{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T13:19:25Z","timestamp":1762521565497},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2013,3,16]],"date-time":"2013-03-16T00:00:00Z","timestamp":1363392000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2013,9]]},"DOI":"10.1007\/s00211-013-0531-y","type":"journal-article","created":{"date-parts":[[2013,3,15]],"date-time":"2013-03-15T19:36:21Z","timestamp":1363376181000},"page":"1-25","source":"Crossref","is-referenced-by-count":7,"title":["Interval bounds on the solutions of semi-explicit index-one DAEs. Part 1: analysis"],"prefix":"10.1007","volume":"125","author":[{"given":"Joseph K.","family":"Scott","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Paul I.","family":"Barton","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2013,3,16]]},"reference":[{"key":"531_CR1","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1023\/A:1024467732637","volume":"4","author":"M Berz","year":"1998","unstructured":"Berz, M., Makino, K.: Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models. Reliab. Comput. 4, 361\u2013369 (1998)","journal-title":"Reliab. Comput."},{"key":"531_CR2","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/0307-904X(92)90083-F","volume":"16","author":"W Blajer","year":"1992","unstructured":"Blajer, W.: Index of differential-algebraic equations governing the dynamics of contrained mechanical systems. Appl. Math. Modell. 16, 70\u201377 (1992)","journal-title":"Appl. Math. Modell."},{"issue":"8","key":"531_CR3","doi-asserted-by":"crossref","first-page":"2235","DOI":"10.1002\/aic.10494","volume":"51","author":"B Chachuat","year":"2005","unstructured":"Chachuat, B., Singer, A.B., Barton, P.I.: Global mixed-integer dynamic optimization. AIChE J. 51(8), 2235\u20132253 (2005)","journal-title":"AIChE J."},{"key":"531_CR4","doi-asserted-by":"crossref","unstructured":"Chutinan, A., Krogh, B.: Verification of polyhedral-invariant hybrid automata using polygonal flow pipe approximations. In: Hybrid Systems: Computation and Control. Lecture Notes in Computer Science 1569, 76\u201390 (1999)","DOI":"10.1007\/3-540-48983-5_10"},{"key":"531_CR5","doi-asserted-by":"crossref","unstructured":"Cross, E.A., Mitchell, I.M.: Level set methods for computing reachable sets of systems with differential algebraic equation dynamics. In: Proceedings of 2008 American Control Conference, pp. 2260\u20132265 (2008)","DOI":"10.1109\/ACC.2008.4586828"},{"key":"531_CR6","doi-asserted-by":"crossref","unstructured":"Girard, A.: Reachability of uncertain linear systems using zonotopes. In: Morari, M., Thiele L. (eds.) Hybrid Systems: Computation and Control. Lecture Notes in Computer Science 3414, 291\u2013305 (2005)","DOI":"10.1007\/978-3-540-31954-2_19"},{"key":"531_CR7","unstructured":"Harrison, G.W.: Dynamic models with uncertain parameters. In: Avula X. (ed.) Proceedings of the First International Conference on Mathematical Modeling, vol. 1, pp. 295\u2013304 (1977)"},{"key":"531_CR8","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719222","volume-title":"Ordinary differential equations","author":"P Hartman","year":"2002","unstructured":"Hartman, P.: Ordinary differential equations, 2nd edn. SIAM, Philidelphia (2002)","edition":"2"},{"key":"531_CR9","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1023\/A:1022858921155","volume":"19","author":"J Hoefkens","year":"2003","unstructured":"Hoefkens, J., Berz, M., Makino, K.: Computing validated solutions of implicit differential equations. Adv. Comput. Math. 19, 231\u2013253 (2003)","journal-title":"Adv. Comput. Math."},{"issue":"1","key":"531_CR10","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1002\/aic.690480110","volume":"48","author":"H Huang","year":"2002","unstructured":"Huang, H., Adjiman, C.S., Shah, N.: Quantitative framework for reliable safety analysis. AIChE J. 48(1), 78\u201396 (2002)","journal-title":"AIChE J."},{"key":"531_CR11","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1016\/S0005-1098(01)00284-9","volume":"38","author":"L Jaulin","year":"2002","unstructured":"Jaulin, L.: Nonlinear bounded-error state estimation of continuous-time systems. Automatica 38, 1079\u20131082 (2002)","journal-title":"Automatica"},{"key":"531_CR12","doi-asserted-by":"crossref","unstructured":"Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations: Analysis and Numerical Solution. European Mathematical Society, Zurich (2006)","DOI":"10.4171\/017"},{"key":"531_CR13","unstructured":"Lin, Y., Stadtherr, M.A.: Deterministic global optimization for parameter estimation of dynamic systems. Ind. Eng. Chem. Res. 45, 8438\u20138448 (2006)"},{"issue":"4","key":"531_CR14","doi-asserted-by":"crossref","first-page":"866","DOI":"10.1002\/aic.11101","volume":"53","author":"Y Lin","year":"2007","unstructured":"Lin, Y., Stadtherr, M.A.: Deterministic global optimization of nonlinear dynamic systems. AIChE J. 53(4), 866\u2013875 (2007)","journal-title":"AIChE J."