Detectability of linear stochastic impulsive systems with state-dependent noises

This paper focuses on the characterization of moment detectability and almost sure detectability for linear stochastic impulsive systems. The concepts and characterizations of the $2p$th moment detectability and almost sure detectability are first introduced. Two expanded systems, the state-expanded stochastic system and the moment equation, are established by utilizing the generalized $H$-representation technique. A series of geometric and algebraic conditions for the $2p$th moment detectability of the linear stochastic impulsive systems are established by analyzing the 2nd moment detectability of the state-expanded system and the complete detectability of the moment equation. Subsequently, by introducing a strictly positive-definite quasi-periodic composite polynomial Lyapunov candidate, almost sure detectability criteria are framed in terms of linear matrix inequalities, showing that almost sure detectability is less conservative than moment detectability. Finally, two numerical examples with comparisons are provided to illustrate the proposed theoretical results.