Uniform null-controllability for the one-dimensional heat equation with rapidly oscillating periodic density

Abstract

We consider the 1-d heat equation with rapidly oscillating periodic density in a bounded interval with Dirichlet boundary conditions. When the period tends to zero and the density weakly converges to its average we prove that the boundary controls converge to a control of the limit, constant coefficient heat equation when the density is C-2.

The proof is based on a control strategy in three steps in which: we first control the low frequencies of the system, we then let the system to evolve freeely and, finally, we control to zero the whole solution. We use the theory of real exponentials to analyze the low frequencies and Carleman inequalities to control the whole solution.

The result is in constrast with the divergent behavior of the null controls for the wave equation with rapidly oscillating coefficients.