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Motion Planning for a nonlinear Stefan Problem

Published online by Cambridge University Press:  15 September 2003

William B. Dunbar ,
Nicolas Petit ,
Pierre Rouchon  and
Philippe Martin
William B. Dunbar
Affiliation:
Control and Dynamical Systems, California Institute of Technology, Mail Code 107-8l, 1200 E California Blvd., Pasadena, CA 91125, USA.
Nicolas Petit
Affiliation:
Centre Automatique et Systèmes, École Nationale Supérieure des Mines de Paris, 60 boulevard Saint-Michel, 75272 Paris Cedex 06, France; petit@cas.ensmp.fr.
Pierre Rouchon
Affiliation:
Centre Automatique et Systèmes, École Nationale Supérieure des Mines de Paris, 60 boulevard Saint-Michel, 75272 Paris Cedex 06, France; petit@cas.ensmp.fr.
Philippe Martin
Affiliation:
Centre Automatique et Systèmes, École Nationale Supérieure des Mines de Paris, 60 boulevard Saint-Michel, 75272 Paris Cedex 06, France; petit@cas.ensmp.fr.
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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.

Keywords

Inverse Stefan problemflatnessmotion planning.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 275 - 296
Copyright
© EDP Sciences, SMAI, 2003

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