Volume 9, March 2003
|Page(s)||275 - 296|
|Published online||15 September 2003|
Motion Planning for a nonlinear Stefan Problem
Control and Dynamical Systems, California Institute of Technology, Mail Code 107-8l, 1200 E California Blvd., Pasadena, CA 91125, USA.
2 Centre Automatique et Systèmes, École Nationale Supérieure des Mines de Paris, 60 boulevard Saint-Michel, 75272 Paris Cedex 06, France; firstname.lastname@example.org.
Revised: 24 January 2003
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.
Mathematics Subject Classification: 93C20 / 80A22 / 80A23
Key words: Inverse Stefan problem / flatness / motion planning.
© EDP Sciences, SMAI, 2003
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