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
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.