Numerous works have addressed the control of multi-robot systems for coverage, mapping, navigation, and task allocation problems. In addition to classical microscopic approaches to multi-robot problems, which model the actions and decisions of individual robots, lately, there has been a focus on macroscopic or Eulerian approaches. In these approaches, the population of robots is represented as a continuum that evolves according to a mean-field model, which is directly designed such that the corresponding robot control policies produce target collective behaviours.



This dissertation presents a control-theoretic analysis of three types of mean-field models proposed in the literature for modelling and control of large-scale multi-agent systems, including robotic swarms. These mean-field models are Kolmogorov forward equations of stochastic processes, and their analysis is motivated by the fact that as the number of agents tends to infinity, the empirical measure associated with the agents converges to the solution of these models. Hence, the problem of transporting a swarm of agents from one distribution to another can be posed as a control problem for the forward equation of the process that determines the time evolution of the swarm density.



First, this thesis considers the case in which the agents' states evolve on a finite state space according to a continuous-time Markov chain (CTMC), and the forward equation is an ordinary differential equation (ODE). Defining the agents' task transition rates as the control parameters, the finite-time controllability, asymptotic controllability, and stabilization of the forward equation are investigated. Second, the controllability and stabilization problem for systems of advection-diffusion-reaction partial differential equations (PDEs) is studied in the case where the control parameters include the agents' velocity as well as transition rates. Third, this thesis considers a controllability and optimal control problem for the forward equation in the more general case where the agent dynamics are given by a nonlinear discrete-time control system. Beyond these theoretical results, this thesis also considers numerical optimal transport for control-affine systems. It is shown that finite-volume approximations of the associated PDEs lead to well-posed transport problems on graphs as long as the control system is controllable everywhere.

This thesis considers two problems in the control of robotic swarms. Firstly, it addresses a trajectory planning and task allocation problem for a swarm of resource-constrained robots that cannot localize or communicate with each other and that exhibit stochasticity in their motion and task switching policies. We model the population dynamics of the robotic swarm as a set of advection-diffusion- reaction (ADR) partial differential equations (PDEs).

Specifically, we consider a linear parabolic PDE model that is bilinear in the robots' velocity and task-switching rates. These parameters constitute a set of time-dependent control variables that can be optimized and transmitted to the robots prior to their deployment or broadcasted in real time. The planning and allocation problem can then be formulated as a PDE-constrained optimization problem, which we solve using techniques from optimal control. Simulations of a commercial pollination scenario validate the ability of our control approach to drive a robotic swarm to achieve predefined spatial distributions of activity over a closed domain, which may contain obstacles. Secondly, we consider a mapping problem wherein a robotic swarm is deployed over a closed domain and it is necessary to reconstruct the unknown spatial distribution of a feature of interest. The ADR-based primitives result in a coefficient identification problem for the corresponding system of PDEs. To deal with the inherent ill-posedness of the problem, we frame it as an optimization problem. We validate our approach through simulations and show that reconstruction of the spatially-dependent coefficient can be achieved with considerable accuracy using temporal information alone.