Explicit general solution of planar linear discrete systems with constant coefficients and weak delays

Josef Diblík,Hana Halfarová

doi:10.1186/1687-1847-2013-50

Josef Diblík, Hana Halfarová

Open Access

https://doi.org/10.1186/1687-1847-2013-50

Copy DOI

Abstract

In this paper, planar linear discrete systems with constant coefficients and two delays are considered where , , are fixed integers and , and are constant matrices. It is assumed that the considered system is one with weak delays. 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 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 weak 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. AMS Subject Classification:39A06, 39A12.

Highlights

IntroductionPreliminary notions and properties We use the following notation: for integers s, q, s ≤ q, we define Zqs := {s, s + ,
Preliminary notions and properties We use the following notation: for integers s, q, s ≤ q, we define Zqs := {s, s +, . . . , q}, where s = –∞ or q = ∞ are admitted, too
We show that a system’s property of being one with weak delays is preserved by every nonsingular linear transformation

Summary

Introduction

Preliminary notions and properties We use the following notation: for integers s, q, s ≤ q, we define Zqs := {s, s + , . Throughout this paper, using notation Zqs , we always assume s ≤ q. We deal with the discrete planar systems x(k + ) = Ax(k) + Bx(k – m) + Cx(k – n), ( ). Together with equation ( ), we consider the initial (Cauchy) problem x(k) = φ(k), where k = –m, –m + , . The existence and uniqueness of the solution of initial problem ( ), ( ) on Z∞ –m is obvious. We recall that the solution x : Z∞ –m → R of ( ), ( ) is defined as an infinite sequence x(–m) = φ(–m), x(–m + ) = φ(–m + ), .

Objectives

Results

Conclusion