Explicit solution of second-order delayed discrete equations
Nazim I. Mahmudov
TL;DR
The paper addresses explicit solvability of inhomogeneous second-order delayed difference equations with noncommutative matrix coefficients by introducing delayed matrix sine/cosine functions defined via a $\mathcal{Z}$-transform framework. It constructs the delayed sine/cosine using a determining matrix $Q(t;s)$ and proves that the solution can be written in closed form as $y(t) = \text{Cos}^{A,B}(t)\varphi(0) + \text{Sin}^{A,B}(t)\Delta\varphi(0) + \sum_{i=-m}^{-1} \text{Sin}^{A,B}(t-i-m-1) B\varphi(i) + \sum_{j=0}^{t-2} \text{Sin}^{A,B}(t-j-1) f(j)$, with the Cos/Sin functions satisfying standard delay-differential-like relations $\Delta\text{Cos}^{A,B}(t) = -A\text{Sin}^{A,B}(t) - B\text{Sin}^{A,B}(t-m)$ and $\Delta\text{Sin}^{A,B}(t) = \text{Cos}^{A,B}(t)$. This provides a concrete tool for analyzing and implementing explicit solutions in iterative learning control and related discrete-time delay systems. The methodology leverages noncommutative binomial expansions and $\mathcal{Z}$-transform techniques to handle matrix delays and noncommuting coefficients, highlighting the theoretical and potential practical impact in control and systems theory.
Abstract
A system of inhomogeneous second-order difference equations with linear parts given by noncommutative matrix coefficients are considered. Closed form of its solution is derived by means of newly defined delayed matrix sine/cosine using the Z-transform and determining function.