Fall 2021

Time/location: Tue 3-5pm and Thu 4-5pm in BF 323 (Bancroft Building).
Instructor: Prof. Boris Khesin

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
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.

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.

MAT357H (recommended)/ MAT337H, MAT351Y (recommended)/APM346H, MAT267H (recommended)/ MAT244H.

PDF version of the course description