Feedforward Boundary Control for the Regulation of a Passive and Diffusive Scalar in Two-Dimensional Unsteady Flows

We study the problem of regulation of a passive and diffusive scalar in two-dimensional unsteady flows. We consider two kinds of outputs: a weighted average of the scalar concentration and the concentration on the boundary of a domain. Flux feedback and feedforward controllers are applied on the boundary. We introduce a method of signal-dependent equilibrium translation to convert the regulation problem into a stabilization problem of the convection diffusion equation. This results in a time-dependent regulator system due to the unsteady flows. We show that the feedforward controllers synthesized from the solution of the regulator system regulate the above defined output to its constant reference and any feedforward controllers give the same convergence rate of the regulation error. Numerical simulations are conducted to further confirm these theoretical results.