Geodesic
Stability of weak solutions of parabolic systems with distributed parameters on the graph
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 15 (2019) no. 4, pp. 457-471 Cet article a éte moissonné depuis la source Math-Net.Ru

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The analysis of the behaviour of the evolutionary equation solution with unlimited time variable has been a subject of discussion in scientific circles for a long time. There are many practical reasons for this when the initial conditions of the equation are specified with a certain error: how the small changes in the initial conditions affect the behaviour of the solution for large values of the time. The paper uses the classical understanding of the stability of the solution of a differential equation or a system of equations that goes back to the works of A. M. Lyapunov: a solution is stable if it little changes under the small perturbations of the initial condition. In the work specified the stability conditions for the solution of an evolutionary parabolic system with distributed parameters on a graph describing the process of transfer of a continuous mass in a spatial network are indicated. The parabolic system is considered in the weak formulation: a weak solution of the system is a summable function that satisfies the integral form identity, which determines the variational formulation for the initial-boundary value problem. By going beyond the classical (smooth) solutions and addressing weak solutions of the problem the authors aim not only to describe more precisely the physical nature of the transfer processes (this takes on particular importance when studying the dynamics of multiphase media) but also to the path analysis processes in multidimensional network-like domains. The used approach is based on a priori estimates of the weak solution and the construction (the Fayedo—Galerkin method with a special basis — the system of eigenfunctions of the elliptic operator of a parabolic equation) of a weakly compact family of approximate solutions in the selected state space. The obtained results underlie the analysis of optimal control problems for differential systems with distributed parameters on a graph, which have interesting analogies with multiphase problems of multidimensional hydrodynamics.
Keywords: evolutionary system of parabolic type, distributed parameters on the graph, a weak solution, stability of a weak solutions. 
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     title = {Stability of weak solutions of parabolic systems with distributed parameters on the graph},
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A. P. Zhabko; A. I. Shindyapin; V. V. Provotorov. Stability of weak solutions of parabolic systems with distributed parameters on the graph. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 15 (2019) no. 4, pp. 457-471. http://geodesic.mathdoc.fr/item/VSPUI_2019_15_4_a3/

[1] V. V. Provotorov, Yu. A. Gnilitskaya, “Boundary control of a wave systems in the space of the generalized solutions”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2013, no. 3, 112–120 (In Russian)

[2] S. L. Podvalny, V. V. Provotorov, “Starting control of a parabolic system with distributed parameters on a graph”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2015, no. 3, 126–142 (In Russian) | MR

[3] V. V. Provotorov, “Boundary control of a parabolic system with distributed parameters on a graph in the class of summable functions”, Automation and Remote Control, 76:2 (2015), 318–322 | DOI | MR | Zbl

[4] 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 (In Russian) | DOI | MR

[5] A. P. Zhabko, O. G. Tikhomirov, O. N. Chizhova, “On stabilization of a class of systems with time proportional delay”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 14:2 (2018), 165–172 | DOI | MR

[6] I. V. Alexandrova, A. P. Zhabko, “A new LKF approach to stability analysis of linear systems with uncertain delays”, Automatica, 91 (2018), 173–178 | DOI | MR | Zbl

[7] V. V. Provotorov, “Boundary control of a parabolic system with delay and distributed parameters on the graph”, International Conference «Stability and Control Processes» in memory of V. I. Zubov (SCP) (Saint Petersburg, 2015), 126–128

[8] S. L. Podvalny, V. V. Provotorov, “The questions of controllability of a parabolic systems with distributed parameters on the graph”, International Conference «Stability and Control Processes» in memory of V. I. Zubov (SCP) (Saint Petersburg, 2015), 117–119 | MR

[9] V. V. Provotorov, A. S. Volkova, Initial boundary value problems with distributed parameters on the graph, Nauchnay kniga Publ, Voronezh, 2014, 188 pp. (In Russian)

[10] V. V. Provotorov, “Eigenfunctions of the Sturm?Liouville problem on astar graph”, Mathematics, 199:10 (2008), 105–126 (In Russian) | MR | Zbl

[11] V. V. Provotorov, “Expansion of eigenfunctions of Sturm?Liouville problem on bundle graph”, Russian Mathematics, 3 (2008), 50–62 (In Russian) | MR | Zbl

[12] O. A. Ladyzhenskaya, Boundary value problems of mathematical physics, Nauka, M., 1973, 407 pp. (In Russian) | MR

[13] Lions J. L., Some methods of solving non-linear boundary value problems, Mir Publ., M., 1972, 587 pp. | MR | MR

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

[15] V. V. Provotorov, E. N. Provotorova, “Optimal control of the linearized Navier?Stokes system in a netlike domain”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 13:4 (2017), 428–441 | DOI | MR

[16] M. A. Artemov, E. S. Baranovskii, A. P. Zhabko, V. V. Provotorov, “On a 3D model of nonisothermal flows in a pipeline network”, Journal of Physics. Conference Series, 1203 (2019), 012094 | DOI

[17] A. Yu. Aleksandrov, A. P. Zhabko, “On stability of solutions to one class of nonlinear difference systems”, Siberian Mathematical Journal, 44:6 (2003), 951–958 (In Russian) | DOI | MR | Zbl

[18] A. Aleksandrov, E. Aleksandrova, A. Zhabko, “Asymptotic stability conditions for certain classes of mechanical systems with time delay”, WSEAS Transactions on Systems and Control, 9 (2014), 388–397 | MR

[19] A. Aleksandrov, E. Aleksandrova, A. Zhabko, “Asymptotic stability conditions of solutions for nonlinear multiconnected time-delay systems”, Circuits Systems and Signal Processing, 35:10 (2016), 3531–3554 | DOI | MR | Zbl

[20] V. V. Karelin, “Penalty functions in the control problem of an observation process”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2010, no. 4, 109–114 (In Russian)

[21] V. V. Karelin, V. M. Bure, M. V. Svirkin, “The generalized model of information dissemination in continuous time”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 13:1 (2017), 74–80 (In Russian) | DOI | MR

[22] A. M. Kamachkin, V. V. Yevstafyeva, “Oscillations in a relay control system at an external disturbance”, Control Applications of Optimization 2000, Proceedings of the 11th IFAC Workshop, v. 2, 2000, 459–462

[23] E. I. Veremey, M. V. Sotnikova, “Plasma stabilization by prediction with stable linear approximation”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2011, no. 1, 116–133 (In Russian)

[24] A. N. Aliseiko, “Lyapunov matrices for a class of systems with an exponential core”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 13:3 (2017), 228–240 (In Russian) | DOI | MR

[25] A. A. Ponomarev, “Approximation of feedback in the ?predictor-corrector? regulator by an explicit function”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 13:2 (2017), 193–208 (In Russian) | DOI | MR

[26] S. E. Kuptsova, S. Yu. Kuptsov, N. A. Stepenko, “On the limiting behavior of a time-delay system's solution”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 14:2 (2018), 173–182 | DOI | MR

[27] S. V. Krasnov, S. M. Sergeev, N. V. Mukhanova, A. N. Grushkin, “Methodical forming business competencies for private label”, 6th International Conference on Reliability, Infocom Technologies and Optimization (Trends and Future Directions), ICRITO 2017, 569–574

[28] 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, 15:1 (2019), 107–117 | DOI | MR