Operator-Theoretic De Bruijn Functional and Robust Safety Control Framework

This paper presents an operator-theoretic formulation of the De Bruijn functional as a unified quadratic metric on L2(R). The framework establishes equivalence between time-domain localization, frequency-domain representation, and positive self-adjoint operator structure. The functional is embedded into a stochastic control framework with belief-state dynamics, coherentrisk measures, stochastic barrier functions, and robust dynamic programming. The resulting formulation provides a unified mathematical structure for safety assessment in safety-critical systems through localized energy, diagnostic residuals, and risk-weighted deviation metrics.