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A verified method for solving piecewise smooth initial value problems


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Acary, V. and Brogliato, B. (2008). Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics, Lecture Notes in Applied and Computational Mechanics, Vol. 35, Springer-Verlag, Berlin/Heidelberg.Search in Google Scholar

Alefeld, G. and Herzberger, J. (1983). Introduction to Interval Computations, Computer Science and Applied Mathematics, Academic Press, New York, NY.Search in Google Scholar

Auer, E., Albassam, H., Kecskem´ethy, A. and Luther,W. (2011). Verified analysis of a model for stance stabilization, in A. Rauh and E.Auer (Eds.), Modeling, Design and Simulation of Systems with Uncertainties, Mathematical Engineering, Springer-Verlag, Berlin/Heidelberg, pp. 294-308.10.1007/978-3-642-15956-5_14Search in Google Scholar

Barboteu, M., Bartosz, K. and Kalita, P. (2013). An analytical and numerical approach to a bilateral contact problem with nonmonotone friction, International Journal of Applied Mathematics and Computer Science 23(2): 263-276, DOI: 10.2478/amcs-2013-0020.10.2478/amcs-2013-0020Search in Google Scholar

Bernardo, M., Budd, C., Champneys, A. and Kowalczyk, P. (2007). Piecewise-smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, Springer-Verlag, London. de Figueiredo, L.H. and Stolfi, J. (2004). Affine arithmetic: Concepts and applications, Numerical Algorithms 37(1-4): 147-158.Search in Google Scholar

Dötschel, T., Auer, E., Rauh, A. and Aschemann, H. (2013). Thermal behavior of high-temperature fuel cells: Reliable parameter identification and interval-based sliding mode control, Soft Computing (17): 1329-1343.10.1007/s00500-013-1003-0Search in Google Scholar

Eble, I. (2007). ¨Uber Taylor-Modelle, Ph.D. thesis, University of Karlsruhe, Karlsruhe.Search in Google Scholar

Eggers, A., Fränzle, M. and Herde, C. (2009). Application of constraint solving and ODE-enclosure methods to the analysis of hybrid systems, in E. Simos, G. Psihoyios and Ch. Tsitouras (Eds.), Numerical Analysis and Applied Mathematics 2009, American Institute of Physics, Melville, NY, pp. 1326-1330.10.1063/1.3241327Search in Google Scholar

Filippov, A. (1988). Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Dordrecht.10.1007/978-94-015-7793-9Search in Google Scholar

Galias, Z. (2012). Rigorous study of the Chua’s circuit spiral attractor, IEEE Transactions on Circuits and Systems 59- I(10): 2374-2382.10.1109/TCSI.2012.2188947Search in Google Scholar

Goldsztejn, A., Mullier, O., Eveillard, D. and Hosobe, H. (2010). Including ordinary differential equations based constraints in the standard CP framework, in D. Cohen (Ed.), Principles and Practice of Constraint Programming CP 2010, Lecture Notes in Computer Science, Vol. 6308, Springer, Berlin, pp. 221-235.10.1007/978-3-642-15396-9_20Search in Google Scholar

Granas, A. and Dugundji, J. (2003). Fixed Point Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, NY.10.1007/978-0-387-21593-8Search in Google Scholar

Hansen, E. and Walster, G. (2004). Global Optimization Using Interval Analysis: Revised and Expanded, Pure and Applied Mathematics, Marcel Dekker, New York, NY.10.1201/9780203026922Search in Google Scholar

Henzinger, T. A., Horowitz, B., Majumdar, R. and Wong-Toi, H. (2000). Beyond HYTECH: Hybrid systems analysis using interval numerical methods, in N.A. Lynch and B.H. Krogh (Eds.), Proceedings of the Third International Workshop on Hybrid Systems: Computation and Control, HSCC’00, Springer-Verlag, London, pp. 130-144.10.1007/3-540-46430-1_14Search in Google Scholar

