scholarly journals Representation of the Solutions of Linear Discrete Systems with Constant Coefficients and Two Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Josef Diblík ◽  
Blanka Morávková
Keyword(s):  
Cauchy Problem ◽  
Closed Form ◽  
Special Function ◽  
Discrete Systems ◽  
Constant Coefficients ◽  
Two Delays ◽  
Unknown Vector ◽  
Linear Discrete Systems

The purpose of this paper is to develop a method for the construction of solutions to initial problems of linear discrete systems with constant coefficients and with two delaysΔx(k)=Bx(k-m)+Cx(k-n)+f(k),wherem,n∈ℕ,m≠n,are fixed,k=0,…,∞,B=(bij),C=(cij)are constantr×rmatrices,fis a givenr×1vector, andxis anr×1unknown vector. Solutions are expressed with the aid of a special function called the discrete matrix delayed exponential for two delays. Such approach results in a possibility to express an initial Cauchy problem in a closed form. Examples are shown illustrating the results obtained.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-37 ◽  
Author(s):  
Josef Diblík ◽  
Hana Halfarová
Keyword(s):  
Dimensional Space ◽  
Discrete Systems ◽  
Infinite Dimensional Space ◽  
Initial Interval ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Linear Differential Systems ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Linear Discrete Systems

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.

Controllability of Linear Discrete Systems with Constant Coefficients and Pure Delay

2008 ◽  
Vol 47 (3) ◽  
pp. 1140-1149 ◽  
Author(s):  
Josef Diblík ◽  
Denys Ya. Khusainov ◽  
M. Růžičková
Keyword(s):  
Discrete Systems ◽  
Constant Coefficients ◽  
Pure Delay ◽  
Linear Discrete Systems
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