Many complex systems operate under intrinsic instability while remaining operational over long time horizons, challenging classical stabilization-based control paradigms. This paper introduces the notion of structural non-stabilizability, in which stabilization is excluded by the structure of the evolution generator rather than by limitations of actuation or information. We show that fixed-point stabilization, invariant mode control, asymptotic convergence, and trajectory tracking are conceptually ill-posed in this regime. Control is reformulated as the regulation of temporal and statistical properties, including residence times, recurrence frequencies, and probabilities of persistent excursions. The framework imposes constraints on admissible control problem formulations rather than prescribing specific control laws. By separating observation, evaluation, and intervention, it provides a general and domain-independent foundation for control under intrinsic instability. The theory explains how sustained operability is possible without stabilization and delineates the structural limits of control.
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