Numerical Approximation of the Boundary Control for the Wave Equation with Periodic Oscillating Coefficients

We consider the boundary control of a string with a rapidly oscillating bounded periodic density of period \(\varepsilon \to 0\) . It is well known that, for some given initial data the control at one extreme might not be uniformly bounded as \(\varepsilon \to 0\) . This is due to the singular behavior of the spectra when the frequency of the associated eigenfunctions is of the order of \(\varepsilon ^{-1}\) . However, if we are only interested in controlling the projection of the solution on the finite dimensional space generated by the low frequencies then the control can be chosen uniformly bounded as \(\varepsilon \to 0\) . It turns out that this control can be approximated by the control associated with a limit system, where the oscillating coefficient is replaced by its average. However, not much is known about this convergence and any possible correctors formula. Here we present a numerical method to approximate this control based on the asymptotics of the spectra.