{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,14]],"date-time":"2026-06-14T21:50:11Z","timestamp":1781473811057,"version":"3.54.1"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2010,9,8]],"date-time":"2010-09-08T00:00:00Z","timestamp":1283904000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Control Signals Syst."],"published-print":{"date-parts":[[2010,10]]},"DOI":"10.1007\/s00498-010-0054-3","type":"journal-article","created":{"date-parts":[[2010,9,7]],"date-time":"2010-09-07T11:55:03Z","timestamp":1283860503000},"page":"155-184","source":"Crossref","is-referenced-by-count":48,"title":["Arc-length-based Lyapunov tests for convergence and stability with applications to systems having a continuum of equilibria"],"prefix":"10.1007","volume":"22","author":[{"given":"Sanjay P.","family":"Bhat","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dennis S.","family":"Bernstein","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2010,9,8]]},"reference":[{"key":"54_CR1","doi-asserted-by":"crossref","first-page":"542","DOI":"10.1137\/0510051","volume":"10","author":"SL Campbell","year":"1979","unstructured":"Campbell SL, Rose NJ (1979) Singular perturbation of autonomous linear systems. SIAM J Math Anal 10: 542\u2013551","journal-title":"SIAM J Math Anal"},{"key":"54_CR2","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1115\/1.2838656","volume":"117","author":"DS Bernstein","year":"1995","unstructured":"Bernstein DS, Bhat SP (1995) Lyapunov stability, semistability, and asymptotic stability of matrix second-order systems. ASME Trans J Vib Acoust 117: 145\u2013153","journal-title":"ASME Trans J Vib Acoust"},{"issue":"5","key":"54_CR3","doi-asserted-by":"crossref","first-page":"1745","DOI":"10.1137\/S0363012902407119","volume":"42","author":"SP Bhat","year":"2003","unstructured":"Bhat SP, Bernstein DS (2003) Nontangency-based Lyapunov tests for convergence and stability in systems having a continuum of equilibria. SIAM J Control Optim 42(5): 1745\u20131775","journal-title":"SIAM J Control Optim"},{"key":"54_CR4","volume-title":"Mathematical models of chemical reactions: theory and applications of deterministic and stochastic models","author":"P Erdi","year":"1988","unstructured":"Erdi P, Toth J (1988) Mathematical models of chemical reactions: theory and applications of deterministic and stochastic models. Princeton University Press, Princeton"},{"key":"54_CR5","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1007\/BF00375614","volume":"132","author":"M Feinberg","year":"1995","unstructured":"Feinberg M (1995) The existence and uniqueness of steady states for a class of chemical reaction networks. Arch Ration Mech Anal 132: 311\u2013370","journal-title":"Arch Ration Mech Anal"},{"key":"54_CR6","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1109\/9.935056","volume":"46","author":"ED Sontag","year":"2001","unstructured":"Sontag ED (2001) Structure and stability of certain chemical networks and applications to the kinetic proofreading model of t-cell receptor signal transduction. IEEE Trans Autom Control 46: 1028\u20131047","journal-title":"IEEE Trans Autom Control"},{"issue":"4","key":"54_CR7","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1109\/MCS.2009.932926","volume":"29","author":"V Chellaboina","year":"2009","unstructured":"Chellaboina V, Bhat SP, Haddad WM, Bernstein DS (2009) Modeling and analysis of mass-action kinetics: nonnegativity, realizability, reducibility, and semistability. IEEE Control Syst Mag 29(4): 60\u201378","journal-title":"IEEE Control Syst Mag"},{"key":"54_CR8","volume-title":"Non-identifier-based high gain adaptive control","author":"A Ilchmann","year":"1993","unstructured":"Ilchmann A (1993) Non-identifier-based high gain adaptive control. Springer, London"},{"issue":"6","key":"54_CR9","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1109\/TAC.1985.1104006","volume":"30","author":"DR Mudgett","year":"1985","unstructured":"Mudgett