Fall 2021

Zoom Meeting ID: 851 3588 3710

No password

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

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.

Students should become familiar with and are expected to adhere to the Code of Behaviour on Academic Matters.

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.