An Input–Output Approach to Structured Stochastic Uncertainty in Continuous Time

ABSTRACT We consider the continuous‐time setting of linear time‐invariant (LTI) systems in feedback with multiplicative stochastic uncertainties. The primary objective of this article is to establish conditions for mean‐square stability using a purely input–output approach, avoiding the need for state space realizations. This approach broadens the applicability to a wider range of models, including infinite‐dimensional systems and systems with delays. The input–output approach leads to uncovering new tools such as stochastic block diagrams, which are closely related to stochastic integral equations rather than the more commonly used stochastic differential equations. We explore various stochastic interpretations, including Itô and Stratonovich, and develop block diagram conversion schemes to transition between these interpretations. The MSS conditions are given in terms of the spectral radius of a matrix operator that takes different forms when different stochastic interpretations are considered.