DR, Morse AS (1985) Adaptive stabilization of linear systems with unknown high-frequency gain. IEEE Trans Autom Control 30(6): 549\u2013554","journal-title":"IEEE Trans Autom Control"},{"key":"54_CR10","volume-title":"Stable adaptive systems","author":"KS Narendra","year":"1989","unstructured":"Narendra KS, Annaswamy AM (1989) Stable adaptive systems. Prentice-Hall, Englewood Cliffs"},{"key":"54_CR11","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/0167-6911(83)90021-X","volume":"3","author":"RD Nussbaum","year":"1983","unstructured":"Nussbaum RD (1983) Some remarks on a conjecture in parameter adaptive control. Syst Control Lett 3: 243\u2013246","journal-title":"Syst Control Lett"},{"key":"54_CR12","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4471-3467-1","volume-title":"Optimization and dynamical systems","author":"U Helmcke","year":"1994","unstructured":"Helmcke U, Moore JB (1994) Optimization and dynamical systems. Springer-Verlag, London"},{"issue":"9","key":"54_CR13","doi-asserted-by":"crossref","first-page":"1520","DOI":"10.1109\/TAC.2004.834113","volume":"49","author":"R Olfati-Saber","year":"2004","unstructured":"Olfati-Saber R, Murray RM (2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Control 49(9): 1520\u20131533","journal-title":"IEEE Trans Autom Control"},{"issue":"8","key":"54_CR14","doi-asserted-by":"crossref","first-page":"1380","DOI":"10.1109\/TAC.2007.902733","volume":"52","author":"M Arcak","year":"2007","unstructured":"Arcak M (2007) Passivity as a design tool for group coordination. IEEE Trans Autom Control 52(8): 1380\u20131390","journal-title":"IEEE Trans Autom Control"},{"issue":"8","key":"54_CR15","doi-asserted-by":"crossref","first-page":"1887","DOI":"10.1109\/TAC.2008.929392","volume":"53","author":"Q Hui","year":"2008","unstructured":"Hui Q, Haddad WM, Bhat SP (2008) Finite-time semistability and consensus for nonlinear dynamical networks. IEEE Trans Autom Control 53(8): 1887\u20131900","journal-title":"IEEE Trans Autom Control"},{"key":"54_CR16","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1016\/0893-6080(89)90018-X","volume":"2","author":"M Hirsch","year":"1989","unstructured":"Hirsch M (1989) Convergent activation dynamics in continuous time networks. Neural Netw 2: 331\u2013349","journal-title":"Neural Netw"},{"issue":"12","key":"54_CR17","doi-asserted-by":"crossref","first-page":"2460","DOI":"10.1109\/TCSI.2004.838143","volume":"51","author":"M Forti","year":"2004","unstructured":"Forti M, Tesi A (2004) Absolute stability of analytic neural networks: an approach based on finite trajectory length. IEEE Trans Circuits Syst I 51(12): 2460\u20132469","journal-title":"IEEE Trans Circuits Syst I"},{"issue":"2","key":"54_CR18","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1137\/040605266","volume":"16","author":"P-A Absil","year":"2005","unstructured":"Absil P-A, Mahony R, Andrews B (2005) Convergence of the iterates of descent methods for analytic cost functions. SIAM J Optim 16(2): 531\u2013547","journal-title":"SIAM J Optim"},{"key":"54_CR19","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0071569","volume-title":"Continuous and discrete dynamics near manifolds of equilibria","author":"B Aulbach","year":"1984","unstructured":"Aulbach B (1984) Continuous and discrete dynamics near manifolds of equilibria. Springer-Verlag, Berlin"},{"key":"54_CR20","unstructured":"\u0141ojasiewicz S (1984) Sur les trajectoires du gradient d\u2019une fonction analytique. In: Seminari di Geometria 1982\u20131983. Dipartimento di Matematica, Istituto di Geometria, Universit\u00e0 di Bologna, Bologna, Italy, pp 115\u2013117"},{"key":"54_CR21","doi-asserted-by":"crossref","first-page":"763","DOI":"10.2307\/2661354","volume":"152","author":"K Kurdyka","year":"2000","unstructured":"Kurdyka K, Mostowski T, Parusi\u0144ski A (2000) Proof of the gradient conjecture of R. Thom. Ann Math 152: 763\u2013792","journal-title":"Ann