Stabilization of weak solutions of parabolic systems with distributed parameters on the graph

A. P. Zhabko ; V. V. Provotorov ; O. R. Balaban

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 15 (2019) no. 2, pp. 187-198

Résumé

In many applications because of the complexity of the mathematical models have to abandon the use of ordinary differential equations in behalf of considering the evolutionary equations with partial derivatives. In addition, most commonly the evolutionary problem study on the finite interval changes of a temporary variable. In practice, where you can solve the problem for arbitrary finite interval changes to a temporary variable it is important to know the behavior of the solution where, when the temporary variable strives to infinity. First of all, this is related to the study of the properties of the stability of the indicated solution and the possibility of constructing the stabilizing control in case of the instability. Precisely this case is the object of the study in this work, in which represent the analysis of the stability of the weak solutions of the evolutionary systems with distributed parameters on the graph with the unlimited growth of the temporary variable, obtain the conditions of the stabilization of the weak solutions. By studying the relevant initial-boundary value problem, we to be beyond the scope of the classical solutions and appeal to the weak solutions of the problem, reflecting more accurately the physical essence of appearance and processes (i. e. consider the initial-boundary value problem in weak formulation). In this case, the choice of the class of weak solutions to be determined one way or the other functional space is at the disposal of the researchers and to meet the demand, above all, conservation of the existence theorems and the uniqueness theorems for the arbitrary finite interval changes to a temporary variable. The fundamental used tool is the representation of a weak solution in the form of a functional series (method Faedo—Galerkin approximation with the special basis-system functions — the eigenfunction system) and the compactness of a many of approximate solutions (thanks to a priori estimates).

Détail

Keywords: an evolutionary system of parabolic type, distributed parameters on the graph, a weak solution, stabilization of a weak solution.

Bibliographie

[1] Zhabko A. P., Tihomirov O. G., Chizhova O. N., “On stability to the class of systems with the proportional delay”, Vestnik of Saint Peterburg University. Applied Mathematics. Computer Science. Control Processes, 14:2 (2018), 165–172 (In Russian) | DOI | MR

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

[3] Provotorov V. V., “Boundary control of a parabolic system with delay and distributed parameters on the graph”, Intern. conference “Stability and Control Processes” in memory of V. I. Zubov, SCP, Saint Petersburg University Publ., Saint Petersburg, 2015, 126–128

[4] Podvalny S. L., Provotorov V. V., “The questions of controllability of a parabolic systems with distributed parameters on the graph”, Intern. conference “Stability and Control Processes” in memory of V. I. Zubov, SCP, Saint Petersburg University Publ., Saint Petersburg, 2015, 117–119

[5] Provotorov V. V., “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

[6] Podvalny S. L., Provotorov V. V., “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

[7] Provotorov V. V., Gnilitskaya Yu. A., “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)

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

[9] Mihlin S. G., Linear partial equation, Vysshaja shkola Publ., M., 1977, 431 pp. (In Russian) | MR

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

[11] Provotorov V. V., “Eigenfunctions of the Sturm–Liouville problem astar graph”, Mathematics, 199:10 (2008), 1523–1545 (In Russian) | MR | Zbl

[12] Provotorov V. V., Provotorova E. N., “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

[13] Lions J.-L., Some methods of solving non-linear boundary value problems, Dunod Gauthier-Villars, Paris, 1968, 587 pp. ; Lions J.-L., Nekotorye metody reshenia nelineinyh kraevyh zadach, Mir Publ., M., 1972, 414 pp. | MR

[14] Provotorov V. V., Ryazhskikh V. I., Gnilitskaya Yu. A., “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] Provotorov V. V., Provotorova E. N., “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] Aleksandrov A. Yu., Zhabko A. P., “On stability of solutions to one class of nonlinear difference systems”, Siberian Mathematical Journal, 44:6 (2003), 951–958 (In Russian) | DOI | MR | Zbl

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

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

[19] Karelin V. V., “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)

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

[21] Veremey E. I., Sotnikova M. V., “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)