Course overview
- Study period
- Semester 2, 2024 (22/07/2024 - 18/11/2024)
- Study level
- Undergraduate
- Location
- St Lucia
- Attendance mode
- In Person
- Units
- 2
- Administrative campus
- St Lucia
- Coordinating unit
- Mathematics & Physics School
Calculus of variations: critical points; Euler equations; transversality; corner conditions; Hamilton equations; Jacobi equations; Legendre sufficient condition; Weierstrass E-function. Control theory: Lagrange, Mayer & Bolza problems; Pontryagin maximal principle, legendre transformations, augmented Hamiltonians, transversality, bang-bang control, linear systems.
Optimisation Theory introduces students to the basic ideas of the Calculus of Variations and of Control Theory. These have applications to many fields including to other areas of mathematics, as well as to physics, engineering, biology, ecologyﾠand economics.
Course requirements
Assumed background
Second Year Calculus and Elementary Differential Equations as set out in the syllabi for MATH2000/2001
Prerequisites
You'll need to complete the following courses before enrolling in this one:
MATH2000 or MATH2001 or MATH2901
Incompatible
You can't enrol in this course if you've already completed the following:
MATH7434 (co-taught).
Jointly taught details
This course is jointly-taught with:
- Another instance of the same course
MATH3404 and MATH7434 are co-badged courses and will share learning activities. MATH7434 will have some differences in assessment to evaluate students at Level 9 (Masters) of the Australian Qualifications Framework.
Course contact
Tutor
Mr Mitchell Jones
Tutor
Mr Marcus Flook
Course staff
Lecturer
Timetable
The timetable for this course is available on the UQ Public Timetable.
Additional timetable information
All classes will be conducted on campus. Consult your personal timetable for times and locations. Students are expected to attend these sessions in person unless they have a valid reason for being unable to attend (such as illness).
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Aims and outcomes
Students should:
Acquire a working knowledge of necessary and sufficient conditions for constrained and unconstrained extremaﾠfor functions of several variables.
Develop a working knowledge of the basic results from the calculus of variations for functions of one variable.
Develop a working knowledge of the basic results from optimal control theory.
Learning outcomes
After successfully completing this course you should be able to:
LO1.
Locate and classify critical points for functions of several variables both with and without constraints;
LO2.
Identify the necessary conditions for extrema for variational problems writing down the appropriate necessary conditions;
LO3.
Identify variational problems in simple cases writing down the appropriate functional to be optimised together with the appropriate subsidiary conditions;
LO4.
Solve the Euler-Lagrange equations including subsidiary conditions to find the critical points of variational problems
LO5.
Find and apply sufficient conditions for extrema in variational problems;
LO6.
Identify optimal control problems in simple cases writing down the appropriate functional to be optimised together with the appropriate subsidiary conditions;
LO7.
Apply Pontryagin's maximal principle to solve linear time optimal control problems in two dimensions identifying the regions of controllability and the associated solution strategy including the optimal control;
LO8.
Apply Pontryagin's maximal principle to solve optimal control problems for linear systems with quadratic cost functionals depending on the end conditions;
LO9.
Apply Pontryagin's maximal principle for other unconstrained and constrained cases using Legendre transformations, augmented Hamiltonians, transversality equations, and bang--bang control
where appropriate.
Assessment
Assessment summary
| Category | Assessment task | Weight | Due date |
|---|---|---|---|
| Tutorial/ Problem Set | Assignment | 40% ,10% each |
Assignment 1: 16/08/2024 5:00 pm Assignment 2: 9/09/2024 5:00 pm Assignment 3: 4/10/2024 5:00 pm Assignment 4: 25/10/2024 5:00 pm |
| Examination |
Final exam
|
60% |
End of Semester Exam Period 2/11/2024 - 16/11/2024 |
Assessment details
Assignment
- Mode
- Written
- Category
- Tutorial/ Problem Set
- Weight
- 40% ,10% each
- Due date
Assignment 1: 16/08/2024 5:00 pm
Assignment 2: 9/09/2024 5:00 pm
Assignment 3: 4/10/2024 5:00 pm
Assignment 4: 25/10/2024 5:00 pm
- Learning outcomes
- L01, L02, L03, L04, L05, L06, L07
Task description
A list of questions to be answered and submitted.
Submission guidelines
Submit this assessment item through the link in Blackboard.
Deferral or extension
You may be able to apply for an extension.
See ADDITIONAL ASSESSMENT INFORMATION for the extension and deferred examination information relating to this assessment item.
Late submission
A penalty of 1 grade for each 24 hour period from time submission is due will apply for up to 7 days. After 7 days you will receive a mark of 0.
You are required to submit assessable items on time. If you fail to meet the submission deadline for any assessment item then the listed penalty will be deducted per day for up to 7 calendar days, at which point any submission will not receive any marks unless an extension has been approved. Each 24-hour block is recorded from the time the submission is due.
Final exam
- In-person
- Mode
- Written
- Category
- Examination
- Weight
- 60%
- Due date
End of Semester Exam Period
2/11/2024 - 16/11/2024
Task description
The final examination in this course will be held during the end-of-semester examination period. It will be an in-person invigilated exam held on campus.
