Quantitative Soft-to-Hard Terminal Constraint Convergence for the Heat Equation

We study an optimal control problem for the heat equation with a prescribed terminal state. To circumvent the difficulty of enforcing a hard terminal constraint, we analyze a penalized formulation and prove that the corresponding optimal controls and terminal states converge to the exact constrained solution as the penalty parameter $\alpha \to \mathrm{\infty }$. We establish explicit quantitative convergence estimates of order $O({\alpha }^{-\theta })$, including the sharp $O(1\mathrm{/}\alpha )$ rate under stronger modal summability assumptions on the terminal mismatch. A finite-dimensional prototype is used to illustrate the underlying projection structure, while numerical illustrations are reported in a companion study.