# Control of multi-agent systems using continuum mechanics modelling

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Dr Anton Selivanov  Applications accepted all year round  Self-Funded PhD Students Only

A multi-agent system is a group of interacting dynamical systems. Famous examples are robot swarms, smart grids, traffic flows, communication networks, and transportation systems. These systems tend to grow rapidly in size and complexity due to the continuous introduction of new agents. As a result, standard control methods fail, and we witness massive system breakdowns such as power blackouts, traffic jams, internet outages, and product delivery failures. Therefore, there is a demand for new scalable solutions for analysing and controlling multi-agent systems.

Partial differential equations (PDEs) are a natural framework for studying multi-agent systems with many agents. When cooperating agents are identical, and their communication topology is uniform, one can take the continuum limit of the ordinary differential equations (ODEs) describing the multi-agent system and obtain a PDE. The form of the PDE depends on the agents’ dynamics and network structure but does not depend on the number of agents. Consequently, such PDEs provide analysis methods whose complexity does not change when the number of agents grows.

The similarities between multi-agent dynamics and PDEs have been observed but not explored. This PhD project will examine the possibility of using PDEs to model, design, analyse, and control multi-agent systems. This problem is inverse to the one solved in numerical methods. Namely, to approximate a solution of a PDE, one discretises it and solves the resulting set of ODEs. Here, we will approximate ODEs with a single PDE to simplify the theoretical analysis. This research topic is full of challenging questions: What is the best way to relate the state of a multi-agent system with the state of a PDE? How many agents justify the usage of PDEs? How to interpret the results derived for the PDE in terms of the original multi-agent system? What conditions on the local dynamics and communication topology make it possible to associate a well-posed PDE with a given multi-agent system?

If you would like to learn about the PhD project more or have any questions, please, feel free to contact the project supervisor, Dr Anton Selivanov, at .

Candidate Requirements

Strong mathematical background (in calculus, linear algebra, ODEs, and functional analysis) and familiarity with PDEs are essential. Experience in control theory is desirable but not mandatory – a mathematically literate candidate can quickly fulfil possible gaps. Most importantly, we are looking for candidates passionate about maths and fundamental research in general.

Applicants are required to hold a BSc/MSc degree in mathematics or engineering. If the degree is not from an English-speaking country, the applicant needs an overall IELTS grade of 6.5 with a minimum of 6.0 in each component (or equivalent). For further details, visit https://www.sheffield.ac.uk/postgraduate/phd/apply.

Learning Environment

The University of Sheffield is a Russell Group university. It is located in the centre of the UK, right next to the Peak District National Park. The Department of Automatic Control and Systems Engineering (ACSE) is the only department in the UK dedicated to Control Engineering. The standard duration of a PhD in the UK is 3.5 years. To learn more about student life in Sheffield, visit https://www.sheffield.ac.uk/sheffield-guide.

Application Process

Informal enquiries are encouraged and should be addressed to Dr Anton Selivanov at . You can apply for this project here: https://www.sheffield.ac.uk/postgradapplication/. Suitable candidates will be invited for an online interview.

## Funding Notes

This is a self-funded research project. We require applicants to have either an undergraduate honours degree (1st) or MSc (Merit or Distinction) in a relevant science or engineering subject from a reputable institution. Full details of how to apply can be found at the following link: View Website

## References

1. A. Selivanov and E. Fridman, “PDE-Based Deployment of Multiagents Measuring Relative Position to One Neighbor,” IEEE Control Systems Letters, vol. 6, pp. 2563–2568, 2022.
2. T. Vicsek and A. Zafeiris, “Collective motion,” Physics Reports, vol. 517, no. 3–4, pp. 71–140, 2012.
3. P. Frihauf and M. Krstic, “Leader-enabled deployment onto planar curves: A PDE-based approach,” IEEE Transactions on Automatic Control, vol. 56, no. 8, pp. 1791–1806, 2011.
4. J. Toner and Y. Tu, “Long-range order in a two-dimensional dynamical XY model: How birds fly together,” Physical Review Letters, vol. 75, no. 23, pp. 4326–4329, 1995.

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