Abstract
This work is devoted to the construction of impulsive sets in \({\mathbb{R}^n}\). In the literature, there are many examples of impulsive dynamical systems whose impulsive sets are chosen in an abstract way, and in this paper we present sufficient conditions to characterize impulsive sets in \({\mathbb{R}^n}\) which satisfy some “tube conditions” and ensure a good behavior of the flow. Moreover, we present some examples to illustrate the theoretical results.
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Bonotto E. M.: Flows of characteristic 0+ in impulsive semidynamical systems. J. Math. Anal. Appl. 332, 81–96 (2007)
E. M. Bonotto, M. C. Bortolan, A. N. Carvalho, and R. Czaja, Global attractors for impulsive dynamical systems – a precompact approach, J. Differential Equations 259 (2015), 2602–2625.
Bonotto E. M., Federson M.: Topological conjugation and asymptotic stability in impulsive semidynamical systems. J. Math. Anal. Appl. 326, 869–881 (2007)
Bonotto E. M., Ferreira J. C.: Uniform attractors of discontinuous semidynamical systems, Collect. Math. 65, 47–59 (2014)
Bruno Bouchard, Ngoc-Minh Dang and Charles-Albert Lehalle, Optimal control of trading algorithms: a general impulse control approach, SIAM J. Financial Math., 2 (2011), 404–438.
Tiziana Cardinali and Raffaella Servadei, Periodic solutions of nonlinear impulsive differential inclusions with constraints, Proc. Amer. Math. Soc., 132 (2004), 2339–2349 (electronic).
Krzysztof Ciesielski, On semicontinuity in impulsive dynamical systems, Bull. Pol. Acad. Sci. Math., 52 (2004), 71–80
Krzysztof Ciesielski, On stability in impulsive dynamical systems, Bull. Pol. Acad. Sci. Math., 52 (2004), 81–91.
Mark H. A. Davis, Xin Guo and Guoliang Wu, Impulse control of multidimensional jump diffusions, SIAM J. Control Optim., 48 (2010), 5276–5293.
Changming Ding, Lyapunov quasi-stable trajectories, Fund. Math., 220 (2013), 139–154.
Morris W. Hirsch, Differential Topology Graduate Texts in Mathematics, No. 33. Springer-Verlag (New York–Heidelberg, 1976).
Saroop Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120–128.
V. Lakshmikantham, D. D. Baĭnov and P. S. Simeonov, Theory of Impulsive Differential Equations, volume 6 of Series in Modern Applied Mathematics, World Scientific Publishing Co., Inc. (Teaneck, NJ, 1989).
Xiaodi Li, Further analysis on uniform stability of impulsive infinite delay differential equations, Appl. Math. Lett., 25 (2012), 133–137.
Lichun Zhao, Lansun Chen and Qingling Zhang, The geometrical analysis of a predator-prey model with two state impulses, Math. Biosci., 238 (2012), 55–64.
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E. M. Bonotto: Partially supported by FAPESP grant 2014/25970-5 and CNPq grant 307317/2013-7.
T. Caraballo: Partially supported by FEDER and Ministerio de Economía y Competitividad (Spain) under grant MTM2015-63723-P, and Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under Proyecto de Excelencia P12-FQM-1492.
R. Collegari: Partially supported by FAPESP grants 2013/23933-2 and 2014/20691-0.
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Bonotto, E.M., Bortolan, M.C., Caraballo, T. et al. Impulsive surfaces on dynamical systems. Acta Math. Hungar. 150, 209–216 (2016). https://doi.org/10.1007/s10474-016-0631-0
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DOI: https://doi.org/10.1007/s10474-016-0631-0