Overview
\u200b\u200b\u200b\u200b\u200b\u200bThis book is devoted to fully developing and comparing\xa0the two main approaches to the numerical approximation of controls for wave propagation phenomena: the continuous and the discrete.\xa0This is accomplished\xa0in the abstract functional setting of conservative semigroups.The main results of the work unify, to a large extent, these two\xa0approaches, which yield\xa0similaralgorithms and convergence rates. The discrete approach, however, gives\xa0not only efficient numerical approximations of the continuous controls, but also ensures some partial controllability properties\xa0of the finite-dimensional approximated dynamics. Moreover, it has\xa0the\xa0advantage 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.\xa0To 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\xa0will be of interest to researchers who deal with wave approximations.\u200b
