Exact Schur-Stability Regions and Computational Controller Selection for Delayed Scalar Discrete-Time Systems

Delayed feedback converts local stabilization of a scalar map with delay τ into a degree-(τ+1) Schur-stability problem. Using the classical Kuruklis criterion as the analytical starting point, we derive exact Schur-stability regions in the multiplier–gain plane for fixed-point feedback control (FPFC), time-delayed feedback control (TDFC), and nonlinear predictive feedback control (NFC) as pullbacks of a universal Schur region. A direct unit-circle parametrization yields the curved boundary and supports exact results on same-parity nesting, large-delay intersections, stabilizable-multiplier projections, delay budgets, and necessary-and-sufficient common-gain conditions under interval uncertainty. These results lead to reproducible algorithms for certified pointwise classification, O(N) boundary generation, endpoint-based robust-gain testing, and gain-constrained controller selection from complete feasible-gain intervals. Non-explicit calculations reduce to monotone scalar bisection, avoiding repeated characteristic-root calculations in routine region construction. Independent root computations at 114,000 validation points, including delays up to τ=100, show no classification mismatch. Logistic-map examples and a non-polynomial tumour-growth map illustrate the local stabilization and controller-selection implications. The resulting framework provides an exact analytical and algorithmic treatment of arbitrary-delay local stabilization in scalar discrete-time systems.