Flatness and null controllability of 1-D parabolic equations

We present a recent result on null controllability of one-dimensional linear parabolic equations with boundary control. The space-varying coefficients in the equation can be fairly irregular, in particular they can present discontinuities, degeneracies or singularities at some isolated points; the boundary conditions at both ends are of generalized Robin-Neumann type.

Given any (fairly irregular) initial condition θ₀ and any final time T, we explicitly construct an open-loop control which steers the system from θ₀ at time 0 to the final state 0 at time T. This control is very regular (namely Gevrey of order s with 1 < s < 2); it is simply zero till some (arbitrary) intermediate time τ, so as to take advantage of the smoothing effect due to diffusion, and then given by a series from τ to the final time T.

We illustrate the effectiveness of the approach on a nontrivial numerical example, namely a degenerate heat equation with control at the degenerate side. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)