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On stability for generalized linear differential equations and applications to impulsive systems

Part of: General theory in ordinary differential equations Stability theory

Published online by Cambridge University Press:  15 February 2023

Claudio A. Gallegos  and
Gonzalo Robledo
Claudio A. Gallegos
Affiliation:
Departamento de Matemáticas, Universidad de Chile, Casilla 653, Santiago, Chile (claudio.gallegos.castro@gmail.com; grobledo@uchile.cl)
Gonzalo Robledo
Affiliation:
Departamento de Matemáticas, Universidad de Chile, Casilla 653, Santiago, Chile (claudio.gallegos.castro@gmail.com; grobledo@uchile.cl)
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Abstract

In this paper, we are interested in investigating notions of stability for generalized linear differential equations (GLDEs). Initially, we propose and revisit several definitions of stability and provide a complete characterization of them in terms of upper bounds and asymptotic behaviour of the transition matrix. In addition, we illustrate our stability results for GLDEs to linear periodic systems and linear impulsive differential equations. Finally, we prove that the well-known definitions of uniform asymptotic stability and variational asymptotic stability are equivalent to the global uniform exponential stability introduced in this article.

Keywords

Generalized ordinary differential equationsKurzweil integralStabilityAsymptotic stabilityImpulsive equations

MSC classification

Primary: 34D20: Stability
Secondary: 34A30: Linear equations and systems, general 34A37: Differential equations with impulses

