MINES ParisTech CAS - Centre automatique et systèmes

Flatness and null controllability of 1-D parabolic equations

Authors: Philippe Martin , Lionel Rosier , Pierre Rouchon, Proc. Appl. Math. Mech., Vol 16, pp. 47 - 50, 25 October 2016, DOI: 10.1002/pamm.201610013
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.
Download PDF
author = {Lionel Rosier Philippe Martin, Pierre Rouchon},
title = {Flatness and null controllability of 1-D parabolic equations},
journal = {Proc. Appl. Math. Mech.},
volume = {16},
number = {},
pages = {47 - 50},
year = {2016},