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.

Chapter 1 introduces some key elements of important topics such as; quantum mechanics,

representation theory of the Lorentz and Poincare groups, and a review of some basic rela- ´

tivistic wave equations that will play an important role in the work to follow. In Chapter 2,

a complex covariant form of the classical Maxwell’s equations in a moving medium or at

rest is introduced. In addition, a compact, Lorentz invariant, form of the energy-momentum

tensor is derived. In chapter 3, the concept of photon helicity is critically analyzed and its

connection with the Pauli-Lubanski vector from the viewpoint of the complex electromag- ´

netic field, E+ iH. To this end, a complex covariant form of Maxwell’s equations is used.

Chapter 4 analyzes basic relativistic wave equations for the classical fields, such as Dirac’s

equation, Weyl’s two-component equation for massless neutrinos and the Proca, Maxwell

and Fierz-Pauli equations, from the viewpoint of the Pauli-Lubanski vector and the Casimir ´

operators of the Poincare group. A connection between the spin of a particle/field and ´

consistency of the corresponding overdetermined system is emphasized in the massless

case. Chapter 5 focuses on the so-called generalized quantum harmonic oscillator, which

is a Schrodinger equation with a time-varying quadratic Hamiltonian operator. The time ¨

evolution of exact wave functions of the generalized harmonic oscillators is determined

in terms of the solutions of certain Ermakov and Riccati-type systems. In addition, it is

shown that the classical Arnold transform is naturally connected with Ehrenfest’s theorem

for generalized harmonic oscillators. In Chapter 6, as an example of the usefulness of the

methods introduced in Chapter 5 a model for the quantization of an electromagnetic field

in a variable media is analyzed. The concept of quantization of an electromagnetic field

in factorizable media is discussed via the Caldirola-Kanai Hamiltonian. A single mode

of radiation for this model is used to find time-dependent photon amplitudes in relation

to Fock states. A multi-parameter family of the squeezed states, photon statistics, and the

uncertainty relation, are explicitly given in terms of the Ermakov-type system.