Math"},{"key":"54_CR22","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1016\/j.sysconle.2006.01.002","volume":"55","author":"P-A Absil","year":"2006","unstructured":"Absil P-A, Kurdyka K (2006) On the stable equilibrium points of gradient systems. Syst Control Lett 55: 573\u2013577","journal-title":"Syst Control Lett"},{"issue":"13\u201314","key":"54_CR23","doi-asserted-by":"crossref","first-page":"1543","DOI":"10.1002\/mana.200410564","volume":"280","author":"C Lageman","year":"2007","unstructured":"Lageman C (2007) Pointwise convergence of gradient-like systems. Math Nachr 280(13\u201314): 1543\u20131558","journal-title":"Math Nachr"},{"issue":"6","key":"54_CR24","doi-asserted-by":"crossref","first-page":"1471","DOI":"10.1109\/TNN.2006.879775","volume":"17","author":"M Forti","year":"2006","unstructured":"Forti M, Nistri P, Quincampoix M (2006) Convergence of neural networks for programming problems via a nonsmooth \u0141ojasiewicz inequality. IEEE Trans Neural Netw 17(6): 1471\u20131486","journal-title":"IEEE Trans Neural Netw"},{"issue":"4","key":"54_CR25","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1137\/050644641","volume":"17","author":"J Bolte","year":"2007","unstructured":"Bolte J, Daniilidis A, Lewis A (2007) The \u0141ojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J Optim 17(4): 1205\u20131223","journal-title":"SIAM J Optim"},{"key":"54_CR26","doi-asserted-by":"crossref","unstructured":"Bhatia NP, Hajek O (1969) Local semi-dynamical systems. In: Lecture notes in mathematics, vol 90. Springer-Verlag, Berlin","DOI":"10.1007\/BFb0079585"},{"key":"54_CR27","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/BF01211748","volume":"9","author":"A Iggidr","year":"1996","unstructured":"Iggidr A, Kalitine B, Outbib R (1996) Semidefinite Lyapunov functions stability and stabilization. Math Control Signal Syst 9: 95\u2013106","journal-title":"Math Control Signal Syst"},{"key":"54_CR28","doi-asserted-by":"crossref","unstructured":"Bhat SP, Bernstein DS (2003) Arc-length-based Lyapunov tests for convergence and stability in systems having a continuum of equilibria. In: Proceedings of American control conference, Denver, CO, June 2003, pp 2961\u20132966","DOI":"10.1109\/ACC.2003.1243775"},{"key":"54_CR29","unstructured":"Bhat SP, Bernstein DS (1999) Lyapunov analysis of semistability. In: Proceedings of the American control conference, San Diego, CA, June 1999, pp 1608\u20131612"},{"key":"54_CR30","volume-title":"Ordinary differential equations","author":"P Hartman","year":"1982","unstructured":"Hartman P (1982) Ordinary differential equations, 2nd edn. Birkh\u00e4user, Boston","edition":"2"},{"key":"54_CR31","volume-title":"Advanced ordinary differential equations","author":"AG Kartsatos","year":"1980","unstructured":"Kartsatos AG (1980) Advanced ordinary differential equations. Mariner Publishing Company, Inc., Tampa"},{"key":"54_CR32","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-9362-7","volume-title":"Stability theory by Liapunov\u2019s direct method. Applied mathematical sciences","author":"N Rouche","year":"1977","unstructured":"Rouche N, Habets P, Laloy M (1977) Stability theory by Liapunov\u2019s direct method. Applied mathematical sciences. Springer-Verlag, New York"},{"key":"54_CR33","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-6376-0","volume-title":"Stability theory and the existence of periodic solutions and almost periodic solutions","author":"T Yoshizawa","year":"1975","unstructured":"Yoshizawa T (1975) Stability theory and the existence of periodic solutions and almost periodic solutions. Springer-Verlag, New York"},{"key":"54_CR34","unstructured":"Apostol TM (1974) Mathematical Analysis, 2nd edn. Addison-Wesley Publishing Company, Inc."