Non-Autonomous Degenerate Parabolic Equations with Robin Boundary Conditions: Carleman Estimates and Null-Controllability

The Earth's climate system naturally adjusts to maintain a balance between the energy received from the Sun and the energy reflected back into space, a concept known as the "Earth's radiation budget". However, this balance has been disrupted by human activities, leading to global warming. Starting from the energy balance model proposed by Budyko and Sellers, and considering a time-dependent diffusion coefficient, we prove the null-controllability of non-autonomous degenerate parabolic problems, in the sense that the Earth achieves a desired temperature, by finding new Carleman estimates for the non-homogeneous adjoint problems. At the degeneracy point, we impose Robin boundary condition which is appropriate for modeling heat transfer at the Earth's surface. Moreover, we provide the equivalence between null-controllability and observability inequality for the non-autonomous case. At the end, we present some extensions and open problems.