ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

New Effective Conditions of Pointwise Completeness For Linear Time-Invariant Delay-Differential Systems


A. A. Korobov

Russia, 630090, Novosibirsk, Ave. of acad. Koptyug, 4
Siberian Division of Russian Academy of Sciences
Sobolev Institute of Mathematics
Section of Theoretical Cybernetics


Complete description of pointwise-degenerate linear time-invariant differential systems of third order is gave. More precisely, defining relations connecting elements of matrices and a delay time is obtained. It is established connection between pointwise degeneracy of such system of sixth order and pointwise completeness of wide class of systems that possess many properties of the conjugate system. Popov's approach for investigation of the strongly regular systems is developed. More precisely, the necessary and sufficient criterion of pointwise degeneracy of such systems of fourth order is obtained. Minyuk's description of canonical forms of pointwise-degenerate linear time-invariant delay-differential systems of fourth order is corrected. New sufficient effective conditions of pointwise completeness of systems with block-triangular matrices is obtained. It is proved that if time-invariant delay-differential equation matrices may be represented as 2x2-matrices with elements in some connected locally bicompact continuous skew field, then such equation is poinwise completeness for any delay time. It is proved that the degeneracy space of time-invariant delay-differential system of order n have dimention at most n-2.

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