Carleman Inequalities for the Heat Equation with Fourier Boundary Conditions: Applications to Null Controllability Problems

In this work, we establish a Carleman inequality for the heat equation with Fourier boundary conditions of the form ${\mathrm{\partial }}_{\nu }y+by=f{1}_{\gamma }$, where the control acts on a small portion $\gamma$ of the boundary. We apply this inequality to address the null controllability problem with boundary control supported on this small region. An explicit solution to this problem is obtained via a system of coupled parabolic equations. Based on these results, we propose an iterative numerical method to solve the coupled system.