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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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/

[1] V. V. Provotorov, S. M. Sergeev, A. A. Part, “Solvability of hyperbolic systems with distributed parameters on the graph in the weak formulation”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 14:1 (2019), 107–117 | MR

[2] O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, Nauka Publ., Moscow, 1973, 407 pp. | MR

[3] A. P. Zhabko, A. I. Shindyapin, V. V. Provotorov, “Stability of weak solutions of parabolic systems with distributed parameters on the graph”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 15:4 (2019), 457–471 | MR

[4] V. V. Provotorov, V. I. Ryazhskikh, Yu. A. Gnilitskaya, “Unique weak solvability of nonlinear initial boundary value problem with distributed parameters in the netlike domain”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 13:3 (2017), 264–277 | MR

[5] A. S. Volkova, V. V. Provotorov, “Generalized solutions and generalized eigenfunctions of boundary-value problems on a geometric graph”, Russian Mathematics, 58:3 (2014), 1–13 | DOI | MR | Zbl

[6] V. V. Provotorov, E. N. Provotorova, “Synthesis of optimal boundary control of parabolic systems with delay and distributed parameters on the graph”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 13:2 (2017), 209–224 | MR

[7] V. V. Provotorov, “Expansion of eigenfunctions of Sturm-Liouville problem on astar graph”, Russian Mathematics, 3 (2008), 45–57 | DOI | MR | Zbl

[8] O. A. Ladyzhenskaya, A Mixed Problem for Hyperbolic Equations, GITTL, M., 1953, 282 pp. (In Russian)

[9] A. P. Zhabko, V. V. Provotorov, O. R. Balaban, “Stabilization of weak solutions of parabolic systems with distributed parameters on the graph”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 15:2 (2019), 187–198 | MR

[10] J.-L. Lions, Some Methods of Solving Non-Linear Boundary Value Problems, Mir Publ., Moscow, 1972, 587 pp. | MR

[11] V. V. Provotorov, E. N. Provotorova, “Optimal control of the linearized Navier-Stokes system in a netlike domain”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 13:4 (2017), 431–443 | MR

[12] M. A. Artemov, E. S. Baranovskii, A. P. Zhabko, V. V. Provotorov, “On a 3D model of non-isothermal flows in a pipeline network”, Journal of Physics. Conference Series, 1203 (2019), Article ID 012094

[13] S. L. Podvalny, V. V. Provotorov, “Determining the starting function in the task of observing the parabolic system with distributed parameters on the graph”, Vestnik of Voronezh State Technical University, 10:6 (2014), 29–35