First Variation: computing the first variation, Euler-Lagrange equation, extensions; Applications: brachistochrone, Lagrangian and Hamiltonian dynamics; Second Variation: computing the second variation, Ricatti equation, convexity and minimisers; Multivariable Variational Problems: eigenpairs, minimal surfaces, gradient flows; Optimal Control Theory: time-optimal linear control, Pontryagin Maximum Principle; Applications: linear-quadratic regulator, production-consumption, optimal harvesting; Dynamic Programming: Hamilton-Jacobi-Bellmann equation, general linear-quadratic regulator; Further Topics on Differential Games, Stochastic Control Theory.
For further information see the academic catalog: IAM773 - Dynamic Optimisation - Calculus of Variations and Optimal Control
Course Objectives
At the end of the course, the student will learn:
- the basics of the Calculus of Variations and its Applications
- the Theory of Optimal Controls and its Applications
- Pontryagin Maximum Principle, Dynamic Programming and Hamilton-Jacobi-Bellmann Equation
Course Learning Outcomes
Student, who passed the course satisfactorily will be able to:
- how to calculate the first and second variations
- how to approach and solve basic problems using calculus of variations
- understand the basics of the theory of Optimal Control
- how to approach and solve basic problems in Optimal Control
Instructional Methods
The following instructional methods will be used to achieve the course objectives: Lecture, questioning, discussion, group work, simulation.
Tentative Weekly Outline
- First Variation and its Applications (1 – 3 weeks)
- Second Variation and its Applications (4 – 5 weeks)
- Multivariable Variational Problems (6 – 7 weeks)
- Optimal Control Theory, Pontryagin Maximum Principle and Applications (8 – 10 weeks)
- Dynamic Programming and Hamilton-Jacobi-Bellmann equation (11 – 12 weeks)
- Further Topics on Differential Games, Stochastic Control Theory (13 – 14 weeks)
Course Textbook(s)
- Mathematical Methods for Optimization – dynamic optimization, by L. C. Evans (2021).
- Classical Mechanics with Calculus of Variations and Optimal Control – an intuitive introduction, by M. Levi (2014).
Course Material(s) and Reading(s)
Books (Textbook):
- An Introduction to Mathematical Optimal Control Theory, version 0.2, by L. C. Evans.
- Dynamic Optimization – the Calculus of Variations and Optimal Control in Economics and Management, 2nd edition, by M. I. Kamien and N. L. Schwartz (1991)
Resources:
- python: https://www.python.org/
- Anaconda: https://www.anaconda.com/