Exam details
| Planning time | 10 minutes |
|---|---|
| Duration | 120 minutes |
| Calculator options | (In person) Casio FX82 series only or UQ approved and labelled calculator |
| Open/closed book | Closed Book examination - no written materials permitted |
| Exam platform | Paper based |
| Invigilation | Invigilated in person |
Submission guidelines
Deferral or extension
You may be able to defer this exam.
See ADDITIONAL ASSESSMENT INFORMATION for the extension and deferred examination information relating to this assessment item.
Course grading
Full criteria for each grade is available in the Assessment Procedure.
| Grade | Cut off Percent | Description |
|---|---|---|
| 1 (Low Fail) | 0 - 19 |
Absence of evidence of achievement of course learning outcomes. Course grade description: Student will achieve a final mark between 0% and 19% by demonstrating an extremely poor knowledge/understanding of the basic concepts in the course material. |
| 2 (Fail) | 20 - 44 |
Minimal evidence of achievement of course learning outcomes. Course grade description: Students must achieve a final mark between 20% and 44% by demonstrating some knowledge of the basic concepts of Optimisation Theory including its applications to model problems as discussed in the course and its associated materials. This includes attempts at expressing their deductions and explanations as well as attempts to answer a few questions accurately. |
| 3 (Marginal Fail) | 45 - 49 |
Demonstrated evidence of developing achievement of course learning outcomes Course grade description: Students must achieve a final mark between 45% and 49% by demonstrating some knowledge of the basic concepts of Optimisation Theory including its applications to model problems as discussed in the course and its associated materials. This includes occasionally expressing their deductions and explanations clearly, the use of a few appropriate and efficient mathematical techniques and attempts to answer a few questions and tasks accurately and with appropriate justification. They will have demonstrated knowledge of techniques used to solve problems. Students with a mark of less than 45% of the marks available for the final examination will not be awarded a course grade higher than a 3. |
| 4 (Pass) | 50 - 64 |
Demonstrated evidence of functional achievement of course learning outcomes. Course grade description: Students must achieve a final mark between 50% and 64% by demonstrating an understanding of the basic concepts of Optimisation Theory including its applications to model problems as discussed in the course and its associated materials. This includes occasionally expressing their deductions and explanations clearly, the occasional use of appropriate and efficient mathematical techniques and accurate answers to a few questions and tasks with appropriate justification. They will have demonstrated knowledge of techniques used to solve problems and successfully applied this knowledge in some cases. To receive a grade of 4 or higher, in addition to the above criteria for each grade, students must achieve at least 45% of the marks available on the final examination. |
| 5 (Credit) | 65 - 74 |
Demonstrated evidence of proficient achievement of course learning outcomes. Course grade description: Students must achieve a final mark between 65% and 74% by demonstrating an adequate understanding of Optimisation Theory including its applications to model problems as discussed in the course and its associated materials. This includes clear expression of some of their deductions and explanations, the use of appropriate and efficient mathematical techniques in some situations and accurate answers to some questions and tasks with appropriate justification. They will be able to apply techniques such as solving the Euler-Lagrange equations to find critical points of functionals to solve fundamental problems. |
| 6 (Distinction) | 75 - 84 |
Demonstrated evidence of advanced achievement of course learning outcomes. Course grade description: Students must achieve a final mark between 75% and 84% by demonstrating a comprehensive understanding of Optimisation Theory including its applications to model problems as discussed in the course and its associated materials. This includes clear expression of most of their deductions and explanations, the general use of appropriate and efficient mathematical techniques and accurate answers to most questions and tasks with appropriate justification. They will be able to apply techniques such as solving the Euler-Lagrange equations to find critical points of functionals to partially solve both theoretical and practical problems. |
| 7 (High Distinction) | 85 - 100 |
Demonstrated evidence of exceptional achievement of course learning outcomes. Course grade description: Students must achieve a final mark between 85% and 100% by demonstrating an excellent understanding of all aspects of Optimisation Theory including its applications to model problems as discussed in the course and its associated materials. This includes clear expression of nearly all their deductions and explanations, the use of appropriate and efficient mathematical techniques and accurate answers to nearly all questions and tasks with appropriate justification. They will be able to apply techniques such as solving the Euler-Lagrange equations to find critical points of functionals to completely solve both theoretical and practical problems. |
Additional course grading information
Exam Hurdle:ﾠ
To receive a grade of 4 or higher,ﾠinﾠ addition to the above ﾠcriteria for each grade, ﾠstudents must achieveﾠ at least 45% of the marks available on the final examination.
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Supplementary assessment
Supplementary assessment is available for this course.
Should you fail a course with a grade of 3, you may be eligible for supplementary assessment. Refer to my.UQ for information on supplementary assessment and how to apply.
Supplementary assessment provides an additional opportunity to demonstrate you have achieved all the required learning outcomes for a course.
If you apply and are granted supplementary assessment, the type of supplementary assessment set will consider which learning outcome(s) have not been met.
