Asymptotic properties of stochastic feedback systems: A functional approach

The asymptotic properties of general stochastic feedback systems are examined in an operator theoretic framework. The intent is to extend the existing input-output stability theory in the presence of stochastic disturbances. The development accordingly avoids the use of Lyapunov arguments and the analytic theory of Markov processes as applied to systems with a state realization. The analysis uses the Prohorov topology as a measure of boundedness for the systems, and the criteria developed for the system operators assure the preservation of weak convergence by the feedback system as mapping from input to output. The criteria involve certain induced norms of the system functions as operators on function spaces and in this manner the theory is reminiscient of the deterministic stability theory. An example consisting of a nonlinear feedback system with a white noise gain element is given to illustrate this relationship explicitly.