CONTROL THEORY
MAT1840HF/MAT482H1F
Fall 2021
Time/location: Tue 3-5pm and Thu 4-5pm in BF 323 (Bancroft Building).
Instructor: Prof. Boris Khesin
Email: khesin@math.toronto.edu
Office: BA 6228
Office hours: Tue and Thu 5-5:30pm or as long as needed in BF 323 and
Wed 10am - 12pm online at the address
https://utoronto.zoom.us/j/85135883710
Zoom Meeting ID: 851 3588 3710
No password
Course description:
The course focuses on the key notions of Calculus of Variations and Optimal Control Theory: key
examples of variational problems,
(un)constrained optimization, first- and second-order conditions, Euler-Lagrange equation,
variational problems with constraints, examples of control systems, the maximum principle, the
Hamilton-Jacobi-Bellman equation (time permitting),
holonomic and nonholonomic constraints, Frobenius theorem, Riemannian and sub-Riemannian geodesics.
Textbooks:
1) D. Liberzon ``Calculus of Variations and Optimal Control Theory: A Concise Introduction'' 2012, Princeton Univ. Press (chapters 1,2,3 and partially chapters 4,5,7); preliminary version
2) A. Agrachev, D. Barilari, and U. Boscain ``A Comprehensive Introduction to Sub-Riemannian Geometry'' 2019, Cambridge Univ. Press (chapter 2 and partially chapters 1,3);
preliminary version
Homework Assignments:
There will be 3-4 assignments and a final individual project,
which together constitute the full course mark.
No late assignments will be accepted.
Note: You must write your solutions yourself, in your own words. If your solution is
aided by information from textbooks or online sources, you must properly quote these
references.
Problem Set 1 (due Thursday, Oct.7)
Problem Set 2 (due Thursday, Oct.28)
Problem Set 3 (due Tuesday, Nov.23)
List of miniproject topics -- Problem Set 4 (due Dec. 10)
Code of Behaviour / Plagiarism:
Students should become familiar with and are expected to adhere to the
Code of Behaviour on
Academic Matters.
Course Outline:
The following is a tentative outline of the material to be covered.
Sep 9-16: Introduction: examples, (un)constrained optimization, Lagrange multipliers
first and second variations.
Sep 21 - Oct 7: Calculus of variations: examples (Dido problem, catenary, brachistochrone), weak and strong extrema, Euler-Lagrange equation,
introduction to Hamiltonian formalism, integral and non-integral constraints.
Oct 12-14: From calculus of variations to optimal control: control system, cost functional, target set.
Oct 19 - Nov. 4: The maximum principle: statement, ideas of proof, weak form, examples.
Nov 16 - Dec 8: Lie brackets of vector fields, Frobenius theorem, nonholonomic constraints, examples (ball rolling, car parking), Chow-Rashevskii theorem, sub-Riemannian metrics.
Prerequisites:
MAT357H (recommended)/ MAT337H,
MAT351Y (recommended)/APM346H, MAT267H (recommended)/ MAT244H.
PDF version of the course description