Abstract
A class of linear periodic impulsive systems with a delay is considered. Sufficient conditions of asymptotic stability are established for systems of this class, reducing the study of the original system’s stability to a similar problem for a system of linear impulsive differential equations (without a delay). This auxiliary result is used to study the interval stability of a linear impulsive system with a delay under general assumptions about the dynamic properties of continuous and discrete components. Examples are given to illustrate the effectiveness of the obtained results.
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REFERENCES
J. Hale, Theory of Functional Differential Equations (Springer, New York, 1977).
R. Bellman and K. Cooke, Differential Difference Equations (Academic, New York, 1963).
C. Tunç, “Instability of solutions of vector Lienard equation with constant delay,” Bull. Math. Soc. Sci. Math. Roum. (N.S.) 59, 197–204 (2016).
J. R. Graef, C. Tunc, and S. Sevgin, “Behavior of solutions of nonlinear functional volterra integro-differential equations with multiple delays,” Dynam. Syst. Appl. 25, 39–46 (2016).
A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, Vol. 14 of World Scientific Series on Nonlinear Science, Ser. A: Monographs and Treatises (River Edge, World Scientific, 1995).
V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, Vol. 6 of Series in Modern Applied Mathematics (World Scientific, Teaneck, 1989).
X. Liu and A. Willms, “Stability analysis and applications to large scale impulsive systems: a new approach,” Canad. Appl. Math. Quart. 3, 419–444 (1995).
A. I. Dvirnyi and V. I. Slyn’ko, “Application of Lyapunov’s direct method to the study of the stability of solutions to systems of impulsive differential equations,” Math. Notes. 96, 26–37 (2014).
A. I. Dvirnyi and V. I. Slyn’ko, “Investigating stability using nonlinear quasihomogeneous approximation to differential equations with impulsive action,” Sb.: Math. 205, 862–891 (2014).
A. O. Ignat’ev, O. A. Ignat’ev, and A. A. Soliman, “On the asymptotic stability and instability of solutions of systems with impulse action,” Math. Notes 80, 491–499 (2006).
A. O. Ignatyev, “On the stability of invariant sets of systems with impulse effect,” Nonlin. Anal. 69, 53–72 (2008).
I. Stamova, Stability Analysis of Impulsive Functional Differential Equations, Vol. 52 of De Gruyter Expositions in Mathematics (Walter De Gruyter, Berlin, 2009).
V. I. Slyn’ko, “Stability conditions for linear impulsive systems with delay,” Int. Appl. Mech. 41, 697–703 (2005).
W. H. Chen, Zh. Ruan, and W. X. Zheng, “Stability and L2-gain analysis for impulsive delay systems: an impulse-time-dependent discretized Lyapunov functional method,” Autom. J. IFAC 86, 129–137 (2017).
M. A. Davoa, A. Banos, F. Gouaisbaut, S. Tarbouriechc, and A. Seuret, “Stability analysis of linear impulsive delay dynamical systems via looped-functionals,” Autom. J. IFAC 81, 107–114 (2017).
I. L. Ivanov and V. I. Slyn’ko, “A stability criterion for autonomous linear time-lagged systems subject to periodic impulsive force,” Int. Appl. Mech. 49, 732–742 (2013).
I. L. Ivanov and V. I. Slyn’ko, “Stability criterion of linear systems with delay and two-periodic impulse excitation,” Autom. Remote Control 73, 1456 (2012).
V. L. Kharitonov, “On asymptotic stability of the balance position of a family of systems of linear differential equations,” Differ. Uravn. Primen 14, 2086–2088 (1978).
B. Liu, X. Liu, and X. Liao, “Robust stability of uncertain impulsive dynamical systems,” J. Math. Anal. Appl. 290, 519–533 (2004).
V. I. Slyn’ko and V. S. Denisenko, “Robust stability of systems of linear differential equations with periodic impulsive influence,” Autom. Remote Control 73, 1005 (2012).
V. S. Denisenko and V. I. Slyn’ko, “Interval stability of linear impulsive systems,” J. Comput. Syst. Sci. Int. 54, 1–12 (2015).
N. Dunford and J. T. Schwartz, Linear Operators: General Theory (Interscience, New York, London, 1958).
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in a Banach Space (Nauka, Moscow, 1970).
L. S. Pontryagin, “On the zeros of some elementary transcendental functions,” Am. Math. Soc. Transl. 2, 95–110 (1955).
N. G. Chebotarev and N. N. Meiman, “The Routh-Hurwitz problem for polynomials and entire functions,” Tr. Mat. Inst. Steklov 26, 3–331 (1949).
Yu. M. Gusev, V. N. Yefanov, V. G. Krymskiy, and V. Yu. Rutkovskiy, “Analysis and synthesis of linear interval dynamical systems (the state of the problem). II. Analysis of the stability of interval matrices and synthesis of robust regulators,” Sov. J. Comput. Syst. Sci. 30 (2), 26–52 (1992).
Funding
The study was completed with the support of The Scientific and Technological Research Council of Turkey (2221–scholarships for scientists–2221–2017/2 period), when Vitaliy Slynko was an invited scientist at Van Yuzuncu Yil University in Van, Turkey. This work was also partially supported by the Ministry of Education and Science of Ukraine (project no. 0116U004691).
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Slynko, V.I., Tunç, C. & Erdur, S. Sufficient Conditions of Interval Stability of a Class of Linear Impulsive Systems with a Delay. J. Comput. Syst. Sci. Int. 59, 8–18 (2020). https://doi.org/10.1134/S1064230719060145
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DOI: https://doi.org/10.1134/S1064230719060145