# Motion planning for a nonlinear Stefan problem

William B. Dunbar; Nicolas Petit; Pierre Rouchon; Philippe Martin

ESAIM: Control, Optimisation and Calculus of Variations (2003)

- Volume: 9, page 275-296
- ISSN: 1292-8119

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topDunbar, William B., et al. "Motion planning for a nonlinear Stefan problem." ESAIM: Control, Optimisation and Calculus of Variations 9 (2003): 275-296. <http://eudml.org/doc/245697>.

@article{Dunbar2003,

abstract = {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.},

author = {Dunbar, William B., Petit, Nicolas, Rouchon, Pierre, Martin, Philippe},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {inverse Stefan problem; flatness; motion planning; motion planning.},

language = {eng},

pages = {275-296},

publisher = {EDP-Sciences},

title = {Motion planning for a nonlinear Stefan problem},

url = {http://eudml.org/doc/245697},

volume = {9},

year = {2003},

}

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

PB - EDP-Sciences

VL - 9

SP - 275

EP - 296

AB - 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.

LA - eng

KW - inverse Stefan problem; flatness; motion planning; motion planning.

UR - http://eudml.org/doc/245697

ER -

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