The velocity tracking problem for Navier-Stokes flows with boundary control. (English) Zbl 0991.49002

Gunzburger, M. D.; Manservisi, S.

SIAM J. Control Optimization 39, No. 2, 594-634 (2000).

The authors consider optimal flow control problems where velocity is controlled by suction-blowing on part of the boundary. There are no state or control constraints and the aim is to minimize a quadratic functional measuring deviation or a target velocity and incorporating a penalization term to limit the size of the control. Results include existence, first order optimality conditions and numerics (by discretization).
Among recent works on the subject, see J. Málek and T. Roubíček [Sequeira, Adélia (ed.) et al., Applied Nonlinear Analysis. In honor of the 70th birthday of Professor Jindřich Necǎs, New York, NY: Kluwer Academic/Plenum Publishers (ISBN 0-306-46303-2/hbk). 355-372 (1999; Zbl 0962.49017)], where, although the control is distributed, the cost functionals are more general and the necessary conditions are shown to be sufficient in certain cases. For the reduction of boundary control to distributed control even for time dependent flows see the reviewer and S. S. Sritharan [Proc. R. Soc. Lond., Ser. A 439, No. 1, 81-102 (1992; Zbl 0786.76063)].

Reviewer: Hector O.Fattorini (Los Angeles)

MSC:

49J20	Existence theories for optimal control problems involving partial differential equations
35Q30	Navier-Stokes equations
76D05	Navier-Stokes equations for incompressible viscous fluids
35B37	PDE in connection with control problems (MSC2000)
65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

optimal control; fluid flow; Navier-Stokes equations

Citations:

Zbl 0962.49017; Zbl 0786.76063