To the memory of Sergey Konstantinovich
Persidsky
Linear relations
in a
problem of decomposition of causal 2-D systems
V.T.Borukhov
Surganov str., 11,
220072,
O.M.Kvetko
As is known, binary relations are used in decision
analysis, in the complex engineering systems simulation and synthesis and in
other branches of science and technology. A binary linear
relations were defined by R.Arens as a
subspace of the direct product X´X, where X is a vector space. A notion of the linear relation extends a
notion of a graph of a linear operator. In particular a product of relations is
defined for two linear relations. And for operators it coincides with the definition
of a product of operators.
Applying linear relations in a theory of linear
dynamic systems is based on the fact that typical properties of linear
dynamical systems are invariant with respect to the action of the Brunovsky group (or the feedback group) or of the Morse group
in a general case. The action of the Morse group on the set of linear
'input-state-output' systems is equivalent to the action of the general linear
group GL(X) on the set of linear relations which are associated with the
'input-state-output' systems. It substantially allows to
simplify the methods of research structure properties of the linear
dynamic systems. For example, controllability condition, invertibility
condition and other conditions are more simple in
terms of associated linear relations than in initial terms of linear system.
In the paper a linear relations approach to
problems of decomposition and classification (with respect to the Brunovsky group) of a set of causal 2-D systems is
described. There are the results obtained for the problems of decomposition of
linear relations and linear discrete 'input-state' systems for arbitrary infinite-dimensional
vector spaces. A peculiarity of the approach we consider is in applying the
ordinals theory for a parametrization of the orbits
set of the action of the Brunovsky group on the
linear 'input-state-output' systems set.
The necessary and sufficient conditions for a causal linear 2-D systems decomposability into an
attainable and free subsystems in general case are obtained in this paper. The sequence of ordinals for 2-D systems which are invariant
with respect to the actions of the Brunovsky group
are described. A problem of stabilization of these systems is considered too. A
criterion of stabilizability of causal 2-D systems in
general case is obtained in this paper.
Preliminary information to the theory of
systems of the form (1) is given in Section 1. In Section 2 we adduce the main
results of investigation of the associated decomposition problem for the linear
relations and the discrete linear dynamic systems in infinite-dimensional
vector spaces. Detailed proofs of these results were obtained early. The
necessary and sufficient conditions for a decomposing the causal linear 2-D systems into an
attainable and free subsystems in general case are obtained in Section
3. The typical properties of 2-D
systems which are invariant with respect to the action of the Brunovsky group are described. A stabilizability
criterion for a given class of 2-D
systems is presented.
It is appropriate mention here that Professor S.K.Persidsky carried out series of fundamental research in
the area of modeling and stability of discrete systems in addition to far-famed
research of stability of nonlinear differential systems. In particular he
proved the effective criteria of exponential stability of non-linear difference
equations. S.K.Persidsky and his school devised mathematical
methods which find a use for different areas of engineering science.
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