Uniform null-controllability for the one-dimensional heat equation with, rapidly oscillating coefficients

Abstract

We consider the 1-d heat equation with a rapidly oscillating periodic, C-2 density in a bounded interval with Dirichlet boundary conditions. We analyze the problem of the null-controllability with a control acting on one of the extremes of the interval. We prove that, when the period tends to zero and the density weakly converges to its average, the boundary controls converge to a control of the limit, constant coefficient, heat equation. This result is in contrast with the divergent behavior of the null controls for the wave equation with rapidly oscillating coefficients. The proof combines Carleman's inequalities, the theory of series of real exponentials and a control estrategy in three steps in which: we first control the low frequencies, then we let the equation to evolve freely and, finally, we control to zero the whole solution.