On the internal stability and asymptotic behavior of 2-D positive systems

Two-dimensional (2-D) positive systems are 2-D state-space models whose variables take only nonnegative values and, hence, are described by a family of nonnegative matrices. The free state evolution of these systems is completely determined by the set of initial conditions and by the pair of nonnegative matrices, (A.B), that represent the shift operators along the coordinate axes. In this paper, internal stability of 2-D positive systems is analyzed and related to the spectral properties of the matrix sum A+B. Also, some aspects of the asymptotic behavior are considered, and conditions guaranteeing that all local states on the same separation set Ct assume the same direction as t goes to infinity, are provided. Finally, some results on the free evolution of positive systems corresponding to homogeneous distributions of the initial local states around a finite mean value, are presented.