Stability, convergence, and performance of an adaptive control algorithm applied to a randomly varying system

The stability and performance of a stochastic adaptive control algorithm applied to a randomly varying linear system is investigated. Using techniques from the theory of Markov chains, it is shown that loss functions on the input-output process converge to their expectation with respect to an invariant probability. It is shown that the convergence is geometric, establishing a form of stochastic exponential asymptotic stability for the closed-loop system. Further results include central limit theorems and the law of large numbers for the input-output and parameter processes and near consistency and optimality in the case where the disturbances are small.<>