In many control problems there is only limited information about the actions that will be available at future stages. We introduce a framework where the Controller chooses actions a0, a1, ..., one at a time. Her goal is to maximize the probability that the infinite sequence is an element of a given subset G. The set G, called the goal, is assumed to be a Borel tail set and the Controller's choices at every given time are restricted by an exogeneous homogeneous random process.

We consider several information structures defined by how far ahead into the future the Controller knows what actions will be available. In the special case where all the action sets are singletons (and thus the Controller is a dummy), Kolmogorov’s 0-1 law says that the probability for the goal to be reached is 0 or 1. We construct a number of counterexamples to show that in general the value of the control problem can be strictly between 0 and 1, and derive several sufficient conditions for the 0-1 ``law" to hold.

Joint work with Janos Flesch (Maastricht University), Arkadi Predtetchinski (Maastricht University) and William Sudderth (University of Minnesota).