We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of

A correlation between fuzzy logic systems (FLS) and zeroing neural networks (ZNN) design is investigated. It is shown that the gain parameter included in ZNN design can be dynamically adjusted over time by means of an appropriate value derived as the output of a properly defined FLS, which includes appropriately defined membership functions and fuzzy logic rules. Dynamical systems which are applicable to time-varying rank-deficient matrices are proposed. Convergence properties are investigated and illustrative simulation experiments are performed. Presented simulation experiments confirm the superiority of the FLS proposed in this article with respect to previously proposed FLS for dynamic adjustment of gain parameters. Furthermore, the superiority of the FLS-based ZNN model over the corresponding ZNN models based on the classical approach in defining the varying-gain parameter is demonstrated.

This paper presents a recurrent neural network (RNN) for computing the Drazin inverse of a real matrix in real time. This recurrent neural network (RNN) is composed of n independent parts (subnetworks), where n is the order of the input matrix. These subnetworks can operate concurrently, so parallel and distributed processing can be achieved. In this way, the computational advantages over the existing sequential algorithms can be attained in real-time applications. The RNN defined in this paper is convenient for an implementation in an electronic circuit. The number of neurons in the neural network is the same as the number of elements in the output matrix, which represents the Drazin inverse. The difference between the proposed RNN and the existing ones for the Drazin inverse computation lies in their network architecture and dynamics. The conditions that ensure the stability of the defined RNN as well as its convergence toward the Drazin inverse are considered. In addition, illustrative examples and examples of application to the practical engineering problems are discussed to show the efficacy of the proposed neural network.

The problem of solving linear equations is considered as one of the fundamental problems commonly encountered in science and engineering. In this article, the complex-valued time-varying linear matrix equation (CVTV-LME) problem is investigated. Then, by employing a complex-valued, time-varying QR (CVTVQR) decomposition, the zeroing neural network (ZNN) method, equivalent transformations, Kronecker product, and vectorization techniques, we propose and study a CVTVQR decomposition-based linear matrix equation (CVTVQR-LME) model. In addition to the usage of the QR decomposition, the further advantage of the CVTVQR-LME model is reflected in the fact that it can handle a linear system with square or rectangular coefficient matrix in both the matrix and vector cases. Its efficacy in solving the CVTV-LME problems have been tested in a variety of numerical simulations as well as in two applications, one in robotic motion tracking and the other in angle-of-arrival localization.