  In this work, we study dynamic programming (DP) algorithms for partially
observable Markov decision processes with jointly continuous and discrete
state-spaces. We consider a class of stochastic systems which have coupled
discrete and continuous systems, where only the continuous state is observable.
Such a family of systems includes many real world systems, for example,
Markovian jump linear systems and physical systems interacting with humans. A
finite history of observations is used as a new information state, and the
convergence of the corresponding DP algorithms is proved. In particular, we
prove that the DP iterations converge to a certain bounded set around an
optimal solution. Although deterministic DP algorithms are studied in this
paper, it is expected that this fundamental work lays foundations for advanced
studies on reinforcement learning algorithms under the same family of systems.