Ishii, D. (2010). Simulation and Verification of Hybrid Systems Based on Interval Analysis and Constraint Programming, Ph.D. thesis, Waseda University, Tokyo.Search in Google Scholar

Ishii, D., Ueda, K. and Hosobe, H. (2011). An interval-based SAT modulo ODE solver for model checking nonlinear hybrid systems, International Journal on Software Tools for Technology Transfer 13(5): 449-461.10.1007/s10009-011-0193-ySearch in Google Scholar

Jaulin, L., Kieffer, M., Didrit, O. and Walter, E. (2001). Applied Interval Analysis, Springer-Verlag, London.10.1007/978-1-4471-0249-6Search in Google Scholar

Kearfott, R.B. (1996). Rigorous Global Search: Continuous Problems, Kluwer, Boston, MA.10.1007/978-1-4757-2495-0Search in Google Scholar

Kofman, E. (2004). Discrete event simulation of hybrid systems, SIAM Journal on Scientific Computing 25(5): 1771-1797.10.1137/S1064827502418379Search in Google Scholar

Kunze, M. (2000). Non-Smooth Dynamical Systems, Springer, Berlin/Heidelberg.10.1007/BFb0103843Search in Google Scholar

Lohner, R. (1988). Einschließung der L¨osung gew¨ohnlicher Anfangs- und Randwertaufgaben und Anwendungen, Ph.D. thesis, Universit¨at Karlsruhe, Karlsruhe.Search in Google Scholar

Lunze, J. and Lamnabhi-Lagarrigue, F. (2009). Handbook of Hybrid Systems Control-Theory, Tools, Applications, Cambridge University Press, Cambridge.10.1017/CBO9780511807930Search in Google Scholar

Magnus, K. and Popp, K. (2005). Schwingungen, Leitf¨aden der angewandten Mathematik und Mechanik, Teubner, Wiesbaden.10.1007/978-3-322-99701-2Search in Google Scholar

Mahmoud, S. and Chen, X. (2008). A verified inexact implicit Runge-Kutta method for nonsmooth ODEs, Numerical Algorithms 47(3): 275-290.10.1007/s11075-008-9180-0Search in Google Scholar

Makino, K. (1998). Rigorous Analysis of Nonlinear Motion in Particle Accelerators, Ph.D. thesis, Michigan State University, East Lansing, MI.Search in Google Scholar

Mannshardt, R. (1978). One-step methods of any order for ordinary differential equations with discontinuous right-hand sides, Nimerische Mathematik 31(2): 131-152.10.1007/BF01397472Search in Google Scholar

McLeod, R. M. (1964/1965). Mean value theorems for vector valued functions, Proceedings of the Edinburgh Mathematical Society 14: 197-209.10.1017/S0013091500008786Search in Google Scholar

Moore, R. (1966). Interval Arithmetic, Prentice-Hall, Englewood Cliffs, NJ.Search in Google Scholar

Munoz, H. and Kearfott, R.B. (2004). Slope intervals, generalized gradients, semigradients, slant derivatives, and csets, Reliable Computing 10(3): 163-193.10.1023/B:REOM.0000032107.85627.45Search in Google Scholar

Myślsński, A. (2012). Topology optimization of quasistatic contact problems, International Journal of Applied Mathematics and Computer Science 22(2): 269-280, DOI: 10.2478/v10006-012-0020-y.10.2478/v10006-012-0020-ySearch in Google Scholar

Nedialkov, N.S. (2002). The Design and Implementation of an Object-Oriented Validated ODE Solver, Kluwer Academic Publishers, Dordrecht.Search in Google Scholar

Nedialkov, N. and von Mohrenschildt, M. (2002). Rigorous simulation of hybrid dynamic systems with symbolic and interval methods, Proceedings of the American Control Conference, Anchorage, AK, USA, Vol. 1, pp. 140-147.Search in Google Scholar

Orlov, Y. (2004). Finite time stability and robust control synthesis of uncertain switched systems, SIAM Journal on Control and Optimization 43(4): 1253-1271.10.1137/S0363012903425593Search in Google Scholar

