On a linear autonomous descriptor equation with discrete time. II. Canonical representation and structural properties

We consider a linear homogeneous autonomous descriptor equation with discrete time
$$B_0g(k+1)+\sum_{i=1}^mB_ig(k+1-i)=0,\,k=m,m+1,\ldots,$$
with rectangular (in general case) matrices $ B_i. $ Such an equation arises in the study of the most important control problems for systems with many commensurate delays in control: the $0$-controllability problem, the synthesis problem of the feedback-type regulator, which provides calming to the solution of the original system, the modal controllability problem (controllability to the coefficients of characteristic quasipolynomial), the spectral reduction problem and the synthesis problem observers for dual surveillance system. The main method of the presented study is based on replacing the original equation with an equivalent equation in the “expanded” state space, which allows one to match the new equation of the beam of matrices. This made it possible to study a number of structural properties of the original equation by using the canonical form of the beam of matrices, and express the results in terms of minimal indices and elementary divisors. In the article, a criterion is obtained for the existence of a nontrivial admissible initial condition for the original equation, the verification of which is based on the calculation of the minimum indices and elementary divisors of the beam of matrices. The following problem was studied: it is required to construct a solution to the original equation in the form $g (k + 1) = T \psi (k + 1)$, $k = 1,2 \ldots, $ where $ T $ is some matrix, the sequence of vectors $ \psi (k + 1)$, $k = 1,2, \ldots, $ satisfies the equation $ \psi (k + 1) = S \psi (k)$, $k = 1,2,\ldots,$ and the square matrix $ S $ has a predetermined spectrum (or part of the spectrum). The results obtained make it possible to construct solutions of the initial descriptor equation with predetermined asymptotic properties, for example, uniformly asymptotically stable.