ECE 553 Home Page

ECE 553: Optimum Control Systems (Spring 2022)

"Since the building of the universe is perfect and is created by the wisdom creator, nothing arises in the universe in which one cannot see the sense of some maximum or minimum." (Leonhard Euler)

This is a graduate-level course on optimal control systems. It presents a rigorous introduction to the theory of calculus of variations, the maximum principle, and the HJB equation. The course deals mainly with general nonlinear systems, but the linear theory will be examined in detail towards the end.


Lecture notes and other materials:

Lecture notes (preliminary copy of the textbook)

Recorded lectures from Fall 2020

Additional reading

Schedule: Tue Thu 2:00-3:20pm, 4070 ECE Building.

Prerequisites: ECE 515 (Linear Systems) or equivalent is the basic prerequisite. Previous or concurrent enrollment in ECE 528 (Nonlinear Systems) is desirable. Proficiency in mathematical analysis, at the level of Math 447 or a course where analysis is used (such as ECE 490 or 528), is also essential. Students with strong mathematical background (functional analysis, differential equations) but without prior knowledge of control theory are also welcome.

Instructor: Daniel Liberzon
Office: 144 CSL
Email: liberzon at
Office hours: see me after class

Homework TA: Guilherme Vicinansa
Email: gs16 at
Office hours: Mon 3:00-4:00 on Zoom

Text: D. Liberzon, Calculus of Variations and Optimal Control Theory: A Concise Introduction, Princeton University Press, 2012. ISBN 978-0-691-15187-8.
For other reserve materials for ECE 553 in Grainger library, search here by course number.

Assignments and grading policy: There will be homework and a final exam. Details will be explained in class.

Brief course outline:

1. Introduction (1.5 weeks)
The goals of the course; path optimization vs. point optimization; basic facts from finite-dimensional optimization.

2. Calculus of variations (3 weeks)
Examples of variational problems; Euler-Lagrange equation; Hamiltonial formalism and Legendre transformation; mechanical interpretation; constraints; second variation and Legendre's necessary condition; weak and strong extrema; conjugate points and sufficient conditions.

3. The maximum principle (5 weeks)
Statement of the optimal control problem; variational argument and preview of the maximum principle; statement and proof of the maximum principle; relation to Lie brackets; bang-bang and singular optimal controls.

4. Hamilton-Jacobi-Bellman equation (2 weeks)
Dynamic programming; sufficient conditions for optimality; viscosity solutions of the HJB equation.

5. LQR problems (1 week)

6. Other topics (time permitting)