Navier-stokes Equations In A Class Of Optimal Control Problem

As is known to all, Navier-Stokes equations are fundamentally important in describing the motion for viscous incompressible fluids. For a long time, these equations have been deeply investigated by a lot of scientific workers, including many famous mathematicians. So far, remarkable progress has been achieved for these equations in many aspects, such as the existence and uniqueness of solutions, the regularity of solutions, the asymptotic behavior, the stability, the corresponding optimal control problems, etc. Based on the previous papers, we will consider a class of optimal control problems governed by the evolutionary Navier-Stokes equation in this paper.This paper includes three parts. In the first part, we investigate the velocity tracking problem for Navier-Stokes flows and obtain the first order necessary conditions. It is worth pointing that the permisson control set is allowed to be unconvex.In the second part, we prove that the weak solution of the Navier-Stokes equation is stable, namely, if the initial data and the external force make a small perturbation, the perturbation of L2 - norm of the solution is small too. This provides the important support for the numerical analysis.In the last part, we show that the solution of the evolutionary Navier-Stokes equation converges to the solution of the stable Navier-Stokes equation, provided that a proper feedback is selected.