EML 6934 - Optimal Control
Spring 2022


Name: Dr. Anil V. Rao
Office: MAE-A 314
E-mail: anilvrao@ufl.edu
Tel: 352-392-5523 (Office); (352) 672-1529 (Mobile).
YouTube Channel: https://www.youtube.com/user/anilvrao2

All contact methods are acceptable! Note for fastest response it is best to reach me on my mobile phone.


Class Period: MWF 1:55 — 2:45 PM (Period 7). Attendance is expected.
Lecture Room: NEB 102
Online Videos:
Class will be recorded via UF EDGE
Office Hours:
As Needed MWF 3:00 PM to 4:00 PM
Office Hours Zoom Link:
Meeting ID: 923 3814 1900
Click here for Zoom link or copy and paste the following URL into your browser: https://ufl.zoom.us/j/92338141900.


Calculus of variations, calculus of variations applied to optimal control, nonlinear optimization, numerical Methods for optimal control


Differential and integral calculus, ordinary and partial differential equations


Develop the fundamentals of calculus of variations
Develop fundamentals of optimal control theory using calculus of variations
Introduce major numerical methods for solving constrained optimal control problems


I consider it an honor and a privilege to be able to teach all of you, and I intend to provide the best instruction possible in order to enable you to learn the material well. If you cannot make office hours, please contact me and we will set up a time for you to get help. Regardless of how busy I am with other things, I will do what I am able to make myself available.



Review of Minima and Maxima of Functions

Calculus of Variations

Calculus of Variations
Applied to
Optimal Control Problems

Nonlinear Optimization

Numerical Methods for Optimal Control

Material Covered

Optimization of Function of One Variable

Develop Theory of Calculus of Variations

Optimality Conditions, Linear Quadratic Control, Pontryagin's
Minimum Principle, Bang — Bang Optimal Control,
Singular Arcs

Unconstrained and Constrained
Finite-Dimensional Optimization

Indirect Methods and Direct Methods


Week 1

Weeks 2 and 3

Weeks 4 through 8

Week 9

Week 10 through 13 


Note: if for some reason you are unable to make my office hours, you can always schedule an appointment at a time that is mutually agreeable to both you and I.



Anil V. Rao


3:00 PM to 4:00 PM

Meeting Location and Contact

Zoom: https://ufl.zoom.us/j/92338141900
(Meeting ID: 923 3814 1900)


I have found that often students want to talk with me about topics other than the course. Sometimes it is just to get career directions and advice, other times to find out about opportunities to work in my research group as an undergraduate or graduate student. Because students would like to have conversations on such topics (and other topics), each week I will hold what I call "personal hours". If you are interested in just having a conversation with me that is not specific to the course material, please join me for personal hours. I will try to make these hours actual in person hours because I feel it is the best way to have non-technical conversations. As a result, I will hold personal hours in the Reitz Union food court.

Personal Office Hours Times

4:00 PM to 5:00 PM

Meeting Location

Reitz Union Food Court
(Next to Starbucks)

Note: I am happy to schedule other times for personal meetings about topics not related to the course. Please feel free to ask.


Kirk, D. E., Optimal Control Theory: An Introduction, Dover Publications, 1970.


All homework in this class is optional. Nothing is graded, but I am happy to discuss and go over any and all problems on the assignments.


One of the key assignments in this class is a course project. The project can be found by clicking here. The instructions are as follows. Choose two optimal control problems, one from the "elementary" pptimal control problems list and one from the "advanced" optimal control problems list. For the problem you choose from "elementary" list you must formulate the complete set of first-order optimality conditions for your problem using the calculus of variations. You must then solve your problem using an indirect numerical method and a direct numerical method that we have studied in the course. Next, for the problem you choose from "advanced" list you must choose a single method (indirect or direct but not both). You must then analyze the quality of your solution. In your analysis, consider the following questions. Can you determine the proximity of your numerical solution to the “true” optimal solution. If you are unable to ascertain how close your solution is to the true optimal, how do you know you have obtained a reasonable approximation? What is the computational efficiency of the methods you applied to solve your problem? What are the limitations of the methods you have chosen on your problem. Given your analysis, what numerical method would you seek in order to overcome the deficiencies you found with the methods you chose? It is highly recommended that you now wait until the last minute to think of a problem for your project! The course project is due on the last day of class. Click here to download a copy of the course project.


