The Wave Equation: Control and Numerics∗

In these Notes we make a self-contained presentation of the theory that has been developed recently for the numerical analysis of the controllability properties of wave propagation phenomena and, in particular, for the constant coefficient wave equation. We develop the so-called discrete approach. In other words, we analyze to which extent the semidiscrete or fully discrete dynamics arising when discretizing the wave equation by means of the most classical scheme of numerical analysis, shear the property of being controllable, uniformly with respect to the mesh-size parameters and if the corresponding controls converge to the continuous ones as the mesh-size tends to zero. We focus mainly on finite-difference approximation schemes for the one-dimensional constant coefficient wave equation. Using the well known equivalence of the control problem with the observation one, we analyze carefully the second one, which consists in determining the total energy of solutions out of partial measurements. We show how spectral analysis and the theory of non-harmonic Fourier series allows, first, to show that high frequency wave packets may behave in a pathological manner and, second, to design efficient filtering mechanisms. We also develop the multiplier approach that allows to provide energy identities relating the total energy of solutions and the energy concentrated Sylvain Ervedoza CNRS ; Institut de Mathématiques de Toulouse UMR 5219 F-31062 Toulouse, France, Université de Toulouse ; UPS, INSA, INP, ISAE, UT1, UTM ; IMT F-31062 Toulouse, France e-mail: [email protected] Enrique Zuazua BCAM Basque Center for Applied Mathematics, Bizkaia Technology Park, B.500, E48160 Derio, Basque Country, Spain. Ikerbasque Research Professor, Ikerbasque Basque Foundation for Science, E48011 Bilbao, Basque Country, Spain. e-mail: [email protected]. ∗ This work was supported by the ERC Advanced Grant FP7-246775 NUMERIWAVES, the Grant PI2010-04 of the Basque Government, the ESF Research Networking Program OPTPDE and Grant MTM2008-03541 of the MICINN, Spain. The first author acknowledges the hospitality and support of the Basque Center for Applied Mathematics where part of this work was done.