On subspace decompositions of finite horizon dynamic programming problems

We consider finite horizon dynamic programming problems where the dynamics are linear over a finite dimensional state-space and where the cost function depends only on the state. Starting from a decomposition of the state-space under the relevant linear transformation, we derive conditions under which the dynamic programming problem decomposes into a set of smaller problems that can be solved independently, with their combined solutions yielding the optimal solution of the original problem. For linear systems over finite fields, we show that the resulting reduction in complexity is exponential.