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 2 , April 2024 , pp. 381 - 407
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Adrianova, L. Y.. Introduction to Linear Systems of Differential Equations, Translations of the AMS, Vol. 146 (AMS, Providence, 1991).Google Scholar
Andrade da Silva, F., Federson, M. and Toon, E.. Stability: boundedness and controllability of solutions of measure functional differential equations. J. Differ. Eq. 307 (2022), 160–210.CrossRefGoogle Scholar
Bainov, D. and Simeonov, P.. Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics Vol. 66 (Harlow: Longman Scientific $\& $ Technical, New York, John Wiley & Sons, 1993).Google Scholar
Barabanov, E. A. and Konyukh, A. V.. Bohl exponents of linear differential equations. Mem. Differential Equations Math. Phys. 24 (2001), 151–158.Google Scholar
Barreira, L. and Valls, C.. Stability of Nonautonomous Differential Equations, Lecture Notes in Mathematics Vol. 1926 (Springer, Berlin, 2008).Google Scholar
Bonotto, E., Federson, M. and Santos, F. L.. Dichotomies for generalized ordinary differential equations and applications. J. Differ. Eq. 264 (2018), 3131–3173.CrossRefGoogle Scholar
Bonotto, E., Federson, M. and Santos, F. L.. Robustness of exponential dichotomies for generalized ordinary differential equations. J. Dynam. Differ. Equ. 32 (2020), 2021–2060.CrossRefGoogle Scholar
Briend, J. Y.. Petit Traité d'Intégration, EDP–Sciences, Grenoble, 2014.Google Scholar
Burton, T. A.. Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Mathematics in Science and Engineering, Vol. 178 (Academic Press, Orlando, 1985).Google Scholar
Federson, M. and Schwabik, Š.. Generalized ODE approach to impulsive retarded functional differential equations. Differ. Integral Equ. 19 (2006), 1201–1234.Google Scholar
Federson, M., Grau, R. and Mesquita, J. G.. Prolongation of solutions of measure differential equations and dynamic equations on time scales. Math. Nachr. 292 (2019), 22–55.CrossRefGoogle Scholar
Federson, M., Grau, R., Mesquita, J. G. and Toon, E.. Lyapunov stability for measure differential equations and dynamic equations on time scales. J. Differ. Equ. 267 (2019), 4192–4223.Google Scholar
Federson, M., Mesquita, J. G. and Slavík, A.. Measure functional differential equations and dynamic equations on time scales. J. Differ. Equ. 252 (2012), 3816–3847.Google Scholar
Fraňková, D.. Regulated functions with values in Banach space. Math. Bohem. 144 (2019), 437–456.Google Scholar
Gallegos, C. A., Grau, R. and Mesquita, J. G.. Stability, asymptotic and exponential stability for various types of equations with discontinuous solutions via Lyapunov functionals. J. Differ. Equ. 299 (2021), 256–283.CrossRefGoogle Scholar
Hahn, W.. Stability of Motion (Berlin, Heidelberg: Springer–Verlag, 1967).CrossRefGoogle Scholar
Halanay, A. and Wexler, D.. Qualitative Theory of Impulse Systems. (Romanian) ( Bucarest (Editura Academiei Republicii Socialiste Romania, 1968).Google Scholar
Hinrichsen, D. and Prithcard, A. J.. Mathematical Systems Theory I Modelling, State Space Analysis, Stability and Robustness. Texts in Applied Mathematics, Vol. 48 (Springer, Berlin–Heidelberg 2005).Google Scholar
Imaz, C. and Vorel, Z.. Generalized ordinary differential equations in Banach spaces and applications to functional equations. Bol. Soc. Mat. Mexicana 11 (1966), 47–59.Google Scholar
Kurtz, D. S. and Swartz, C. W.. Theories of Integration: The integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane. Ser. Real Anal., Vol. 13 (Singapore, World Scientific, 2012).Google Scholar
Kurzweil, J.. Addition to my paper ‘Generalized ordinary differential equations and continuous dependence on a parameter’. Czech. Math. J. 9 (1959), 564–573.Google Scholar
Kurzweil, J.. Generalized ordinary differential equations and continuous dependence on a parameter. Czech. Math. J. 7 (1957), 418–448.CrossRefGoogle Scholar
Kurzweil, J.. Generalized ordinary differential equations. Czech. Math. J. 8 (1958), 360–388.Google Scholar
Kurzweil, J.. Unicity of solutions of generalized differential equations. Czech. Math. J. 8 (1958), 502–509.Google Scholar
Lakshmikantham, V., Baĭnov, D. D. and Simeonov, P. S.. Theory of impulsive differential equations. Series in Modern Applied Mathematics, Vol. 6 (Teaneck, World Scientific, 1989).Google Scholar
Milev, N. V. and Bainov, D. D.. Stability of linear impulsive equations. Comput. Math. Appl. 20 (1990), 35–41.Google Scholar
Monteiro, G. A. and Tvrdý, M.. Generalized linear differential equations in a Banach space: continuous dependence on a parameter. Discrete Contin. Dyn. Syst. 33 (2013), 283–303.CrossRefGoogle Scholar
Monteiro, G. A. and Tvrdý, M.. On Kurzweil–Stieltjes integral in Banach space. Math. Bohem. 137 (2012), 365–381.Google Scholar
Oliva, F. and Vorel, Z.. Functional equations and generalized ordinary differential equations. Bol. Soc. Mat. Mexicana 11 (1966), 40–46.Google Scholar
Perko, L.. Differential Equations and Dynamical Systems (New York, Springer–Verlag, 2001).Google Scholar
Samoilenko, A. and Perestyuk, N.. Impulsive differential equations, World Scientific Series on Nonlinear Science (Singapore, World Scientific, 1995).CrossRefGoogle Scholar
Schwabik, Š.. Abstract Perron-Stieltjes integral. Math. Bohem. 121 (1996), 425–447.Google Scholar
Schwabik, Š.. Floquetova teorie pro zobecněné diferenciální rovnice. Časopis pěst. Mat. 98 (1973), 416–418.CrossRefGoogle Scholar
Schwabik, Š.. Generalized Ordinary Differential Equations, Ser. Real Anal., Vol. 5 (Singapore, World Scientific, 1992).CrossRefGoogle Scholar
Schwabik, Š.. Variational stability for generalized differential equations. Časopis pěst. Mat. 109 (1984), 389–420.CrossRefGoogle Scholar
Schwabik, Š.. Verallgemeinerte lineare differentialgleichungssysteme. Časopis pěst. Mat. 96 (1971), 183–211.Google Scholar
Schwabik, Š. and Tvrdý, M.. On the generalized linear ordinary differential equation. Časopis pěst. Mat. 98 (1973), 206–211.Google Scholar
Schwabik, Š., Tvrdý, M. and Vejvoda, O.. Differential and Integral Equations. Boundary Value Problems and Adjoints (Reidel Dordrecht, Academia Praha, 1979).Google Scholar
Slavík, A.. Dynamic equations on time scales and generalized ordinary differential equations. J. Math. Anal. Appl. 385 (2012), 534–550.Google Scholar
Zhang, J., Chang, X. and Wang, J.. Existence of robustness of nonuniform $(h,\, k,\, \mu,\, \nu )$–dichotomies for nonautonomous impulsive differential equations. J. Math. Anal. Appl. 400 (2013), 710–723.Google Scholar