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

This paper introduces a new theoretical framework for the analysis of dynamical
and control systems, composed of two main concepts: (i) Relativistic Stability, in
which stability is explicitly treated as a property that may depend on reference-frame
transformations; and (ii) Control Space, a natural or structural mechanism that induces
stability without explicitly designed feedback controllers.
On the Relativistic Stability side, we formalize reference transformations as smooth
diffeomorphisms acting on the state space and show, using Lyapunov methods and
energy-based reasoning, that classical Lyapunov stability is in general not invariant
under non-orthogonal transformations. A central theorem characterizes how the Jaco
bian 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 a
distorted frame. This provides a mathematically precise notion of observer-dependent
stability that complements contraction, incremental stability, and passivity-based in
variance results.