  We consider the optimal control of linear systems over wireless MIMO fading
channels, where the MIMO wireless fading and random access of the remote
controller may cause intermittent controllability or uncontrollability of the
closed-loop control system. We formulate the optimal control design over random
access MIMO fading channels as an infinite horizon average cost Markov decision
process (MDP), and we propose a novel state reduction technique such that the
optimality condition is transformed into a time-invariant reduced-state Bellman
optimality equation. We provide the closed-form characterizations on the
existence and uniqueness of the optimal control solution via analyzing the
reduced-state Bellman optimality equation. Specifically, in the case that the
closed-loop system is almost surely controllable, we show that the optimal
control solution always exists and is unique. In the case that MIMO fading
channels and the random access of the remote controller destroy the closed-loop
controllability, we propose a novel controllable and uncontrollable positive
semidefinite (PSD) cone decomposition induced by the singular value
decomposition (SVD) of the MIMO fading channel contaminated control input
matrix. Based on the decomposed fine-grained reduced-state Bellman optimality
equation, we further propose a closed-form sufficient condition for both the
existence and the uniqueness of the optimal control solution. The closed-form
sufficient condition reveals the fact that the optimal control action may still
exist even if the closed-loop system suffers from intermittent controllability
or almost sure uncontrollability.
