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We study the stability of a class of networked control systems with hard bounds on the control authority. The plant dynamics are discrete-time, linear, and time-invariant, with stochastic process noise and measurement noise. The controller is designed as a norm-bounded causal history-dependent function of the past outputs perturbed by bounded noise. The resulting control signals are assumed to be transmitted through a lossy channel with packet dropouts. We show that under mild assumptions on the system matrices, the statistics of the process and measurement noise sequences, and the probability of dropouts, it is possible to ensure bounded variance of the system in closed-loop.