NUMERICAL-SOLUTION OF OPTIMAL DISTRIBUTED CONTROL-PROBLEMS FOR INCOMPRESSIBLE FLOWS

We study the numerical solution of optimal control problems associated with the two-dimensional Viscous incompressible flows which are gover ned by the Navier-Stokes equations. Although the techniques apply to m ore general settings, the presentation is confined to the objectives o f minimizing the vorticity in the steady-state case with distributed c ontrols and tracking the velocity field in the nonstationary case with piecewise distributed controls. In the steady-state case, we develop a systematic way to use the Lagrange multiplier rules to derive an opt imality system of equations from which an optimal solution can be comp uted; finite element methods are used to find approximate solutions fo r the optimality system of equations. In the time-dependent case, a pi ecewise-in-time optimal control approach is proposed and the fully dis crete approximation algorithm for solving the piecewise optimal contro l problem is defined. Numrical results are presented for both the stea dy-state and time-dependent optimal control problems.