Robust Stability of Implicit Dynamic Equations on Time Scales
Main Article Content
Abstract
This paper studies the robust stability of implicit dynamic equations on time scales, which is a general form of differential algebraic equations and implicit difference equations. The paper discusses the reservation of exponential stability of these equations under small Lipschitz perturbations.
Keywords:
Uniform stability, time scales, implicit dynamic equations, Lipschitz perturbations.
References
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[2] M. Bracke, On stability radii of parametrized linear differential-algebraic systems, Ph.D. Thesis, University of Kaiserslautern, 2000.
[3] P. Kunkel, V. Mehrmann, Differential-Algebraic Equations, Analysis and Numerical Solution. EMS Publishing House, Z¨urich, Switzerland, 2006.
[4] V.H. Linh, N.T.T. Nga, D.D. Thuan, Exponential stability and robust stability for linear time-varying singular systems of second-order difference equations, SIAM J. Matrix Anal. Appl. 39 (2018) 204–233.
[5] D.G. Luenberger, Dynamic equations in descriptor form IEEE, Trans. Automat. Control. 22(1977) 312–322.
[6] Z. Bartosiewicz, Linear positive control systems on time scales, Math. Control Signals Syst., 25 (2013) 327–343.
[7] J.J. DaCunha, J.M. Davis, A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems., J. Differential Equations 251(2011) 2987–3027.
[8] E. Griepentrog, R. M¨arz, Differential-algebraic equations and their numerical treatment, Teubner-Texte zur Mathematik, Leibzig 1986.
[9] R. M¨arz, Extra-ordinary differential equation attempts to an analysis of differential algebraic system, Progress in Mathematics, 168 (1998) 313-334.
[10] T. Berger, A. Ilchmann, On stability of time-varying linear differential-algebraic equations, International Journal of Control, 86 (2013) 1060–1076.
[11] A.A. Shcheglova, V.F. Chistyakov, Stability of linear differential-algebraic systems, Differential Equations 40(1) (2004) 50–62.
[12] M. Bohner, A. Peterson, Dynamic equations on time scales: An Introduction with Applications, Birkh¨auser, Boston, 2001.
[13] N.H. Du, T.K. Duy, V.T. Viet, Degenerate cocycle with index-1 and Lyapunov exponent, Stochatics and Dynamics 7(2)(2007) 229-245.
[14] N.H.Du, N.C. Liem, Linear transformations and Floquet theorem for linear implicit dynamic equations on time scales, Asian-European Journal of Mathematics, 6(1)(2013), 1350004 (21 pages).