Optimal control of linearized compressible Navier-Stokes equations
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 2, pp. 587-615.

We study an optimal boundary control problem for the two dimensional unsteady linearized compressible Navier-Stokes equations in a rectangle. The control acts through the Dirichlet boundary condition. We first establish the existence and uniqueness of the solution for the two-dimensional unsteady linearized compressible Navier-Stokes equations in a rectangle with inhomogeneous Dirichlet boundary data, not necessarily smooth. Then, we prove the existence and uniqueness of the optimal solution over the control set. Finally we derive an optimality system from which the optimal solution can be determined.

DOI : 10.1051/cocv/2012023
Classification : 49J20, 49K20, 35Q30, 76N25
Mots-clés : optimal control, linearized compressible Navier-Stokes equations, boundary control, optimality system 
@article{COCV_2013__19_2_587_0,
     author = {Chowdhury, Shirshendu and Ramaswamy, Mythily},
     title = {Optimal control of linearized compressible {Navier-Stokes} equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {587--615},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {2},
     year = {2013},
     doi = {10.1051/cocv/2012023},
     mrnumber = {3049725},
     zbl = {1266.49006},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2012023/}
}
TY  - JOUR
AU  - Chowdhury, Shirshendu
AU  - Ramaswamy, Mythily
TI  - Optimal control of linearized compressible Navier-Stokes equations
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2013
SP  - 587
EP  - 615
VL  - 19
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2012023/
DO  - 10.1051/cocv/2012023
LA  - en
ID  - COCV_2013__19_2_587_0
ER  - 
%0 Journal Article
%A Chowdhury, Shirshendu
%A Ramaswamy, Mythily
%T Optimal control of linearized compressible Navier-Stokes equations
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2013
%P 587-615
%V 19
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2012023/
%R 10.1051/cocv/2012023
%G en
%F COCV_2013__19_2_587_0
Chowdhury, Shirshendu; Ramaswamy, Mythily. Optimal control of linearized compressible Navier-Stokes equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 2, pp. 587-615. doi : 10.1051/cocv/2012023. http://archive.numdam.org/articles/10.1051/cocv/2012023/

[1] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd edition. Birkhäuser (2006). | MR | Zbl

[2] R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology, in Evolution Problems. I. With the collaboration of M. Artola, M. Cessenat and H. Lanchon. Translated from the French by A. Craig. Springer-Verlag, Berlin 5 (1992). | MR | Zbl

[3] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). | MR | Zbl

[4] G. Geymonat and P. Leyland, Transport and propagation of a perturbation of a flow of a compressible fluid in a bounded region. Arch. Rational Mech. Anal. 100 (1987) 53-81. | MR | Zbl

[5] V. Girinon, Quelques problémes aux limites pour les équations de Navier-Stokes compressibles. Ph.D. thesis, Université de Toulouse (2008).

[6] 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

[7] V.I. Judovič, A two-dimensional problem of unsteady flow of an ideal incompressible fluid across a given domain. Amer. Math. Soc. Trans. 57 (1966) 277-304 [previously in Mat. Sb. (N.S.) 64 (1964) 562-588 (in Russian)]. | MR

[8] J. Neustupa, A semigroup generated by the linearized Navier-Stokes equations for compressible fluid and its uniform growth bound in Hölder spaces. Navier-Stokes equations: theory and numerical methods (Varenna, 1997), Pitman. Research Notes Math. Ser. 388 (1998) 86-100. | MR | Zbl

[9] J.P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions. Ann. Inst. Henri Poincaré Anal. Non Linéaire 24 (2007) 921-951. | MR | Zbl

[10] J.P. Raymond and A.P. Nguyen, Control localized on thin structures for the linearized Boussinesq system. J. Optim. Theory Appl. 141 (2009) 147-165. | MR | Zbl

[11] A. Valli and W.M. Zajczkowski, Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case. Commun. Math. Phys. 103 (1986) 259-296. | MR | Zbl

Cité par Sources :