  In this paper, we study the problem of control of discrete-time nonlinear
systems in Lure form over erasure channels at the input and output. The input
and output channel uncertainties are modeled as Bernoulli random variables. The
main results of this paper provide sufficient condition for the mean square
exponential stability of the closed loop system expressed in terms of
statistics of channel uncertainty and plant characteristics. We also provide
synthesis method for the design of observer-based controller that is robust to
channel uncertainty. To prove the main results of this paper, we discover a
stochastic variant of the well known Positive Real Lemma and principle of
separation for stochastic nonlinear system. Application of the results for the
stabilization of system in Lure form over packet-drop network is discussed.
Finally a result for state feedback control of a Lure system with a general
multiplicative uncertainty at actuation is discussed.
