Exact Boundary Controllability of the Spherical Heat Equation: A Rigorous Verification via the Method of Manufactured Solutions

This work investigates the exact boundary controllability of the heat equation posed in a spherical domain, using the method of moments. Starting from the spectral decomposition of the radial Laplacian in the weighted space \( L^2_{r^2}(0,R) \), we derive a sequence of moment equations whose solvability is established via biorthogonal sequences, following the Fattorini–Russell theory. The resulting control function is expressed as a series expansion whose convergence in \( L^2(0,t_f) \) is rigorously proved for any initial temperature distribution in \( L^2_{r^2}(0,R) \) and any final time tf exceeding a minimal threshold. The numerical implementation employs linear finite elements in space and the implicit Euler scheme in time. The accuracy of the solver is rigorously verified through the Method of Manufactured Solutions (MMS), confirming optimal convergence rates: second-order in space and first-order in time. Numerical experiments show that the computed control drives the temperature to zero with a residual norm of 1.96×10⁻⁴, consistent with the spatial discretization error. A comparative analysis with two alternative approaches—the Hilbert Uniqueness Method (HUM) and gradient-based optimization—demonstrates that the proposed moment-based strategy achieves an 18.9% reduction in total control energy relative to HUM, making it particularly attractive for energy-constrained thermal control applications.