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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; [email protected].
Pierre Rouchon
Affiliation:
Centre Automatique et Systèmes, École Nationale Supérieure des Mines de Paris, 60 boulevard Saint-Michel, 75272 Paris Cedex 06, France; [email protected].
Philippe Martin
Affiliation:
Centre Automatique et Systèmes, École Nationale Supérieure des Mines de Paris, 60 boulevard Saint-Michel, 75272 Paris Cedex 06, France; [email protected].
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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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References

J.R. Cannon, The one-dimensional heat equation. Addison-Wesley Publishing Company, Encyclopedia Math. Appl. 23 (1984).
Fila, M. and Souplet, P., Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem. Interfaces Free Boundaries 3 (2001) 337-344.
Fliess, M., Lévine, J., Martin, Ph. and Rouchon, P., Flatness and defect of nonlinear systems: Introductory theory and examples. Int. J. Control 61 (1995) 1327-1361. CrossRef
Fliess, M., Lévine, J., Martin, Ph. and Rouchon, P., Lie-Bäcklund, A approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 44 (1999) 922-937. CrossRef
Friedman, A. and Hu, B., Stefan, A problem for multidimensional reaction-diffusion systems. SIAM J. Math. Anal. 27 (1996) 1212-1234. CrossRef
Gevrey, M., La nature analytique des solutions des équations aux dérivées partielles. Ann. Sci. École Norm. Sup. 25 (1918) 129-190. CrossRef
Hill, C.D., Parabolic equations in one space variable and the non-characteristic Cauchy problem. Comm. Pure Appl. Math. 20 (1967) 619-633. CrossRef
Chen Hua and L. Rodino, General theory of partial differential equations and microlocal analysis, in Proc. of the workshop on General theory of PDEs and Microlocal Analysis, International Centre for Theoretical Physics, Trieste, edited by Qi Min-You and L. Rodino. Longman (1995) 6-81.
Laroche, B., Martin, Ph. and Rouchon, P., Motion planing for the heat equation. Int. J. Robust Nonlinear Control 10 (2000) 629-643. 3.0.CO;2-N>CrossRef
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.
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).
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).
M. Petkovsek, H.S. Wilf and D. Zeilberger, A = B. Wellesley (1996).
L.I. Rubinstein, The Stefan problem. AMS, Providence, Rhode Island, Transl. Math. Monogr. 27 (1971).
W. Rudin, Real and Complex Analysis. McGraw-Hill International Editions, Third Edition (1987).