Non-Bypassability and Adversarial Unreachability in Continuous-Time Control Systems

This record collects three closely related mathematical works on non-bypassability and safety enforcement in continuous-time control systems. The first document develops a purely continuous mathematical framework in which a distinguished anchor state cannot be bypassed for a defined class of high-impact controls. Non-bypassability is formalised as unreachability of a forbidden state–control set and, equivalently, as viability and invariance of an admissible graph under a differential inclusion. The construction relies on set-valued dynamics, viability theory, and optional equivalent formulations via projection, quadratic programming, and sweeping processes. The second document studies state-dependent actuator saturation as a physical enforcement mechanism. By imposing a state-dependent actuator feasible set, it is shown that high-impact control inputs are physically unreachable outside a gated state region, even under adversarial disturbances and actuator-side input injection. The resulting safety guarantee is enforced at the physical layer and is independent of controller design or software correctness. The third document presents a closed-loop formulation of non-bypassability, where a realised control operator ensures that a forbidden bypass set is unreachable, equivalently that an admissible state–control set is forward invariant. The result is expressed entirely in terms of closed-loop feasibility and set-valued constraints, with no implementation narrative. Taken together, these works provide a unified continuous-time perspective on non-bypassability, unreachability, and safety enforcement, spanning abstract differential inclusions, closed-loop invariance, and physical actuator constraints.