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Optimal control of linearized compressible Navier–Stokes equations

Published online by Cambridge University Press:  21 February 2013

Shirshendu Chowdhury  and
Mythily Ramaswamy
Shirshendu Chowdhury
Affiliation:
T.I.F.R Centre for Applicable Mathematics, Post Bag No. 6503, GKVK Post Office, 560065 Bangalore, India. shirshendu@math.tifrbng.res.in; mythily@math.tifrbng.res.in
Mythily Ramaswamy
Affiliation:
T.I.F.R Centre for Applicable Mathematics, Post Bag No. 6503, GKVK Post Office, 560065 Bangalore, India. shirshendu@math.tifrbng.res.in; mythily@math.tifrbng.res.in
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Abstract

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.

Keywords

Optimal controllinearized compressible Navier–Stokes equationsboundary controloptimality system
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 2 , April 2013 , pp. 587 - 615
Copyright
© EDP Sciences, SMAI, 2013

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References

A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd edition. Birkhäuser (2006).
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).
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992).
Geymonat, G. and Leyland, P., 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. Google Scholar
V. Girinon, Quelques problémes aux limites pour les équations de Navier–Stokes compressibles. Ph.D. thesis, Université de Toulouse (2008).
Gunzburger, M.D. and Manservisi, S., The velocity tracking problem for Navier–Stokes flows with boundary control. SIAM J. Control Optim. 39 (2000) 594–634. Google Scholar
Judovič, V.I., 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)]. Google Scholar
Neustupa, J., 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. Google Scholar
Raymond, J.P., Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions. Ann. Inst. Henri Poincaré Anal. Non Linéaire 24 (2007) 921–951. Google Scholar
Raymond, J.P. and Nguyen, A.P., Control localized on thin structures for the linearized Boussinesq system. J. Optim. Theory Appl. 141 (2009) 147–165. Google Scholar
Valli, A. and Zajczkowski, W.M., 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. Google Scholar