Relativistic Stability and Control Space: A Framework for Modern Control Theory

This paper introduces a new theoretical framework for the analysis of dynamicaland control systems, composed of two main concepts: (i) Relativistic Stability, inwhich stability is explicitly treated as a property that may depend on reference-frametransformations; and (ii) Control Space, a natural or structural mechanism that inducesstability without explicitly designed feedback controllers.On the Relativistic Stability side, we formalize reference transformations as smoothdiffeomorphisms acting on the state space and show, using Lyapunov methods andenergy-based reasoning, that classical Lyapunov stability is in general not invariantunder non-orthogonal transformations. A central theorem characterizes how the Jacobian geometry of the transformation can change the sign of the Lyapunov derivative,so that a system that is stable in its natural coordinates may appear unstable in adistorted frame. This provides a mathematically precise notion of observer-dependentstability that complements contraction, incremental stability, and passivity-based invariance results.