Boundary value problems and optimal boundary control for the Navier-Stokes system: The two-dimensional case

SIAM Journal on Control and Optimization

Volumn 36, Issue 3, 1998, Pages 852-894

  Fursikov, A V ^a   Gunzburger, M D ^b,c   Hou, L S ^d  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

^b VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

^c IOWA STATE UNIVERSITY   (United States)

^d YORK UNIVERSITY   (Canada)

Author keywords

Boundary value problem; Drag minimization; Navier Stokes equations; Optimal control

Indexed keywords

BOUNDARY CONDITIONS; BOUNDARY VALUE PROBLEMS; DRAG; LAGRANGE MULTIPLIERS; NAVIER STOKES EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS;

DRAG MINIMIZATION;

OPTIMAL CONTROL SYSTEMS;

EID: 0032074249     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012994273374     Document Type: Article

References (29)

1. R. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
2. P. ABERGEL AND R. TEMAM, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynamics, 1 (1990) pp. 303-325.
3. V. ALEKSEEV, V. TIKHOMIROV, AND S. FOMIN, Optimal Control, Consultants Bureau, New York, 1987.
4. O. BESOV, V. IL'IN, AND S. NIKOL'SKII, Integral Representations of Functions and Imbedding Theorems, Winston, Washington, D.C., 1979.
5. P. CONSTANTIN AND C. FOIAS, Navier-Stokes Equations, Univ. of Chicago Press, Chicago, IL, 1988.
6. C. CUVERLIER, Optimal control of a system governed by the Navier-Stokes equations coupled with the heat equation, in New Developments in Differential Equations, W. Eckhaus, ed., North-Holland, Amsterdam, 1976, pp. 81-98.
7. H. FATTORINI AND S. SRITHARAN, Existence of optimal controls for viscous flow problems, Proc. Roy. Soc. London Ser. A, 439 (1992), pp. 81-102.
8. H. FATTORINI AND S. SRITHARAN, Necessary and sufficient conditions for optimal controls in viscous flow problems, Proc. Roy. Soc. London Ser. A, 124 (1994), pp. 211-251.
9. H. FATTORINI AND S. SRITHARAN, Optimal chattering control for viscous flows, Nonlinear Anal., 25 (1995), pp. 763-797.
10. H. FATTORINI AND S. SRITHARAN, Optimal control theory for viscous flow problems, to appear.
11. A. FURSIKOV, On some control problems and results concerning the unique solvability of a mixed boundary value problem for the three-dimensional Navier-Stokes and Euler systems, Soviet Math. Dokl., 21 (1980), pp. 889-893.
12. A. FURSIKOV, Control problems and theorems concerning the unique solvability of a mixed boundary value problem for the three-dimensional Navier-Stokes and Euler equations, Math USSR Sb., 43 (1982), pp. 281-307.
13. A. FURSIKOV, Properties of solutions of some extremal problems connected with the Navier-Stokes system. Math USSR Sb., 46 (1983), pp. 323-351.
14. V. GIRAULT AND P.-A. RAVIART, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986.
15. M. GUNZBURGER, L. HOU, AND T. SVOBODNY, Numerical approximation of an optimal control problem associated with the Navier-Stokes equations, Appl. Math. Lett., 2 (1989), pp. 29-31.
16. M. GUNZBURGER, L. HOU, AND T. SVOBODNY, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls, Model. Math. Anal. Numér., 25 (1991), pp. 711-748.
17. M. GUNZBURGER, L. HOU, AND T. SVOBODNY, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls, Math. Comp., 57 (1991), pp. 123-151.
18. M. GUNZBURGER, L. HOU, AND T. SVOBODNY, Boundary velocity control of incompressible flow with an application to viscous drag reduction, SIAM J. Control Optim., 30 (1992), pp. 167-181.
19. O. LADYZHENSKAYA, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1963.
20. J.-L. LIONS, Control of Distributed Singular systems, Bordas, Paris, 1985.
21. J.-L. LIONS AND E. MAGENES, Non-Homogeneous Boundary Value Problems and Applications I, Springer-Verlag, Berlin, 1972.
22. P. PANAGIOTOPOULOS, Inequality Problems in Mechanics and Applications, Birkhäuser, Boston, 1985.
23. J. SERRIN, Mathematical principles of classical fluid mechanics, in Handbuch der Physik, VIII/1, S. Flügge and C. Truesdell, eds., 1959, pp. 1-125.
24. S. SRITHARAN, An optimal control problem in exterior hydrodynamics, in Distributed Parameter Control Systems, New Trends and Applications, G. Chen, B. Lee, W. Littman, and L. Markus, eds., Marcel Dekker, New York, 1991, pp. 385-417.
25. S. SRITHARAN, Dynamic programming of the Navier-Stokes equations, System Control Lett., 16 (1991), pp. 299-307.
26. S. SRITHARAN, An optimal control problem in exterior hydrodynamics, Roy. Soc. Edinburgh Proc. A, 121 (1992), pp. 5-32.
27. S. SRITHARAN, On Hamilton-Jacobi equation in infinite dimensions, in Nonlinear Problems in Aviation and Aerospace, S. Sivasundaram, ed., Embry-Riddle University Press, Daytona Beach, FL, 1994, pp. 631-638.
28. R. TEMAM, Navier-Stokes Equations, North-Holland, Amsterdam, 1979.
29. M. TRIEBEL, Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher Verlag der Wissenschaften, Berlin, 1978; also, North-Holland, Amsterdam, New York, 1978.