{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/64377"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/64377","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Some Controllability and Stabilization Problems of Surface Waves on Water with Surface tension","abstract":"The thesis consists of two parts. The first part discusses the initial value problem of a fifth-order Korteweg-de Vries type of equation wt + wxxx - wxxxxx - n∑j=1 ajwjwx = 0, w(x, 0) = w0(x) posed on a periodic domain x ∈ [0, 2π] with boundary conditions wix(0, t) = wix(2π, t), i = 0, 2, 3, 4 and an L2-stabilizing feedback control law wx(2π, t) = αwx(0, t) + (1 - α)wxxx(0; t) where n is a fixed positive integer, aj, j = 1, 2, ... n, α are real constants, and |α| < 1. It is shown that for w0(x) ∈ H1α(0, 2π) with the boundary conditions described above, the problem is locally well-posed for w ∈ C([0, T]; H1α(0, 2π)) with a conserved volume of w, [w] = ∫2π0 w(x, t)dx. Moreover, the solution with small initial condition exists globally and approaches to [w0(x)]/(2π) as t → + ∞. The second part concerns wave motions on water in a rectangular basin with a wave generator mounted on a side wall. The linear governing equations are used and it is assumed that the surface tension on the free surface is not zero. Two types of generators are considered, flexible and rigid. For the flexible case, it is shown that the system is exactly controllable. For the rigid case, the system is not exactly controllable in a finite-time interval. However, it is approximately controllable. The stability problem of the system with the rigid generator controlled by a static feedback is also studied and it is proved that the system is strongly stable for this case.","abstract_html":"The thesis consists of two parts. The first part discusses the initial value problem of a fifth-order Korteweg-de Vries type of equation w<sub>t</sub> + w<sub>xxx</sub> - w<sub>xxxxx</sub> - <sup>n</sup>∑<sub>j=1</sub> a<sub>j</sub>w<sup>j</sup>w<sub>x</sub> = 0, w(x, 0) = w<sub>0</sub>(x) posed on a periodic domain x ∈ [0, 2π] with boundary conditions w<sub>ix(</sub>0, t) = w<sub>ix</sub>(2π, t), i = 0, 2, 3, 4 and an L<sup>2</sup>-stabilizing feedback control law w<sub>x</sub>(2π, t) = αw<sub>x</sub>(0, t) + (1 - α)w<sub>xxx</sub>(0; t) where n is a fixed positive integer, a<sub>j</sub>, j = 1, 2, ... n, α are real constants, and |α| < 1. It is shown that for w<sub>0</sub>(x) ∈ H<sup>1</sup><sub>α</sub>(0, 2π) with the boundary conditions described above, the problem is locally well-posed for w ∈ C([0, T]; H<sup>1</sup><sub>α</sub>(0, 2π)) with a conserved volume of w, [w] = ∫<sup>2π</sup><sub>0</sub> w(x, t)dx. Moreover, the solution with small initial condition exists globally and approaches to [w<sub>0</sub>(x)]/(2π) as t → + ∞. The second part concerns wave motions on water in a rectangular basin with a wave generator mounted on a side wall. The linear governing equations are used and it is assumed that the surface tension on the free surface is not zero. Two types of generators are considered, flexible and rigid. For the flexible case, it is shown that the system is exactly controllable. For the rigid case, the system is not exactly controllable in a finite-time interval. However, it is approximately controllable. The stability problem of the system with the rigid generator controlled by a static feedback is also studied and it is proved that the system is strongly stable for this case.","abstract_has_math":false,"creators":["Gao, Guangyue"],"institution":"Virginia Tech","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Sun, Shu-Ming"],"committee_members":["Kim, Jong U.","Renardy, Michael J.","Yue, Pengtao"],"year":2015,"date_issued":"2015-12-23","date_published":"2015-12-23","updated_at":"2026-07-22T22:20:17Z","subjects":["Kawahara Equation","Contraction Mapping Principle","Boundary Control","Hydrodynamics"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:6988"],"render_values":[{"text":"vt_gsexam:6988","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/64377","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Sun, Shu-Ming"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Kim, Jong U.","Renardy, Michael