Exact and Weak Disturbance Rejection in Localized Continuous Linear Systems

Disturbance rejection in localized continuous linear systems remains challenging due to the interplay between spatial constraints, exact invariance conditions, and discretization effects. Existing methods either lack rigorous guarantees for exact rejection in continuous time or fail to address the subtleties of weak rejection in discrete-time implementations. In this work, we first establish necessary and sufficient conditions for exact disturbance rejection in the continuous-time output case, deriving control laws that enforce output invariance. For the discrete-time output scenario, we resolve the weak rejection problem by proving that—under stabilizability assumptions and an optimal discretization step—the system can achieve practical disturbance attenuation. Our results unify these two regimes, providing explicit design criteria for robust control in localized linear systems.