# 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

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topDunbar, 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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- 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).
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- 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).

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