Regular attendance is expected of all students.


Cheating of any kind in this course will be enforced in accordance with the university rules. Any violation of any kind (even something as simple as a single line of code that is identical in the homework of two students) will automatically result in an "E" in the course and will reported as appropriate to the Dean of Students Office.


Because all assignments in this course are not time limited (in the same manner as that a usual in-class exam), make-ups will be provided on a case-by-case basis. If you have an issue (illness, other urgent matter), please discuss it with me and we will work to find a fair and reasonable solution.



Homework Assignments

Midterm Exam



Point Value




100 Points


Grades in this course are determined using the following scale:

Letter Grade













Score Range

95 and Above

90 to less than 95

85 to less than 90

80 to less than 85

75 to less than 80

70 to less than 75

65 to less than 70

60 to less than 65

55 to less than 60

50 to less than 55

45 to less than 50

Less Than 45


The grading scale posted above is not flexible.
Any score on the boundary between two ranges will receive the higher grade (for example, a 94 receives a grade of "A-").
Finally, it is noted that while your individual scores for assignments, exams, and quizzes will be posted on E-learning (Canvas), the Canvas portal may not accurately reflect a student's relative standing in the class. Regardless of the information that is seen in Canvas, computation of final grades will be based on the criteria set forth above and a student's grade will only be final when grades have been computed at the end of the semester.

Any assignment either not submitted or not completed with a good faith effort (where the judgment of "good faith effort" rests wholly with me) will result in a full letter grade deduction in the course. For example, if the final score falls into the category of an "A-" and one homework or quiz is not submitted or is deemed to not have been performed with a good faith effort, the final grade will be a "B-". This policy is not flexible.


Students with disabilities who experience learning barriers and would like to request academic accommodations should connect with the disability Resource Center by visiting https://disability.ufl.edu/students/get-started/. It is important for students to share their accommodation letter with their instructor and discuss their access needs, as early as possible in the semester.


Students are expected to provide professional and respectful feedback on the quality of instruction in this course by completing course evaluations online via GatorEvals. Guidance on how to give feedback in a professional and respectful manner is available at https://gatorevals.aa.ufl.edu/students/. Students will be notified when the evaluation period opens, and can complete evaluations through the email they receive from GatorEvals, in their Canvas course menu under GatorEvals, or via https://ufl.bluera.com/ufl/. Summaries of course evaluation results are available to students at https://gatorevals.aa.ufl.edu/public-results/.


Students are allowed to record video or audio of class lectures. However, the purposes for which these recordings may be used are strictly controlled. The only allowable purposes are (1) for personal educational use, (2) in connection with a complaint to the university, or (3) as evidence in, or in preparation for, a criminal or civil proceeding. All other purposes are prohibited. Specifically, students may not publish recorded lectures without the written consent of the instructor.

A “class lecture” is an educational presentation intended to inform or teach enrolled students about a particular subject, including any instructor-led discussions that form part of the presentation, and delivered by any instructor hired or appointed by the University, or by a guest instructor, as part of a University of Florida course. A class lecture does not include lab sessions, student presentations, clinical presentations such as patient history, academic exercises involving solely student participation, assessments (quizzes, tests, exams), field trips, private conversations between students in the class or between a student and the faculty or lecturer during a class session.

Publication without permission of the instructor is prohibited. To “publish” means to share, transmit, circulate, distribute, or provide access to a recording, regardless of format or medium, to another person (or persons), including but not limited to another student within the same class section. Additionally, a recording, or transcript of a recording, is considered published if it is posted on or uploaded to, in whole or in part, any media platform, including but not limited to social media, book, magazine, newspaper, leaflet, or third party note/tutoring services. A student who publishes a recording without written consent may be subject to a civil cause of action instituted by a person injured by the publication and/or discipline under UF Regulation 4.040 Student Honor Code and Student Conduct Code.
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