},{"issue":"9","key":"531_CR15","doi-asserted-by":"crossref","first-page":"2335","DOI":"10.1002\/aic.11535","volume":"54","author":"Y Lin","year":"2008","unstructured":"Lin, Y., Stadtherr, M.A.: Fault detection in nonlinear continuous-time systems with uncertain parameters. AIChE J. 54(9), 2335\u20132345 (2008)","journal-title":"AIChE J."},{"key":"531_CR16","doi-asserted-by":"crossref","unstructured":"Lygeros, J., Tomlin, C., Sastry, S.: Controllers for reachability specifications for hybrid systems. Automatica 35, 349\u2013370 (1999)","DOI":"10.1016\/S0005-1098(98)00193-9"},{"issue":"1","key":"531_CR17","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1177\/003754978905200105","volume":"52","author":"SE Mattsson","year":"1989","unstructured":"Mattsson, S.E.: On modeling and differential algebraic systems. Simulation 52(1), 24\u201332 (1989)","journal-title":"Simulation"},{"key":"531_CR18","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/BF01580665","volume":"10","author":"GP McCormick","year":"1976","unstructured":"McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I\u2014convex underestimating problems. Math. Program. 10, 147\u2013175 (1976)","journal-title":"Math. Program."},{"key":"531_CR19","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970906","volume-title":"Methods and Applications of Interval Analysis","author":"RE Moore","year":"1979","unstructured":"Moore, R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)"},{"key":"531_CR20","volume-title":"Analysis on Manifolds","author":"JR Munkres","year":"1991","unstructured":"Munkres, J.R.: Analysis on Manifolds. Westview Press, Cambridge (1991)"},{"issue":"1","key":"531_CR21","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1137\/050638448","volume":"45","author":"M Neher","year":"2007","unstructured":"Neher, M., Jackson, K.R., Nedialkov, N.S.: On Taylor model based integration of ODEs. SIAM J. Numer. Anal. 45(1), 236\u2013262 (2007)","journal-title":"SIAM J. Numer. Anal."},{"key":"531_CR22","volume-title":"Interval Methods for Systems of Equations","author":"A Neumaier","year":"1990","unstructured":"Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)"},{"key":"531_CR23","doi-asserted-by":"crossref","unstructured":"Oishi, M., Mitchell, I., Tomlin, C., Saint-Pierre, P.: Computing viable sets and reachable sets to design feedback linearizing control laws under saturation. In: Proceedings of 45th IEEE Conference on Decision and Control, San Diego, pp. 3801\u20133807 (2006)","DOI":"10.1109\/CDC.2006.377614"},{"issue":"1","key":"531_CR24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1023\/A:1016259507911","volume":"24","author":"I Papamichail","year":"2002","unstructured":"Papamichail, I., Adjiman, C.S.: A rigorous global optimization algorithm for problems with ordinary differential equations. J. Glob. Optim. 24(1), 1\u201333 (2002)","journal-title":"J. Glob. Optim."},{"key":"531_CR25","doi-asserted-by":"crossref","first-page":"1771","DOI":"10.1016\/j.automatica.2004.05.006","volume":"40","author":"T Raissi","year":"2004","unstructured":"Raissi, T., Ramdani, N., Candau, Y.: Set membership state and parameter estimation for systems described by nonlinear differential equations. Automatica 40, 1771\u20131777 (2004)","journal-title":"Automatica"},{"issue":"10","key":"531_CR26","doi-asserted-by":"crossref","first-page":"2352","DOI":"10.1109\/TAC.2009.2028974","volume":"54","author":"N Ramdani","year":"2009","unstructured":"Ramdani, N., Meslem, N., Candau, Y.: A hybrid bounding method for computing an over-approximation for the reachable set of uncertain nonlinear systems. IEEE Trans. Automat. Contr. 54(10), 2352\u20132364 (2009)","journal-title":"IEEE Trans. Automat. Contr."},{"key":"531_CR27","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.mbs.2004.07.004","volume":"193","author":"A Rapaport","year":"2005","unstructured":"Rapaport, A., Dochain, D.: Interval observers for biochemical processes with uncertain kinetics and inputs. Math. Biosci. 193, 235\u2013253 (2005)","journal-title":"Math. Biosci."