Abstract
We investigate discrete approximation of the solution set of an impulsive differential inclusions with not fixed time of impulses (jumps) in finite dimensional Euclidean space. The right-hand side is assumed to be almost upper semi continuous and one sided Lipschitz. The fact that the impulsive times are not fixed posses problems and in the paper we study several variants of the Euler method in case of autonomous and not autonomous systems. The accuracy (in appropriate metric) of all considered variants is \(O(\sqrt{h})\). The results can be applied to impulsive optimal control problems.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
![](https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png)
References
Aubin, J.-P.: Impulsive Differential Inclusions and Hybrid Systems: A Viability Approach, Lecture Notes, Univ. Paris (2002)
Benchohra, M., Henderson, J., Ntouyas, S.: Impulsive Differential Equations and Inclusions. Hindawi Publishing Company, New York (in press)
Deimling, K.: Multivalued differential equations. de Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyter & Co., Berlin (1992)
Donchev, T.: Impulsive differential inclusions with constrains. EJDE 66, 1–12 (2006)
Donchev, T., Farkhi, E.: Euler approximation of discontinuous one-sided Lipschitz convex differential inclusions. In: Ioffe, A., Reich, S., Shafrir, I. (eds.) Calculus of Variations and Differential Equations, pp. 101–118. Chapman & Hall/CRC, Boca Raton, New York (1999)
Dontchev, A., Farkhi, E.: Error estimates for discretized differential inclusions. Computing 41, 349–358 (1989)
Dontchev, A., Lempio, F.: Difference methods for differential inclusions: a survey. SIAM Rev. 34, 263–294 (1992)
Grammel, G.: Towards fully discretized differential inclusions. Set Valued Analysis 11, 1–8 (2003)
Lakshmikantham, V., Bainov, D., Simeonov, P.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)
Lempio, F., Veliov, V.: Discrete approximations of differential inclusions. Bayreuther Mathematische Schriften, Heft 54, 149–232 (1998)
Pereira, F., Silva, G.: Necessary Conditions of Optimality for Vector-Valued Impulsive Control Problems. System Control Letters 40, 205–215 (2000)
Plotnikov, V., Kitanov, N.: On Continuous dependence of solutions of impulsive differential inclusions and impulse control problems. Kybern. Syst. Analysis 5, 143–154 (2002) (in Russian) English translation: Cybernetics System Analysis 38, 749–758 (2002)
Plotnikov, V., Plotnikov, A., Vityuk, A.: Differential Equations with Multivalued Right-Hand Side. Asymptotic Methods, Astro Print Odessa (Russian) (1999)
Samoilenko, A., Peresyuk, N.: Differential Equations with Impulsive Effects. World Scientific, Singapore (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Donchev, T. (2008). Approximation of the Solution Set of Impulsive Systems. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2007. Lecture Notes in Computer Science, vol 4818. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78827-0_34
Download citation
DOI: https://doi.org/10.1007/978-3-540-78827-0_34
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78825-6
Online ISBN: 978-3-540-78827-0
eBook Packages: Computer ScienceComputer Science (R0)