Calculus of Variations & Optimal Control


MATH 5545/ CRN 85558 / Fall 2017


11:00 AM - 12:15 PM, Tuesday & Thursday


McBryde Hall 210




OFFICE: ICAM, West Campus Drive / Phone: 231-7667 / E-mail:


OFFICE HOURS: 9:30 - 10:30 TU, TH – 540 McBryde Hall


TEXT: Introduction to the Calculus of Variations and Control with Modern Applications, John A. Burns, Chapman and Hall/CRC, 2014.



          [1] The Calculus of Variations and Optimal Control, G. Leitmann, Plenum Publishing Co., New York, NY, 1981.

          [2] Calculus of Variations with Applications, George M. Ewing, Dover Press, 1985, New York.

          [3] Foundations of Optimal Control Theory, E. B. Lee and L. Markus, John Wiley & Sons, 1967.

          [4] Optimal Control Theory with Aerospace Applications, J. Z. Ben-Asher, AIAA Publishers, 2010.


COURSE CONTENT and PREREQUISITES: The objective of this course is to develop an understanding of the concepts listed in the textbook and their applications to problems of mathematics, social science, physical science, and engineering.  It is assumed that the student has some background in elementary differential equations and analysis.


Numerical work and projects will require MatlabTM.  It is assumed that the student has some background in elementary differential equations and analysis.  Engineering students will have MATLAB in their software bundle. All other majors need to purchase MATLAB from The Software Distribution Office that is located in 3240 Torgersen Hall, see

for more information.  The estimated cost is $25 + tax and this MATLAB license will be valid for as long as you are enrolled at Virginia Tech.  This license also includes annual updates at no extra charge.


EXAMS, GRADING CRITERIA and EVALUATION PROCEDURES: There will be a midterm and a final examination. These exams will be closed book, in-class and will be announced at least one week in advance. Homework will be assigned and selected homework problems may be graded. The final exam will be given on December 19, 2017 at 2:05 PM as indicated in the Time Table of Classes. No make-up exams will be given. Any EXCUSED work will be replaced by a grade determined from a portion (or perhaps all) of the final exam.  Attendance is required.  Anyone missing more than 5 classes in the semester may have their graded lowered by 1 letter grade.  Students auditing the course must take all the exams and will not receive a passing grade if they miss more than 5 classes.


The midterm will determine no more than 40% of the course grade and the final will determine no more than 60% of the course grade. Homework and projects will determine no more than 15% of the course grade. An average grade of 90, 80, 70, 60 guarantees a grade of A-, B-, C- or D-, respectively.


STUDENT RESPONSIBILITY: It is assumed that each student attends the lectures and works all the assigned problems. The student is responsible for ALL material covered in class and any assigned reading. Students may discuss homework problems. However, it will be assumed that all homework problems turned in for a grade will be the students own work. Any messy, sloppy and/or otherwise unreadable work will receive ZERO points. If your name does not appear on the final class roll, then you will not receive a grade for this course.


ACADEMIC INTEGRITY and HONOR CODE: For graded homework assignments, you may discuss solutions with other students, but each student must do individual, independent write-ups to turn in.  Direct copying of solutions or parts of solutions from any source is a violation of the honor code, as is sharing your solutions with others.  All exams are to be done without any assistance from anyone else.  Your work here is at all times governed by the Virginia Tech Undergraduate Honor System as stated in the Virginia Tech Undergraduate Catalog. Failure to follow these restrictions, and giving or receiving unauthorized aid or assistance on in-class quizzes or exams are Virginia Tech Honor System violations and cases will be filed.


The Undergraduate Honor Code pledge that each member of the university community agrees to abide by states:

          “As a Hokie, I will conduct myself with honor and integrity at all times. I will not lie, cheat, or steal, nor will I accept the actions of those who do.”

Students enrolled in this course are responsible for abiding by the Honor Code. A student who has doubts about how the Honor Code applies to any assignment is responsible for obtaining specific guidance from the course instructor before submitting the assignment for evaluation. Ignorance of the rules does not exclude any member of the University community from the requirements and expectations of the Honor Code.  For more information about the Honor Code, please see


If you have questions or are unclear about what constitutes academic misconduct on an assignment, please speak with me. I take the Honor Code very seriously in this course. The normal sanction I will recommend for a violation of the Honor Code is an F* sanction as your final course grade. The F represents failure in the course. The “*” is intended to identify a student who has failed to uphold the values of academic integrity at Virginia Tech. A student who receives a sanction of F* as their final course grade shall have it documented on their transcript with the notation “FAILURE DUE TO ACADEMIC HONOR CODE VIOLATION.” You would be required to complete an education program administered by the Honor System in order to have the “*” and notation “FAILURE DUE TO ACADEMIC HONOR CODE VIOLATION” removed from your transcript. The “F” however would be permanently on your transcript.”