Boundary control of a nonlinear Stefan problem

The classical Stefan problem is a linear one-dimensional heat equation with a free boundary at one end, modelling a column of liquid (e.g. water) in contact with an infinite strip of solid (ice). Given the fixed boundary conditions, the column temperature and free boundary motion can be uniquely determined. In the inverse problem, one specifies the free boundary motion, say from one steady-state length to another, and seeks to determine the column temperature and fixed boundary conditions, or boundary control. This motion planning problem is a simplified version of a crystal growth problem. In this paper, we consider motion planning of the free boundary (Stefan) problem with a quadratic nonlinear reaction term. The treatment here is a first step towards treating higher order nonlinearities as observed in crystal growth furnaces. Convergence of a series solution is proven and a detailed parametric study on the series radius of convergence given. Moreover, we prove that the parametrization can indeed be used for motion planning purposes; computation of the open loop motion planning is straightforward and we give simulation results.

[1]  W. Rudin.  Real and complex analysis , 1968 .

[2]  C. Denson Hill,et al.  Parabolic equations in one space variable and the non-characteristic cauchy problem , 1967 .

[3]  Felix E. Browder,et al.  The One-Dimensional Heat Equation: Preface , 1984 .

[4]  M. Fliess,et al.  Flatness and defect of non-linear systems: introductory theory and examples , 1995 .

[5]  A Stefan problem for multidimensional reaction-diffusion systems , 1996 .

[6]  General theory of partial differential equations and microlocal analysis : proceedings of the Workshop on General Theory of PDEs and Microlocal Analysis, International Centre for Theoretical Physics, Trieste 1995 , 1996 .

[7]  Philippe Martin,et al.  A Lie-Backlund approach to equivalence and flatness of nonlinear systems , 1999, IEEE Trans. Autom. Control..

[8]  Philippe Martin,et al.  Motion planning for the heat equation , 2000 .

[9]  Mark B. Milam,et al.  A new computational approach to real-time trajectory generation for constrained mechanical systems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[10]  Philippe Souplet,et al.  Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem , 2001 .

[11]  J. Rudolph,et al.  Flatness-based boundary control of a nonlinear parabolic equation modelling a tubular reactor , 2001 .

[12]  Mark B. Milam,et al.  A NEW COMPUTATIONAL METHOD FOR OPTIMAL CONTROL OF A CLASS OF CONSTRAINED SYSTEMS GOVERNED BY PARTIAL DIFFERENTIAL EQUATIONS , 2002 .

[13]  Philippe Martin,et al.  MOTION PLANNING FOR A NONLINEAR STEFAN PROBLEM , 2003 .