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

TL;DR

This paper demonstrates that in optimal control of evolution equations, localized spatial perturbations have only local effects, with exponential damping ensuring robustness, supported by theoretical proofs and numerical examples.

Contribution

It establishes domain-uniform stabilizability and detectability conditions for transport and wave equations, revealing exponential damping effects in optimal control.

Findings

Localized perturbations only affect the solution locally

Exponential damping occurs even for unitary semigroups

Numerical examples confirm theoretical results

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.