Flatness and null controllability of 1‐D parabolic equations. Issue 1 (October 2016) - Martin, Philippe; Rosier, Lionel; Rouchon, Pierre
Abstract: 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)