*"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:**

- Click here for final exam info.
- The current plan is that this course will be taught in-person only (no online option). Of course we will follow all university regulations and Covid protocols. So, during the first week classes will be taught on Zoom, and then we'll switch to regular classes as soon as the university allows it. I will email the Zoom link to all registered students a day or so before the first class (please email me if you don't receive the Zoom link by Tuesday morning). Video recordings of lectures from a previous semester are also available (see below) for those of you who might be occasionally unable to attend class.

**Lecture notes and other materials:**

Lecture notes (preliminary copy of the textbook)

Recorded lectures from Fall 2020

**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 illinois.edu

Office hours: see me after class

**Homework TA: **Guilherme Vicinansa

Email: gs16 at illinois.edu

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)