sensitivity analysis , exponential localization , optimal control of partial differential equations
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.
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