Abstract
We study the problem of optimal boundary control of two-dimensional steady-state flow of a viscous inhomogeneous incompressible fluid. The role of control is played by the values of the velocity on a part of the boundary of the domain considered. On the remaining part of the boundary, the vector of flow velocity and the fluid density are given. We seek the fluid density as a scalar function (determined by the initial data) of the stream function, study the solvability of the problem, and obtain necessary optimality conditions.
Similar content being viewed by others
REFERENCES
N. N. Frolov, “Solvability of the boundary-value problem for motion of an inhomogeneous fluid,” Mat. Zametki [Math. Notes], 53 (1993), no. 6, 130-140.
N. N. Frolov, “The boundary-value problem describing the motion of an inhomogeneous fluid,” Sibirsk. Mat. Zh. [Siberian Math. J.], 37 (1996), no. 2, 433-451.
A. Yu. Chebotarev, “Stationary variational inequalities in the inhomogeneous fluid model,” Sibirsk. Mat. Zh. [Siberian Math. J.], 38 (1997), no. 5, 1185-1193.
A. V. Fursikov, “Properties of solutions of several extremum problems related to the Navier-Stokes system,” Mat. Sb. [Math. USSR-Sb.], 118 (180) (1982), no. 3 (7), 323-349.
A. V. Fursikov, “Control problems and theorems related to unique solvability of the mixed boundaryvalue problem for three-dimensional Navier-Stokes equations,” Mat. Sb. [Math. USSR-Sb.], 115 (1981), no. 2, 281-306.
J.-L. Lions, Contrôle des systèmes distribués singuliers, Méthodes mathématiques de l'Informatique, no. 13, Gauthier-Villars, Paris, 1983.
M. D. Gunzburger, L. Hou, and T. P. Svobodny, “Boundary velocity control of incompressible flow with application to viscous drag reduction,” SIAM J. Contr. Optim., 30 (1992), no. 1, 167-182.
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, 1977.
V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow, 1980.
C. Conca, F. Murat, and O. Pironneau, “The Stokes and Navier-Stokes equation with boundary conditions involving the pressure,” Japan J. Math., 20 (1994), no. 2, 279-318.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Illarionov, A.I. Optimal Boundary Control of Steady-State Flow of a Viscous Inhomogeneous Incompressible Fluid. Mathematical Notes 69, 614–624 (2001). https://doi.org/10.1023/A:1010297424324
Issue Date:
DOI: https://doi.org/10.1023/A:1010297424324