BOUNDARY-VALUE-PROBLEMS AND OPTIMAL BOUNDARY CONTROL FOR THE NAVIER-STOKES SYSTEM - THE 2-DIMENSIONAL CASE

We study optimal boundary control problems for the two-dimensional Nav ier-Stokes equations in an unbounded domain. Control is effected throu gh the Dirichlet boundary condition and is sought in a subset of the t race space of velocity fields with minimal regularity satisfying the e nergy estimates. An objective of interest is the drag functional. We f irst establish three important results for inhomogeneous boundary valu e problems for the Navier-Stokes equations; namely, we identify the tr ace space for the velocity fields possessing finite energy, we prove t he existence of a solution for the Navier-Stokes equations with bounda ry data belonging to the trace space, and we identify the space in whi ch the stress vector (along the boundary) of admissible solutions is w ell defined. Then, we prove the existence of an optimal solution over the control set. Finally, we justify the use of Lagrange multiplier pr inciples, derive an optimality system of equations in the weak sense f rom which optimal states and controls may be determined, and prove tha t the optimality system of equations satisfies in appropriate senses a system of partial differential equations with boundary values.