### Teaching Staff

- M.-A. Belabbas, Office: CSL 166, email:
`belabbas@illinois.edu`

- Office hours: on Zoom (same link as for lectures) , Thursday 330PM to 430PM or by appointment.

### Meeting times

The class meets Tuesday and Thursday 2:00pm to 3:20pm on Zoom. Link

We are on Piazza

Find the lectures online: subscribe to our channel

### Homeworks

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.

Homework 1: Do Exercise 2.8 to 2.12 in the notes.

### Exams

- There is no final exam nor midterms, but a final project with written report (4-8 pages) and oral presentation (30 mins). The goal of the project is to lead to publishable work, or learn something not covered in the lectures.

### Textbook

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

### Prerequisistes

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.

### Notes

Lecture notes will be posted here. They are a work in progress, so please check back for updated versions.

*Table of Contents:*

- Review from Linear Systems
- Elements of Differential Geometry
- Elements of Riemannian Geometry
- Distributions, Frobenius Theorem, Chow-Rashevski theorem and Controllability
- Observability
- Degree theory, Feedback linearization and Brockett’s theorem
- Elements of SubRiemannian Geometry
- Control on Lie groups
- Elements of Geometric Mechanics
- Bilinear Systems
- Feedback Linearizations and feedback invariants
- Optimization on Manifolds.

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)

- Week 1: Lecture 1: Overview of the course and discussion of logistics

Lecture 2: Review of linear systems (fundamental solution, Peano-baker Series, Variation of constants formula) - Week 2: Lecture 1: Controllability for linear systems; Intro to differential geometry: definition of manifold. (Reading: [2], chapter 1-3) Notes

Lecture 2: Intro to differential geometry: abstract manifold, tangent space. Notes - Week 3: Lecture 1: Vector fields, tangent space and Lie derivatives. Notes

Lecture 2: Lie derivatives, Controllability (Frobenius Theorem) Notes - Week 4: Lecture 1: Controllability (Frobenius theorem)

Lecture 2: Controllability: Frobenius Notes - Week 5: Lecture 1: Project discussion and Controllability Notes

Lecture 2: Controllability: Chow-Rashevski Theorem Notes - Week 6: Lecture 1: Chow’s theorem Notes — Feedback control (Topological preliminaries) Notes

Lecture 2: Brockett Theorem Notes - Week 7: Lecture 1: Cotangent space and exterior forms Notes

Lecture 2: Exterior algebra and Cartan Calculus Notes - Week 8: Lecture 1: Cartan Calculus Notes

Lecture 2: Riemannian geometry P1 Notes - Week 9: Lecture 1: Riemannian geometry P2 (Length, Energy, geodesics) Notes

Lecture 2: Riemannian geometry P3 (geodesics, connections) Notes - Week 10: Lecture 1: Riemannian geometry p4 (Connections and holonomy) Notes

Lecture 2: Feedback invariants and feedback linearization Notes - Week 11: Lecture 1: Feedback invariants and feedback linearization Notes + Frobenius theorem redux Notes

Week 2: no lecture — project time! - Week 12: Lecture 1: No instruction day, which I learned of Tuesday at 2pm

Week 2: Frobenius theorem and differential ideals Notes + Feedback linearization Notes - Week 13: Lecture 1: Observability 1 Notes

Lecture 2: Observability codistribution Notes - Week 14: Lecture 1: Gradient flows on manifolds Notes

Lecture 2: Gradient flows on manifolds Notes

### Projects

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

### References

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.

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