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

Дифференциальные Уравнения
и
Процессы Управления

Frequency Domain Conditions for the Existence of Finite-dimensional Projectors and Determining Observations of Attractors

Автор(ы):

Sergei Albertovich Popov

Saint-Petersburg State University

serg.pobeda@gmail.com

Volker Reitmann

70 corp.3, Botanicheskaya st,
Peterhof, Saint-Petersburg,
198516, Russia
Saint-Petersburg State University
professor of the Department of Applied Cybernetics
Prof. Dr.

vreitmann@aol.com

Аннотация:

Frequency domain conditions for the existence of finite-dimensional projectors and determining observations for attractors of semi-dynamical systems in Hilbert spaces are derived. Evolutionary variational equations are considered as control systems in a rigged Hilbert space structure. As an example we investigate a coupled system of Maxwell's equations and the heat equation in one-space dimension. We show the controllability of the linear part and the frequency domain conditions for this example.

Ссылки:

  1. Yu. M. Berezansky, Expansions in Eigenfunctions of Selfadjoint Operators (In Russian), Naukova Dumka, Kiev, English translation: Amer. Math. Soc. Transl., 17, (1968), Providence, R. I
  2. V. A. Boichenko, G. A. Leonov, and V. Reitmann, Dimension Theory for Ordinary Differential Equations, Teubner, Stuttgart - Leipzig, 2004
  3. H. Brezis, Problemes unilateraux, J. Math. Pures. Appl, 51, (1972), 1 - 168
  4. V. A. Brusin, The Lurie equations in Hilbert space and its solvability, Prikl. Math. Mekh., 40, 5, (1976), 947-955 (in Russian)
  5. R. Datko, Extending a theorem of A. M. Liapunov to Hilbert spaces, J. Math. Anal. Appl, 32, (1970), 610 - 616
  6. G. Duvant and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976
  7. I. Ermakov, Y. Kalinin and V. Reitmann, Determining modes and almost periodic integrals for cocycles, J. Differential Equations, (2011)
  8. C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolution equations, J. Diff. Eq., 73, (1988), 309 -353
  9. D. Kalinichenko, V. Reitmann and S. Skopinov, Asymptotic behaviour of solutions of a coupled system of Maxwell's equation and a controlled differential inclusions, Proc. 9AIMS Conference on Dynamical Systems, Differential Equations and Applications, (2012), Orlando, Florida, USA
  10. Y. N. Kalinin and V. Reitmann, Almost periodic solutions in control systems with monotone nonlinearities, e-journal Differential Equations and Control Processes, 4, (2012), 40-68. www. math. spbu. ru/diffjournal/RU/numbers/2012. 4/article. 1. 4. html
  11. Y. Kalinin, V. Reitmann and N. Yumaguzin, Asymptotic behavior of Maxwell's equation in one-dimension space with termal effect, Discrete and Cont. Dyn. Sys. - Supplement 2011 2, (2011), 754 - 762
  12. H. Kantz and V. Reitmann, Reconstructing attractors of infinite-dimensional dynamical systems from low-dimensional projections, Workshop on Multivaluate Time Series Analysis, IWH Heidelberg (2004)
  13. O. A. Ladyzhenskaya, On estimates of the fractal dimension and the number of determining modes for invariant sets of dynamical systems, Zapiski nauchnich seminarov LOMI. 163, (1987), 105 - 129
  14. J. Louis and D. Wexler, The Hilbert space regulator problem and operator Riccati equation under stabilizability, Annales de la Societe Scientifique de Bruxelles, 1 105, 4 (1991), 137-165
  15. A. L. Likhtarnikov, Absolute stability criteria for nonlinear operator equations, Izv. Akad. Nauk SSSR Ser. Mat., 41, 5, (1977), 1064 - 1083
  16. A. L. Likhtarnikov and V. A. Yakubovich, The frequency theorem for equations of evolutionary type, Siberian Math. ~J. 17, 5, (1976), 790 - 803
  17. A. L. Likhtarnikov and V. A. Yakubovich, Dichotomy and stability of uncertain nonlinear systems in Hilbert spaces, Algebra and Analysis, 9, 6, (1997), 132 - 155
  18. R. V. Manoranjan and H. -M. Yin, On two-phase Stefan problem arising from a microwave heating process, J. Continuous and Discrete Dynamical Systems 15, (2006), 1155 - 1168
  19. A. Pankov, Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations, Kluwer Academic Publishers, 1990
  20. S. Popov, Takens time delay embedding theorem for dynamical systems on infinite-dimensional manifolds, Proc. International student conference "Science and Progress", St. Petersburg-Peterhof (2011)
  21. J. C. Robinson, Inertial manifolds and the cone condition, Dyn. Syst. Appl. 2, 3, (1993), 311 - 330
  22. G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 1990
  23. R. A. Smith, Orbital stability of ordinary differential equations, J. Differential Equations 69, (1986), 265 -287

Полный текст (pdf)