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

• Published in: Applied mathematics & optimization Vol. 84; no. Suppl 1; pp. 1001 - 1054
• Main Authors: Benner, Peter, Trautwein, Christoph
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2021; Springer Nature B.V
• 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; 93E20; Nonconvex optimization; 49J20; 76D55; Stochastic control; Maximum principle; Stochastic Navier–Stokes equations; 35Q30
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-021-09792-6

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.