Bill Burke but I do. Manifolds and Differential Geometry by Jeffrey Lee is a good book, as are the texts by John Lee mentioned above.

Milnor's books on Differential Topology and on Morse Theory are mathematical and quite nice. Or you may want to read about general relativity if you are inclined to learn some physics as a bonus in the process.

I fear that if you try to learn differential geometry from the control literature, your progress will be slow and intuition incomplete. But that's just a very personal opinion.

Geometric control theory is a study of a family of tangent vector fields on a manifold. Therefore, the necessary background includes the concept of a manifold, tangent vector field, and tangent space. Arnol'd's "Ordinary differential equations" will give you the basic background on manifolds, tangent spaces, Lie bracket, and on the study of one vector field on a manifold. (The same author's "Mathematical methods of classical mechanics" gives a somewhat more comprehensive introduction to manifolds, the tangent bundle, and Lie groups.)

If you want an extremely thorough introduction to manifolds and tangent spaces, there is a chapter in the 2nd volume of Zorich's "Mathematical analysis", although he loves to use a lot of notation, so see if it's to your taste.

Once you have a solid grasp on the geometric theory of ODE (that's Arnol'd:), i.e., on the study of one tangent vector field on a manifold, I next recommend reading, not a book, but a paper: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.649.6659&rep=rep1&type=pdf This paper should help you see the high-level picture of geometric control theory and gauge what material (and, hence, what sources on differential geometry and geometric control) would be best to go to next.