2024/08/31 by Chemetov, Nikolai, Cipriano, Fernanda · 1 citation
#49K45 #60H15 #6D55 #93E20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR)
paper · doi:10.48550/arxiv.2409.00479
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 non-homogeneous 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.