Optimal Mixing by Feedback in Pipe Flow

We design a Lyapunov based boundary feedback controller for achieving mixing in a D pipe ow governed by Navier Stokes equations We show that the control law maximizes a measure of mixing that incorporates stretching and folding of material elements while at the same time minimi zing the control e ort and the sensing e ort The penalty on sensing results in a static output feedback control law rather than full state feedback W e also derive a lower bound on the gain from the control e ort to the mixing measure Furthermore we establish input output to state stability properties for the open loop system These results show a form of detectability of mixing in the interior of the pipe from the chosen outputs on the wall The e ectiveness of the optimal control in achieving mixing enhancement is demonstrated in numerical simulations