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.
Keywords: linear retarded systems, stability, generalized Bohl-Perron principle
@article{COCV_2012__18_3_877_0, author = {Gil, Michael Iosif}, title = {Stability of retarded systems with slowly varying coefficient}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {877--888}, publisher = {EDP-Sciences}, volume = {18}, number = {3}, year = {2012}, doi = {10.1051/cocv/2011185}, mrnumber = {3041668}, zbl = {1268.34134}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2011185/} }
TY - JOUR AU - Gil, Michael Iosif TI - Stability of retarded systems with slowly varying coefficient JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 877 EP - 888 VL - 18 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2011185/ DO - 10.1051/cocv/2011185 LA - en ID - COCV_2012__18_3_877_0 ER -
%0 Journal Article %A Gil, Michael Iosif %T Stability of retarded systems with slowly varying coefficient %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 877-888 %V 18 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2011185/ %R 10.1051/cocv/2011185 %G en %F COCV_2012__18_3_877_0
Gil, Michael Iosif. Stability of retarded systems with slowly varying coefficient. ESAIM: Control, Optimisation and Calculus of Variations, Volume 18 (2012) no. 3, pp. 877-888. doi : 10.1051/cocv/2011185. http://archive.numdam.org/articles/10.1051/cocv/2011185/
[1] The Theory of Lyapunov Exponents. Nauka, Moscow (1966) (in Russian).
, , and[2] Inequalities. A Jorney into Linear Analysis. Cambridge, Cambridge Univesity Press (2007). | MR | Zbl
,[3] Stability of Finite and Infinite Dimensional Systems. Kluwer, NewYork (1998). | MR | Zbl
,[4] The Aizerman-Myshkis problem for functional-differential equations with causal nonlinearities. Functional Differential Equations 11 (2005) 175-185. | MR | Zbl
,[5] Differential Equations: Stability. Oscillation, Time Lags. Academic Press, NY (1966) | MR | Zbl
,[6] Introduction to Functional Differential Equations. Springer, New York (1993). | MR | Zbl
and ,[7] Linear systems of ordinary differential equations. Itogi Nauki i Tekhniki. Mat. Analis. 12 (1974) 71-146 (Russian). | MR | Zbl
,[8] Applied Theory of Functional Differential Equations. Kluwer (1999). | Zbl
and ,[9] Linear Equations in a Banach Space. Nauka, Moscow (1971) (in Russian). | MR | Zbl
,[10] A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Boston (1964). | MR | Zbl
and ,[11] Time-delay systems: an overview of some recent advances and open problems. Automatica 39 (2003) 1667-1694. | MR | Zbl
,[12] An improved estimate in the method of freezing. Proc. Amer. Soc. 89 (1983) 125-129. | MR | Zbl
,[13] Delay-independent stability conditions for time-varying nonlinear uncertain systems. IEEE Trans. Automat. Contr. 51 (2006) 1482-1485. | MR
and ,[14] Sharp bounds for Lyapunov exponents and stability conditions for uncertain systems with delays. IEEE Trans. Automat. Contr. 55 (2010) 1249-1253. | MR
and ,Cited by Sources: