# Optimal control methods for agent-based models

## Project Description

### Existing background work

Agent-based models (ABM), often described in terms of systems of interacting diffusions (multidimensional stochastic differential equations) arise in many applications, including mathematical biology, e.g. models for chemotaxis, epidemiology, and the social sciences, for example in models for opinion formation, pedestrian dynamics and macroeconomics. Such systems often exhibit interesting collective behaviour as a result of the interaction between agents. Several different macroscopic configurations of the ABM are possible, for different choices of the parameters in the system, such as the interaction strength. The transition between different macroscopic configurations can have dramatic effects on the behaviour of the ABM

Greg Pavliotis is professor of Applied Mathematics at the department of Mathematics at Imperial College. His main research interests include statistical mechanics, multiscale systems, inference for stochastic processes and data-driven methods in applied mathematics. He has studied phase transitions for agent-based models (ABM) and interacting particle systems (IPS) and he has developed efficient inference methodologies for IPS and their mean field limit. He has also developed data-driven approaches to the study of ABMs.

Dante Kalise is Reader in Optimisation and Control at the Department of Mathematics at Imperial College. His main research interests involve scientific computation and machine learning, optimal control, high-dimensional approximation and agent-based models across scales. He has developed optimal control methods for high-dimensional problems which constitute the basis of this project, and developed applications in swarm robotics, pedestrian motion, and global optimization.

### Main objectives of the project

The goal of the proposed project is to develop efficient control methodologies for steering the ABM dynamics towards desired macroscopic configurations. Different approaches, such as optimal control for PDEs and data driven approaches based on model predictive control will be developed.

### Details of Software/Data Deliverables

Software deliverables will include:

An ABM simulator for trajectory generation. To date, there’s no universal standard/benchmarks for ABM simulation, despite a generic structure based on the interaction forces in the system. We will develop a trajectory simulator which is fundamental for the study of data-driven methods and for optimisation and control purposes. Such simulator will provide sufficient freedom for prescribing interaction forces, thus being useful for different applications including opinion dynamics, swarm robotics, and pedestrian motion, among others. The simulator will be constructed using state-of-the-art methods for accurate and fast approximation of large-scale SDEs.

An optimal control toolbox for ABM. There are several optimal control solvers readily available for traditional control engineering applications (robotics, power electronics), however, none of them scale to high-dimensional settings which are natural in ABMs. Moreover, they do not exploit model structure and eventually resort to treating the optimal control problem as a large-scale, model-free, nonlinear optimization problem. Instead, we will develop a toolbox for control of ABMs based on adjoint calculus, making extensive use of the particular model structure of ABMs and circumventing calls to external nonlinear optimization solvers.