The paper considers evolutionary problems underlying the mathematical description of oscillatory
and hydrodynamic processes in network-like objects (waveguides, hydraulic networks, etc.). The main
attention is paid to the analysis of the properties of the elliptic operator (the one-dimensional Laplace
operator) with distributed parameters on the network, establishing the spectral completeness of the system
of eigenfunctions in the class of square-integrable functions. Conditions are obtained that guarantee
Neumann stability (spectral stability) of difference schemes for evolutionary problems; a solution to the
moment method control problem is presented. The methods for studying evolutionary problems are based on
the properties of a positive definite elliptic operator: a system of eigenfunctions forms a basis in the space of
functions summable with a square; series in the system of eigenfunctions admit a priori estimates of the
solutions of the evolutionary problem; approximation of an elliptic operator reduces it to a finite-dimensional
operator in a finite-dimensional space of grid functions with a natural Euclidean norm, which (a finitedimensional operator) approximates the original with any predetermined accuracy in the sense of the norm
of the space of functions summable squared. For evolutionary problems, an explicit first-order
approximation scheme on the graph grid (parabolic system) and an explicit second-order approximation
scheme (hyperbolic system) are used. The oscillatory properties of the obtained operators are established,
similar to the classical oscillatory properties. For difference schemes of parabolic and hyperbolic systems of
equations, conditions are obtained that guarantee countable spectral stability (stability in the sense of
Neumann) and, therefore, the possibility of obtaining analogues of A.F. Filippova on the convergence of
difference schemes in terms of approximation steps of a graph grid. To illustrate the applicability of the
approach used, the control problem is considered - the translation of evolutionary systems of parabolic and
hyperbolic types from given initial to given final states; conditions are obtained that guarantee the
controllability of the systems under study.
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Balaban Olesya Ruslanovna
Email: bal-olesya@mail.ru
VUNC Air Force «Military Academy named after Professor N.Y. Zhukovsky and Y.A. Gagarin»
Voronezh, Russian Federation
