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Bracketing the solutions of an ordinary differential equation with uncertain initial conditions. (English) Zbl 1426.93022
Summary: In this paper, we present a new method for bracketing (i.e., characterizing from inside and from outside) all solutions of an ordinary differential equation in the case where the initial time is inside an interval and the initial state is inside a box. The principle of the approach is to cast the problem into bracketing the largest positive invariant set which is included inside a given set \(\mathbb{X} \). Although there exists an efficient algorithm to solve this problem when \(\mathbb{X}\) is bounded, we need to adapt it to deal with cases where \(\mathbb{X}\) is unbounded.

MSC:
93B03 Attainable sets, reachability
34A34 Nonlinear ordinary differential equations and systems
34C45 Invariant manifolds for ordinary differential equations
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[1] Araya, I.; Trombettoni, G.; Neveu, B., A contractor based on convex interval Taylor, Proceedings of CPAIOR, LNCS 7298, 1-16, (2012), Springer
[2] Asarin, E.; Dang, T.; Girard, A., Reachability analysis of nonlinear systems using conservative approximation, Hybrid Systems: Computation and Control, 20-35, (2003), Springer Berlin Heidelberg · Zbl 1032.93034
[3] Blanchini, F.; Miani, S., Set-Theoretic Methods in Control, (2007), Springer Science & Business Media · Zbl 0996.93040
[4] Chabert, G.; Jaulin, L., Contractor programming, Artificial. Intell., 173, 1079-1100, (2009) · Zbl 1191.68628
[5] Collins, P.; Goldsztejn, A., The reach-and-evolve algorithm for reachability analysis of nonlinear dynamical systems, Electron. Notes Theor.Comput. Sci., 223, 87-102, (2008) · Zbl 1337.93018
[6] dit Sandretto, J. A.; Chapoutot, A., Validated explicit and implicit Runge-Kutta methods, Reliab. Comput., 22, 79, (2016)
[7] Goubault, E.; Mullier, O.; Putot, S.; Kieffer, M., Inner approximated reachability analysis, Proceedings of the 17th International Conference on Hybrid Systems: Computation and Control, HSCC ’14, Berlin, Germany, 163-172, (2014) · Zbl 1362.93016
[8] Hansen, E. R., A generalized interval arithmetic, (Nickel, K., Interval Mathematics 1975, (1975), Springer-Verlag), 7-18
[9] Hladik, M., Enclosures for the solution set of parametric interval linear systems, Int. J. Appl. Math. Comput. Sci., 22, 3, 561-574, (2012) · Zbl 1310.65051
[10] Kaynama, S.; Maidens, J.; Oishi, M.; Mitchell, I. M.; Dumont, G. A., Computing the viability kernel using maximal reachable sets, Proceedings of the 15th ACM International Conference on Hybrid Systems: Computation and Control, HSCC ’12, 55-64, (2012), ACM New York, NY, USA · Zbl 1362.93017
[11] Konečnỳ, M.; Taha, W.; Bartha, F. A.; Duracz, J.; Duracz, A.; Ames, A. D., Enclosing the behavior of a hybrid automaton up to and beyond a Zeno point, Nonlinear Anal.: Hybrid Syst., 20, 1-20, (2016) · Zbl 1336.93085
[12] Lhommeau, M.; Jaulin, L.; Hardouin, L., Capture basin approximation using interval analysis, Int. J. Adapt. Control. Signal Process., 25, 3, 264-272, (2011) · Zbl 1225.93057
[13] Lhommeau, M.; Jaulin, L.; Hardouin, L., Inner and outer approximation of capture basin using interval analysis., Proceedings of ICINCO-SPSMC, 5-9, (2007)
[14] Mackworth, A. K., Consistency in networks of relations, Artif. Intell., 8, 1, 99-118, (1977) · Zbl 0341.68061
[15] Maidens, J. N.; Kaynama, S.; Mitchell, I. M.; Oishi, M. M.K.; Dumont, G. A., Lagrangian methods for approximating the viability kernel in high-dimensional systems, Automatica, 49, 7, 2017-2029, (2013) · Zbl 1364.93057
[16] Menec, S. L., Linear differential game with two pursuers and one evader, (Breton, M.; Szajowski, K., Advances in Dynamic Games, Vol. 11, (2011)), 209-226 · Zbl 1218.91026
[17] Mézo, T. L.; Jaulin, L.; Zerr, B., Inner approximation of a capture basin of a dynamical system, Proceedings of Workshop on Abstracts of the 9th Summer Interval Methods SWIM’2016, (2016), Lyon, France, June 19-22
[18] Mézo, T. L.; Jaulin, L.; Zerr, B., An interval approach to solve an initial value problem, AIP Conf. Proc., 1776, 1, (2016)
[19] Mézo, T. L.; Jaulin, L.; Zerr, B., An interval approach to compute invariant sets, IEEE Trans. Autom. Control, PP, 99, 1, (2018)
[20] Nedialkov, N. S.; Jackson, K. R.; Corliss, G. F., Validated solutions of initial value problems for ordinary differential equations, Appl. Math. Comput., 105, 1, 21-68, (1999) · Zbl 0934.65073
[21] Ramdani, N.; Nedialkov, N., Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint propagation techniques, Nonlinear Anal.: Hybrid Syst., 5, 2, 149-162, (2011) · Zbl 1225.93026
[22] Sergeyev, Y. D., Arithmetic of Infinity, (2003), Kindle Edition · Zbl 1076.03048
[23] Walawska, I.; Wilczak, D., An implicit algorithm for validated enclosures of the solutions to variational equations for {ODEs}, Appl. Math. Comput., 291, 303-322, (2016) · Zbl 1410.65259
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