Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T22:10:59.283Z Has data issue: false hasContentIssue false
  • English
  • Français

Distributed control for multistate modified Navier-Stokesequations

Published online by Cambridge University Press:  02 May 2012

Nadir Arada
Nadir Arada*
Affiliation:
Departamento de Matemática, Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal. [email protected]
Get access

Abstract

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

Keywords

Optimal controlmultistate Navier-Stokes equationsshear-dependent viscositynecessary optimality conditions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 1 , January 2013 , pp. 219 - 238
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abergel, F. and Casas, E., Some optimal control problems of multistate equations appearing in fluid mechanics. RAIRO Modél. Math. Anal. Numér. 27 (1993) 223–247. Google Scholar
N. Arada, Optimal Control of shear-thickening flows. Departamento de Matemática, FCT-UNL, Portugal, Technical Report 3 (2012).
Casas, E., Boundary control problems for quasi-linear elliptic equations : a Pontryagin’s principle. Appl. Math. Optim. 33 (1996) 265–291. Google Scholar
E. Casas and L.A. Fernández, Boundary control of quasilinear elliptic equations. INRIA, Rapport de Recherche 782 (1988).
Casas, E. and Fernández, L.A., Distributed control of systems governed by a general class of quasilinear elliptic equations. J. Differ. Equ. 35 (1993) 20–47. Google Scholar
De Los Reyes, J.C. and Griesse, R., State-constrained optimal control of the three-dimensional stationary Navier-Stokes equations. J. Math. Anal. Appl. 343 (2008) 257–272. Google Scholar
Frehse, J., Málek, J. and Steinhauer, M., An existence result for fluids with shear dependent viscosity-steady flows. Nonlinear. Anal. 30 (1997) 3041–3049. Google Scholar
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).
Gunzburger, M.D. and Trenchea, C., 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. Google Scholar
Gunzburger, M.D., Hou, L. and Svobodny, T.P., Boundary velocity control of incompressible flow with an application to viscous drag reduction. SIAM J. Control Optim. 30 (1992) 167–181. Google Scholar
Horgan, C.O., Korn’s inequalities and their applications in continuum mechanics. SIAM Rev. 37 (1995) 491–511. Google Scholar
O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow. Gordon and Beach, New York (1969).
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris (1969).
Kaplický, P., Málek, J. and Stará, J., C 1,α-solutions to a class of nonlinear fluids in two dimensions-stationary Dirichlet problem. Zap. Nauchn. Sem. POMI 259 (1999) 89–121. Google Scholar
Kunisch, K. and Marduel, X., Optimal control of non-isothermal viscoelastic fluid flow. J. Non-Newton. Fluid Mech. 88 (2000) 261–301. Google Scholar
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).
Roubcíěk, T. and Tröltzsch, F., Lipschitz stability of optimal controls for the steady-state Navier-Stokes equations. Control Cybernet. 32 (2003) 683–705. Google Scholar
Slawig, T., Distributed control for a class of non-Newtonian fluids. J. Differ. Equ. 219 (2005) 116–143. Google Scholar
Wachsmuth, D. and Roubcíěk, T., Optimal control of incompressible non-Newtonian fluids. Z. Anal. Anwend. 29 (2010) 351–376. Google Scholar