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

S. S. Ravindran

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 3, page 840-863
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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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 20.3 (2014): 840-863. <http://eudml.org/doc/272927>.

@article{Ravindran2014,
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.},
author = {Ravindran, S. S.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {boundary penalty; dirichlet boundary control; Navier–stokes type system; soret convection; Navier-Stokes type system; Soret convection; Dirichlet boundary control},
language = {eng},
number = {3},
pages = {840-863},
publisher = {EDP-Sciences},
title = {Dirichlet control of unsteady Navier–Stokes type system related to Soret convection by boundary penalty method},
url = {http://eudml.org/doc/272927},
volume = {20},
year = {2014},
}

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
PB - EDP-Sciences
VL - 20
IS - 3
SP - 840
EP - 863
AB - 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.
LA - eng
KW - boundary penalty; dirichlet boundary control; Navier–stokes type system; soret convection; Navier-Stokes type system; Soret convection; Dirichlet boundary control
UR - http://eudml.org/doc/272927
ER -

References

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  1. [1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). Zbl1098.46001MR450957
  2. [2] V.K. Andreev and I.I. Ryzhkov, Symmetry classification and exact solutions of the thermal diffusion equations. Differ. Eqs.41 (2005) 538–547. Zbl1087.35077MR2200621
  3. [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. Zbl1217.49005MR2752771
  4. [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. Zbl1043.35014MR1992281
  5. [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. Zbl0964.76534
  6. [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 Dynamics11 (1998) 93–112. Zbl0939.76026MR1682723
  7. [7] E. Casas and M. Mateos and J.P. Raymond, Penalization of Dirichlet optimal control problems, ESAIM: COCV 15 (2009) 782–809. Zbl1175.49027MR2567245
  8. [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. Zbl0786.76063MR1188854
  9. [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. Zbl0910.76011MR1613873
  10. [10] M. Gad-el-Hak, A. Pollard and J. P. Bonnet, Flow Control, Fundamentals and Practices. Lect. Notes Phys. Springer, Berlin (1998). Zbl0896.76001
  11. [11] E. Gagliardo, Proprieta di alcune classi di funzioni in piu variabili. Ricerche. Mat.7 (1958) 102–137 Zbl0089.09401MR102740
  12. [12] V. Girault and P.A. Raviart, Finite Element Method for Navier–Stokes Equations. Springer, Berlin (1986). Zbl0585.65077MR851383
  13. [13] M.D. Gunzburger, Flow Control, IMA 68. Springer-Verlag, New York (1995). Zbl0816.00037MR1348639
  14. [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. Zbl0991.49002MR1788073
  15. [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. Zbl0737.76045MR1135991
  16. [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. Zbl1042.35047MR2038338
  17. [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. Zbl0917.49003MR1632548
  18. [18] G. Hellwig, Differential Operators of Mathematical Physics: An Introduction. Addison-Wesley, Reading, MA (1967). Zbl0163.11801MR211292
  19. [19] D.T.J. Hurle and E. Jakeman, Soret driven thermo-solutal convection. J. Fluid Mech.47 (1971) 667–687. 
  20. [20] K. Ito and S.S. Ravindran, Optimal control of thermally convected fluid flows. SIAM J. Sci. Comput.19 (1998) 1847–1869. Zbl0918.49004MR1638072
  21. [21] J.L. Lions and E. Magnes, Problemes aux limits Non Homogeneous et Applications, Vol. II. Dunod, Paris (1968). Zbl0165.10801MR247244
  22. [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. Zbl1225.76107
  23. [23] J. Necas, Les Méthods Directes en Théorie des Équations Elliptiques. Masson et Cie, Paris (1967). Zbl1225.35003MR227584
  24. [24] L. Nirenberg, On elliptic partial differential equations. Annul. Sc. Norm. Sup. Pisa13 (1959) 116–162. Zbl0088.07601MR109940
  25. [25] S.S. Ravindran, Convergence of Extrapolated BDF2 Finite Element Schemes For Unsteady Penetrative Convection Model. Numer. Funct. Anal. Opt.33 (2012) 48–79. Zbl1237.76074MR2870491
  26. [26] V.M. Shevtsova, D.E. Melnikov and J.C. Legros, Onset of convection in Soret-driven instability. Phys. Rev. E 73 (2006) 047302. Zbl1189.76162
  27. [27] J. Simon, Compact sets in the space Lp(0,T;B) Annali di Matematika Pura ed Applicata (IV) 146 (1987) 65–96. Zbl0629.46031MR916688
  28. [28] J. Singer and H.H. Bau, Active control of convection. Phys. Fluids A3 (1991) 2859–2865. Zbl0745.76078
  29. [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. Zbl1045.76014
  30. [30] S.S. Sritharan, Optimal Control of Viscous Flows. SIAM, Philadelphia (1998). Zbl0920.76004MR1632418
  31. [31] R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis. North-Holland (1977). Zbl0568.35002
  32. [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. Zbl0926.65097MR1616154

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