Stochastic weak passivity for weakly stabilizing stochastic systems with nonvanishing noise

A stochastic system loses stability in the sense of probability if its noise port is nonvanishing at the desired state. This further leads to the loss of stochastic passivity if the storage function is simultaneously expected to be positive definite. For this reason, the current work defines a kind of novel stability, called stochastic weak stability covering the convergence of distribution and ergodicity, for characterizing stable behaviors of such a class of stochastic systems. We also define the concept of stochastic weak passivity that admits stochastic passivity only outside a neighborhood of the desired state rather than in the whole state space. Based on it, the stochastic weak passivity theorems are developed to guide control design for stabilizing the stochastic systems with persistent noise in the sense of stochastic weak stability. We demonstrate the validity of these results by applying them to a nonlinear chemical process system. •Provide conditions under which stochastic systems are not stochastically passive.•Propose a weak stability notion, including convergence in distribution & ergodicity.•Propose stochastic weak passivity for systems that are not stochastically passive.•This weak passivity requires passivity outside a neighborhood of a desired point.•Stochastic weak passivity is associated with the weak stability notion.