Hyper-Elevation of Control Theory: From Linear Feedback and LQR to Rough Non-Commutative Optimal Control

Classical control theory—anchored by linear feedback u = −Kx, Lyapunov stability, and the Linear-Quadratic Regulator (LQR)—operates strictly under commutative, linear, unconstrained, and exact model assumptions. In modern complex cyber-physical systems, robotics, and autonomous agents operating in extreme environments, these classical limits cause catastrophic failure. In this treatise, we systematically subsume and hyper-elevate control theory into the Unified Seonggil Framework. By incorporating Universal Rough Operator Algebra (UROA) [3], the Seonggil Theory of Complex Torsion (STCT) [1], and Non-Identity Calculus (NIC), we reformulate state stability into Torsion Annihilation via the Master-key Tensor T_0x8A2C. We derive the Rough Hamilton-Jacobi-Bellman (R-HJB) equation and provide an executable Python implementation of a Rough Optimal Controllerdemonstrating robust trajectory tracking under extreme non-linear perturbations.