{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:34:51Z","timestamp":1787322891319,"version":"build-2736575974"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"name":"MIUR-PRIN"},{"name":"GNAMPA of INdAM"},{"DOI":"10.13039\/501100006476","name":"Academia Rom\u00e3na","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100006476","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004462","name":"Consiglio Nazionale delle Ricerche","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004462","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004963","name":"Seventh Framework Programme","doi-asserted-by":"publisher","award":["FP7-IDEAS-ERC-StG 256872"],"award-info":[{"award-number":["FP7-IDEAS-ERC-StG 256872"]}],"id":[{"id":"10.13039\/501100004963","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006595","name":"Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii","doi-asserted-by":"publisher","award":["PNII-ID-PCE-2011-3-0027"],"award-info":[{"award-number":["PNII-ID-PCE-2011-3-0027"]}],"id":[{"id":"10.13039\/501100006595","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>In the present contribution the sliding mode control (SMC) problem for a phase-field model of Caginalp type is considered. First we prove the well-posedness and some regularity results for the phase-field type state systems modified by the state-feedback control laws. Then, we show that the chosen SMC laws force the system to reach within finite time the sliding manifold (that we chose in order that one of the physical variables or a combination of them remains constant in time). We study three different types of feedback control laws: the first one appears in the internal energy balance and forces a linear combination of the temperature and the phase to reach a given (space dependent) value, while the second and third ones are added in the phase relation and lead the phase onto a prescribed target $\\phi^*$. While the control law is nonlocal in space for the first two problems, it is local in the third one, i.e., its value at any point and any time just depends on the value of the state.<\/jats:p>","DOI":"10.1137\/15m102424x","type":"journal-article","created":{"date-parts":[[2017,6,29]],"date-time":"2017-06-29T10:07:33Z","timestamp":1498730853000},"page":"2108-2133","source":"Crossref","is-referenced-by-count":11,"title":["Sliding Mode Control for a Nonlinear Phase-Field System"],"prefix":"10.1137","volume":"55","author":[{"given":"Viorel","family":"Barbu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pierluigi","family":"Colli","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gianni","family":"Gilardi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gabriela","family":"Marinoschi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Elisabetta","family":"Rocca","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,6,29]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"G. Bartolini, L. Fridman, A. Pisano, and E. Usai, eds.\n                      Modern Sliding Mode Control Theory New Perspectives and Applications\n                      , Lecture Notes in Control and Inform. Sci. 375, Springer, New York, 2008.","DOI":"10.1007\/978-3-540-79016-7"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"V. Barbu,\n                      Nonlinear Differential Equations of Monotone Types in Banach Spaces\n                      , Springer, New York, 2010.","DOI":"10.1007\/978-1-4419-5542-5"},{"key":"atypb3","unstructured":"H. Brezis,\n                      Ope\u0301rateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert\n                      , Math. Stud. 5, North-Holland, Amsterdam, 1973."},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"M. Brokate and J. Sprekels,\n                      Hysteresis and Phase Transitions\n                      , Springer, New York, 1996.","DOI":"10.1007\/978-1-4612-4048-8"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/BF00254827"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2010.10.045"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2015.09.011"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.3934\/mcrf.2016.6.95"},{"key":"atypb9","first-page":"A1461","volume":"276","author":"Duvaut G.","year":"1973","journal-title":"C. R. Acad. Sci. Paris Ser. A-B"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"C. Edwards, E. Fossas Colet, and L. Fridman, eds.\n                      Advances in Variable Structure and Sliding Mode Control\n                      , Lecture Notes in Control and Inform. Sci. 334, Springer-Verlag, New York, 2006.","DOI":"10.1007\/11612735"},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"C. Edwards and S. Spurgeon,\n                      Sliding Mode Control: Theory and Applications\n                      , Taylor and Francis, London, 1999.","DOI":"10.1201\/9781498701822"},{"key":"atypb12","first-page":"46","author":"Elliott C. M.","year":"1990","journal-title":"Switzerland"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"M. Fre\u0301mond,\n                      Non-smooth Thermomechanics\n                      , Springer, New York, 2002.","DOI":"10.1007\/978-3-662-04800-9"},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"L. Fridman, J. Moreno, and R. Iriarte, eds.\n                      Sliding Modes After the First Decade of the $21$st Century: State of the Art\n                      , Lecture Notes in Control and Inform. Sci. 412, Springer, New York, 2011.","DOI":"10.1007\/978-3-642-22164-4"},{"key":"atypb15","unstructured":"D. Gilbarg and N. S. Trudinger,\n                      Elliptic Partial Differential Equations of Second Order\n                      , Springer, New York, 1998."},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.4171\/ZAA\/1277"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1080\/01630569208816458"},{"key":"atypb18","first-page":"305","volume":"17","author":"Hoffmann K.-H.","year":"2007","journal-title":"Adv. Math. Sci. Appl."},{"key":"atypb19","unstructured":"U. Itkis,\n                      Control Systems of Variable Structure\n                      , Wiley, New York, 1976."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1016\/0362-546X(94)90235-6"},{"key":"atypb21","volume":"23","author":"Ladyz\u030censkaja O.A.","journal-title":"Trans. Amer. Math. 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Showalter,\n                      Monotone Operators in Banach Space and Nonlinear Partial Differential Equations\n                      , Math. Surveys Monogr. 49, AMS, Providence, RI, 1991."},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1007\/BF01762360"},{"key":"atypb33","doi-asserted-by":"crossref","unstructured":"V. Utkin,\n                      Sliding Modes in Control and Optimization\n                      , Comm. Control Engrg. Ser., Springer, New York, 1992.","DOI":"10.1007\/978-3-642-84379-2"},{"key":"atypb34","unstructured":"V. Utkin, J. Guldner, and J. Shi,\n                      Sliding Mode Control in Electro-Mechanical Systems\n                      , 2nd ed., Autom. Control Eng., CRC Press, BocaRaton, FL, 2009."},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfranklin.2012.12.007"},{"key":"atypb36","doi-asserted-by":"crossref","unstructured":"K. D. Young and U\u0308. 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