Geodesic
Stability of a weak solution for a hyperbolic system with distributed parameters on a graph
Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 133, pp. 55-67

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In the work, the stability conditions for a solution of an evolutionary hyperbolic system with distributed parameters on a graph describing the oscillating process of continuous medium in a spatial network are indicated. The hyperbolic system is considered in the weak formulation: a weak solution of the system is a summable function that satisfies the integral identity which determines the variational formulation for the initial-boundary value problem. The basic idea, that has determined the content of this work, is to present a weak solution in the form of a generalized Fourier series and continue with an analysis of the convergence of this series and the series obtained by its single termwise differentiation. The used approach is based on a priori estimates of a weak solution and the construction (by the Fayedo–Galerkin method with a special basis, the system of eigenfunctions of the elliptic operator of a hyperbolic equation) of a weakly compact family of approximate solutions in the selected state space. The obtained results underlie the analysis of optimal control problems of oscillations of netset-like industrial constructions which have interesting analogies with multi-phase problems of multidimensional hydrodynamics.
Keywords: hyperbolic system; distributed parameters on a graph; weak solution; stability. 
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     author = {V. V. Provotorov and A. P. Zhabko},
     title = {Stability of a weak solution for a hyperbolic system with distributed parameters on a graph},
     journal = {Vestnik rossijskih universitetov. Matematika},
     pages = {55--67},
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     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTAMU_2021_26_133_a5/}
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V. V. Provotorov; A. P. Zhabko. Stability of a weak solution for a hyperbolic system with distributed parameters on a graph. Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 133, pp. 55-67. http://geodesic.mathdoc.fr/item/VTAMU_2021_26_133_a5/