Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise
We consider the controlled stochastic Navier–Stokes equations in a bounded multidimensional domain, where the noise term allows jumps. In order to prove existence and uniqueness of an optimal control w.r.t. a given control problem, we first need to show the existence and uniqueness of a local mild s...
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| Published in | Mathematische Nachrichten Vol. 292; no. 7; pp. 1444 - 1461 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Weinheim
Wiley Subscription Services, Inc
01.07.2019
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| Online Access | Get full text |
| ISSN | 0025-584X 1522-2616 |
| DOI | 10.1002/mana.201700185 |
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| Abstract | We consider the controlled stochastic Navier–Stokes equations in a bounded multidimensional domain, where the noise term allows jumps. In order to prove existence and uniqueness of an optimal control w.r.t. a given control problem, we first need to show the existence and uniqueness of a local mild solution of the considered controlled stochastic Navier–Stokes equations. We then discuss the control problem, where the related cost functional includes stopping times dependent on controls. Based on the continuity of the cost functional, we can apply existence and uniqueness results provided in [4], which enables us to show that a unique optimal control exists. |
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| AbstractList | We consider the controlled stochastic Navier–Stokes equations in a bounded multidimensional domain, where the noise term allows jumps. In order to prove existence and uniqueness of an optimal control w.r.t. a given control problem, we first need to show the existence and uniqueness of a local mild solution of the considered controlled stochastic Navier–Stokes equations. We then discuss the control problem, where the related cost functional includes stopping times dependent on controls. Based on the continuity of the cost functional, we can apply existence and uniqueness results provided in [4], which enables us to show that a unique optimal control exists. We consider the controlled stochastic Navier–Stokes equations in a bounded multidimensional domain, where the noise term allows jumps. In order to prove existence and uniqueness of an optimal control w.r.t. a given control problem, we first need to show the existence and uniqueness of a local mild solution of the considered controlled stochastic Navier–Stokes equations. We then discuss the control problem, where the related cost functional includes stopping times dependent on controls. Based on the continuity of the cost functional, we can apply existence and uniqueness results provided in [4], which enables us to show that a unique optimal control exists. |
| Author | Trautwein, Christoph Benner, Peter |
| Author_xml | – sequence: 1 givenname: Peter surname: Benner fullname: Benner, Peter organization: Max Planck Institute for Dynamics of Complex Technical Systems – sequence: 2 givenname: Christoph surname: Trautwein fullname: Trautwein, Christoph email: trautwein@mpi-magdeburg.mpg.de organization: Max Planck Institute for Dynamics of Complex Technical Systems |
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| Cites_doi | 10.1002/mana.201300248 10.1137/030601259 10.1007/BF01214869 10.3934/eect.2017021 10.1007/s00245-002-0734-6 10.3792/pja/1195526510 10.1155/S1048953300000228 10.1051/m2an:2000140 10.1006/jfan.1994.1140 10.1017/CBO9780511662829 10.4064/sm-44-1-47-60 10.1002/cpa.20077 10.1080/07362990701673047 10.1007/BF01192467 10.1007/s00440-004-0392-5 10.1007/978-3-663-13911-9 10.1007/978-1-4612-5561-1 10.1017/CBO9780511809781 10.1002/mana.19790930106 10.4171/RMI/250 10.1017/CBO9780511721373 10.2977/prims/1195182014 10.1016/j.cam.2016.04.030 10.1016/S0167-6911(02)00274-8 10.1017/CBO9781107295513 10.1080/07362994.2013.759482 10.3792/pja/1195521686 10.1016/j.na.2012.10.011 10.1016/j.spa.2006.04.001 10.1007/s00021-015-0234-5 10.1051/m2an:2000151 10.1007/BF00276188 10.1007/BF00276875 10.1002/mana.3210040121 |
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| SubjectTerms | 35Q30 49J20 93E20 Continuity (mathematics) Control theory Fluid dynamics Fluid flow local mild solution Lévy process Mathematical analysis Navier-Stokes equations Optimal control stochastic Navier–Stokes equations stochastic optimal control Uniqueness |
| Title | Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise |
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