The Wave Equation: Control and Numerics

Abstract

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

Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis

“…To be more precise, in [20], solutions of (1.5) were constructed, with frequencies of the order of 1/h, localized around the rays of geometric optics given by the projection on the physical space of the bicharacteristic curves provided by the Hamiltonian 10) or equivalently…”

“…Besides, the control v in (4.5) is the one of minimal L 2 (0, T )-norm. The method above can be adapted to ensure that controls are regular for regular data (see [10,11]). To do that, given a smooth function η : [0, T ] → [0, 1] flat at t = 0 and at t = T so that the following observability inequality is satisfied…”

“…As before, the observability property (4.6) yields the coercivity and strict convexity of J, which therefore has a unique minimizer 8) which is the control of minimal L 2 (0, T ; dt/η(t))-norm among all admissible controls. This method, being well-known to perform better ( [10,11]), is the one we will use in our numerical simulations below.…”

“…But this process may fail to converge as J h may not be uniformly coercive with respect to h. Actually, even worse, as a consequence of Banach Steinhaus theorem, one can show that if the discrete systems (1.5) are not uniformly observable, then there exists initial data in L 2 (0, 1) × H −1 (0, 1) such that the sequence of corresponding discrete controls v h in (4.10) is unbounded as h → 0, see [10,Theorem 8]. It turns out that in the case of a uniform mesh, that is when g(x) = x, uniform observability fails [16].…”

“…We refer to the survey articles [10,28] for an account of the most recent developments. But, until now, most often, the analysis was restricted to uniform meshes.…”