Energy Decay in Measure Time: HUM Observability, Product-Exponential Envelopes, and GCC Calibration

Abstract

We prove that for impulsive exposure patterns there is no uniform exponential energy law in wall-clock time t, which explains why past t-based unifications of continuous damping with impulses fail. We therefore replace t by a measure-valued clock, sigma, that aggregates absolutely continuous exposure and atomic doses within a single Lyapunov ledger. On this ledger we prove an observability-dissipation principle in the sense of the Hilbert Uniqueness Method (HUM): there exists a structural constant c_sigma > 0 such that the energy decays at least at a product-exponential rate with respect to sigma. When sigma = t, the statement reduces to classical exponential stabilization with the same constant. For the damped wave under the Geometric Control Condition (GCC), the constant is calibrated by the usual observability and geometric factors. The framework yields a monotonicity principle ("more sigma-mass implies faster decay") and unifies intermittent regimes where quiescent intervals are punctuated by impulses. As robustness, secondary to the main contribution, the same decay law persists under structure-compatible discretizations and along compact variational limits; a stochastic extension supplies expectation and pathwise envelopes via the compensator. The contribution is a qualitative dynamics backbone: observability implies sigma-exponential decay with sharp constants.