Feedback solution to a bilinear fluid mixing control problem

Mixing problems are common in science and engineering, the aim being to combine fluids as quickly and efficiently as possible. We consider the design of a simple controller to promote mixing in a Stokes´ fluid flow. In this paper, a controlled stirring motion is represented by a velocity field consisting of the superposition of a steady base flow and a second field modulated by a saturating, time-dependent control variable. The problem can be formulated as an optimal control one, but the presence of a nonlinearity in the state dynamics and an input constraint make the construction of a feedback law difficult. The size of the problem currently renders receding horizon schemes revolving around real-time optimization of even a simplified model not feasible for fast applications. To address these problems, we exploit theory for the control of bilinear systems to propose a novel, well-performing, explicit feedback law.