In Control theory from the geometric viewpoint by Agrachev and Sachkov, the authors mention the concept of control-affine (affine in control $u_i$) systems: $$\dot{x} = (f_0 + \sum_{i=1}^{m} u_i f_i)x $$

I would like to have some sources on the topic, preferably a survey of the field, overview or a textbook. My primary questions are:

  • What are main results?
  • When can a CA system be linearized?
  • What are specific criteria for controllability, observability, optimality of control?
  • Other generalizations of results for linear systems.
  • Some good examples.

4 Answers 4


You may want to take a look at the book "Nonlinear Control Systems" by Alberto Isidori. This book deals with control-affine systems.

Regarding controllability, strong statements can be made for the "drift free" case ($f_{0}\equiv 0$) such as the Chow-Rashevsky theorem. The controllability issues for general case are more delicate and topics of on-going research.


Try the online ICTP lecture notes by Witold Respondek here: http://users.ictp.it/~pub_off/lectures/vol8.html Those notes provide a very nice introduction to feedback linearization for CA systems.

The notes by Bronislaw Jakubczyk treat controllability for nonlinear systems but the focus is on CA systems.

Respondek's notes also treat nonlinear observability, but the final word about nonlinear observability is the book "Deterministic Observation Theory" by Jean-Paul Gauthier and Ivan Kupka.

The texts by Agrachev-Sachkov and Jurdjevic treat the topic of optimal control rather carefully; some of the computations related to optimal controls become easier for CA systems but the difference isn't much and the theory doesn't change significantly.


For controllability issues and many good examples (in a setting even more general than control affine systems), I suggest you the following book:

Jurdjevic, Velimir, Geometric control theory, Cambridge Studies in Advanced Mathematics. 52. Cambridge: Cambridge Univ. Press. xviii, 492 p. (1997). ZBL0940.93005.

I also suggest you to have a look at:

Coron, Jean-Michel, Control and nonlinearity., Mathematical Surveys and Monographs 136. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-3668-2/hbk). xiv, 426 p. (2007). ZBL1140.93002.

  • $\begingroup$ Thank you. My question is based on a similar book, and I'd like to have more specific sources for contol-affine systems. $\endgroup$ Commented Jul 18, 2018 at 9:53

For some examples of driftless control-affine systems, you can see this paper "The geometry of Lie algebroids and applications to optimal control" https://arxiv.org/abs/1302.5212

  • 1
    $\begingroup$ First, you should have pointed out that the article is authored by you. Second, after having taken a look ay it, it is not clear how it answer the paper: only a small part of it is dedicated to a particular case of control-affine systems (the majority of the article being about the geometry of Lie algebroids), and this small part does not address the OP's questions. I claim that this post shameless self-promotion, and should therefore be deleted. $\endgroup$
    – Alex M.
    Commented Dec 23, 2018 at 14:01
  • $\begingroup$ Let's give the OP a chance to answer the question; "shameless self-promotion" might be premature. The remark about link-only pages is generally valid, although less likely in the case of the arXiv. $\endgroup$
    – Todd Trimble
    Commented Dec 23, 2018 at 14:38

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