Properties and computation of continuous-time solutions to linear systems

According to the traditional notation, C m ?n (resp. R m ?n ) indicate m ? n complex (resp. real) matrices. Further, rank (A ) , A *, R (A ) and N (A ) denote the rank, the conjugate transpose, the range (column space) and the null space of A ? C m ?n . The index of A ? C n ?n is the minimal k determined by rank ( A k ) = rank ( A k +1 ) and termed as ind (A ) . About the notation and main properties of generalized inverses, we suggest monographs [2,30,42] . The Drazin inverse of We investigate solutions to the system of linear equations (SoLE) in both the time-varying and time-invariant cases, using both gradient neural network (GNN) and Zhang neural network (ZNN) designs. Two major limitations should be overcome. The first limitation is the inapplicability of GNN models in time-varying environment, while the second constraint is the possibility of using the ZNN design only under the presence of invertible coefficient matrix. In this paper, by overcoming the possible limitations, we suggest, in all possible cases, a suitable solution for a consistent or inconsistent linear system. Convergence properties are investigated as well as exact solutions. ? 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates solutions to linear equations using gradient neural network and Zhang neural network designs, overcoming two major limitations. By proposing a method to overcome these limitations, a suitable solution for consistent or inconsistent linear systems is found in all possible cases.