This paper analyzes the integration of neural networks and linear systems for the identification, state estimation and output feedback control of weakly nonlinear systems. Considering previous knowledge about the system given by approximated linear state-space models, linear observers and linear controllers, training algorithms for the neuro-identification, state neuro-estimation and output feedback neuro-control were derived considering the dynamics of the nonlinear system. It was found that the integrated linear-neuro model can identify the dynamics of the system much more accurately than a purely linear model or a purely neuro model. It was also found that the state estimation and vibration isolation performance of the system with integrated linear-neuro output feedback control is better than the system with linear control or neuro-control.