General Explicit Solution of Planar Weakly Delayed Linear Discrete Systems and Pasting Its Solutions

Abstract

Planar linear discrete systems with constant coefficients and delays $x(k+\mathrm{1})=Ax(k)+{\sum }_{l=\mathrm{1}}^n\mathrm{‍}{B}^l{x}_l(k-m_l)$ are considered where $k{\in \mathbb{Z}}_{\mathrm{0}}^{\mathrm{\infty }}:=\{\mathrm{0,1},\dots ,\mathrm{\infty }\}$, $m_{\mathrm{1}},m_{\mathrm{2}},\dots ,m_n$ are constant integer delays, , $A,B^{\mathrm{1}},\dots ,B^n$ are constant $\mathrm{2}\times \mathrm{2}$ matrices, and $x:{\mathbb{Z}}_{{-m}_n}^{\mathrm{\infty }}{\to ℝ}^{\mathrm{2}}$. It is assumed that the considered system is weakly delayed. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension $\mathrm{2}(m_n+\mathrm{1})$ is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and special delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained.