GRAVITY-TYPE INTERACTIVE MARKOV-MODELS .2. LYAPUNOV STABILITY OF STEADY-STATES

In Part I of this paper (Smith and Hsieh, 1997) a programming formulat ion of steady states was developed for gravity-type interactive Markov chains in terms of their associated spatial-flow chains. These result s are here applied to analyze the stability properties of interactive Markov chains. In particular, the objective function for this programm ing formulation is shown to constitute a Lyapunov function for an appr opriately defined continuous-time version of spatial-flow chains. The Lyapunov stability properties of these spatial flows are then shown to yield corresponding stability properties for the continuous-time vers ions of interactive Markov chains. In particular, these processes alwa ys exhibit global convergence to steady states. Finally, it is shown t hat when steady states are unique, these convergence results are inher ited by those interactive Markov chains that are 'sufficiently close' to their continuous-time versions.