  This paper explores the fundamental limits of a simple system, inspired by
the intermittent Kalman filtering model, where the actuation direction is drawn
uniformly from the unit hypersphere. The model allows us to focus on a
fundamental tension in the control of underactuated vector systems -- the need
to balance the growth of the system in different dimensions.
  We characterize the stabilizability of $d$-dimensional systems with symmetric
gain matrices by providing tight necessary and sufficient conditions that
depend on the eigenvalues of the system. The proof technique is slightly
different from the standard dynamic programming approach and relies on the fact
that the second moment stability of the system can also be understood by
examining any arbitrary weighted two-norm of the state.
