FAR EASTERN BRANCH OF THE RUSSIAN ACADEMY OF SCIENCES
INSTITUTE OF APPLIED MATHEMATICS KHABAROVSK DIVISION
Far Eastern Mathematical Journal
Optimal control in non well posed problem for Stokes equations |
V. A. Annenkov |
2003, issue 1, P. 18–26 |
Abstract |
| This paper deals with the optimal control problem for Stokes system in bounded domain $\Omega$. The bound of $\Omega$ is a union of two smooth disjoint parts: $\partial \Omega = \Gamma_0 \cup \Gamma_1$, $\Gamma_0 \cap \Gamma_1 = \emptyset$. Fluid velocity and stress vector on the part $\Gamma_0$ on the boundary simultaneously play a role of control parameters. So, we deal with a system governed by non well posed problem. Extremal condition on the part $\Gamma_0$ leads to the necessary a priory estimation for velocity. We use penalization tecnique and study convergence of penalized problem to original problem when penalization parameter tends to zero. The tecnique we use was developed by J.-L. Lions. One of possible applications of obtained result is obtaining the conditions for optimal state (singular optimality system). |
Keywords: |
Download the article (PDF-file) |
References |
| [1] Zh.-L. Lions, Upravlenie singulyarnymi raspredelennymi sistemami, Nauka, M., 1987. [2] M. M. Lavrent'ev, L. Ya. Savel'ev, Linejnye operatory i nekorrektnye zadachi, Nauka, M., 1991. [3] Zh.-L. Lions, Optimal'noe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi, Mir, M., 1972. [4] F. Abergel, Non-well Posed Problem in Convex Optimal Control, Applied Mathematics and Optimization, 1988. [5] M. D. Gunzburger, L. S. Hou, T. P. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet conditions, Math. Modelling Numer. Anal., 25 (1991), 711–748. [6] M. D. Gunzburger, L. S. Hou, T. P. Svobodny, Boundary velocity control of incompressible flow with application to viscous drag reduction, SIAM J. Control Optim., 30:1 (1992), 167–182. [7] Gilles Fourestey, Marwan Moubachir, Optimal control of Navier-Stokes equations using Lagrange-Galerkin methods, Rapport de recherche, 4609, October (2002), 88 pp., {http://www.iam.dvo.ru:8100/Redirect/www.inria.fr/rrrt/rr-4609.html}. [8] V. A. Annenkov, Obobshhennye resheniya kraevoj zadachi dlya sistemy Stoksa s zadannym vektorom napryazhenij, Dal'nevostochnyj matematicheskij sbornik, 8, 1999, 23–31. [9] G. V. Alekseev, V. V. Malykin, Chislennoe issledovanie stacionarnyx e'kstremal'nyx zadach dlya dvumernyx uravnenij vyazkoj zhidkosti, Vychislitel'nye texnologii, 2:5 (1993), 5–16. [10] R. Temam, Uravneniya Nav'e – Stoksa. Teoriya i chislennyj analiz, Mir, M., 1981. [11] O. A. Ladyzhenskaya, Matematicheskie voprosy dinamiki vyazkoj neszhimaemoj zhidkosti, Nauka, M., 1970. [12] A. Yu. Chebotarev, Zadachi granichnogo optimal'nogo upravleniya stacionarnymi techeniyami vyazkoj zhidkosti, Preprint IPM DVO RAN, Vladivostok, 1992, 31 s. |