},{"key":"54_CR35","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-62006-5","volume-title":"Stability theory of dynamical systems","author":"NP Bhatia","year":"1970","unstructured":"Bhatia NP, Szeg\u00f6 GP (1970) Stability theory of dynamical systems. Springer-Verlag, Berlin"},{"issue":"4","key":"54_CR36","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1109\/TCT.1960.1086720","volume":"CT-7","author":"JP LaSalle","year":"1960","unstructured":"LaSalle JP (1960) Some extensions of Liapunov\u2019s second method. IRE Trans Circuit Theory CT-7(4): 520\u2013527","journal-title":"IRE Trans Circuit Theory"},{"key":"54_CR37","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1109\/9.388676","volume":"40","author":"CI Byrnes","year":"1995","unstructured":"Byrnes CI, Martin CF (1995) An integral-invariance principle for nonlinear systems. IEEE Trans Autom Control 40: 983\u2013994","journal-title":"IEEE Trans Autom Control"},{"key":"54_CR38","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/s004980200007","volume":"15","author":"A Teel","year":"2002","unstructured":"Teel A, Panteley E, Lor\u00eda A (2002) Integral characterizations of uniform asymptotic and exponential stability with applications. Math Control Signal Syst 15: 177\u2013201","journal-title":"Math Control Signal Syst"},{"key":"54_CR39","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-8176-4606-6","volume-title":"Set-theoretic methods in control","author":"F Blanchini","year":"2008","unstructured":"Blanchini F, Miani S (2008) Set-theoretic methods in control. Birkh\u00e4user, Boston"},{"key":"54_CR40","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511755217","volume-title":"Inequalities: a journey into linear analysis","author":"DJH Garling","year":"2007","unstructured":"Garling DJH (2007) Inequalities: a journey into linear analysis. Cambridge University Press, Cambridge"},{"key":"54_CR41","volume-title":"Topology a first course","author":"JR Munkres","year":"1975","unstructured":"Munkres JR (1975) Topology a first course. Prentice-Hall, Englewood Cliffs"},{"key":"54_CR42","first-page":"181","volume-title":"Differential geometric control theory","author":"RW Brockett","year":"1983","unstructured":"Brockett RW (1983) Asymptotic stability and feedback stabilization. In: Millman RS, Brockett RW, Sussmann HJ (eds) Differential geometric control theory. Birkh\u00e4user, Boston, pp 181\u2013191"},{"key":"54_CR43","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-015-7793-9","volume-title":"Differential equations with discontinuous right-hand sides","author":"AF Filippov","year":"1988","unstructured":"Filippov AF (1988) Differential equations with discontinuous right-hand sides. Kluwer, Dordrecht"},{"key":"54_CR44","unstructured":"Aubin JP, Frankowska H (1990) Set-valued analysis. In: Systems and control: foundations and applications, vol 2. Birkh\u00e4user, Boston"},{"key":"54_CR45","doi-asserted-by":"crossref","unstructured":"Rockafellar RT, Wets RJ-B (1998) Variational analysis. In: Comprehensive studies in mathematics, vol 317. Springer-Verlag, Berlin","DOI":"10.1007\/978-3-642-02431-3"},{"key":"54_CR46","doi-asserted-by":"crossref","DOI":"10.1515\/9781400873173","volume-title":"Convex analysis","author":"RT Rockafellar","year":"1970","unstructured":"Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton"}],"container-title":["Mathematics of Control, Signals, and Systems"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00498-010-0054-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00498-010-0054-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00498-010-0054-3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,4]],"date-time":"2019-06-04T00:51:11Z","timestamp":1559609471000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00498-010-0054-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,9,8]]},"references-count":46,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2010,10]]}},"alternative-id":["54"],"URL":"https:\/\/doi.org\/10.1007\/s00498-010-0054-3","relation":{},"ISSN":["0932-4194","1435-568X"],"issn-type":[{"value":"0932-4194","type":"print"},{"value":"1435-568X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,9,8]]}}}