Stochastic adaptive control and Martingale limit theory

Recently, S.P. Meyn and P.E. Caines (ibid., vol.AC-32, p.220-6, 1987) have used ergodic theory for Markov processes to give the first asymptotic stability analysis of a nontrivial stochastic adaptive control problem. By nontrivial is meant a stochastic adaptive control problem whose parameter variation has finite nonzero power. They correctly observed that the stochastic Lyapunov function methods fail here, because there is no almost sure parameter convergence. It is shown here how Martingale asymptotics can be used to produce many results close to those of Meyn and Caines, as well as to supply some new observations. Strengths and weaknesses of both approaches are discussed