Control Theory Meets POMDPs: A Hybrid Systems Approach

Partially observable Markov decision processes(POMDPs) provide a modeling framework for a variety of sequential decision making under uncertainty scenarios in artificial intelligence (AI). Since the states are not directly observable ina POMDP, decision making has to be performed based on the output of a Bayesian filter (continuous beliefs); hence, making POMDPs intractable to solve and analyze. To overcome the complexity challenge of POMDPs, we apply techniques from control theory. Our contributions are fourfold: (i) We begin by casting the problem of analyzing a POMDP into analyzing the behavior of a discrete-time switched system. Then, (ii) in order to estimate the reachable belief space of a POMDP, i.e., the set of all possible evolutions given an initial belief distribution over the states and a set of actions and observations, we find over-approximations in terms of sub-level sets of Lyapunov-like functions. Furthermore, (iii) in order to verify safety and performance requirements of a given POMDP, we formulate a barrier certificate theorem