Based on a series of lectures given over the past four years in France, Hungary and the US. The first part examines exact boundary controllability problems using the Hilbert Uniqueness Method. The latter half deals with stabilizability. Of special note: the multiplier method, applied systematically, is remarkably elementary and efficient.

Unlike abstract approaches to advanced control theory, this volume presents key concepts through concrete examples. Once the basic fundamentals are established, readers can apply them to solve other control problems of partial differential equations.

This book begins with an introduction on continuum mechanics and a derivation of the linear partial differential equations for sound waves in fluids and elastic waves in solids. There is a brief chapter on the wave equations of electrodynamics. This is followed by a description of plane wave solutions and a discussion of concepts like reflection, refraction, polarization and the role of boundary conditions.The second part of the book deals with the theory and applications of distributions and Fourier transforms. Furthermore, dispersion, the method of stationary phase, Kramers-Kronig relations and various examples including surface waves on liquids are discussed.This text is unique because it emphasizes the use of distributions to analyze the solutions of the wave equation. The treatment of continuum mechanics is self-contained, as well as the discussion on distributions and Fourier transforms. In addition, many classical methods of theoretical physics are thoroughly discussed, e.g. the use of Green functions and multipole expansions.

​​​​​​This book is devoted to fully developing and comparing the two main approaches to the numerical approximation of controls for wave propagation phenomena: the continuous and the discrete. This is accomplished in the abstract functional setting of conservative semigroups.The main results of the work unify, to a large extent, these two approaches, which yield similaralgorithms and convergence rates. The discrete approach, however, gives not only efficient numerical approximations of the continuous controls, but also ensures some partial controllability properties of the finite-dimensional approximated dynamics. Moreover, it has the advantage of leading to iterative approximation processes that converge without a limiting threshold in the number of iterations. Such a threshold, which is hard to compute and estimate in practice, is a drawback of the methods emanating from the continuous approach. To complement this theory, the book provides convergence results for the discrete wave equation when discretized using finite differences and proves the convergence of the discrete wave equation with non-homogeneous Dirichlet conditions. The first book to explore these topics in depth, "On the Numerical Approximations of Controls for Waves" has rich applications to data assimilation problems and will be of interest to researchers who deal with wave approximations.​