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

TL;DR

The work addresses robustness of optimal control for general evolution equations to spatially localized perturbations by introducing domain-uniform stabilizability and detectability, and proves that perturbations decay exponentially in space when these properties hold uniformly across domain sizes. It develops a comprehensive transport-equation analysis (constant and space-dependent velocity) and extends the results to the wave equation via a damped-transport representation, providing explicit necessary and sufficient conditions for domain-uniform stabilization in terms of control-domain geometry (intervals and measures). Theoretical results are complemented by numerical examples in one spatial dimension, illustrating how equidistant, distributed control regions yield domain-size–independent decay, while localized single-interval damping does not. The findings offer a principled framework for robust, scalable optimal control of hyperbolic PDEs with localized disturbances, with potential applications to sensor/actuator network design and predictive control. Future work suggests extending the framework to PDE networks and more general perturbation models, leveraging the domain-uniform stability/detectability paradigm to guide discretization and control synthesis.

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.

Figures (5)

• Figure 1: Initial condition function (blue) of \ref{['TransportOCP']} with single control interval (red) and distance $| (\Omega \setminus (\Omega _c \cup \mathrm{supp}(\varepsilon))|$ between perturbation and control domain (grey)
• Figure 2: Optimal state (blue, space variable $\omega$) for two different control domains (red) and domain sizes ($T = 2.5$, $\alpha = 0.125$)
• Figure 3: Spatial decay of the optimal state on different domains for a single control interval (top) and equidistant control intervals (bottom)
• Figure 4: Relation between the domain size and the $L^2(0,T;[0,L])$-norm of the optimal state (blue) respectively costate (green, dotted) for parameters $T = 5$, $\alpha = 0.125$
• Figure 5: Optimal state (top) and control (bottom) for different control weights with equidistant control intervals (red) for $T = 5$

Theorems & Definitions (72)

• Definition 1
• Remark 2
• Definition 3: Exponential localization
• Example 5: Distributed control
• Example 7: First-order problems: One-dimensional transport equation
• Example 8: Second-order problems: Multi-dimensional wave equation
• Remark 9
• Theorem 10
• proof
• Remark 11