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.

Keywords: boundary penalty, dirichlet boundary control, Navier-stokes type system, soret convection

@article{COCV_2014__20_3_840_0, author = {Ravindran, S. S.}, title = {Dirichlet control of unsteady {Navier-Stokes} type system related to {Soret} convection by boundary penalty method}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {840--863}, publisher = {EDP-Sciences}, volume = {20}, number = {3}, year = {2014}, doi = {10.1051/cocv/2013086}, mrnumber = {3264226}, zbl = {1301.35093}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2013086/} }

TY - JOUR AU - Ravindran, S. S. TI - Dirichlet control of unsteady Navier-Stokes type system related to Soret convection by boundary penalty method JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 840 EP - 863 VL - 20 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2013086/ DO - 10.1051/cocv/2013086 LA - en ID - COCV_2014__20_3_840_0 ER -

%0 Journal Article %A Ravindran, S. S. %T Dirichlet control of unsteady Navier-Stokes type system related to Soret convection by boundary penalty method %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 840-863 %V 20 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2013086/ %R 10.1051/cocv/2013086 %G en %F COCV_2014__20_3_840_0

Ravindran, S. S. Dirichlet control of unsteady Navier-Stokes type system related to Soret convection by boundary penalty method. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 3, pp. 840-863. doi : 10.1051/cocv/2013086. http://archive.numdam.org/articles/10.1051/cocv/2013086/

[1] Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl

,[2] Symmetry classification and exact solutions of the thermal diffusion equations. Differ. Eqs. 41 (2005) 538-547. | MR | Zbl

and ,[3] Dirichlet boundary control for a parabolic equation with final observation: A space-time mixed formulation and penalization. Asympotic Anal. 71 (2011) 101-121. | MR | Zbl

, and ,[4] A penalized approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions. Asymptotic Anal. 34 (2003) 121-136. | MR | Zbl

, and ,[5] Simulation of two dimensional thermosolutal convection in liquid metals induced by horizontal temperature and species gradients. Int. J. Heat Mass Transfer 39 (1996) 2883. | Zbl

and ,[6] Feedback control of thermal fluid using state estimation, Flow Control and Optimization. Int. J. Comput. Fluid Dynamics 11 (1998) 93-112. | MR | Zbl

, and ,[7] E. Casas and M. Mateos and J.P. Raymond, Penalization of Dirichlet optimal control problems, ESAIM: COCV 15 (2009) 782-809. | Numdam | MR | Zbl

[8] Existence of optimal controls for viscous flow problems. Proc. Royal Soc. London, Ser. A 439 (1992) 81-102. | MR | Zbl

and ,[9] Boundary value problems and optimal boundary control for the Navier-Stokes system: the two-dimensional case. SIAM J. Control Optim. 36 (1998) 852-894. | MR | Zbl

, and ,[10] M. Gad-el-Hak, A. Pollard and J. P. Bonnet, Flow Control, Fundamentals and Practices. Lect. Notes Phys. Springer, Berlin (1998). | Zbl

[11] Proprieta di alcune classi di funzioni in piu variabili. Ricerche. Mat. 7 (1958) 102-137 | MR | Zbl

,[12] Finite Element Method for Navier-Stokes Equations. Springer, Berlin (1986). | MR | Zbl

and ,[13] Flow Control, IMA 68. Springer-Verlag, New York (1995). | MR | Zbl

,[14] The velocity tracking problem for Navier-Stokes flows with boundary control. SIAM J. Control Optim. 39 (2000) 594-634. | MR | Zbl

and ,[15] Analysis and finite approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet control. Math. Model. Numer. Anal. 25 (1990) 711-748. | Numdam | MR | Zbl

, and ,[16] Second order methods for boundary control of the instationary Navier-Stokes system. Z. Angew. Math. Mech. 84 (2004) 171-187. | MR | Zbl

and ,[17] A Penalized Neumann Control Approach for Solving an Optimal Dirichlet Control Problem for the Navier-Stokes Equations. SIAM J. Control and Optim. 36 (1998) 1795-1814. | MR | Zbl

and ,[18] Differential Operators of Mathematical Physics: An Introduction. Addison-Wesley, Reading, MA (1967). | MR | Zbl

,[19] Soret driven thermo-solutal convection. J. Fluid Mech. 47 (1971) 667-687.

and ,[20] Optimal control of thermally convected fluid flows. SIAM J. Sci. Comput. 19 (1998) 1847-1869. | MR | Zbl

and ,[21] Problemes aux limits Non Homogeneous et Applications, Vol. II. Dunod, Paris (1968). | MR | Zbl

and ,[22] Convections, anti-convections and multi-convections in binary fluid convection. J. Fluid Mech. 667 (2011) 586-606. | Zbl

, , and ,[23] Directes en Théorie des Équations Elliptiques. Masson et Cie, Paris (1967). | MR | Zbl

,[24] On elliptic partial differential equations. Annul. Sc. Norm. Sup. Pisa 13 (1959) 116-162. | Numdam | MR | Zbl

,[25] Convergence of Extrapolated BDF2 Finite Element Schemes For Unsteady Penetrative Convection Model. Numer. Funct. Anal. Opt. 33 (2012) 48-79. | MR | Zbl

,[26] Onset of convection in Soret-driven instability. Phys. Rev. E 73 (2006) 047302.

, and ,[27] Compact sets in the space Lp(0,T;B) Annali di Matematika Pura ed Applicata (IV) 146 (1987) 65-96. | MR | Zbl

,[28] Active control of convection. Phys. Fluids A 3 (1991) 2859-2865. | Zbl

and ,[29] Convection of a binary mixture under conditions of thermal diffusion and variable temperature gradient. J. Appl. Mech. Tech. Phys. 43 (2002) 217-223. | Zbl

,[30] Optimal Control of Viscous Flows. SIAM, Philadelphia (1998). | MR | Zbl

,[31] Navier-Stokes Equations: Theory and Numerical Analysis. North-Holland (1977). | Zbl

,[32] The adjoint method for an inverse design problem in the directional solidification of binary alloys. J. Comput. Phys. 40 (1998) 432-452. | MR | Zbl

and ,*Cited by Sources: *