On stochastic stabilization of sampled systems

Abstract

Sample-and-hold refers to implementation of control policies in a digital mode, in which the dynamical system is treated continuous, whereas the control actions are held constant in predefined time steps. In such a setup, special attention should be paid to the sample-to-sample behavior of the involved Lyapunov function. This paper extends on the stochastic stability results specifically to address for the sampleand-hold mode. We show that if a Markov policy stabilizes the system in a suitable sense, then it also practically stabilizes it in the sample-and-hold sense. This establishes a bridge from an idealized continuous application of the policy to its digital implementation. The central results applies to dynamical systems described by stochastic differential equations driven by the standard Brownian motion. Some generalizations are discussed, including the case of non-smooth Lyapunov functions, given that the controls are bounded or, alternatively, the driving noise possesses bounded realizations. A brief overview of models of such processes is given.