The paper studies the qualitative behavior of solutions of linear discrete systems with constant coefficients and a constant time-delay under the influence of a control function. In particular, it deals with a non-trivial task of transferring an arbitrary solution to several prescribed positions (trajectory controllability), being more general and different from the problem of relative controllability frequently solved in the literature. A Kalmann-type controllability criterion is proved and, in addition, previous results are discussed and illustrative examples given.

In recent papers, a discrete matrix delayed exponential for a single delay was defined and its main property connected with the solution of linear discrete systems with a single delay was proved. In the present paper, a generalization of the concept of discrete matrix delayed exponential is designed for two delays and its main property is proved as well.

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.