Higher order Runge-Kutta methods for impulsive differential systems. (English) Zbl 1278.65105

Summary: This paper studies higher order approximations of solutions of differential equations with non-fixed times of impulses. We assume that the right-hand side is sufficiently smooth. Using a Runge-Kutta method of higher order and natural assumptions on the impulsive surfaces and the impulses, we calculate good approximations of the jump times, which enables us to extend the classical results for higher order of convergence of Runge-Kutta methods to more complicated systems.


65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations


Full Text: DOI


[1] J.-P. Aubin, Impulsive Differential Inclusions and Hybrid Systems: A Viability Approach, Lecture Notes, Univ. Paris, 2002.
[2] Baier, R.; Donchev, T., Discrete approximations of impulsive differential inclusions, Numer. funct. anal. optim., 31, 653-678, (2010) · Zbl 1205.34012
[3] Bainov, D.; Simeonov, P., Impulsive differential equations: asymptotic properties of the solutions, Series on advances in mathematics for applied sciences 28, (1995), World Scientific River Edge, NJ · Zbl 0828.34002
[4] Bellen, A.; Zennaro, M., Numerical methods for delay differential equations, (2003), Oxford University Press · Zbl 0749.65042
[5] Benchohra, M.; Henderson, J.; Ntouyas, S., Impulsive differential equations and inclusions, (2006), Hindawi Publishing Company New York · Zbl 1130.34003
[6] Brogliato, B., Nonsmooth impact mechanics. models, dynamics and control, Lecture notes in control and information sciences, vol. 220, (1996), Springer London
[7] Butcher, J., Numerical methods for ordinary differential equations, (2003), Wiley West Sussex · Zbl 1040.65057
[8] Donchev, T., Impulsive differential inclusions with constrains, Ejde, 66, 1-12, (2006)
[9] Filippova, T.; Vzdornova, O., State estimation for linear impulsive control systems, WSEAS trans. syst., 4, 974-979, (2005) · Zbl 1124.93036
[10] Hamzah, N.; Mamat, M.; Kavikumar, J.; Chong, L.; Ahmad, N., Impulsive differential equations by using Euler method, Appl. math. sci., 4, 3219-3232, (2010) · Zbl 1239.65048
[11] Hossainzadeh, H.; Afrouzi, G.; Yazdani, A., Application of Adomian decomposition method for solving impulsive differential equations, J. math. comput. sci., 2, 672-681, (2011)
[12] R. Knapp, M. Sofroniou, S. Wolfram, et al., Advanced Numerical Differential Equation Solving in Mathematica, Wolfram Mathematica Documentation Center, Wolfram Research, Inc., 2012. <http://reference.wolfram.com/mathematica/tutorial/NDSolveOverview.html>.
[13] Kostousova, E., State estimation for linear impulsive differential systems through polyhedral techniques, Discrete contin. dynam. syst. suppl., 2009, 466-475, (2009), (special issue) · Zbl 1184.93011
[14] Lakshmikantham, V.; Bainov, D.; Simeonov, P., Theory of impulsive differential equations, Series in modern applied mathematics, vol. 6, (1989), World Scientific Teaneck, NJ · Zbl 0719.34002
[15] Perestyuk, N.; Plotnikov, V.; Samoıˇlenko, A.; Skripnik, N., Differential equations with impulse effects. multivalued right-hand sides and discontinuities, De gruyter studies in mathematics, vol. 40, (2011), De Gruyter Berlin, Russian original: Impulsive Differential Equations with a Multivalued and Discontinuous Right-Hand Side, Inst. Math. NAS Ukraine, Kiev, 2007 · Zbl 1234.34002
[16] Plotnikov, V.; Kitanov, N., On continuous dependence of solutions of impulsive differential inclusions and impulse control problems, Cybernet. system anal., 38, 749-758, (2002) · Zbl 1024.49029
[17] Plotnikov, V.; Plotnikov, A.; Vitiuk, A., Differential equations with multivalued right-hand side, asymptotical methods, (1999), Astro Print Odessa, (in Russian)
[18] Ran, X.; Liu, M.; Zhu, Q., Numerical methods for impulsive differential equation, Math. comput. model., 48, 46-55, (2008) · Zbl 1145.65317
[19] Samoıˇlenko, A.; Perestyuk, N., Impulsive differential equations, World scientific series on nonlinear science. series A: monographs and treatises, vol. 14, (1995), World Scientific River Edge, NJ, Russian original: Differential Equations with Impulse Effects, Visča Škola, Kiev, 1987
[20] Schirotzek, W., Nonsmooth analysis, (2007), Springer-Verlag Berlin-Heidelberg · Zbl 1120.49001
[21] Wolenski, P.R.; Žabić, S., A sampling method and approximation results for impulsive systems, SIAM J. control optim., 46, 983-998, (2007) · Zbl 1157.34006
[22] Yang, T., Impulsive control theory, Lecture notes in control and information sciences, vol. 272, (2001), Springer Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.