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

Josef Diblík,Hana Halfarová

doi:10.1155/2014/627295

Abstract

Planar linear discrete systems with constant coefficients and delaysx(k+1)=Ax(k)+∑l=1n‍Blxl(k-ml)are considered wherek∈ℤ0∞:={0,1,…,∞},m1,m2,…,mnare constant integer delays,0<m1<m2<⋯<mn,A,B1,…,Bnare constant2×2matrices, andx:ℤ-mn∞→ℝ2. 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 dimension2(mn+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.

Highlights

IntroductionThe aim of this paper is to give explicit formulas for solutions of weakly delayed systems and to show that, after several steps, the dimension of the space of all solutions, being initially equal to the dimension 2(mn + 1) of the space of initial data (3) generated by discrete functions φ, is reduced to a dimension less than the initial one on an interval of the form Z∞ s with an s > 0
Since all the possible cases of the planar system (1) with weak delay have been analysed, we are ready to formulate results concerning the dimension of the space of solutions of (1) assuming that initial condition (3) is variable
The weakly delayed systems analyzed in this paper can be simplified and their solutions can be found in explicit analytical forms

Summary

Introduction

The aim of this paper is to give explicit formulas for solutions of weakly delayed systems and to show that, after several steps, the dimension of the space of all solutions, being initially equal to the dimension 2(mn + 1) of the space of initial data (3) generated by discrete functions φ, is reduced to a dimension less than the initial one on an interval of the form Z∞ s with an s > 0. If (1) is a weakly delayed system, the corresponding characteristic equation has only two eigenvalues instead of 2(mn + 1) eigenvalues in the case of systems with nonweak delays This explains why the dimension of the space of solutions becomes less than the initial one. For example, to [4,5,6,7,8] and to relevant references therein

General Solution of Weakly Delayed System

Dimension of the Set of Solutions

Concluding Remarks