Boundary mediated position control of traveling waves

Reaction control is an essential task in biological systems and chemical process industry. Often, the excitable medium supporting wave propagation exhibits an irregular shape and/or is limited in size. In particular, the analytic treatment of wave phenomena is notoriously difficult due to the spatial modulation of the domain's. Recently [S. Martens et al., PRE \textbf{91}, 022902; JCP \textbf{145}, 094108], we have provided a first systematic treatment by applying asymptotic perturbation analysis leading to an approximate description that involves a reduction of dimensionality; the $3$D RD equation with spatially dependent NFBCs on the reactants reduces to a $1$D reaction-diffusion-advection equation. Here, we present a novel method to control the position $\phi(t)$ of traveling waves in modulated domains according to a prespecified protocol of motion. Given this protocol, the ``optimal'' geometry of reactive domains $Q(x)$ is found as the solution of the perturbatively derived equation of motion. Noteworthy, such a boundary control can be expressed in terms of the uncontrolled wave profile and its propagation velocity, rendering detailed knowledge of the reaction kinetics unnecessary.