Nonnegative singular control systems using the Drazin projector
Abstract In this work we study conditions for guaranteeing the nonnegativity of a discrete-time singular control system. A first approach can be found in the literature for general systems, using the whole coefficient matrices. Also, the particular case of matrices of index 1 has been treated by using a block decomposition and the group-projector of the matrix that gives the singularity to the system. In order to complete this study, an analysis of the nonnegativity of a singular control system for matrices having arbitrary index is done by means of the core–nilpotent decomposition. This technique allows us to reduce the size of the original matrices, improving the results where the whole coefficients are involved.
We experimentally realized a waveguide device with alternating positive and negative coupling and show that this geometry is an optical simulator of the conditions found for a massless relativistic particle described by the one-dimensional Dirac-equations.