Dirichlet control of unsteady Navier–Stokes type system related to Soret convection by boundary penalty method

Department of Mathematical Sciences, 201C Shelby Center for Science and Technology, The University of Alabama in Huntsville, Huntsville, AL 35899, USA
ravinds@uah.edu

Received: 19 March 2013
Revised: 15 November 2013

Abstract

In this paper, we study the boundary penalty method for optimal control of unsteady Navier–Stokes type system that has been proposed as an alternative for Dirichlet boundary control. Existence and uniqueness of solutions are demonstrated and existence of optimal control for a class of optimal control problems is established. The asymptotic behavior of solution, with respect to the penalty parameter ϵ, is studied. In particular, we prove convergence of solutions of penalized control problem to the corresponding solutions of the Dirichlet control problem, as the penalty parameter goes to zero. We also derive an optimality system and determine optimal solutions. In order to illustrate the theoretical results and the practical utility of control, we numerically address the problem of controlling unsteady convection with Soret effect using a gradient-based method. Numerical results show the effectiveness of the approach.