Representation of solutions of discrete delayed system x(k +1) = Ax(k)+Bx(k −m)+f (k) with commutative matrices

In the investigation performed we give, on half-infinity discrete intervals, formulas for solution of initial problem of linear discrete systems x(k + 1) = Ax(k) + Bx(k − m) + f (k) with constant square matrices A, B such that AB = BA, detA = 0 and with a vector function f (k). Corresponding representations are obtained with the aid of so-called discrete matrix delayed exponential, which permits to represent solutions in a matrix form similarly as for ordinary differential systems with constant matrices, or as well as for differential systems with constant matrices and constant delay. © 2005 Elsevier Inc. All rights reserved.