Motion planning for a nonlinear Stefan problem
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 275-296.

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.

DOI : https://doi.org/10.1051/cocv:2003013
Classification : 93C20,  80A22,  80A23
Mots clés : inverse Stefan problem, flatness, motion planning
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author = {Dunbar, William B. and Petit, Nicolas and Rouchon, Pierre and Martin, Philippe A.},
title = {Motion planning for a nonlinear {Stefan} problem},
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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. http://www.numdam.org/articles/10.1051/cocv:2003013/

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