no. 3
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.

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.

DOI : 10.1051/cocv/2013086
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},
     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, Tome 20 (2014) no. 3, pp. 840-863. doi : 10.1051/cocv/2013086. http://archive.numdam.org/articles/10.1051/cocv/2013086/

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

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

[3] F. Ben Belgacem, C. Bernardi and H. El Fekih, 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

[4] F. Ben Belgacem, H. El Fekih and J.P. Raymond, A penalized approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions. Asymptotic Anal. 34 (2003) 121-136. | MR | Zbl

[5] L. Bergman and M.T. Hyun, 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

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

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

[9] A. Fursikov, M.D. Gunzburger and L.S. Hou, 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

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

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

[12] V. Girault and P.A. Raviart, Finite Element Method for Navier-Stokes Equations. Springer, Berlin (1986). | MR | Zbl

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

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

[15] M.D. Gunzburger, L.S. Hou and Th.P. Svobodny, 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

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

[17] L.S. Hou and S.S. Ravindran, 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

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

[19] D.T.J. Hurle and E. Jakeman, Soret driven thermo-solutal convection. J. Fluid Mech. 47 (1971) 667-687.

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

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

[22] I. Mercader, O. Batiste, A. Alonso and E. Knoblock, Convections, anti-convections and multi-convections in binary fluid convection. J. Fluid Mech. 667 (2011) 586-606. | Zbl

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

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

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

[26] V.M. Shevtsova, D.E. Melnikov and J.C. Legros, Onset of convection in Soret-driven instability. Phys. Rev. E 73 (2006) 047302.

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

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

[29] B.L. Smorodin, 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] S.S. Sritharan, Optimal Control of Viscous Flows. SIAM, Philadelphia (1998). | MR | Zbl

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

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

Cité par Sources :