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ECE 553: Optimum Control Systems (Spring 2010)

"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.


Final exam questions are now posted. Exam date: Thursday, May 6 (reading day), in usual classroom (Everitt 241), 10am - 7pm (you will be assigned a more specific time slot)

Chapter 6 homework is due Thursday, April 22

All pending homework problems are due Tuesday, April 27

Updated version of Chapter 6 is now posted, and updated class notes (Chapters 1-6) will be sold in Everitt Lab room 243

A very rough draft version of Chapter 7 is now posted

Lecture notes and other materials:

Lecture notes (preliminary copy of a textbook to be published by Princeton University Press in 2011)

Final exam questions

Additional reading

Schedule: Tue Thu 12:30-1:50pm, room 241, Everitt Lab.

Prerequisites: ECE 515 (Linear Systems) or equivalent is required. 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.

Instructor: Daniel Liberzon
Office: 144 CSL
Email: liberzon at
Office hours: Thursdays 3:30-5:00pm (subject to change)

Text: Class Notes by the instructor, to be available for purchase in room 243, Everitt Lab starting on the first day of classes.
For other reserve materials for ECE 553 in Grainger library, search here by course number.

Assignments and grading policy: 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 (2 weeks)