MATHEMATICAL OPTIMISATION - 2026/7

Module Overview

In many practical settings it is important to be able to deduce optimal states and/or solutions to problems that can be posed mathematically. Early such problems include determining the brachistochrone, which is the shape of a curve in space that that minimises the time taken for a particle to move between two fixed points (first posed by Johann Bernoulli and solved shortly afterwards by Isaac Newton and other great scholars of the day.) In more modern applications, we might want to control a dynamical system (e.g. the motion of a spacecraft) in such a way as to minimize an associated cost (e.g. fuel usage), and so determine an optimal `path' (e.g. a flight trajectory). In still other settings, we might be concerned with the optimal transport of mass from one region to another, which in mathematical terms is a type of constrained minimisation problem. 

In this module, students are introduced to ideas, results and techniques from the calculus of variations, optimal control theory and optimal mass transport. The underpinning theory will be developed in detail and illustrated through examples wherever possible.

Module provider

Mathematics & Physics

Module Leader

BEVAN Jonathan (Maths & Phys)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 6

Module cap (Maximum number of students): N/A

Module Availability

Semester 2

Prerequisites / Co-requisites

N/A

Module content

Indicative content includes: 

• Calculus of Variations: the role of integral functionals in modelling problems, the Euler-Lagrange equations and convexity in low dimensional problems. 
• Optimal control of linear equations: general formulation, the variation of constants formula, the controllability matrix and a criterion for controllability, the bang-bang principle and extremal controls. 
• Linear time-optimal controls: the existence of time-optimal controls and Pontryagin's maximum principle. 
• Nonlinear dynamic programming:  the payoff functional, the value function as a solution of the Hamilton-Jacobi-Bellman equation, use of the value function as a means to design optimal feedback control(s). 
• Optimal mass transport: introduction to the problem as formulated by Monge, transport maps and the interpretation of Monge-Kantorovich duality as a transshipment problem. 

Assessment pattern

Alternative Assessment

N/A

Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

•  Understanding and knowledge of important definitions, key theorems and proofs, and related mathematical concepts in Optimisation and Optimal Control. 
• The ability to apply subject knowledge to the analysis of unseen problems. 

Thus, the summative assessment for this module consists of: 

• One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1,2 and 3. 
• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to Learning Outcomes 1, 2, 3, 4 and 5. 

Formative assessment

There will be formative unassessed coursework that is designed to consolidate student learning. 

Feedback

Students will receive individual written feedback on formative unassessed coursework and the in-semester test. The former supports preparation for the latter, while the latter, together with former, supports preparation for the final exam.

Module aims

• Introduce students to the fundamentals of the calculus of variations, to include the Euler-Lagrange equations, convexity, and the combination of the two to prove minimality in cases such as the brachistochrone
• Introduce students to the optimal control of linear equations, the bang-bang principle and extremality
• Give the theory of linear time-optimal controls, including the existence of optimal controls and Pontryagin's maximum principle
• Introduce students to (nonlinear) dynamic programming and the Hamilton-Jacobi-Bellman equation
• Introduce students to the Monge mass transfer problem, transport plans and the Monge-Kantorovich duality (statement only).

Learning outcomes

Methods of Teaching / Learning

The learning and teaching strategy is designed to: 

Be an introduction to the topics outlined above, which represent a selection from a much wider range of possible optimisation and control results.  Students will meet theory alongside examples and they will develop the ability to recognise different types of optimisation problems and to apply suitable techniques to solve them.  Students gain familiarity with the key mathematical techniques used to investigate these concepts through the examples presented in the lectures as well as through unassessed coursework.  This helps students to develop good mathematical practice. 

The learning and teaching methods include: 

• Three one-hour lectures for eleven weeks, with sketch module notes provided to complement the lectures, together with supplementary material as appropriate.  These lectures provide a structured learning environment and opportunities for students to ask questions and to practice the methods taught. 
• Five one-hour tutorials per semester. These tutorials provide an opportunity for students to gain feedback and assistance with examples. 
• Unassessed coursework enables students to consolidate learning. Students receive feedback to guide their progress and understanding. 
• Lectures may be recorded or the equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending lectures. 

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Reading list

https://readinglists.surrey.ac.uk
Upon accessing the reading list, please search for the module using the module code: MAT3057

Other information

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2026/7 academic year.