On Decompositions of Finite Horizon DP Problems with Linear Dynamics

We consider finite horizon dynamic programming problems with linear dynamics over a finite dimensional, but otherwise arbitrary, state-space. For the case where the cost function is a function only of the state, we consider a decomposition of the underlying state-space under the system transformation: We begin by refining a previously introduced notion of decomposition of the original DP problem in terms of families of suitably defined smaller DP problems. We then introduce a second notion of decomposition, and we derive necessary and sufficient conditions for the existence of such decompositions. Finally, we investigate the relations between these two notions of decompositions, and we show that they are equivalent in certain instances that we explicitly identify.