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Русская версия

**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*.