Motion planning for a class of partial differential equations with boundary control

We study the heat equation with one space dimension and control on the boundary. We give an explicit parametrization of the trajectories as a power series in the space variable with coefficients involving time derivatives of the "flat" output. This series is convergent when the flat output is restricted to be a Gevrey function (i.e., a smooth function with a "not too divergent" Taylor expansion). This parametrization provides a new proof of approximate controllability, and above all an explicit open-loop control achieving the desired motion. We then extend some of these results to the general linear diffusion equation.