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Optimal control of newtonian fluids in a stochastic environment
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Orientador(es)
Resumo(s)
We consider a velocity tracking problem for stochastic Navier-Stokes equations in a 2D-bounded domain. The control acts on the boundary through an injection-suction device with uncertainty, which acts in accordance with the nonhomogeneous Navier-slip boundary conditions. After establishing a suitable stability result for the solution of the stochastic state equation, we prove the well-posedness of the stochastic linearized state equation and show that the Gâteaux derivative of the control-to-state mapping corresponds to the unique solution of the linearized equation. Next, we study the stochastic backward adjoint equation and establish a duality relation between the solutions of the forward linearized equation and the backward adjoint equation. Finally, we derive the first-order optimality conditions.
Descrição
The work of the first author was supported by FAPESP (Fundacc\u00E3o de Amparo \u00E0 Pesquisa do Estado de S\u00E3o Paulo), project 2021/03758-8, ``Mathematical problems in fluid dynamics\"\". Also the collaboration scientific work of the authors was supported by FAPESP (Fundac c\u00E3o de Amparo \u00E0 Pesquisa do Estado de S\u00E3o Paulo), through the Visiting Professor Project 2023/05271-4, ``Problema de controlo otimo atrav\u00E9s da fronteira para equacc\u00F5es de Navier-Stokes estoc\u00E1sticas\"\".
Publisher Copyright:
© 2025 Society for Industrial and Applied Mathematics.
Palavras-chave
first-order optimality conditions Navier-slip boundary conditions stochastic backward equation stochastic Navier-Stokes equations Analysis Computational Mathematics Applied Mathematics