  This paper investigates a class of optimal control problems associated with
Markov processes with local state information. The decision-maker has only
local access to a subset of a state vector information as often encountered in
decentralized control problems in multi-agent systems. Under this information
structure, part of the state vector cannot be observed. We leverage ab initio
principles and find a new form of Bellman equations to characterize the optimal
policies of the control problem under local information structures. The dynamic
programming solutions feature a mixture of dynamics associated unobservable
state components and the local state-feedback policy based on the observable
local information. We further characterize the optimal local-state feedback
policy using linear programming methods. To reduce the computational complexity
of the optimal policy, we propose an approximate algorithm based on virtual
beliefs to find a sub-optimal policy. We show the performance bounds on the
sub-optimal solution and corroborate the results with numerical case studies.
