Stabilizability of Vector Systems with Uniform Actuation Unpredictability

Control strategies for vector systems typically depend on the controller's ability to plan out future control actions. However, in the case where model parameters are random and time-varying, this planning might not be possible. 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.