Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

• Published in: Mathematische Nachrichten Vol. 292; no. 7; pp. 1444 - 1461
• Main Authors: Benner, Peter, Trautwein, Christoph
• Format: Journal Article
• Language: English
• Published: Weinheim Wiley Subscription Services, Inc 01.07.2019
• Subjects: 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
• ISSN: 0025-584X; 1522-2616
• DOI: 10.1002/mana.201700185

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.