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In this paper we consider a free boundary problem for a nonlinear parabolic partial differential equation. In particular, we are concerned with the inverse problem, which means we know the behavior of the free boundary a priori and would like a solution, e.g. a convergent series, in order to determine what the trajectories of the system should be for steady-state to steady-state boundary control. In this paper we combine two issues: the free boundary (Stefan) problem with a quadratic nonlinearity. We prove convergence of a series solution and give a detailed parametric study on the series radius of convergence. Moreover, we prove that the parametrization can indeed can be used for motion planning purposes; computation of the open loop motion planning is straightforward. Simulation results are given and we prove some important properties about the solution. Namely, a weak maximum principle is derived for the dynamics, stating that the maximum is on the boundary. Also, we prove asymptotic positiveness of the solution, a physical requirement over the entire domain, as the transient time from one steady-state to another gets large.
@article{COCV_2003__9__275_0,
author = {Dunbar, William B. and Petit, Nicolas and Rouchon, Pierre and Martin, Philippe},
title = {Motion planning for a nonlinear {Stefan} problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {275--296},
publisher = {EDP-Sciences},
volume = {9},
year = {2003},
doi = {10.1051/cocv:2003013},
mrnumber = {1966534},
zbl = {1063.93021},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2003013/}
}
TY - JOUR AU - Dunbar, William B. AU - Petit, Nicolas AU - Rouchon, Pierre AU - Martin, Philippe TI - Motion planning for a nonlinear Stefan problem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2003 SP - 275 EP - 296 VL - 9 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv:2003013/ DO - 10.1051/cocv:2003013 LA - en ID - COCV_2003__9__275_0 ER -
%0 Journal Article %A Dunbar, William B. %A Petit, Nicolas %A Rouchon, Pierre %A Martin, Philippe %T Motion planning for a nonlinear Stefan problem %J ESAIM: Control, Optimisation and Calculus of Variations %D 2003 %P 275-296 %V 9 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv:2003013/ %R 10.1051/cocv:2003013 %G en %F COCV_2003__9__275_0
Dunbar, William B.; Petit, Nicolas; Rouchon, Pierre; Martin, Philippe. Motion planning for a nonlinear Stefan problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 275-296. doi: 10.1051/cocv:2003013
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