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In this paper we report the control and synchronization of chaos in a Memristive Murali-Lakshmanan-Chua circuit. This circuit, introduced by the present authors in 2013, is basically a non-smooth system having two discontinuity boundaries by virtue of it having a flux controlled active memristor as its nonlinear element. While the control of chaos has been effected using state feedback techniques, the concept of adaptive synchronization and observer based approaches have been used to effect synchronization of chaos. Both of these techniques are based on state space representation theory which is well known in the field of control engineering. As in our earlier works on this circuit, we have derived the Poincaré Discontinuity Mapping (PDM) and Zero Time Discontinuity Mapping (ZDM) corrections, both of which are essential for realizing the true dynamics of non-smooth systems. Further we have constructed the observer and controller based canonical forms of the state space representations, have set up the Luenberger observer, derived the controller gain vector to implement state feedback control and calculated the gain matrices for switch feed back and finally performed parameter estimation for effecting observer based adaptive synchronization. Our results obtained by numerical simulation include time plots, phase portraits, estimation of the parameters and convergence of errors graphs and phase plots showing complete synchronization.