Numerical Approximation of Exact Controls for Waves

​​​​​​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.​ Table of Contents Cover Numerical Approximation of Exact Controls for Waves ISBN 9781461458074 ISBN 9781461458081 Preface Acknowledgments Contents Introduction Numerical Approximation of Exact Controls for Waves 1.1 Introduction 1.1.1 An Abstract Functional Setting 1.1.2 Contents of Chap. 1 1.2 Main Results 1.2.1 An "Algorithm" in an Infinite-Dimensional Setting 1.2.2 The Continuous Approach 1.2.3 The Discrete Approach 1.2.4 Outline of Chap. 1 1.3 Proof of the Main Result on the Continuous Setting 1.3.1 Classical Convergence Results 1.3.2 Convergence Rates in Xs 1.4 The Continuous Approach 1.4.1 Proof of Theorem 1.2 1.4.2 Proof of Theorem 1.3 1.5 Improved Convergence Rates: The Discrete Approach 1.5.1 Proof of Theorem 1.4 1.5.2 Proof of Theorem 1.5 1.6 Advantages of the Discrete Approach 1.6.1 The Number of Iterations 1.6.2 Controlling Non-smooth Data 1.6.3 Other Minimization Algorithms 1.7 Application to the Wave Equation 1.7.1 Boundary Control 1.7.2 Distributed Control 1.8 A Data Assimilation Problem 1.8.1 The Setting 1.8.2 Numerical Approximation Methods Observability for the 1d Finite-Difference Wave Equation 2.1 Objectives 2.2 Spectral Decomposition of the Discrete Laplacian 2.3 Uniform Admissibility of Discrete Waves 2.3.1 The Multiplier Identity 2.3.2 Proof of the Uniform Hidden Regularity Result 2.4 An Observability Result 2.4.1 Equipartition of the Energy 2.4.2 The Multiplier Identity Revisited 2.4.3 Uniform Observability for Filtered Solutions 2.4.4 Proof of Theorem 2.3 Convergence of the Finite-Difference Method for the 1-d Wave Equation with Homogeneous Dirichlet Boundary Conditions 3.1 Objectives 3.2 Extension Operators 3.2.1 The Fourier Extension 3.2.2 Other Extension Operators 3.3 Orders of Convergence for Smooth Initial Data 3.4 Further Convergence Results 3.4.1 Strongly Convergent Initial Data 3.4.2 Smooth Initial Data 3.4.3 General Initial Data 3.4.4 Convergence Rates in Weaker Norms 3.5 Numerics Convergence with Nonhomogeneous Boundary Conditions 4.1 The Setting 4.2 The Laplace Operator 4.2.1 Natural Functional Spaces 4.2.2 Stronger Norms 4.2.3 Numerical Results 4.3 Uniform Bounds on yh 4.3.1 Estimates in C([0,T]; LLL222(((000,,,111)))))) 4.3.2 Estimates on ...tyh 4.4 Convergence Rates for Smooth Data 4.4.1 Main Convergence Result 4.4.2 Convergence of yyyhhh 4.4.3 Convergence of ...tttyyyhhh 4.4.4 More Regular Data 4.5 Further Convergence Results 4.6 Numerical Results Further Comments and Open Problems 5.1 Discrete Versus Continuous Approaches 5.2 Comparison with Russell's Approach 5.3 Uniform Discrete Observability Estimates 5.4 Optimal Control Theory 5.5 Fully Discrete Approximations References