ECE 557: Geometric Control Theory Sp 2023

Teaching Staff

  • M.-A. Belabbas, Office: CSL 166, email:
  • Office hours: Tuesday 12.30PM to 1.30PM or by appointment.

Meeting times

The class meets Tuesday and Thursday 11:00am to 12:20pm in ECEB 3015

We are on Piazza.

Find the lectures online: lectures will be recorded and posted online.


There will be roughly 4-5 homeworks assigned throughout the course. The homework are designed to help you verify and further your understanding of the material covered in the class, or of material not-covered.


  • There is no final exam nor midterms, but a final project with written report (4-8 pages). The goal of the project is to lead to publishable work, or learn something not covered in the lectures.


There are no mandatory textbooks. Lecture notes will be posted as we go along.


The official prerequisites for this course are ECE 515 (Control system theory and design) and ECE 528 (Nonlinear systems). In practice, a good understanding of linear algebra, multivariable calculus  and the willingness to do additional work on the way to cover possible gaps in your background should be enough. If you fall in this latter category, please discuss first with me.


Lecture notes will be posted here. Below, you can find the 2021 version that will be progressively updated.

Table of Contents:

  1. Review from Linear Systems
  2. Elements of Differential Geometry
  3. Elements of Riemannian Geometry
  4. Distributions, Frobenius Theorem, Chow-Rashevski theorem and Controllability
  5. Observability
  6. Degree theory, Feedback linearization and Brockett’s theorem
  7. Elements of SubRiemannian Geometry
  8. Control on Lie groups
  9. Elements of Geometric Mechanics
  10. Bilinear Systems
  11. Feedback Linearizations and feedback invariants
  12. Optimization on Manifolds.


  1. Week 1: Tuesday Jan 17: Course overview; Preliminaries.
    Thursday Jan 19: Preliminaries (Topology, differential equations)
  2. Week 2: Tuesday Jan 24: Review of linear system theory
    Thursday Jan 26: Controllability of linear systems. The evaluation map. Lie brackets.

Lecture Notes

  1. Preliminaries
  2. Review of Linear System Theory
  3. Controllability I
  4. Manifolds in R^n
  5. Frobenius Theorem
  6. Controllability II
  7. Riemannian Geometry I
  8. Differential Forms
  9. Lie groups
  10. Optimization and gradient flows
  11. Feedback stabilization and Brockett’s theorem
  12. Feedback invariants and feedback linearization
  13. Observability of nonlinear systems
  14. Riemannian Geometry II (p1)


  1. Robotics: Topological classification of singularities (F. Leve) – Topology of motion planning (Farber) – Soft Robotics (ask me)
  2. Quantum control: Time optimal control of spin systems (Khaneja-Brockett-Glaser)
  3. Subriemannian geometry: Singular Riemannian Geometry (Brockett) – SubRiemannian Geometry and vision , 2, (Boscain-Chertovskih-Gauthier-Remizov) Book
  4. Topological data analysis: Topology and data (Carlson);
  5. Nonholonomic motion planning (Summarize the general idea) Book (Available on Springer Link via University Library)
  6. Topological obstructions (1,2)
  7. Ensemble control (1,2)


The paper “Early days of geometric control”, by Brockett, is a good read on the historical development of the field. (for the published version, follow this link; you can access it if on campus or using the campus VPN)

The following books are helpful in understanding parts of the material covered.

  1. W. Boothby, Introduction to Differential Geometry and Lie Groups, Academic Press, N.Y., 1976.
  2. John Lee, Introduction to Smooth Manifolds, Springer, 2012 (Available online with UIUC Library account)
  3. Frank Warner, Foundations of Differentiable Manifolds, Springer, New York, 1983.
  4. Alberto Isidori Nonlinear Control Systems: An Introduction, Second Ed. Springer,New York, 1989.
  5. R. Abraham and J. Marsden, Foundations of Mechanics (Second Ed.) Addison- Wesley, Reading, Mass., 1979.
  6. V.I. Arnold, Mathematical Methods in Classical Mechanics, Springer-Verlag New York, 1989.
  7. Velimir Jurdjevic, Geometric Control Theory, Cambridge University Press, Cambridge, England, 1997.
  8. Anthony Bloch, Nonholonomic Mechanics and Control, Springer-Verlag NY, 2003
  9. R.W. Brockett, Finite Dimensional Linear Systems, J. Wiley, N.Y., 1970.
  10. Francesco Bullo and Andrew D. Lewis, Geometric Control of Mechanical Systems, Springer-Verlag,  2004  

Old notes

Lecture 1: Preliminaries and review from Linear Systems Theory. (One Column version)

Lecture 2: Introduction to Differential Geometry (One Column version)

Lecture 4: Controllability: Frobenius’ and Chow-Rashevski’s theorems (One Column version)

Lecture 6: Feedback stabilization and Brockett’s Theorem. (One Column version)

Lecture 3: Riemannian Geometry (One Column version)

Lecture 10: Feedback invariants and feedback linearization (One Column version)

Lecture 5: Observability (One Column version)

Lecture 11: Optimization on manifolds (One Column version)