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

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

Determining Observations for Stability and Bifurcation on a Finite Time Interval in Variational Control Systems with a Parameter

Author(s):

D. Yu. Kalinchenko

Abstract:

Stability and bifurcation on a finite time interval for a thermovisco-elastoplastic contact problem are considered. To describe such a type of contact Coulomb's law for dry friction, which is written as a variational inequality is used. The contact problem is presented as a parameter dependent variational system. Phase spaces for the system are given by scales of Hilbert spaces. Determining observation operators for bifurcation of the system and output convergence are introduced. The frequency theorem is applied in order to describe stability and the bifurcation which is understood as a loss of stability on a finite time interval. Frequency-domain conditions for the existence of determining observations and for absolute dichotomy of a variational equation are given. The connection between the frequency-domain condition and the completeness defect of the observation operator is considered.

Keywords

References:

  1. Andrews K. T., Kuttler K. L., Shillor M. On the dynamic behaviour of a thermoviscoelastic body in frictional contact with a rigid obstacle. Euro. Jnl. Appl. Math. , 1997, vol. 8, pp. 417-436
  2. Berezanskii Yu. M. Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov [Expansion in eigenfunctions of self-adjoint operators]. Kiev, Naukova dumka Publ., 1965, 799 p. (In Russian)
  3. Duvant G., Lions J. -L. Inequalities in mechanics and physics. Berlin, Springer-Verlag, 1976, 397 p
  4. Ermakov I. V., Kalinin Yu. N., Reitmann V. Determining modes and almost periodic integrals for cocycles. Differential Equations, 2011, vol. 47, no. 13, pp. 1837-1852
  5. Ermakov I. V., Reitmann V. [Determining functionals for a microwave heating system]. Vestnik Sankt-Peterburgskogo universiteta. Ser. 1. Mat., mekh., astron. , 2012, no. 4, pp. 13-17. (In Russian)
  6. Kalinichenko D., Reitmann V. Bifurcation on a finite time interval in nonlinear hyperbolic-parabolic parameter dependent control systems. Proc. The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Madrid, 2014, p. 211
  7. Kalinichenko D., Reitmann V., Skopinov S. Asymptotic behavior of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion. Discrete and Continuous Dynamical Systems, Supplement, 2013, pp. 407-414
  8. Likhtarnikov A. L., Yakubovich V. A. [Dichotomy and stability of uncertain nonlinear systems in Hilbert spaces]. Algebra i analiz, 1997, vol. 9, no. 6, pp. 132-155. (In Russian)
  9. Likhtarnikov A. L., Yakubovich V. A. [The frequency theorem for equations of evolutionary type]. Sibirskii matematicheskii zhurnal, 1976, vol. 17, no. 5, pp. 1069-1085. (In Russian)
  10. Pankov A. A. Ogranichennye i pochti periodicheskie resheniya nelineinykh differentsial'no-operatornykh uravneii [Bounded and almost periodic solutions of nonlinear differential-operator equations]. Kiev, Naukova dumka Publ., 1985, 182 p. (In Russian)
  11. Reitmann V. Frequency domain conditions for the existence of almost periodic solutions in evolutionary variational inequalities. Stochastics and Dynamics, 2004, vol. 4, pp. 483-499

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