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In this paper, a novel radial basis function (RBF) neural network is proposed and applied successively for online stable identification and control of nonlinear discrete-time systems. The proposed RBF network is a one hidden layer neural network (NN) with its all parameters being adaptable. The RBF network parameters are optimized by gradient descent method with stable learning rate whose stable convergence behavior is proved by Lyapunov stability approach. The parameter update is succeeded by a new strategy adapted from Levenberg-Marquardth (LM) method. The aim of construction of the proposed RBF network is to combine power of the networks which have different mapping abilities. These networks are auto-regressive exogenous input model, nonlinear static NN model and nonlinear dynamic NN model. To apply the model to control of the nonlinear systems, a known sliding mode control is applied to generate input of the system. From simulations; it is sown that the proposed network is an alternative model for identification and control of nonlinear systems with accurate results.