Robust Internal-Time Viability

Internal-Time Viability Theory establishes that, for a class of dynamical systems with moving safety constraints, admissible-set invariance cannot be recovered when constraints are indexed by external clock time. Instead, viability requires an endogenous internal time that governs phase-indexed constraint evolution. An open question, however, concerns the robustness of this formulation in the presence of uncertainty. This paper investigates internal-time viability under bounded disturbances. We consider control systems subject to unknown but bounded perturbations and study whether phase- indexed admissible-set invariance can be preserved. We show that robustness can be achieved by introducing disturbance-aware internal-time dynamics, in which internal-time progression is adaptively slowed near safety boundaries through a tempo margin. This mechanism compensates for worst-case disturbance effects without tightening admissible sets or modifying constraint geometry. A sufficient robust viability condition is derived in the form of a boundary inequality that explicitly accounts for disturbance bounds. A smooth minimal example demonstrates how external-time formulations fail to admit robustness guarantees, while internal-time indexing enables simple and interpretable phase-indexed invariance. The analysis is conservative and focuses on structural safety preservation rather than optimality. This paper is a companion Part II to "Internal-Time Viability Theory"and extends the framework to bounded disturbances. https://doi.org/10.5281/zenodo.18211574