It is known that strictly unstable linear systems that are subject to nonvanishing additive stochastic noise with unbounded supports cannot be stabilized by using deterministically bounded control inputs. In this paper, we explore similar impossibility results for scenarios where the expected value of the squared control input norm is subject to constraints and the support of the noise distribution is not necessarily unbounded. Specifically, we consider the stabilization problem with control policies that have bounded time-averaged second moments. We obtain values of such average second moment bounds, below which stabilization is not possible and the second moment of the state diverges regardless of the choice of the control policy and the initial state distribution. The results are illustrated with a numerical example.
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