On Stability and Stabilization of Perturbed Time Scale Systems with Gronwall Inequalities
DOI:
https://doi.org/10.15407/mag11.03.207Анотація
У статті представлено нові нелінійні інтегральні нерівності на часовій шкалі. За допомогою явних інтегральних оцінок одержано достатні умови для рівномірної асимптотичної стійкості збурених систем на часових шкалах. Далі, ґрунтуючись на прямому методі Ляпунова, розроблено необхідний тип стабільності.
Mathematics Subject Classification: 34N, 34D, 76E, 34E.
Ключові слова:
збурені системи, часова шкала, нерівності Гронуолла, квадратична стабільністьПосилання
V.M. Alekseev, An Estimate for the Perturbations of Solutions of Ordinary Differential Equations. — Vestnik Moskov. Univ. Ser. I. Mat. Meh. (1961), No. 2, 28–36. (Russian)
B. Aulbach and S. Hilger, Linear Dynamic Processes with Inhomogeneous Time Scale, Nonlinear Dynamics and Quantum Dynamical Systems. (Gaussing, 1990), Math. Res. Vol. 59, Akademie Verlag, Berlin, 1990, p. 9–20.
S.V. Babenko and A.A. Martynyuk, Nonlinear Dynamic Inequalities and Stability of Quasilinear Systems on Time Scales. — Nonlinear Dynamics and Systems Theory 13 (2013), No. 1, 13–23.
S.V. Babenko, Revisiting the Theory of Stability on Time Scales of the Class of Linear Systems with Structural perturbations. — Internationnal Applied Mechanics 47 (2011), No. 1, 86–96.
A. Ben Abdallah, M. Dlala, and M.A. Hammami, A New Lyapunov Function for Stability of Time-Varying Nonlinear Perturbed Systems. — Syst. Control Lett. 56 (2007), No. 3, 179–187.
M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser Boston, 2001. https://doi.org/10.1007/978-1-4612-0201-1
M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, 2003. https://doi.org/10.1007/978-0-8176-8230-9
M. Bohner and A.A. Martynyuk, Elements of Stability Theory of A.M. Liapunov for Dynamic Equations on Time Scales. — Nonlinear Dynamics and Systems Theory 7 (2007), No. 3, 225–251.
M. Bohner, Some Oscillation Criteria for First Order Delay Dynamic Equations. — Far East J. Appl. Math. 18 (2005), No. 3, 289–304.
F. Brauer and J.A. Nohel, The Qualitative Theory of Ordinary Differential Equations. An Introduction. New York, NY, USA, Dover Publications, 1989. https://doi.org/10.1007/978-1-349-19661-6_1
S.K. Choi, D.M. Im, and N. Koo, Stability of Linear Dynamic Systems on Time Scales. Advances in Difference Equations. Article ID 670203, 2008.
J.J. DaCunha, Transition Matrix and Generalized Matrix Exponential Via the Peano–Baker Series. — J. Difference Equ. Appl. 11 (2005), No. 15, 1245–1264.
J.J. DaCunha, Stability for Time Varying Linear Dynamic Systems on Time Scales. — J. Comput. Appl. Math. 176 (2005), No. 2, 381–410.
M. Dlala and M.A. Hammami, Uniform Exponential Practical Stability of Impulsive Perturbed Systems. — J. Dynamical and Control Systems 13 (2007), No. 3, 373–386.
B. Ghanmi, N. Hadj Taieb, and M.A. Hammami, Growth Conditions for Exponential Stability of Time-Varying Perturbed Systems. — Int. J. Control 86 (2013), No. 6, 1086–1097.
S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Ph. D. thesis, Universität Würzburg, 1988.
S. Hilger, Analysis on Mesure Chains −a Unified Approach to Continuous and Discrete Calculus. — Results Math. 18 (1990), No. 1/2, 18–56.
S. Hilger and P.E. Kloeden, Comparative Time Grainyness and Asymptotic Stability of Dynamical Systems. — Autom. Remote Control 55 (1994), No. 1, 1293–1298.
J. Hoffacker and C.C. Tisdell, Stability and Instability for Dynamic Equations on Time Scales. — Comput. Math. Appl. 49 (2005), No. 9/10, 1327–1334.
B. Kaymakcalan, Lyapunov Stability Theory for Dynamic Systems on Time Scales. — J. Appl. Math. Stochastic Anal. 5 (1992), No. 3, 275–281.
V. Lakshmikantham, S. Sivasundram, and B. Kaymakcalan, Dynamic Systems on Measure Chains. Kluwer Academic Publishers, Boston, Mass, USA, 1996. https://doi.org/10.1007/978-1-4757-2449-3
V. Lakshmikantham and S. Leela, Differential and Integral Inequalities. Vol. 1. New York, NY, USA, Academic Press, 1969.
Lei Li and Shi Huang Hong, Exponential Stability for Set Dynamic Equations on Time Scales. — J. Comput. Appl. Math. 235 (2011), No. 17, 4916–4924.
W.N. Li, Some Pachpatte Type Inequalities on Time Scales. — Comput. Math. Appl. 57 (2009), No. 2, 275–282.
Li Yin and Feng Qi, Some Integral Inequalities on Time Scales. — Results Math. 64 (2013), No. 3/4, 371–381.
M.E. Lord and A.R. Mitchell, A New Approach to the Method of Nonlinear Variation of Parameters. — J. Appl. Math. Com. 4 (1978), No. 2, 95–105.
A.A. Martynyuk, On the Exponential Stability of a Dynamical System on a Time Scale. — Dokl. Akad. Nauk 421 (2008), No. 3, 312–317. (Russian)
Yu.A. Martynyuk-Chernienko and L.N. Chernetskaya, Analyzing the Exponential Stability of Motion on Time Scales. — Int. Appl. Mech. 46 (2010), No. 4, 467–473.
B.G. Pachpatte, Inequalities for Differential and Integral Equations. Mathematics in Science and Engineering. 197, Academic Press Inc. San Diego CA, 1998.
D.B. Pachpatte, Explicit Estimates on Integral Inequalities with Time Scale. — J. Inequal. Pure Appl. Math. 7 (2006), No. 4, Article 143.
A. Peterson and R.F. Raffoul, Exponential Stability of Dynamic Equations on Time Scales. — Adv. Differ. Equ. Appl. 2 (2005), 133–144.
Yjun Ma and Jitao Sun, Uniform Eventual Lipshitz Stability of Impulsive Systemson Time Scales. — J. Comput. Appl. Math. 211 (2009), No. 1, 246–250.
