Stability radii of positive higher order difference systems under fractional perturbations (Q987438)

The authors consider the robustness of stability of the positive higher order difference systems described by higher order difference equations of the form \[ A_{N}x(t+N)=A_{N-1}x(t+N-1)+\dots+A_0x(t),\quad t\in\mathbb N, \] where they suppose \(\det A_N\neq0\), under fractional perturbations of the form \[ A_i\to A_i+D_i(I-\Delta_i M_i)^{-1}\Delta_iE_i \] and \[ A_i\to A_i+\sum_{j=1}^{K}\delta_{ij}B_{ij}, i\in\bar N \] where \(B_{ij}, i\in\bar N:=\{1,2,\dots,N\}, j\in\bar K:=\{1,2,\dots,K\}\) are given matrices and \(B_{ij}, i\in\bar N, j\in\bar K\) are scalar parameters, where the matrices \(D_{i}, E_{i}, M_{i}, i\in\bar N\) are given matrices defining the structure of the perturbations and \(\Delta_{i}, i\in\bar N; \delta_{ij}, i\in\bar N, j\in\bar K\) are unknown matrices and scalar parameters, respectively. It is shown that real and complex stability radii coincide and can be computed by a simple formula. An example is given to illustrate the results.