Local Stability Analysis of Discrete-Time, Continuous-State, Complex-Valued Recurrent Neural Networks With Inner State Feedback

Recurrent neural networks (RNNs) are well known for their capability to minimize suitable cost functions without the need for a training phase. This is possible because they can be Lyapunov stable. Although the global stability analysis has attracted a lot of interest, local stability is desirable for specific applications. In this brief, we investigate the local asymptotical stability of two classes of discrete-time, continuous-state, complex-valued RNNs with parallel update and inner state feedback. We show that many already known results are special cases of the results obtained here. We also generalize some known results from the real-valued case to the complex-valued one. Finally, we investigate the stability in the presence of time-variant activation functions. Complex-valued activation functions in this brief are separable with respect to the real and imaginary parts.