Conservation laws are evolution equations that describe fundamental laws of nature and are therefore central to our understanding of the physical world. Moreover, a mathematical handle on these equations would allow for the design of mechanisms to be used in controlling real-world systems. An important subclass of conservation laws is that of nonlocal transport equations with nonlinear mobility (NTNs). They commonly arise as large population limits of complex phenomena manifested in interacting systems that exist at different scales of organization—from subatomic entities in physical and biological processes to planetary systems. NTNs are also used on network structures to study, e.g., transportation and information networks, and supply chains, with the intent of optimizing said networks. Despite the abundance of mathematical tools available for studying specific subclasses of conservation laws, these tools are not directly applicable to NTNs. For this reason, new tools have been developed in the past years, and are still being developed, to analyze them. Building upon recent progress in the analysis and geometric understanding of NTNs that makes use of a discrete-to-continuum approach, this project, which is part of an NWO M1 grant, proposes a novel and mathematically justified framework for studying optimal control problems for NTNs and NTNs on networks.  

Together with the supervisor, the PhD student will focus on the development of an adjoint-based optimal control framework (analytical and numerical) for NTNs, where NTNs are consistently approximated by so-called deterministic particle approximations. This new framework not only unifies various optimal control theories for conservation laws but also enables the development of efficient numerical algorithms for the design of control mechanisms, such as smart traffic light controls for transportation networks. 

Next to these research activities, you will participate in educational activities performed at the Department of Mathematics and Computer Science, which include assisting in the teaching of undergraduates and master's students.

We are looking for a candidate that meets the following requirements:

• A Master's degree in Mathematics, Applied Mathematics, or related fields;
• Strong knowledge of calculus of variations, measure theory and functional analysis is highly desirable;
• An interest in cross-topic collaboration between different fields of mathematics;
• Affinity with high-level, mathematics-oriented programming languages (Python, etc.);
• Good communicative skills in English, both in speaking and in writing;
• Candidates from non-Dutch or non-English speaking countries should be prepared to prove their English language skills.

A meaningful job in a dynamic and ambitious university, in an interdisciplinary setting and within an international network. You will work on a beautiful, green campus within walking distance of the central train station. In addition, we offer you:

• Full-time employment for four years, with an intermediate evaluation (go/no-go) after nine months. You will spend 10% of your employment on teaching tasks.
• Salary and benefits (such as a pension scheme, paid pregnancy and maternity leave, partially paid parental leave) in accordance with the Collective Labour Agreement for Dutch Universities, scale P (2,541 per month in the first year increasing up to €3,247 in the fourth year).
• A year-end bonus of 8.3% and annual vacation pay of 8%.
• High-quality training programs and other support to grow into a self-aware, autonomous scientific researcher. At TU/e we challenge you to take charge of your own learning process.
• An excellent technical infrastructure, on-campus children's day care and sports facilities.
• An allowance for commuting, working from home and internet costs.
• A Staff Immigration Team and a tax compensation scheme (the 30% facility) for international candidates.