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## Archive of Issues

Belarus Grodno
Year
2020
Volume
30
Issue
2
Pages
290-311
 Section Mathematics Title On a linear autonomous descriptor equation with discrete time. I. Application to the 0-controllability problem Author(-s) Khartovskii V.E.a Affiliations Grodno State Universitya Abstract We consider a linear homogeneous autonomous descriptor equation with discrete time $$B_0g(k+1)+\sum_{i=1}^mB_ig(k+1-i)=0,\quad k=m,m+1,\ldots,$$ 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 of the coefficients of characteristic quasipolynomial), the spectral reduction problem and the problem of observers' synthesis for a dual surveillance system. For the studied descriptor equation with discrete time, a subspace of initial conditions for which this equation is solvable is described based on the solution of a finite chain of homogeneous algebraic systems. The representation of all its solutions is obtained in the form of some explicit recurrent formula convenient for the organization of the computational process. Some properties of this equation that are used in the problems of regulator synthesis for continuous systems with many commensurate delays in control are studied. A distinctive feature of the presented study of the object under consideration is the use of an approach that does not require the construction of transformations reducing the matrices of the original equation to different canonical forms. Keywords linear systems with multiple delays, linear descriptor autonomous equation with discrete time, subspace of initial conditions, representation of the solution UDC 517.977 MSC 93B99, 93C55 DOI 10.35634/vm200211 Received 22 April 2020 Language Russian Citation Khartovskii V.E. On a linear autonomous descriptor equation with discrete time. I. Application to the 0-controllability problem, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 2, pp. 290-311. References Krasovskii N.N. Optimal processes in delay systems, Statisticheskie metody: Tr. II Mezhdunar. kongressa IFAK (Bazel’, 1963) (Statistic Methods. Proc. II IFAC Congress, Basel, 1963), Moscow, 1965, vol. 2, pp. 201-210 (in Russian). Khartovskii V.E. 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