Control and Homogenization of a System of Coupled Parabolic Equations With an Oscillating Coefficient

This article investigates the uniform null controllability problem for a system of coupled parabolic equations with a periodic oscillating coefficient. Our approach combines spectral analysis and Carleman estimates. First, we analyze the spectral properties of an elliptic operator with an oscillating coefficient to control the low frequencies. Then the system is allowed to evolve freely to achieve the required decay. In the third step, we establish a Carleman estimate that leads to a suitable observability result. By combining the three steps, we prove the uniform null controllability of the system, which is then used to homogenize the associated coupled parabolic system.