Spatial exponential decay of perturbations in optimal control of general evolution equations
Simone Göttlich, Benedikt Oppeneiger, Manuel Schaller, Karl Worthmann
Published: 2025/1/21
Abstract
We analyze the robustness of optimally controlled evolution equations with respect to spatially localized perturbations. We prove that if the involved operators are domain-uniformly stabilizable and detectable, then these localized perturbations only have a local effect on the optimal solution. We characterize this domain-uniform stabilizability and detectability for the transport equation with constant transport velocity, showing that even for unitary semigroups, optimality implies exponential damping. We extend this result to the case of a space-dependent transport velocity. Finally we leverage the results for the transport equation to characterize domain-uniform stabilizability of the wave equation. Numerical examples in one space dimension complement the theoretical results.