Advertisement

Journal of Applied and Industrial Mathematics

, Volume 13, Issue 2, pp 261–269 | Cite as

On the Differential Realization of a Second-Order Bilinear System in a Hilbert Space

  • A. V. LakeyevEmail author
  • Yu. E. LinkeEmail author
  • V. A. RusanovEmail author
Article

Abstract

We study the necessary and sufficient conditions for the existence of a nonlinear differential realization of a continuous infinite-dimensional behaviorist system in the class of nonstationary second-order bilinear ordinary differential (in particular, hyperbolic) equations in a separable Hilbert space. The obtained conditions rely upon the tensor products of Hilbert spaces. In passing, we analytically justify some topological-metrical continuity conditions for the projectivization of the Rayleigh—Ritz operator with the calculation of the fundamental group of its image.

Keywords

inverse problem of nonlinear system analysis bilinear differential realization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. C. Willems, “System Models for the Analysis of Physical Systems,” Ric. Aut. No. 10, 71–106 (1979).Google Scholar
  2. 2.
    R. E. Kalman, P. L. Falb, and M. A. Arbib, Topics in Mathematical System Theory (McGraw Hill, New York, 1969; Mir, Moscow, 1971).zbMATHGoogle Scholar
  3. 3.
    V. A. Voronov, A. V. Lakeyev, Yu. E. Linke, and V. A. Rusanov, “Accuracy Estimate in the Readjustment Process of an Identification Matrix,” Problemy Upravleniya i Informatiki 4, 16–26 (2015).Google Scholar
  4. 4.
    A. V. Daneev, V. A. Rusanov, and M. V. Rusanov, “From Kalman—Mesarovich Realization to a Normal-Hyperbolic Linear Model,” Kibernet. Sistem. Anal. 41 (6), 137–157 (2005) [Cybernetics and Systems Analysis 41 (6), 909–923 (2005)].MathSciNetzbMATHGoogle Scholar
  5. 5.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis (Academic Press, Oxford, 1980; Mir, Moscow, 1977).zbMATHGoogle Scholar
  6. 6.
    L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nauka, Moscow, 1984; Pergamon, Oxford, 2014).zbMATHGoogle Scholar
  7. 7.
    V. A. Rusanov, A. V. Daneev, A. V. Lakeyev, and Yu. E. Linke, “On the Differential Realization Theory of Nonlinear Dynamic Processes in Hilbert Space,” Far East J. Math. Sci. 97 (4), 495–532 (2015).zbMATHGoogle Scholar
  8. 8.
    V. A. Rusanov, A. V. Daneev, and Yu. E. Linke, “To the Geometrical Theory of Differential Realization of Dynamic Processes in a Hilbert Space,” Kibernet. Sistem. Anal. 53 (4), 71–83 (2017) [Cybernetics and Systems Analysis 53 (4), 554–564 (2017)].MathSciNetzbMATHGoogle Scholar
  9. 9.
    S. I. Kabanikhin, Inverse and Ill-Posed Problems (Sibir. Nauchn. Izd., Novosibirsk, 2009) [in Russian].Google Scholar
  10. 10.
    Yu. E. Anikonov and M. V. Neshchadim, “On Analytical Methods in the Theory of Inverse Problems for Hyperbolic Equations. I, II” Sibir. Zh. Industr. Mat. 14 (1), 27–39 (2011) [J. Appl. Indust. Math. 5 (4), 506–518 (2011)]; Sibir. Zh. Industr. Mat. 14 (2), 28–33 (2011) [J. Appl. Indust. Math. 6 (1), 6–11 (2012)].zbMATHGoogle Scholar
  11. 11.
    V. A. Rusanov, A. V. Daneev, A. V. Lakeyev, and Yu. E. Linke, “On Solvability of the Identification-Inverse Problem for Operator-Functions of a Nonlinear Regulator of a Nonstationary Hyperbolic System,” Advances in Differential Equations and Control Processes 16 (2), 71–84 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. V. Lakeyev, Yu. E. Linke, and V. A. Rusanov, “Realization of a Polylinear Controller as a Second-Order Differential System in a Hilbert Space,” Differentsial’nye Uravneniya 53 (8), 1098–1109 (2017) [Differential Equations 53 (8), 1070–1081 (2017)].MathSciNetzbMATHGoogle Scholar
  13. 13.
    V. V. Prasolov, Elements of Combinatorial and Differential Topology (Izd. MTsNMO, Moscow, 2014) [in Russian].zbMATHGoogle Scholar
  14. 14.
    A. A. Kirillov, Elements of the Theory of Representations (Nauka, Moscow, 1978) [in Russian].Google Scholar
  15. 15.
    V. A. Rusanov, A. V. Lakeyev, and Yu. E. Linke, “Existence of a Differential Realization of a Dynamical System in a Banach Space in the Constructions of Extensions to M p-Operators,” Differentsial’nye Uravneniya 49 (3), 358–370 (2013) [Differential Equations 49 (3), 346–358 (2013)].zbMATHGoogle Scholar
  16. 16.
    M. I. D’yachenko and P. L. Ul’yanov, Measure and Integral (Faktorial, Moscow, 1998) [in Russian].Google Scholar
  17. 17.
    A. V. Lakeyev, Yu. E. Linke, and V. A. Rusanov, “On a Criterion for Continuity of the Rayleigh—Ritz Operator,” Vestnik Buryat. Gos. Univ. Mat. i Informat. No. 3, 3–13 (2018).Google Scholar
  18. 18.
    R. Engelking, General Topology (Heldermann, Berlin, 1989; Mir, Moscow, 1986).zbMATHGoogle Scholar
  19. 19.
    A. V. Daneev, V. A. Rusanov, M. V. Rusanov, and V. N. Sizykh, “To a Posteriori Modeling of Unsteady Hyperbolic Systems,” Izv. Samar. Nauchn. Tsentr. Ross. Akad. Nauk 20 (1), 106–113 (2018).Google Scholar
  20. 20.
    V. A. Rusanov, A. V. Daneev, Yu. E. Linke, V. N. Sizykh, and V. A. Voronov, “System-Theoretical Foundation for Identification of Dynamic Systems. I,” Far East J. Math. Sci. 106 (1), 1–42 (2018).Google Scholar
  21. 21.
    E. Kaiser, J. N. Kutz, and S. L. Brunton, “Sparse Identification of Nonlinear Dynamics for Model Predictive Control in the Low-Data Limit,” arXiv:1711.05501v2 [math.OC] September 30 (2018).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Matrosov Institute for System Dynamics and Control TheoryIrkutskRussia
  2. 2.Irkutsk National Research Technical UniversityIrkutskRussia

Personalised recommendations