Numerical Approximation of Exact Controls for Waves

1. Front Matter

   Pages i-xvii

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2. Numerical Approximation of Exact Controls for Waves

   • Sylvain Ervedoza, Enrique Zuazua

   Pages 1-48

3. Observability for the 1—d Finite-Difference Wave Equation

   • Sylvain Ervedoza, Enrique Zuazua

   Pages 49-58

4. Convergence of the Finite-Difference Method for the 1—d Wave Equation with Homogeneous Dirichlet Boundary Conditions

   • Sylvain Ervedoza, Enrique Zuazua

   Pages 59-78

5. Convergence with Nonhomogeneous Boundary Conditions

   • Sylvain Ervedoza, Enrique Zuazua

   Pages 79-114

6. Further Comments and Open Problems

   • Sylvain Ervedoza, Enrique Zuazua

   Pages 115-118

7. Back Matter

   Pages 119-122

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​​​​​​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.​

• controllability
• convergence results
• finite differences
• numerical approximation
• observability
• wave equation
• partial differential equations

• Université Paul Sabatier & CNRS, Equipe MIP, Institut de Mathématique de Toulouse, Toulouse Cedex 9, France

  Sylvain Ervedoza

• BCAM-Basque Center for Applied Mathemati, Bilbao, Spain

  Enrique Zuazua

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