An Introduction to the Controllability of Partial Differential Equations

These notes are a written abridged version of a course that both authors have delivered in the last five years in a number of schools and doctoral programs. Our main goal is to introduce some of the main results and tools of the modern theory of controllability of Partial Differential Equations (PDE). The notes are by no means complete. We focus on the most elementary material by making a particular choice of the problems under consideration. Roughly speaking, the controllability problem may be formulated as follows. Consider an evolution system (either described in terms of Partial or Ordinary Differential Equations (PDE/ODE)). We are allowed to act on the trajectories of the system by means of a suitable control (the right hand side of the system, the boundary conditions, etc.). Then, given a time interval t ∈ (0, T ), and initial and final states we have to find a control such that the solution matches both the initial state at time t = 0 and the final one at time t = T . This is a classical problem in Control Theory and there is a large literature on the topic. We refer for instance to the book by Lee and Marcus [44] for an introduction in the context of finite-dimensional systems. We also refer to the survey paper by Russell [55] and to the book of Lions [45] for an introduction to the controllability of PDE, also referred to as Distributed Parameter Systems. Research in this area has been very intensive in the last two decades and it would be impossible to report on the main progresses that have been made within these notes. For this reason we have chosen to collect some of the most relevant introductory material at the prize of not reaching the best results that