Decentralized minimax control problems with partial history sharing

We consider a decentralized minimax control problem with the partial history sharing information structure. The partial history sharing model is a general decentralized model where (i) controllers sequentially share part of their past data (past observations and control) with each other by means of a shared memory; and (ii) all controllers have perfect recall of the shared data (common information). Instead of modeling the noise variables in dynamics and observations as random variables, we model them as uncertain quantities that take values in some fixed and known finite sets. The objective is to find control strategies that minimize the worst-case cost. We first consider a terminal cost problem. We provide a common information based dynamic program for this decentralized problem. The information state in the dynamic program is the set of feasible values of the current state and local information consistent with the information that is commonly known to all controllers. We then extend the terminal cost problem to incorporate additive costs and common observations.