Analysis of the wave phenomenon stability in network-like industrial constructions

Abstract

Weak solutions of the initial boundary value problem for the second order hyperbolic equation with distributed parameters on an arbitrary oriented graph are considered. Weak solutions are determined by an integral identity, replacing the equation, the initial and boundary conditions. This indicates the spaces in which the search a weak solution is proposed, the conditions of weak solvability of such a problem and the consequent analysis of the stability of the solution are given. The central idea that defined all the content of the work is to apply the Fourier method to analysis evolutionary initial boundary value problems. Namely, the presentation of the solution in the form of a generalized series of Fourier, followed by an analysis of the convergence of this series and series, obtained by his one-time member differentiation. A essential role will play the system of generalized eigenfunctions and spectral characteristics of the elliptical operator of the hyperbolic equation. This considers homogeneous boundary conditions of the first genus (Dirichle conditions), the analysis of stability for the boundary conditions of the second and third genus is similar. The results are fundamental in the study of the control problems of oscillation of network-like industrial constructions, first of oll under studying the question of damping antenna constructions of different types.