Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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A Method for Obtaining an Explicit Solution of Second-order Matrix ODE Based on Diagonalization the Matrices and the Kronecker Matrix Algebra

Author(s):

Alexander Georgievich Madera

Scientific Research Institute for System Analysis of
Russia Academy of Sciences (SRISA RAS)
Head of Department
«Mathematical modeling for processes in complex technical systems»,
Dr.Sci, Professor
118218, Moscow, Nahimovskiy prosp., 36, k.1

agmprof@mail.ru

Abstract:

A method for obtaining an explicit solution of matrix differential equations in second-order ordinary derivatives with constant matrices is considered. The method allows one to reduce an initial system of interconnected differential equations to a system of independent differential equations that can be easily solved analytically. The method developed in the article is based on the diagonalization of all matrices included in the equation, which is carried out by using the spectral decomposition of the matrices and the Kronecker matrix algebra. An example of the application of the developed method is given.

Keywords

References:

  1. Afanas'ev V. N., Kolmanovskii V. B., Nosov V. R. Matematicheskaya teoriya konstruirovaniya sistem upravleniya [Mathematical Theory of Control Systems Design]. Moscow, Vysshaya shkola, 2003, 615 p
  2. Chua L. O., Pen-Min Lin Computer-aided analysis оf electronic circuits. New Jersey: Prentice‐ Hall, Englewood Cliffs, 1975. 737 p
  3. Landau L. D., Lifshits E. M. Курс теоретической механики. Mekhanika. Том 1 [Course of theoretical physics, Volume 1, Mechanics]. Moscow, FIZMATLIT, 2017, 216 p
  4. Frazer R. A., Duncan W. J., Collar A. R. Elementary matrices and some applications to dynamics and differential equations. New York: Cambridge University Press, 1960. 416 p
  5. Duffin R. J. A minimax theory for overdamped networks. Journal of Rational Mechanics and Analysis; 1955; Vol. 4: 221-233
  6. MacDuffee C. C. The theory matrices. N. Y. : Dover Publications, 2004, 128 p
  7. Figotin A., Welters A. Lagrangian Framework for Systems Composed of High-Loss and Lossless Components. Available at: arXiv:1401. 0230v2 01. 05. 2014. (accessed 10. 2018)
  8. Veselić K. Modal approximations to damped linear systems. Available at: arXiv:0907. 0167v1 01. 07. 2009. (accessed October 01. 07. 2009)
  9. Bellman R. Introduction to matrix analysis, New York: McGraw-Hill, 1960, 368 p
  10. Lankaster P. Theory of matrix. New York - London, Academic Press, 1969. 280 p
  11. Horn R. A., Johnson C. R. Matrix analysis. England, Cambridge: Cambridge University Press, 1986. 656 p
  12. Brewer J. W. Kronecker products and matrix calculus in system theory. IEEE Transactions on Circuits and Systems; 1978; Vol. CAS-25, No. 9, September: 772 - 781
  13. Marcus M., Minc H. A survey of matrix theory and matrix inequalities. USA, Dover Publications, 2010
  14. Kamke E. Spravochnik po obyknovennym differentsial'nym uravneniyam [Differentialgleichungen Lö sungsmethoden und Lö sungen]. Moscow, Nauka Publ., 1971. 576 p

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