*"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:**

- Homework 1 (Exercises 1.1-1.7 in textbook Chapter 1) is due in class on Monday, Feb. 5
- Homework 2 (Exercises 2.1-2.7 in textbook Chapter 2) is due in class on Wednesday, Feb. 14
- Homework 3 (Exercises 2.8-2.14 in textbook Chapter 2) is due in class on Monday, Feb. 26
- Homework 1 resubmission is due in James's office hour on Tuesday, Feb. 27
- Homework 2 resubmission is due in James's office hour on Tuesday, Mar. 13
- Homework 4 (Exercises 3.1-3.8 in textbook Chapter 3) is due in class on Wednesday, Mar. 14
- Homework 3 resubmission is due in James's office hour on Tuesday, Apr. 3
- Homework 5 (Exercises 4.1, 4.3, 4.4, 4.7, 4.8 in textbook Chapter 4) is due in class on Wednesday, Apr. 4
- Homework 4 resubmission is due in James's office hour on Tuesday, Apr. 10
- Homework 6 (Exercises 4.10, 4.11, 4.12 in textbook Chapter 4) is due in class on Wednesday, Apr. 11
- Homework 7 (Exercises 5.1, 5.2 in textbook Chapter 5) is due in class on Wednesday, Apr. 18
- Homework 8 (Exercises 5.3, 5.5, 6.1 in textbook) is due in class on Wednesday, Apr. 25
- Homework 9 (Exercises 6.2, 6.4 in textbook) is due on Friday, May 4 at noon (You should submit your homework to CSL 355. If no one is there, you can put your homework under the door.)
- Homework 5-8 resubmission is due on Friday, May 4 at noon (You should submit your homework to CSL 355. If no one is there, you can put your homework under the door.)
- Final exam is due on May 10th at noon. I’ll collect the exam on May 10th during 10 am-12 pm at CSL 360. When submitting the exam, you should also send me an electronic copy (please scan your solution if it is handwritten) by email.

**Lecture notes and other materials:**

Lecture notes (preliminary copy of the textbook)

**Schedule: **Mon Wed 9:30-10:50pm, 3013 ECE Building.

**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:** Xiaobin Gao

Office: 360 CSL

Email: xgao16@illinois.edu

Office hours: Monday 11:00 am - 12:00 pm at ECEB 5034

**Homework TA:** James Schmidt

Email: ajschmd2@illinois.edu

Office hours: Tuesday 4:30 pm - 5:30 pm at CSL 141

**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 homeworks (**50%** of the final grade) and a final exam (**50%** of the final grade). Each homework problem can be resubmitted **once**.

**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 (6 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 (2.5 weeks)

Dynamic programming; sufficient conditions for optimality; viscosity solutions of the HJB equation.

5. LQR problems (1.5 week)