Spatial exponential decay of perturbations in optimal control of general evolution equations

¹ Chair of Scientific Computing, School of Business Informatics and Mathematics, University of Mannheim, Germany
² Optimization-based Control Group, Institute of Mathematics, Technische Universität Ilmenau, Germany
³ Faculty of Mathematics, Chemnitz University of Technology, Germany

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 21 January 2025
Accepted: 29 December 2025

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.