ENEE 664 - Optimal Control

ENEE 664 - Optimal Control

Spring 2012

Last Update Saturday April 14, 2012

Course Information

Special Announcements

(19) Homework 8 posted. Problems 1-4 due Thursday April 19. Problem 5 due Tuesday April 24.

(18) IMPORTANT: MID-TERM EXAM II (closed book) will be held on Tuesday, May 1.

(17) Solutions to Homework Set 6 emailed. Homework Set 7 posted.

(16) Addendum to Lecture 7 posted (on transversality condition)

(15) Update to Lecture Notes 5(a) posted - page number 5 of the notes replaced. The change is the norm on C[0,1] is the infinity norm. Otherwise in the exercise stated on page 4 the claim that the Gateaux differential formula given is also the Frechet differential formula is not true. In the L_1 norm you do not get Frechet differentiability (take g(t, x) = x^2 as a counter-example - thanks to your classmate Gulcu). Accordingly in Homework Set 6, problem 1, work with the infinity norm. Sorry for the error. Remember to download the revised last page for Lecture Notes 5(a).

(14) Update to Lecture Notes 5(c) posted - cleaned up proof of key result that leads up to Lagrange mutiplier theorem

(13) Homework Assignment 6 uploaded (also on Thursday March 15 Homework Assignment 7 will be posted)

(12) Solutions to Problem Set 5 mailed.

(11) MID-TERM EXAM I (closed book) - during regular class hours (12:30 - 1:45 p.m.), Tuesday March 6 in EGR 0110 basement of Engineering Building (Martin Hall). Based on material upto and including Lecture Notes 4, and Problem Set 5.

(10) CORRECTION: In Problem 1 of Homework set 5, sent to you earlier (Friday), the definition of the function f, for the condition x_1 not equal to 0, has in the DENOMINATOR only square terms. Instead the denominator should read (x_1)^2 + (x_2)^4 . (This correction was also emailed to you on Saturday afternoon.)

(9) Homework Set 5 posted (due back in class, Thursday March 1). A fix to page 12 of Lecture 4 notes posted.

(8) Two yellow stickies added to page 5 of Lecture Notes 3 and one yellow stickie added to page 7 of the same set, to fix small "typoes" that I thought I had pointed out in class. Solutions to Homework 3 emailed.

(7) Homework Set 4 sent by email and posted (due 02/23/2012)

(6) Homework Set 2 solutions sent by email; Homework 3 assignment - see link below (also emailed), due back in class on Thursday, February 16, 2012 (extended to Tuesday, February 21, 2012) Note - at the bottom of the web-page there are links to (a) book by Daniel Liberzon on Calculus of Variations and Optimal Control (Princeton U Press 2012) (b) online accessible (PDF) Real Analysis Book by Cinlar and Vanderbei - (material useful in Systems courses)

For Homework Set 3 start reading Lecture Notes 3. Warning: In these notes, page 5 bottom of Theorem statement part (c) there is a term B(t)v(t) missing on the r.h.s of the differential equation. Also a missing (-L) term in the Riccati equation above it.

Additional required reading: Understand Theorem 2.2 on page 16 of Professor Andre Tits' Lecture Notes. Of course read the rest of his Chapter 2 as well - corresponds to what we have done in Lecture Notes 1 - 3.

(5) Homework assignment 2 posted and emailed. Solution to homework set 1 emailed.

(4) Homework assignment 1 posted.

(3) The Sussmann-Willems paper can be obtained by clicking on one of the links at the bottom of this page.

(2) There is no required textbook for this course.

(1) All lecture notes (initial version) are posted. Updates will be posted on a regular basis.

Weekly Lecture Notes by P. S. Krishnaprasad

Survey Lecture on Linear Systems and link to ENEE 660 System Theory Notes


Lecture 1

Lecture 2

Lecture 3

Lecture 4 and an addendum

Lecture 4 Page 12 fix

Lecture 5(a), updated; Lecture 5(b); Lecture 5(c) Lecture 5(c) Update

Lecture 6

Lecture 7 and solution to Queen Dido's problem

Lecture 7 addendum (on transversality condition)

Lecture 8 on fixed point problems

Lecture 9(a) on Newton's method and additional material (lecture 9(b)) on

mean value theorem

Lecture 10(a) on Newton's method and rate of convergence and

Lecture 10(b) on iterative minimization

Lecture 11(a) on second order necessary conditions

Lecture 11(b) on Taylor's theorem

Lecture 11(c) on second order necessary conditions in the calculus of variations (Legendre)

Lecture 12 on maximum principle

Lecture 13 on Hamilton Jacobi Bellman Equation


Lecture Notes by Professor Andre L. Tits


Homework Assignments

Problem Set 1

Problem Set 2

Problem Set 3

Problem Set 4

Problem Set 5

Problem Set 6

Problem Set 7

Problem Set 8


Homework solutions are sent by email


Some interesting resources on the web

Riccati had an interesting life. Some historical remarks on his life and work due to Sergio Bittanti are available here . See also the page on Riccati at the St. Andrews University archive (also a source of biographical information on other mathematicians).

The Brachystochrone problem was originally set by Johann Bernoulli in June 1696. The paper of Hector J. Sussmann and Jan C. Willems in the IEEE Control Systems Magazine, June 1997, pp 32-44, celebrates this event as a beginning of optimal control theory.

Solution (based on calculus) of Queen Dido's problem by P. D. Lax from American Mathematical Monthly, vol. 102, No. 2, February 1995, pp. 158-159

Book by John T. Betts Practical Methods for Optimal Control and Estimation using Nonlinear Programming

Link to book by Daniel Liberzon on Calculus of Variations and Optimal Control

Real Analysis Book by Cinlar and Vanderbei - material useful in Systems courses

About C. Caratheodory

Impact of Control Technology vignettes, and, full report (warning 35 MB)