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UCSB “Contraction Theory for Dynamical Systems”
ME 225FB, Winter 2026
Instructor: Francesco Bullo
This is the website for the UCSB course ME 225FB “Contraction Theory for Dynamical Systems”,
Winter 2026. This website's URL is
http://motion.me.ucsb.edu/ME225FB-Winter2026.
Course Description
The course provides a mathematical introduction to contraction theory for
dynamical systems. Contractivity is a robust and computationally-friendly
stability theory. Special emphasis is given to continuous-time differential
equations arising in the study of network multi-agent systems, neural
networks, gradient dynamical systems, monotone dynamics, and
semi-contracting systems. The course is intended primarily for graduate
students interested in nonlinear dynamical systems, control, computation
and optimization.
Prerequisites
Competency in linear algebra, nonlinear dynamical systems, and linear control systems.
Lecture Time and Place
Mon+Wed 2pm-3:15pm in ENGR II 2243 (ME Classroom)
Due to the closure of Engineering Building II, the first three lectures
(Jan Mon 5th, Wed 7th, and Mon 12th) will be held in ESB 2003, concurrent
with a research group meeting by Professor Bullo. (We will make
arrangements for Wed 14th)
Course credit
Units: 3. Letter grade.
Textbook
The course will follow the text: https://fbullo.github.io/ctds
and the slides: https://fbullo.github.io/talks/2023-09-FBullo-ContractionTheory-Tutorial.pdf
Instructor
Professor Francesco Bullo
Department of Mechanical Engineering
Email: bullo-at-ucsb.edu
Website: https://fbullo.github.io
Office hours = Discussion times
Place: Professor Bullo's zoom room (I might simply be in my office anyway, room EII 2325 )
Time: Tuesdays 3:30pm-5pm
Please be sure to turn on your video camera when you come to my zoom room,
as I will have mine on. I encourage you to use a background.
If you have any questions about the course, please send me email. I will
try to respond as quickly as possible. Additionally, I will share
questions that are particularly good (and their answers) with the rest of
the class by broadcasting my answer to the entire class.
If you come to office hours with questions about homework, please be
prepared to show attempts at solving the problem prepared before coming.
Grading
Class grade will be determined by:
In exceptional cases, I reserve the right to give extra points for
excellent performance on the final report and presentation. Please, do not
count on it as a way to avoid doing homework assignments.
Homework Self-Grading Scheme
Homework is due one week after the corresponding chapter is fully covered in class; precise deadlines will be announced in class.
Please write one exercise per page and write clearly for partial credit.
To turn in the homework, please scan solutions in a single PDF file and email me the file by the deadline.
After the homework deadline, I will email you the solutions (after which no late homework will be accepted).
I will ask you to please self-grade your homework and
email to me both: (1) the marked homework in PDF form, and (2) the grade.
Please self-grade within a week of receiving the solutions.
Each exercise in each homework assignment is worth 10 points, unless otherwise marked.
Final Report and Final Presentations
Instructions on the report and presentation will be discussed in class.
Here are some
instructions in the file Instructions-FinalPresentationsReport-ME225FB-Winter26.pdf.
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