Stability of stochastic nonlinear systems in cascade with not necessarily unbounded decay rates

This paper develops tools to verify stability and robustness of cascaded nonlinear stochastic systems based on Lyapunov functions. Constituent systems are formulated in terms of integral input-to-state stability (iISS) and input-to-state stability (ISS) which are popular notions for both stochastic and deterministic systems. This paper highlights differences between the stochastic and the deterministic cases. In contrast to deterministic systems, it is demonstrated that assuming ISS systems having unbounded decay rates in dissipation inequalities is restrictive. Taking this fact into account, stability criteria are formulated without assuming unboundedness of decay rates, so that ISS systems with bounded decay rates and iISS systems which are not ISS are covered in a unified manner. Stability criteria for stochastic cascades involve the growth rate conditions at connecting channels. This paper clarifies how noise diffusion fields affect the growth rate conditions and the influence depends on definition of stochastic robustness.