On the Limits of Robust Control under Adversarial Disturbances

This paper addresses a fundamental and important question in control: under what conditions does no robust control policy exist that can guarantee constraint satisfaction or finite-time target reachability for a constrained linear system under bounded disturbances? This issue has direct implications for actuator and sensor specification, feasibility of reference tracking, and the design of adversarial attacks in cyber-physical systems. While existing work largely focuses on computing control invariant sets via optimization to certify feasibility, we take the complementary perspective of characterizing explicit sufficient conditions for infeasibility of robust invariance and robust reachability. We derive explicit, closed-form sufficient conditions for infeasibility that relate the admissible disturbance magnitude—modelled as a scaling of a basic set—to the system's spectral structure and the geometry of the constraint sets. These conditions yield analytical insight into how instability, constraint geometry, and disturbance structure interact to preclude robust control, and provide new and practical tools for system analysis.