On subspace decompositions of finite horizon DP problems with switched linear dynamics

We consider finite horizon dynamic programming problems for switched linear dynamics over a finite dimensional but otherwise arbitrary state-space, where the cost function depends only on the state. We start from a decomposition of the underlying state-space as the sum of invariant subspaces under all linear transformations induced by the input, and pose the following question: Assuming the cost function has an additive structure compatible with the state-space decomposition, under what conditions does the cost-to-go function also exhibit an additive structure? We begin by deriving a necessary and sufficient condition for the existence of this additive structure. We then propose a sufficient condition, expressed in terms of the cardinality of the input set: While not necessary, this condition has the advantage of being readily verifiable. Finally, we characterize the resulting reduction in complexity for the case of systems with finite state-spaces, and we conclude with a simple illustrative example.