We consider the problem of optimal mixing control. Our objective is to best enhance mixing by flow advection while the flow is optimized in the sense that it is almost steady and is of the least magnitude and the least rotation. For this we de. ne a mixing efficiency functional by penalizing the average of variance of a diffusive scalar, the average of the flow, and the average of its acceleration and strain tensor. By variational principles, we prove the existence of an optimal flow and derive optimality conditions that consist of a system of nonlinear advection-diffusion equations, wave equations, and Laplace's equation.