Flatness and null controllability of 1-D parabolic equations

Philippe Martin (1) , Lionel Rosier (1) , Pierre Rouchon (1, 2)

1 CAS - Centre Automatique et Systèmes
2 QUANTIC - QUANTum Information Circuits

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.

Domaines: Automatique / Robotique

Philippe Martin, Lionel Rosier, Pierre Rouchon. Flatness and null controllability of 1-D parabolic equations. PAMM, 2016, 16 (1), pp.47-50. ⟨10.1002/pamm.201610013⟩. ⟨hal-01486915⟩