Uniform null controllability of a parabolic equation with rapidly oscillating periodic coefficients

Abstract

We consider a parabolic equation with fast oscillating periodic coefficients, and an interior control in a bounded domain. First, we prove sharp convergence estimates depending explicitly on the initial data for the corresponding uncontrolled equation; these estimates are new in a bounded domain, and their proof relies on a judicious smoothing of the initial data. Then we use those estimates to prove that the original equation is uniformly null controllable, provided a carefully chosen extra vanishing interior control is added to that equation. This uniform null controllability result is the first in the multidimensional setting for parabolic equations with oscillating coefficients. Finally, we prove that the sequence of null controls converges to the optimal null control of the homogenized equation when the period tends to zero.