Sufficient Conditions for Local Observability on Time Scales

We extend Sontag’s continuous-dependence and differentiability theorem and his observable linearization theorem to nonlinear systems on an arbitrary Hilger time scale, in a delta-Caratheodory framework. We prove a variation-of-constants formula requiring neither regressivity, nor rd-continuity, nor bounded graininess. We establish continuous dependence on initial data and inputs, and Frechet differentiability of the state map. We prove the observable linearization theorem: if the linearization along a trajectory is observable, then the nonlinear system is locally observable at the trajectory’s initial state. The input that generates the trajectory is the witness input: it distinguishes every pair of distinct nearby states at one of finitely many times fixed in advance. A Gramian test and two rank conditions make the linearization hypothesis checkable. Their hypotheses are weaker than those in the literature in four directions: regressivity, rd-continuity, bounded graininess and time-scale unboundedness. Both rank conditions carry an in-window sampling hypothesis that the published rank conditions omit. We show by counterexample that it cannot be omitted. The linearization test requires only a first derivative in the state and the input. It remains available where the iterated output derivatives required by the Hermann–Krener rank test and its time-scale generalizations do not exist.