{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:34:20Z","timestamp":1787322860082,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"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\/100009112","name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100009112","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003407","name":"Ministero dell'Istruzione, dell'Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["2015PA5MP7"],"award-info":[{"award-number":["2015PA5MP7"]}],"id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>In this paper, we investigate a distributed optimal control problem for a convective viscous Cahn--Hilliard system with dynamic boundary conditions. Such systems govern phase separation processes between two phases taking place in an incompressible fluid in a container and, at the same time, on the container boundary. The cost functional is of standard tracking type, while the control is exerted by the velocity of the fluid in the bulk. In this way, the coupling between the state (given by the associated order parameter and chemical potential) and control variables in the governing system of nonlinear partial differential equations is bilinear, which presents an additional difficulty for the analysis. The nonlinearities in the bulk and surface free energies are of logarithmic type, which entails that the thermodynamic forces driving the phase separation process may become singular. We show existence for the optimal control problem under investigation, prove the Fr\u00e9chet differentiability of the associated control-to-state mapping in suitable Banach spaces, and derive the first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint system. Due to the strong nonlinear couplings between state variables and control, the corresponding proofs require a considerable analytical effort.<\/jats:p>","DOI":"10.1137\/17m1146786","type":"journal-article","created":{"date-parts":[[2018,5,15]],"date-time":"2018-05-15T10:56:42Z","timestamp":1526381802000},"page":"1665-1691","source":"Crossref","is-referenced-by-count":28,"title":["Optimal Velocity Control of a Viscous Cahn--Hilliard System with Convection and Dynamic Boundary Conditions"],"prefix":"10.1137","volume":"56","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7921-5041","authenticated-orcid":true,"given":"Pierluigi","family":"Colli","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gianni","family":"Gilardi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J\u00fcrgen","family":"Sprekels","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,5,15]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/140984749"},{"key":"atypb2","first-page":"41","volume":"7","author":"Colli P.","year":"2015","journal-title":"Ann. Acad. Rom. Sci. Ser. Math. Appl."},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2015.04.057"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2015.07.011"},{"key":"atypb5","first-page":"487","volume":"2","author":"Colli P.","year":"1997","journal-title":"Adv. Differential Equations"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6544\/aa6e5f"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1515\/anona-2015-0035"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.3934\/Math.2016.3.225"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-015-9299-z"},{"key":"atypb10","first-page":"5","volume":"10","author":"Colli P.","year":"2017","journal-title":"Ser. Mat. Model. Program."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.3934\/eect.2017003"},{"key":"atypb12","first-page":"10","author":"Colli P.","year":"2018","journal-title":"Ann. Mat. Pura Appl. (4), ("},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1080\/03605309908820706"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/140994800"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"T. Fukao and N. Yamazaki,\n                      A boundary control problem for the equation and dynamic boundary condition of Cahn-Hilliard type\n                      , in Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels, eds., Springer INdAM Series, 22, Springer, Milan, 2017, pp. 255-280.","DOI":"10.1007\/978-3-319-64489-9_10"},{"key":"atypb16","first-page":"1","author":"Hintermu\u0308ller M.","year":"2016","journal-title":"Berlin"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/15M1025128"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1137\/110824152"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/120865628"},{"key":"atypb20","unstructured":"M. Hintermu\u0308ller and D. Wegner,\n                      Distributed and boundary control problems for the semidiscrete Cahn-Hilliard\/Navier-Stokes system with nonsmooth Ginzburg-Landau energies\n                      , Isaac Newton Institute Preprint Series, NI14042-FRB, 2014, pp. 1-29."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcrysgro.2012.11.049"},{"key":"atypb22","doi-asserted-by":"crossref","unstructured":"A. Lunardi,\n                      Analytic Semigroups and Optimal Regularity in Parabolic Problems\n                      , Birkha\u0308user Verlag, Basel, Switzerland, 1995.","DOI":"10.1007\/978-3-0348-0557-5"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1137\/140964308"},{"key":"atypb24","unstructured":"S\u030c. Schwabik, M. Tvrdy\u0301, and O. Vejvoda,\n                      Differential and Integral Equations: Boundary Value Problems and Adjoints\n                      , D. Reidel Publishing Co., Dordrecht, Boston, London, 1979."},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/BF01762360"},{"key":"atypb26","first-page":"172","volume":"1128","author":"Wang Q.-F.","year":"2000","journal-title":"Surikaisekikenkyusho Kokyuroku"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1080\/00036811.2011.643786"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-013-9234-0"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1007\/s10883-014-9259-y"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1146786","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:29:53Z","timestamp":1787318993000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1146786"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":29,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M1146786"],"URL":"https:\/\/doi.org\/10.1137\/17m1146786","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}