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

Stabilizability of Vector Systems with Uniform Actuation Unpredictability