Distributed control for multistate modified Navier-Stokes equations
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 1, pp. 219-238.

The aim of this paper is to establish necessary optimality conditions for optimal control problems governed by steady, incompressible Navier-Stokes equations with shear-dependent viscosity. The main difficulty derives from the fact that equations of this type may exhibit non-uniqueness of weak solutions, and is overcome by introducing a family of approximate control problems governed by well posed generalized Stokes systems and by passing to the limit in the corresponding optimality conditions.

DOI : 10.1051/cocv/2012007
Classification : 49K20, 76D55, 76A05
Mots-clés : optimal control, multistate Navier-Stokes equations, shear-dependent viscosity, necessary optimality conditions 
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     title = {Distributed control for multistate modified {Navier-Stokes} equations},
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     pages = {219--238},
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Arada, Nadir. Distributed control for multistate modified Navier-Stokes equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 1, pp. 219-238. doi : 10.1051/cocv/2012007. http://www.numdam.org/articles/10.1051/cocv/2012007/

[1] F. Abergel and E. Casas, Some optimal control problems of multistate equations appearing in fluid mechanics. RAIRO Modél. Math. Anal. Numér. 27 (1993) 223-247. | Numdam | MR | Zbl

[2] N. Arada, Optimal Control of shear-thickening flows. Departamento de Matemática, FCT-UNL, Portugal, Technical Report 3 (2012). | MR | Zbl

[3] E. Casas, Boundary control problems for quasi-linear elliptic equations : a Pontryagin's principle. Appl. Math. Optim. 33 (1996) 265-291. | MR | Zbl

[4] E. Casas and L.A. Fernández, Boundary control of quasilinear elliptic equations. INRIA, Rapport de Recherche 782 (1988).

[5] E. Casas and L.A. Fernández, Distributed control of systems governed by a general class of quasilinear elliptic equations. J. Differ. Equ. 35 (1993) 20-47. | MR | Zbl

[6] J.C. De Los Reyes and R. Griesse, State-constrained optimal control of the three-dimensional stationary Navier-Stokes equations. J. Math. Anal. Appl. 343 (2008) 257-272. | MR | Zbl

[7] J. Frehse, J. Málek and M. Steinhauer, An existence result for fluids with shear dependent viscosity-steady flows. Nonlinear. Anal. 30 (1997) 3041-3049. | MR | Zbl

[8] G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, I, II, 2nd edition. Springer-Verlag, New York. Springer Tracts in Natural Philosophy 38, 39 (1998). | MR | Zbl

[9] M.D. Gunzburger and C. Trenchea, Analysis of an optimal control problem for the three-dimensional coupled modified Navier-Stokes and maxwell equations. J. Math. Anal. Appl. 333 (2007) 295-310. | MR | Zbl

[10] M.D. Gunzburger, L. Hou and T.P. Svobodny, Boundary velocity control of incompressible flow with an application to viscous drag reduction. SIAM J. Control Optim. 30 (1992) 167-181. | MR | Zbl

[11] C.O. Horgan, Korn's inequalities and their applications in continuum mechanics. SIAM Rev. 37 (1995) 491-511. | Zbl

[12] O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow. Gordon and Beach, New York (1969). | MR | Zbl

[13] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris (1969). | MR | Zbl

[14] P. Kaplický, J. Málek and J. Stará, C1,α-solutions to a class of nonlinear fluids in two dimensions-stationary Dirichlet problem. Zap. Nauchn. Sem. POMI 259 (1999) 89-121. | Zbl

[15] K. Kunisch and X. Marduel, Optimal control of non-isothermal viscoelastic fluid flow. J. Non-Newton. Fluid Mech. 88 (2000) 261-301. | Zbl

[16] J. Nečas, J. Málek, J. Rokyta and M. Ružička, Weak and measure-valued solutions to evolutionary partial differential equations, Chapmann and Hall, London. Appl. Math. Math. Comput. 13 (1996). | Zbl

[17] T. Roubcíěk and F. Tröltzsch, Lipschitz stability of optimal controls for the steady-state Navier-Stokes equations. Control Cybernet. 32 (2003) 683-705. | Zbl

[18] T. Slawig, Distributed control for a class of non-Newtonian fluids. J. Differ. Equ. 219 (2005) 116-143. | MR | Zbl

[19] D. Wachsmuth and T. Roubcíěk, Optimal control of incompressible non-Newtonian fluids. Z. Anal. Anwend. 29 (2010) 351-376. | MR

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