AbstractThis paper deals with the optimal control of a coefficient in the modification of Navier-Stokes equations. Namely, the motion of the viscous incompressible fluid for a small gradient of velocity is described by Navier-Stokes equations where the coefficient of the kinematic viscosity ν is the positive constant (ν 0). For a greater gradient of velocity the coefficient of kinematic viscosity is a positive function of the gradient of velocity, that is ν (|∇u|). In our case ν (|∇u|) = ν 0 + ν 1 a (|∇u|) where ν 0, ν 1 ∈ ℝ+. The function a is positive and monotone and it is taken as a control variable. The existence of a solution of the optimal control problem is proved. Further, the approximation of the control problem by the finite-dimensional control problem is performed. The proof of the existence of a solution of that aproximate problem has been brought into light. Finally, the connection between the solution of the control problem and the solution of the approximate control problem is established.
Numerical aproximation of an optimal control problem associated with the Navier-Stokes equations
