| Summary: | 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.
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