Stability of retarded systems with slowly varying coefficient

Michael Iosif Gil

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 3, page 877-888
  • ISSN: 1292-8119

Abstract

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The “freezing” method for ordinary differential equations is extended to multivariable retarded systems with distributed delays and slowly varying coefficients. Explicit stability conditions are derived. The main tool of the paper is a combined usage of the generalized Bohl-Perron principle and norm estimates for the fundamental solutions of the considered equations.

How to cite

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Gil, Michael Iosif. "Stability of retarded systems with slowly varying coefficient." ESAIM: Control, Optimisation and Calculus of Variations 18.3 (2012): 877-888. <http://eudml.org/doc/244352>.

@article{Gil2012,
abstract = {The “freezing” method for ordinary differential equations is extended to multivariable retarded systems with distributed delays and slowly varying coefficients. Explicit stability conditions are derived. The main tool of the paper is a combined usage of the generalized Bohl-Perron principle and norm estimates for the fundamental solutions of the considered equations. },
author = {Gil, Michael Iosif},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Linear retarded systems; stability; generalized Bohl-Perron principle; linear retarded systems},
language = {eng},
month = {11},
number = {3},
pages = {877-888},
publisher = {EDP Sciences},
title = {Stability of retarded systems with slowly varying coefficient},
url = {http://eudml.org/doc/244352},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Gil, Michael Iosif
TI - Stability of retarded systems with slowly varying coefficient
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/11//
PB - EDP Sciences
VL - 18
IS - 3
SP - 877
EP - 888
AB - The “freezing” method for ordinary differential equations is extended to multivariable retarded systems with distributed delays and slowly varying coefficients. Explicit stability conditions are derived. The main tool of the paper is a combined usage of the generalized Bohl-Perron principle and norm estimates for the fundamental solutions of the considered equations.
LA - eng
KW - Linear retarded systems; stability; generalized Bohl-Perron principle; linear retarded systems
UR - http://eudml.org/doc/244352
ER -

References

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  1. B.F. Bylov, B.M. Grobman, V.V. Nemyckii and R.E. Vinograd The Theory of Lyapunov Exponents. Nauka, Moscow (1966) (in Russian).  
  2. D.J. Garling, Inequalities. A Jorney into Linear Analysis. Cambridge, Cambridge Univesity Press (2007).  
  3. M.I. Gil, Stability of Finite and Infinite Dimensional Systems. Kluwer, NewYork (1998).  Zbl0916.93002
  4. M.I. Gil, The Aizerman-Myshkis problem for functional-differential equations with causal nonlinearities. Functional Differential Equations11 (2005) 175–185.  Zbl1084.34063
  5. A. Halanay, Differential Equations: Stability. Oscillation, Time Lags. Academic Press, NY (1966)  Zbl0144.08701
  6. J.K. Hale and S.M.V. Lunel, Introduction to Functional Differential Equations. Springer, New York (1993).  
  7. N.A. Izobov, Linear systems of ordinary differential equations. Itogi Nauki i Tekhniki. Mat. Analis.12 (1974) 71–146 (Russian).  
  8. V. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations. Kluwer (1999).  Zbl0917.34001
  9. S.G. Krein, Linear Equations in a Banach Space. Nauka, Moscow (1971) (in Russian).  
  10. M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Boston (1964).  Zbl0126.02404
  11. J.-P. Richard, Time-delay systems: an overview of some recent advances and open problems. Automatica39 (2003) 1667–1694.  Zbl1145.93302
  12. R. Vinograd, An improved estimate in the method of freezing. Proc. Amer. Soc. 89 (1983) 125–129.  Zbl0525.34040
  13. A. Zevin and M. Pinsky, Delay-independent stability conditions for time-varying nonlinear uncertain systems. IEEE Trans. Automat. Contr.51 (2006) 1482–1485.  
  14. A. Zevin and M. Pinsky, Sharp bounds for Lyapunov exponents and stability conditions for uncertain systems with delays. IEEE Trans. Automat. Contr.55 (2010) 1249–1253. 

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