Exact solutions and convergence of gradient based dynamical systems for computing outer inverses

摘要

This paper investigates convergence properties of gradient neural network (GNN) and GNN-based dynamical systems for computing generalized inverses. The main results are exact analytical solutions for the state matrices of the corresponding GNN and GNN-based dynamical systems. The exact solutions are given in each time instant, which enables to express final results as appropriate limiting expressions. This enables more rigorous convergence analysis in terms of exact solutions. Finally, trajectories of state variables in considered dynamical systems can be generated by avoiding time-consuming numerical solving matrix differential equations inside the selected time interval. Using the fact that the stated dynamical systems are in the essence systems of linear equations, explicit solutions can be obtained using known techniques of ordinary differential equations. Main results of the paper are further transformations of obtained exact solutions using main properties of generalized inverses and linear algebra tools. It is important to mention that the convergence results are expressed in terms of expressions involving outer inverses. Several examples are presented and a closed-form expressions for the solutions are given and implemented in package Mathematica. These are compared with the numerical solutions obtained by Matlab Simulink implementation.