Admissible Region Theory: A Unified Mathematical Framework for Stability, Control, and Optimization

Many mathematical and engineering disciplines analyse systems by studying the behaviour of trajectories within a defined state space. However, these analyses often treat stability, control, optimization, and constraint satisfaction as separate problems. This paper proposes a unifying interpretation: each of these disciplines is fundamentally concerned with identifying and maintaining system trajectories within an admissible region of state space. The admissible region represents the subset of states compatible with governing constraints and system continuation. By formalising this concept, numerical stability, control systems, optimization procedures, signal processing constraints, and phase-space dynamics can be interpreted as different operational perspectives on the same structural object. Description This paper introduces the concept of an admissible region as a unifying mathematical structure underlying several disciplines including numerical stability analysis, control theory, optimization, and dynamical systems. By identifying the subset of state space in which valid system trajectories must remain, the paper demonstrates that many apparently distinct mathematical methods are different operational mechanisms for maintaining system behaviour within admissible constraints. The framework provides a structural interpretation linking stability, control, optimization, and phase-space dynamics within a single conceptual model.