Perturbations in PDE-constrained optimal control decay exponentially in space

For linear-quadratic optimal control problems (OCPs) governed by elliptic and parabolic partial differential equations (PDEs), we investigate the impact of perturbations on optimal solutions. Local perturbations may occur, e.g., due to discretization of the optimality system or {disturbed} problem data. Whereas these perturbations may exhibit global effects in the uncontrolled case, we prove that the ramifications are exponentially damped in space under stabilizability and detectability conditions. To this end, we prove a bound on the optimality condition's solution operator that is uniform in the domain size. Then, this uniformity is used in a scaling argument to show the exponential decay of perturbations in space. We numerically validate and illustrate our results by solving OCPs involving Helmholtz, Poisson, and advection-diffusion-reaction equations.