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 $\partial_νy+by=f1_γ$, where the control acts on a small portion $γ$ 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.
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