A Stochastic Maximum Principle for Control Problems Constrained by the Stochastic Navier–Stokes Equations

We analyze the control problem of the stochastic Navier–Stokes equations in multi-dimensional domains considered in Benner and Trautwein (Math Nachr 292(7):1444–1461, 2019 ) restricted to noise terms defined by a Q-Wiener process. The cost functional related to this control problem is nonconvex. Usi...

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Bibliographic Details
Published in:Applied mathematics & optimization 2021-12, Vol.84 (Suppl 1), p.1001-1054
Main Authors: Benner, Peter, Trautwein, Christoph
Format: Article
Language:English
Subjects:
Applied mathematics
Calculus of Variations and Optimal Control
Optimization
Control
Domains
Fluid flow
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Maximum principle
Navier-Stokes equations
Noise
Numerical and Computational Physics
Optimization
Partial differential equations
Simulation
Stability
Systems Theory
Theoretical
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Summary:We analyze the control problem of the stochastic Navier–Stokes equations in multi-dimensional domains considered in Benner and Trautwein (Math Nachr 292(7):1444–1461, 2019 ) restricted to noise terms defined by a Q-Wiener process. The cost functional related to this control problem is nonconvex. Using a stochastic maximum principle, we derive a necessary optimality condition to obtain explicit formulas the optimal controls have to satisfy. Moreover, we show that the optimal controls satisfy a sufficient optimality condition. As a consequence, we are able to solve uniquely control problems constrained by the stochastic Navier–Stokes equations especially for two-dimensional as well as for three-dimensional domains.
ISSN:0095-4616
1432-0606
DOI:10.1007/s00245-021-09792-6