J.","Yue, Pengtao"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Gao, Guangyue"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2015-12-26T09:06:08Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2015-12-26T09:06:08Z"]},{"key":"dc:date.issued","label":"Date","values":["2015-12-23"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Kawahara Equation","Contraction Mapping Principle","Boundary Control","Hydrodynamics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:6988"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/64377"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The thesis consists of two parts. The first part discusses the initial value problem of a fifth-order Korteweg-de Vries type of equation wt + wxxx - wxxxxx - n∑j=1 ajwjwx = 0, w(x, 0) = w0(x) posed on a periodic domain x ∈ [0, 2π] with boundary conditions wix(0, t) = wix(2π, t), i = 0, 2, 3, 4 and an L2-stabilizing feedback control law wx(2π, t) = αwx(0, t) + (1 - α)wxxx(0; t) where n is a fixed positive integer, aj, j = 1, 2, ... n, α are real constants, and |α| < 1. It is shown that for w0(x) ∈ H1α(0, 2π) with the boundary conditions described above, the problem is locally well-posed for w ∈ C([0, T]; H1α(0, 2π)) with a conserved volume of w, [w] = ∫2π0 w(x, t)dx. Moreover, the solution with small initial condition exists globally and approaches to [w0(x)]/(2π) as t → + ∞. The second part concerns wave motions on water in a rectangular basin with a wave generator mounted on a side wall. The linear governing equations are used and it is assumed that the surface tension on the free surface is not zero. Two types of generators are considered, flexible and rigid. For the flexible case, it is shown that the system is exactly controllable. For the rigid case, the system is not exactly controllable in a finite-time interval. However, it is approximately controllable. The stability problem of the system with the rigid generator controlled by a static feedback is also studied and it is proved that the system is strongly stable for this case."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["Some Controllability and Stabilization Problems of Surface Waves on Water with Surface tension"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Sun, Shu-Ming"],"dc:contributor.committeemember":["Kim, Jong U.","Renardy, Michael J.","Yue, Pengtao"],"dc:contributor.department":["Mathematics"],"dc:creator":["Gao, Guangyue"],"dc:date.accessioned":["2015-12-26T09:06:08Z"],"dc:date.available":["2015-12-26T09:06:08Z"],"dc:date.issued":["2015-12-23"],"dc:description.abstract":["The thesis consists of two parts. The first part discusses the initial value problem of a fifth-order Korteweg-de Vries type of equation wt + wxxx - wxxxxx - n∑j=1 ajwjwx = 0, w(x, 0) = w0(x) posed on a periodic domain x ∈ [0, 2π] with boundary conditions wix(0, t) = wix(2π, t), i = 0, 2, 3, 4 and an L2-stabilizing feedback control law wx(2π, t) = αwx(0, t) + (1 - α)wxxx(0; t) where n is a fixed positive integer, aj, j = 1, 2, ... n, α are real constants, and |α| < 1. It is shown that for w0(x) ∈ H1α(0, 2π) with the boundary conditions described above, the problem is locally well-posed for w ∈ C([0, T]; H1α(0, 2π)) with a conserved volume of w, [w] = ∫2π0 w(x, t)dx. Moreover, the solution with small initial condition exists globally and approaches to [w0(x)]/(2π) as t → + ∞. The second part concerns wave motions on water in a rectangular basin with a wave generator mounted on a side wall. The linear governing equations are used and it is assumed that the surface tension on the free surface is not zero. Two types of generators are considered, flexible and rigid. For the flexible case, it is shown that the system is exactly controllable. For the rigid case, the system is not exactly controllable in a finite-time interval. However, it is approximately controllable. The stability problem of the system with the rigid generator controlled by a static feedback is also studied and it is proved that the system is strongly stable for this case."],"dc:description.degree":["Ph. D."],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:6988"],"dc:identifier.uri":["http://hdl.handle.net/10919/64377"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Kawahara Equation","Contraction Mapping Principle","Boundary Control","Hydrodynamics"],"dc:title":["Some Controllability and Stabilization Problems of Surface Waves on Water with Surface tension"],"dc:type":["Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:20:17Z"}