Supplementary assessment in this course will be a 2-hour examination similar in style to the end-of-semester examination. To receive a passing grade of 3S4, you must obtain a mark of 50% or more on the supplementary assessment.
Additional assessment information
Artificial Intelligence
The assessment tasks in this course evaluate students’ abilities, skills and knowledge without the aid of Artificial Intelligence (AI). Students are advised that the use of AI technologies to develop responses is strictly prohibited and may constitute misconduct under the Student Code of Conduct.
Applications for Extensions to Assessment Due Dates
Extension requests are submitted online via my.UQ – applying for an extension. Extension requests received in any other way will not be approved. Additional details associated with extension requests, including acceptable and unacceptable reasons, may be found at my.UQ.
Please note:
- Requests for an extension to an assessment due date must be submitted through your my.UQ portal and you must provide documentation of your circumstances, as soon as it becomes evident that an extension is needed. Your application must be submitted on or before the assessment item's due date and time.
- Applications for extension can take time to be processed so you should continue to work on your assessment item while awaiting a decision. We recommend that you submit any completed work by the due date, and this will be marked if your application is not approved. Should your application be approved, then you will be able to resubmit by the agreed revised due date.
- If an extension is approved, you will be notified via your my.UQ portal and the new date and time for submission provided. It is important that you check the revised date as it may differ from the date that you requested.
- If the basis of the application is a medical condition, applications should be accompanied by a medical certificate dated prior to the assignment due date. If you are unable to provide documentation to support your application by the due date and time you must still submit your application on time and attach a written statement (Word document) outlining why you cannot provide the documentation. You must then upload the documentation to the portal within 24 hours.
- If an extension is being sought on the basis of exceptional circumstances, it must be accompanied by supporting documentation (eg. Statutory declaration).
- For extensions based on a SAP you may be granted a maximum of 7 days (if no earlier maximum date applies). See the Extension or Deferral availability section of each assessment for details. Your SAP is all that is required as documentation to support your application. However, additional extension requests for the assessment item will require the submission of additional supporting documentation e.g., a medical certificate. All extension requests must be received by the assessment due date and time.
- Students may be asked to submit evidence of work completed to date. Lack of adequate progress on your assessment item may result in an extension being denied.
- If you have been ill or unable to attend class for more than 14 days, you are advised to carefully consider whether you are capable of successfully completing your courses this semester. You might be eligible to withdraw without academic penalty - seek advice from the Faculty that administers your program.
- There are no provisions for exemption from an assessment item within UQ rules. If you are unable to submit an assessment piece then, under special circumstances, you may be granted an exemption, but may be required to submit alternative assessment to ensure all learning outcomes are met.
Applications to defer an exam
In certain circumstances you can apply to take a deferred examination for in-semester and end-of-semester exams. You'll need to demonstrate through supporting documentation how unavoidable circumstances prevented you from sitting your exam. If you can’t, you can apply for a one-off discretionary deferred exam.
Deferred Exam requests are submitted online via mySi-net. Requests received in any other way will not be approved. Additional details associated with deferred examinations, including acceptable and unacceptable reasons may be found at my.UQ.
Please note:
- Applications can be submitted no later than 5 calendar days after the date of the original exam.
- There are no provisions to defer a deferred exam. You need to be available to sit your deferred examination.
- Your deferred examination request(s) must have a status of "submitted" in mySI-net to be assessed.
- All applications for deferred in-semester examinations are assessed by the relevant school. Applications for deferred end-of-semester examinations are assessed by the Academic Services Division.
- You’ll receive an email to your student email account when the status of your application is updated.
- If you have a medical condition, mental health condition or disability and require alternative arrangements for your deferred exam you’ll need to complete the online alternative exam arrangements through my.UQ. This is in addition to your deferred examinations request. You need to submit this request on the same day as your request for a deferred exam or supplementary assessment. Contact Student Services if you need assistance completing your alternative exam arrangements request.
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Learning resources
You'll need the following resources to successfully complete the course. We've indicated below if you need a personal copy of the reading materials or your own item.
Library resources
Find the required and recommended resources for this course on the UQ Library website.
Additional learning resources information
There is no set text for this course. In addition to the references listed above and the lecture notes students should consult the UQﾠLibrary for books with titles including Calculus of Variations, Optimal Control or Optimisation Theory or look on the shelves for books with similar titles and catalogue numbers in the range QA315 and QA316.
Learning activities
The learning activities for this course are outlined below. Learn more about the learning outcomes that apply to this course.
Filter activity type by
Please select
| Learning period | Activity type | Topic |
|---|---|---|
Multiple weeks |
Lecture |
Weeks 1-13 All lectures will be given by Min-Chun Hong |
Tutorial |
Weeks 2-13 One tutorial each week |
Policies and procedures
University policies and procedures apply to all aspects of student life. As a UQ student, you must comply with University-wide and program-specific requirements, including the:
- Student Code of Conduct Policy
- Student Integrity and Misconduct Policy and Procedure
- Assessment Procedure
- Examinations Procedure
- Reasonable Adjustments - Students Policy and Procedure
Learn more about UQ policies on my.UQ and the Policy and Procedure Library.