Patton, R.J., Chen, L. and Klinkhieo, S. (2012). An LPV pole-placement approach to friction compensation as an FTC problem, International Journal of Applied Mathematics and Computer Science 22(1): 149-160, DOI: 10.2478/v10006-012-0011-z.10.2478/v10006-012-0011-zSearch in Google Scholar

Ramdani, N. and Nedialkov, N.S. (2011). Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint-propagation techniques, Nonlinear Analysis: Hybrid Systems 5(2): 149-162.10.1016/j.nahs.2010.05.010Search in Google Scholar

Ratschan, S. (2012). An algorithm for formal safety verification of complex heterogeneous systems, Proceedings of REC 2012, Brno, Czech Republic, pp. 457-468.Search in Google Scholar

Rauh, A. and Auer, E. (2011). Verified simulation of ODEs and DAEs in VALENCIA-IVP, Reliable Computing 5(4): 370-381.Search in Google Scholar

Rauh, A., Brill, M. and G¨unther, C. (2009). A novel interval arithmetic approach for solving differential-algebraic equations with VALENCIA-IVP, International Journal of Applied Mathematics and Computer Science 19(3): 381-397, DOI: 10.2478/v10006-009-0032-4.10.2478/v10006-009-0032-4Search in Google Scholar

Rauh, A., Kletting, M., Aschemann, H. and Hofer, E.P. (2006). Interval methods for simulation of dynamical systems with state-dependent switching characteristics, IEEE CCA 2006, Munich, Germany, pp. 355-360.Search in Google Scholar

Rauh, A., Siebert, C. and Aschemann, H. (2011). Verified simulation and optimization of dynamic systems with friction and hysteresis, Proceedings of ENOC 2011, Rome, Italy.Search in Google Scholar

Rihm, R. (1992). Enclosing solutions with switching points in ordinary differential equations, in L. Atanassova and J. Herzberger (Eds.), Computer Arithmetic and Enclosure Methods. Proceedings of SCAN 91, North-Holland, Amsterdam, pp. 419-425.Search in Google Scholar

Rihm, R. (1993). ¨Uber Einschließungsverfahren f¨ur gew¨ohnliche Anfangswertprobleme und ihre Anwendung auf Differentialgleichungen mit unstetiger rechter Seite, Ph.D. thesis, Universit¨at Karlsruhe, Karlsruhe.Search in Google Scholar

Rihm, R. (1998). Implicit methods for enclosing solutions of ODEs, Journal of Universal Computer Science 4(2): 202-209.Search in Google Scholar

Schnurr, M. (2007). Steigungen h¨oherer Ordnung zur verifizierten globalen Optimierung, Ph.D. thesis, Universit¨at Karlsruhe, Karlsruhe.Search in Google Scholar

Smirnov, G. (2002). Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI.10.1090/gsm/041Search in Google Scholar

Stauning, O. (1997). Automatic Validation of Numerical Solutions, Ph.D. thesis, Technical University of Denmark, Kgs. Lyngby.Search in Google Scholar

Stewart, D. (1990). A high accuracy method for solving ODEs with discontinuous right-hand side, Numerische Mathematik 58(1): 299-328.10.1007/BF01385627Search in Google Scholar

Walter, W. (1972). Gew¨ohnliche Differentialgleichungen, Springer, Berlin/Heidelberg/New York, NY.Search in Google Scholar

Zgliczynski, P. and Kapela, T. (2009). A Lohner-type algorithm for control systems and ordinary differential inclusions, Discrete and Continuous Dynamical Systems B 11(2): 365-385. 10.3934/dcdsb.2009.11.365Search in Google Scholar

ISSN:
1641-876X
Sprache:
Englisch
Zeitrahmen der Veröffentlichung:
4 Hefte pro Jahr
Fachgebiete der Zeitschrift:
Mathematik, Angewandte Mathematik