Optimal control problems for the Navier-Stokes system coupled with the - turbulence model

Optimal control of fluid-dynamics systems has gained attention in the last several years from the scientific community because of its potential use in design of new engineering devices and optimization of existing ones. Many research works have extensively studied the optimal control for the system of Navier-Stokes but the problem of turbulence in these works is usually not taken into account because of the many difficulties arising from the numerical implementation and solution of the optimality system. In this work turbulence is considered by coupling the - two-equation turbulence model with the averaged Navier-Stokes system. The complete optimality system is derived and the existence of a weak solution proven. Some numerical examples are reported.