The dynamical discrepancy method in problems of reconstructing unknown characteristics of a second-order system

M. S. Blizorukova

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 53 (2019), pp. 48-60.

Résumé

This paper considers two problems of dynamical reconstruction of unknown characteristics of a system of nonlinear equations describing the process of innovation diffusion through inaccurate measurements of phase states. A dynamical variant for solving these problems is designed. The system is assumed to operate on a given finite time interval. The evolution of the system's phase state, i.e., the solution of the system, is determined by an unknown input. A precise reconstruction of the real input (acting on the system) is, generally speaking, impossible due to inaccurate measurements. Therefore, some approximation to this input is constructed which provides an arbitrary smallness to the real input if the measurement errors and the step of incoming information are sufficiently small. Based on the dynamical version of the discrepancy method, two algorithms for solving the problems in question are specified. One of them is oriented to the case of measuring all coordinates of the phase vector, and the other, to the case of incomplete measurements. The algorithms suggested are stable with respect to informational noises and computational errors. Actually, they are special regularizing algorithms from the theory of dynamic inverse problems.

Détail

Mots-clés : dynamical reconstruction
Keywords: part of coordinates, nonlinear differential equations.

Bibliographie

[1] Osipov Yu. S., Kryazhimskii A. V., Inverse problems for ordinary differential equations: Dynamical solutions, Gordon and Breach, Basel, 1995, 625 pp. | MR | Zbl

[2] Osipov Yu. S., Kryazhimskii A. V., Maksimov V. I., The methods of dynamical reconstruction of inputs, Izd-vo Ural. Otd. Ross. Akad. Nauk, Yekaterinburg, 2011, 292 pp.

[3] Maksimov V. I., Dynamic inverse problems of distributed systems, VSP, Utrecht, 2002, 269 pp. | MR

[4] Osipov Yu.S., Kryazhimskii A. V., Maksimov V. I., “N.N. Krasovskii's extremal shift method and problems of boundary control”, Automation and Remote Control, 70:4 (2009), 577–588 | DOI | MR | Zbl

[5] Maksimov V. I., Osipov Yu. S., “Infinite-horizon boundary control of distributed systems”, Computational Mathematics and Mathematical Physics, 56:1 (2016), 14–25 | DOI | DOI | MR | Zbl

[6] Osipov Yu., Pandolfi L., Maksimov V., “Problems of dynamical reconstruction and robust boundary control: the case of Dirichlet boundary conditions”, J. Inverse and Ill-Posed Problems, 9:2 (2001), 149–162 | DOI | MR | Zbl

[7] Kryazhimskii A. V., Osipov Yu. S., Transactions, Ural. Otd. Akad. Nauk USSR, Sverdlovsk, 1988 (in Russian) | MR

[8] Maksimov V. I., “The dynamical decoupled method in the input reconstruction problem”, Computational Mathematics and Mathematical Physics, 44:2 (2004), 278–288 | MR | Zbl

[9] Blizorukova M. S., “Input modeling in time-delayed systems”, Computational Mathematics and Modeling, 12:2 (2001), 174–185 | DOI | MR | Zbl

[10] Maksimov V. I., “Positional modeling of controls and initial functions for Volterra systems”, Differ. Uravn., 23:4 (1987), 618–629 (in Russian) | MR

[11] Maksimov V. I., Blizorukova M. S., “On robust on-line parameter reconstruction technique”, Appl. Comput. Math., 7:2 (2008), 223–234 http://acmij.az/view.php?lang=az&menu=journal&id=216 | MR | Zbl

[12] Maksimov V. I., “Simulation of unknown perturbations in nonlinear parabolic systems of variational inequalities,”, Tekhnicheskhaya Kibernetika, 1992, no. 1, 157–162 (in Russian) | Zbl

[13] Maksimov V. I., “Some dynamical inverse problems for hyperbolic systems”, Control and Cybernetics, 25:3 (1996), 465–481 http://control.ibspan.waw.pl:3000/contents/export?filename=1996-3-06_maksimov.pdf | MR | Zbl

[14] Bertuglia C. S., Lombado S., Nijkamp P., Innovation behavior in space and time, Springer, Berlin, 1997

[15] Maksimov V. I., Rosenberg V. L., Methods of mathematical modeling of dynamic processes associated with the environment, Izd-vo Ural. Otd. Ross. Akad. Nauk, Yekaterinburg, 2009, 195 pp.

[16] Kryazhimskii A., Maksimov V., “On identification of nonobservable contamination inputs”, Environ. Model. Softw., 20 (2005), 1057–1061 | DOI

[17] Blizorukova M. S., Kapustyan V. E., Maksimov V. I., “Application of feedback control methods to two models in environmental economics”, Computational Mathematics and Modeling, 25:4 (2014), 459–469 | DOI | MR | Zbl

[18] Blizorukova M. S., “On a modification of the dynamical discrepancy method”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2008, no. 2, 21–22 (in Russian) | DOI

[19] Blizorukova M. S., “On the reconstruction of the trajectory and control in a nonlinear second-order system”, Proceedings of the Steklov Institute of Mathematics, 275, suppl. 1 (2011), 1–11 | DOI | MR | Zbl

[20] Avvakumov S. N., Kiselev Yu.N., “Optimal control laws for the model of information diffusion in a social group”, Computational Mathematics and Modeling, 22:3 (2011), 288–341 | DOI | MR | Zbl

[21] Avvakumov S. N., Kiselev Yu. N., “Models of information diffusion in a social group: construction of optimal programs”, Computational Mathematics and Modeling, 27:3 (2016), 327–350 | DOI | MR | Zbl

[22] Avvakumov S. N., Kiselev Yu. N., “Information diffusion model in a social group with possible special regimes on an infinite planning horizon”, Transactions, Problems of dynamic control, 6, MAKS Press, M., 2012, 189–196 (in Russian)