},{"issue":"3","key":"531_CR28","doi-asserted-by":"crossref","first-page":"381","DOI":"10.2478\/v10006-009-0032-4","volume":"19","author":"A Rauh","year":"2009","unstructured":"Rauh, A., Brill, M., Gunther, C.: A novel interval arithmetic approach for solving differential-algebraic equations with Valencia-IVP. Int. J. Appl. Math. Comput. Sci. 19(3), 381\u2013397 (2009)","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"531_CR29","volume-title":"Principles of Mathematical Analysis","author":"W Rudin","year":"1964","unstructured":"Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1964)","edition":"3"},{"key":"531_CR30","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1016\/j.compchemeng.2009.11.021","volume":"34","author":"JK Scott","year":"2010","unstructured":"Scott, J.K., Barton, P.I.: Tight, efficient bounds on the solutions of chemical kinetics models. Comput Chem Eng 34, 717\u2013731 (2010)","journal-title":"Comput Chem Eng"},{"key":"531_CR31","doi-asserted-by":"crossref","unstructured":"Scott, J.K., Barton, P.I.: Bounds on the reachable sets of nonlinear control systems. Automatica 49, 93\u2013100 (2013)","DOI":"10.1016\/j.automatica.2012.09.020"},{"key":"531_CR32","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1007\/s10898-011-9664-7","volume":"51","author":"JK Scott","year":"2011","unstructured":"Scott, J.K., Stuber, M.D., Barton, P.I.: Generalized McCormick relaxations. J. Glob. Optim. 51, 569\u2013606 (2011). doi: 10.1007\/s10898-011-9664-7","journal-title":"J. Glob. Optim."},{"key":"531_CR33","doi-asserted-by":"crossref","first-page":"2167","DOI":"10.1137\/040604388","volume":"27","author":"AB Singer","year":"2006","unstructured":"Singer, A.B., Barton, P.I.: Bounding the solutions of parameter dependent nonlinear ordinary differential equations. SIAM J. Sci. Comput. 27, 2167\u20132182 (2006)","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"531_CR34","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1021\/jp0548873","volume":"110","author":"AB Singer","year":"2006","unstructured":"Singer, A.B., Barton, P.I.: Global dynamic optimization for parameter estimation in chemical kinetics. J. Phys. Chem. A 110(3), 971\u2013976 (2006)","journal-title":"J. Phys. Chem. A"},{"key":"531_CR35","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/s10898-005-7074-4","volume":"34","author":"AB Singer","year":"2006","unstructured":"Singer, A.B., Barton, P.I.: Global optimization with nonlinear ordinary differential equations. J. Glob. Optim. 34, 159\u2013190 (2006)","journal-title":"J. Glob. Optim."},{"key":"531_CR36","volume-title":"Linear Algebra and its Applications","author":"G Strang","year":"2006","unstructured":"Strang, G.: Linear Algebra and its Applications, 4th edn. Thomson Brooks\/Cole, Belmont (2006)","edition":"4"},{"key":"531_CR37","volume-title":"Differential Inequalities","author":"J Szarski","year":"1965","unstructured":"Szarski, J.: Differential Inequalities. Polish Scientific Publishers, Warszawa (1965)"},{"key":"531_CR38","doi-asserted-by":"crossref","unstructured":"Tomlin, C., Mitchell, I., Bayen, A.M., Oishi, M.: Computational techniques for the verification of hybrid systems. In: Proceedings of the IEEE, vol. 91(7), 986\u20131001 (2003)","DOI":"10.1109\/JPROC.2003.814621"},{"key":"531_CR39","doi-asserted-by":"crossref","first-page":"831","DOI":"10.2307\/2315025","volume":"72","author":"DE Varberg","year":"1965","unstructured":"Varberg, D.E.: On absolutely continuous functions. Am. Math. Mon. 72, 831\u2013941 (1965)","journal-title":"Am. Math. Mon."},{"key":"531_CR40","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-86405-6","volume-title":"Differential and Integral Inequalities","author":"W Walter","year":"1970","unstructured":"Walter, W.: Differential and Integral Inequalities. Springer, New York (1970)"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-013-0531-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-013-0531-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-013-0531-y","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,7,11]],"date-time":"2019-07-11T01:11:03Z","timestamp":1562807463000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-013-0531-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,3,16]]},"references-count":40,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2013,9]]}},"alternative-id":["531"],"URL":"https:\/\/doi.org\/10.1007\/s00211-013-0531-y","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,3,16]]}}}