On The Convergence of Stability Domain on Time Scales
Main Article Content
Abstract
This paper studies convergence of the stability domains for a sequence of time scales. It is proved that if the sequence of time scales (Tn) converges to a time scale T in Hausdorff topology then their stability domains UTn will converge to the stability domain UT of T.
Keywords: Implicit dynamic equations, time scales, convergence, stability domain, stability radius.
References
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[2] H. Attouch, R. Lucchetti, R.J.-B. Wets, The topology of ρ-Hausdorff distance, Ann. Mat. Pur. Appl., CLX (1991) 303-320.
[3] F. Memoli, Some properties of Gromov-Hausdorff distance, Discrete Comput. Geom. 48 (2012) 416-440.
[4] M. Bohner and A. Peterson, Dynamic equations on time scales: An Introduction with Applications, Birkhäuser, Boston, 2001.
[5] M. Bohner and A. Peterson, Advances in dynamic equations on time scales, Birkhäuser, Boston, 2003.
[6] T.S. Doan, R. Kalauch and S. Sieghmund, Exponentially Stability of Linear Time-Invariant System on Time Scales, Nonliear Dyamics and Systems Theory 9 (1) (2009) 37-50 .
[7] N.H. Du, D.D. Thuan and N.C. Liem, Stability radius of implicit dynamic equations with constant coefficients on time scales, Systems Control Lett. 60 (2011) 596-603.
[8] N. H. Du, Stability radii of differential-algebraic equations with structured perturbations, Systems Control Lett. 57 (2008) 546-553.
[9] R. Agarwal, M. Bohner, D. O'Regan, A. Peterson, Dynamic equations on time scales: a survey, J. Comput. Appl. Math. 141 (2002) 1-26.
[10] M. Bracke, On stability radii of parametrized linear differential-algebraic systems, Ph.D thesis, University of Kaiserslautern, 2000.
[11] T.S. Doan, A. Kalauch, S. Siegmund and F. Wirth, Stability radii for positive linear time-invariant systems on time scales, Systems Control Lett. 59 (2010) 173-179.
[12] S. Hilger, “Ein Makettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten”, Ph.D thesis, Universität Würzburg, 1988.
[13] S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results. Math. 18 (1990) 18-56.
[14] D. Hinrichsen and A. J. Pritchard, Stability radii of linear systems, Systems Control Lett. 7 (1986) 1-10.
[15] D. Hinrichsen, A. J. Pritchard, Stability radius for structured perturbations and the algebraic Riccati equation, Systems Control Lett. 8 (1986) 105-113.