A Survey On Optimal Control Problems Of Navier-Stokes Equations

The Navier-Stokes equations are a widely accepted model for the behavior of viscous incompressible fluids, which have very strong applied background. The analysis of controlling these equations has a more recent history , it appears that such problems received increased attention since the 1980s and obtain a series of research result. In this paper, we will summarize the optimal control problems of the the Navier-Stokes equations. It includes the time optimal control of the Navier-Stokes equations, Pontryagin's maximum principle for optimal control of the Navier-Stokes equations, second-order optimality conditions for the Navier-Stokes equations, and an optimal Dirichlet control problem for the Navier-Stokes equations. This paper consists of six sections. Section 1 is preliminary in which we give the simple introduction of the optimal control and the physics background of the Navier-Stokes equations. Section 2 is also preliminary in which we give definitions and theorems used later . Section 3 presents the time optimal control of the Navier-Stokes equations. Chaper4 presents Pontryagin's maximum principle. Section 5 presents second-order optimality conditions . Section 6 is concerned with an optimal Dirichlet control problem for the Navier-Stokes equations. Here, we list the main conclusions and the simple introduction of the main method, for the readers can have a preliminary comprehension about this problems.