Abstract. We study optimal boundary control problems for the two-dimensional Navier-Stokes equations in an unbounded domain. Control is effected through the Dirichlet boundary condition and is sought in a subset of the trace space of velocity fields with minimal regularity satisfying the energy estimates. An objective of interest is the drag functional. We first establish three important results for inhomogeneous boundary value problems for the Navier-Stokes equations; namely, we identify the trace space for the velocity fields possessing finite energy, we prove the existence of a solution for the Navier-Stokes equations with boundary data belonging to the trace space, and we identify the space in which the stress vector (along the boundary) of admissible solutions is well defined. Then, we prove the existence of an optimal solution over the control set. Finally, we justify the use of Lagrange multiplier principles, derive an optimality system of equations in the weak sense from which optimal states and controls may be determined, and prove that the optimality system of equations satisfies in appropriate senses a system of partial differential equations with boundary values.Key words. optimal control, Navier-Stokes equations, boundary value problem, drag minimization
AMS subject classifications. 76D05, 49J20, 49K20, 35K50PII. S0363012994273374 1. Introduction. Optimal control problems for fluid flows have been a subject of interest to experimenters and designers since at least the time of Prandtl. In more recent times, they have also become of substantial interest to mathematicians and computational scientists. For the steady state Navier-Stokes system, complete and systematic mathematical and numerical analyses of optimal control problems of different types (e.g., having Dirichlet, Neumann, and distributed controls and also finite-dimensional controls) were given in [15,16,17,18]. Mathematical treatments of optimal control problems for the time-dependent Navier-Stokes system were given in [2], [6,7,8,9,10,11,12,13], [20], and [24, 25, 26, 27]. In [6], free convection problems with boundary heat flux controls were considered; the existence of optimal solutions was proved and necessary conditions that characterize optimal controls and states were derived. In [11,12,13], the existence of optimal distributed controls was shown, an optimality system of equations was derived, and the question of the uniqueness