Optimal mixing enhancement in 3-D pipe flow

We design a Lyapunov based-boundary feedback controller for achieving mixing in a three-dimensional (3-D) pipe flow governed by Navier-Stokes equations. We show that the control law maximizes a measure related to mixing (that incorporates stretching and folding of material elements), while at the same time minimizing the control effort and the sensing effort. The penalty on sensing results in a static output-feedback control law (rather than full-state feedback). We also derive a lower bound on the gain from the control effort 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 effectiveness of the optimal control in achieving mixing enhancement is demonstrated in numerical simulations.