Detectability under impulse differential inclusions

This paper adapts to the case of impulse and hybrid control systems the results obtained by Aubin, Biechi & Pancanti on "detectability" of solutions of usual control systems. Measurements of the state, described by a detectability tube, that may be quantized, are gathered along time. The detector associates at each time with any state satisfying the given measurement the (possibly) empty set of the initial states from which starts a solution that arrives at this state while satisfying the measurements. This detector is then studied by tools of viability theory, and shown to be a solution to a system of Hamilton-Jacobi-Bellman partial differential inclusions satisfying supplementary conditions (that can be regarded as the vectorial analogue of Bensoussan-Lions "quasi-variational inequalities" in impulse optimal control. The derivatives of the detector provide the regulation map governing the motives of the detectable runs.