*"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.

**Announcements:**

- This course will be taught online. Details will be discussed during the first lecture on Tue Aug 25. Lectures will be recorded, but real-time participation is strongly encouraged.

**Lecture notes and other materials:**

Lecture notes (preliminary copy of the textbook)

**Schedule: **Tue Thu 2:00-3:20pm, online. Zoom link will be emailed to registered students prior to the first meeting. If you are not registered (yet), please contact the instructor by email.

**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 illinois.edu

Office hours: after class or by appointment

**Homework TA: **Guilherme Vicinansa

Email: gs16 at illinois.edu

Office hours: Wed 4:15-5:15pm, Zoom link sent to registered students by email

**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)