Motion Planning for a nonlinear Stefan Problem

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

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

Abstract

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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.

How to cite

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Dunbar, William B., et al. "Motion Planning for a nonlinear Stefan Problem." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 275-296. <http://eudml.org/doc/90696>.

@article{Dunbar2010,
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.; inverse Stefan problem},
language = {eng},
month = {3},
pages = {275-296},
publisher = {EDP Sciences},
title = {Motion Planning for a nonlinear Stefan Problem},
url = {http://eudml.org/doc/90696},
volume = {9},
year = {2010},
}

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
DA - 2010/3//
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.; inverse Stefan problem
UR - http://eudml.org/doc/90696
ER -

References

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  10. A.F. Lynch and J. Rudolph, Flatness-based boundary control of a nonlinear parabolic equation modelling a tubular reactor, edited by A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek. Springer, Lecture Notes in Control Inform. Sci. 259: Nonlinear Control in the Year 2000, Vol. 2. Springer (2000) 45-54.  
  11. M.B. Milam, K. Mushambi and R.M. Murray, A new computational approach to real-time trajectory generation for constrained mechanical systems, in IEEE Conference on Decision and Control (2000).  
  12. N. Petit, M.B. Milam and R.M. Murray, A new computational method for optimal control of a class of constrained systems governed by partial differential equations, in Proc. of the 15th IFAC World Congress (2002).  
  13. M. Petkovsek, H.S. Wilf and D. Zeilberger, A = B. Wellesley (1996).  
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  15. W. Rudin, Real and Complex Analysis. McGraw-Hill International Editions, Third Edition (1987).  

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