Proper Policies in Infinite-State Stochastic Shortest Path Problems

We consider stochastic shortest path problems with infinite state and control spaces, a nonnegative cost per stage, and a termination state. We extend the notion of a proper policy, a policy that terminates within a finite expected number of steps, from the context of finite state space to the context of infinite state space. We consider the optimal cost function J*, and the optimal cost function Ĵ over just the proper policies. We show that J* and Ĵ are the smallest and largest solutions of Bellman’s equation, respectively, within a suitable class of Lyapounov-like functions. If the cost per stage is bounded, these functions are those that are bounded over the effective domain of Ĵ . The standard value iteration algorithm may be attracted to either J* or Ĵ , depending on the initial condition.