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.
Classification : 35Q30, 35B40, 76B75, 49J20, 65M60, 76R99
Mots clés : 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}, zbl = {1301.35093}, mrnumber = {3264226}, language = {en}, url = {archive.numdam.org/item/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, Tome 20 (2014) no. 3, pp. 840-863. doi : 10.1051/cocv/2013086. http://archive.numdam.org/item/COCV_2014__20_3_840_0/
[1] Sobolev Spaces. Academic Press, New York (1975). | MR 450957 | Zbl 1098.46001
,[2] Symmetry classification and exact solutions of the thermal diffusion equations. Differ. Eqs. 41 (2005) 538-547. | MR 2200621 | Zbl 1087.35077
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 2752771 | Zbl 1217.49005
, and ,[4] A penalized approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions. Asymptotic Anal. 34 (2003) 121-136. | MR 1992281 | Zbl 1043.35014
, 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 0964.76534
and ,[6] Feedback control of thermal fluid using state estimation, Flow Control and Optimization. Int. J. Comput. Fluid Dynamics 11 (1998) 93-112. | MR 1682723 | Zbl 0939.76026
, 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 2567245 | Zbl 1175.49027
[8] Existence of optimal controls for viscous flow problems. Proc. Royal Soc. London, Ser. A 439 (1992) 81-102. | MR 1188854 | Zbl 0786.76063
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 1613873 | Zbl 0910.76011
, and ,[10] M. Gad-el-Hak, A. Pollard and J. P. Bonnet, Flow Control, Fundamentals and Practices. Lect. Notes Phys. Springer, Berlin (1998). | Zbl 0896.76001
[11] Proprieta di alcune classi di funzioni in piu variabili. Ricerche. Mat. 7 (1958) 102-137 | MR 102740 | Zbl 0089.09401
,[12] Finite Element Method for Navier-Stokes Equations. Springer, Berlin (1986). | MR 851383 | Zbl 0585.65077
and ,[13] Flow Control, IMA 68. Springer-Verlag, New York (1995). | MR 1348639 | Zbl 0816.00037
,[14] The velocity tracking problem for Navier-Stokes flows with boundary control. SIAM J. Control Optim. 39 (2000) 594-634. | MR 1788073 | Zbl 0991.49002
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 1135991 | Zbl 0737.76045
, and ,[16] Second order methods for boundary control of the instationary Navier-Stokes system. Z. Angew. Math. Mech. 84 (2004) 171-187. | MR 2038338 | Zbl 1042.35047
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 1632548 | Zbl 0917.49003
and ,[18] Differential Operators of Mathematical Physics: An Introduction. Addison-Wesley, Reading, MA (1967). | MR 211292 | Zbl 0163.11801
,[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 1638072 | Zbl 0918.49004
and ,[21] Problemes aux limits Non Homogeneous et Applications, Vol. II. Dunod, Paris (1968). | MR 247244 | Zbl 0165.10801
and ,[22] Convections, anti-convections and multi-convections in binary fluid convection. J. Fluid Mech. 667 (2011) 586-606. | Zbl 1225.76107
, , and ,[23] Directes en Théorie des Équations Elliptiques. Masson et Cie, Paris (1967). | MR 227584 | Zbl 1225.35003
,[24] On elliptic partial differential equations. Annul. Sc. Norm. Sup. Pisa 13 (1959) 116-162. | Numdam | MR 109940 | Zbl 0088.07601
,[25] Convergence of Extrapolated BDF2 Finite Element Schemes For Unsteady Penetrative Convection Model. Numer. Funct. Anal. Opt. 33 (2012) 48-79. | MR 2870491 | Zbl 1237.76074
,[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 916688 | Zbl 0629.46031
,[28] Active control of convection. Phys. Fluids A 3 (1991) 2859-2865. | Zbl 0745.76078
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 1045.76014
,[30] Optimal Control of Viscous Flows. SIAM, Philadelphia (1998). | MR 1632418 | Zbl 0920.76004
,[31] Navier-Stokes Equations: Theory and Numerical Analysis. North-Holland (1977). | Zbl 0568.35002
,[32] The adjoint method for an inverse design problem in the directional solidification of binary alloys. J. Comput. Phys. 40 (1998) 432-452. | MR 1616154 | Zbl 0926.65097
and ,