Stability of multidimensional nonnegative systems (Q1582329)

The stability of the linear mD system \[ x(i) =\sum^m_{l=1} A_lx(i- e_l), \] where \(x(i)\in \mathbb{R}^n\), \(A_l\in \mathbb{R}^{n\times n}\), \(i= (i_1,\dots, i_m)\), \(i- e_l= (i_1,\dots, i_l- 1,\dots, i_m)\) is analyzed by the use of a localized Lyapunov function. It is shown that if the spectral radius of \(B= \sum^m_{l= 1}|A_l|\), \(\rho(B)<1\), then the system is diagonally strongly stable and any state transition of the system decreases step-by-step with respect to an appropriately constructed localized diagonal Lyapunov function. It is an extension of a known result on the stability of nonnegative matrices.