Abstract
We consider systems modeled by a differential inclusion subject to impulsive, set valued state resets. We study existence of solutions for this class of systems and derive conditions for a set of states to be viable. From the point of view of hybrid systems, of central interest is the fact that the class of systems and the solution concept considered allow any finite number of left and right accumulation points of the impulse times; in other words, very complex Zeno type behaviors. The results are demonstrated on simple examples that exhibit such behaviors.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bernhard, P., El Farouq, N., Thiery, S.: An impusive game arising in finance with interesting singularities. In: Haurie, A., et al. (eds.) Advances in Dynamic Games: Applications to Economics, Management Science, Engineering, and Environmental Management. Number 8 in Annals of the International Society of Dynamic Games, pp. 335–364. Birkhäuser, Boston (2006)
Dal Maso, G., Rampazzo, F.: On systems of ordinary differential equations with measures as controls. Differential and Integral Equations 4, 739–765 (1991)
Silva, G., Vinter, R.: Measure driven differential inclusions. Journal of Mathematical Analysis and Applications 202, 727–746 (1996)
Mota, M., Rampazzo, F.: Dynamic programming for nonlinear systems driven by ordinary and impulsive controls. SIAM Journal on Control and Optimization 34(1), 199–225 (1996)
Silva, G., Vinter, R.: Necessary conditions for optimal impulsive control problems. SIAM Journal on Control and Optimization 35(6), 1829–1846 (1997)
Miller, J.: Decidability and complexity results for timed automata and semi-linear hybrid automata. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 296–309. Springer, Heidelberg (2000)
Imura, J., van der Schaft, A.J.: Characterization of well-posedness of piecewise linear systems. IEEE Transactions on Automatic Control 45(9), 1600–1619 (2000)
Aubin, J.P., et al.: Impulse differential inclusions: A viability approach to hybrid systems. IEEE Transactions on Automatic Control 47(1), 2–20 (2002)
Lygeros, J., et al.: Dynamical properties of hybrid automata. IEEE Transactions on Automatic Control 48(1), 2–17 (2003)
Imura, J.: Well-posedness analysis of switch-driven piecewise affine systems. IEEE Transactions on Automatic Control 48(11), 1926–1935 (2003)
Pogromsky, A.Y., Heemels, W., Nijmeijer, H.: On solution concepts and well-posedness of linear relay systems. Automatica 39, 2139–2147 (2003)
Goebel, R., et al.: Hybrid systems: Generalized solutions and robust stability. In: IFAC Symposium on Nonlinear Control Systems (2004)
Johansson, K., et al.: On the regularization of Zeno hybrid automata. Systems and Control Letters 38(3), 141–150 (1999)
Zheng, H., Lee, E., Ames, A.: Beyond Zeno: Get on with it? In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 568–582. Springer, Heidelberg (2006)
Ames, A., et al.: Is there life after Zeno? Taking executions past the breaking (Zeno) point. In: American Control Conference (2006)
Filippov, A.F.: Differential equations with discontinuous right-hand sides. Kluwer Academic Publishers, Dordrecht (1988)
Camlibel, M., et al.: Solution concepts for hybrid dynamical systems. In: IFAC World Congress (2002)
Aubin, J.P.: Viability Theory. Birkhäuser, Boston (1991)
Quincampoix, M., Veliov, V.: Viability with target: Theory and applications. In: Cheshankov, B., Todorov, M. (eds.) Applications of Mathematics in Engineering, pp. 47–54. Heron Press, Sofia (1998)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Lygeros, J., Quincampoix, M., Rzezuchowski, T. (2007). Impulse Differential Inclusions Driven by Discrete Measures. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds) Hybrid Systems: Computation and Control. HSCC 2007. Lecture Notes in Computer Science, vol 4416. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71493-4_31
Download citation
DOI: https://doi.org/10.1007/978-3-540-71493-4_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-71492-7
Online ISBN: 978-3-540-71493-4
eBook Packages: Computer ScienceComputer Science (R0)