Motion planning for a nonlinear Stefan problem
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 275-296.

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.

DOI : 10.1051/cocv:2003013
Classification : 93C20, 80A22, 80A23
Mots clés : inverse Stefan problem, flatness, motion planning
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     author = {Dunbar, William B. and Petit, Nicolas and Rouchon, Pierre and Martin, Philippe},
     title = {Motion planning for a nonlinear {Stefan} problem},
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     pages = {275--296},
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Dunbar, William B.; Petit, Nicolas; Rouchon, Pierre; Martin, Philippe. Motion planning for a nonlinear Stefan problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 275-296. doi : 10.1051/cocv:2003013. http://www.numdam.org/articles/10.1051/cocv:2003013/

[1] J.R. Cannon, The one-dimensional heat equation. Addison-Wesley Publishing Company, Encyclopedia Math. Appl. 23 (1984). | MR | Zbl

[2] M. Fila and P. Souplet, Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem. Interfaces Free Boundaries 3 (2001) 337-344. | MR | Zbl

[3] M. Fliess, J. Lévine, Ph. Martin and P. Rouchon, Flatness and defect of nonlinear systems: Introductory theory and examples. Int. J. Control 61 (1995) 1327-1361. | MR | Zbl

[4] M. Fliess, J. Lévine, Ph. Martin and P. Rouchon, A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 44 (1999) 922-937. | Zbl

[5] A. Friedman and B. Hu, A Stefan problem for multidimensional reaction-diffusion systems. SIAM J. Math. Anal. 27 (1996) 1212-1234. | MR | Zbl

[6] M. Gevrey, La nature analytique des solutions des équations aux dérivées partielles. Ann. Sci. École Norm. Sup. 25 (1918) 129-190. | EuDML | JFM | Numdam

[7] C.D. Hill, Parabolic equations in one space variable and the non-characteristic Cauchy problem. Comm. Pure Appl. Math. 20 (1967) 619-633. | MR | Zbl

[8] 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. | MR | Zbl

[9] B. Laroche, Ph. Martin and P. Rouchon, Motion planing for the heat equation. Int. J. Robust Nonlinear Control 10 (2000) 629-643. | MR | Zbl

[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. | Zbl

[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 equations2002).

[13] M. Petkovsek, H.S. Wilf and D. Zeilberger, A = B. Wellesley (1996). | MR

[14] L.I. Rubinstein, The Stefan problem. AMS, Providence, Rhode Island, Transl. Math. Monogr. 27 (1971). | Zbl

[15] W. Rudin, Real and Complex Analysis. McGraw-Hill International Editions, Third Edition (1987). | MR | Zbl

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