Frechet differentiability in Besov spaces in the optimal control of parabolic free boundary problems

@article{Abdulla2016FrechetDI,
  title={Frechet differentiability in Besov spaces in the optimal control of parabolic free boundary problems},
  author={Ugur G. Abdulla and Jonathan Goldfarb},
  journal={Journal of Inverse and Ill-posed Problems},
  year={2016},
  volume={26},
  pages={211 - 227}
}
Abstract We consider the inverse Stefan type free boundary problem, where information on the boundary heat flux and the density of the sources are missing and must be found along with the temperature and the free boundary. We pursue the optimal control framework analyzed in [1, 2], where the boundary heat flux, the density of the sources, and the free boundary are components of the control vector. We prove the Frechet differentiability in Besov spaces, and derive the formula for the Frechet… 
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TLDR
Computational analysis of the inverse Stefan type free boundary problem, where information on the boundary heat flux is missing and must be found along with the temperature and the free boundary, and optimal control framework introduced in Abdulla is pursued.
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References

SHOWING 1-10 OF 55 REFERENCES
ON THE OPTIMAL CONTROL OF THE FREE BOUNDARY PROBLEMS FOR THE SECOND ORDER PARABOLIC EQUATIONS. I. WELL-POSEDNESS AND CONVERGENCE OF THE METHOD OF LINES
We develop a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free
On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations. II.Convergence of the Method of Finite Differences
We consider a variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free
Real-time control of the free boundary in a two-phase stefan problem
A two-phase Stefan-problem is considered where the progress of the free boundary is observed by fully automatic real-time controls (thermostats or photo-electric cells). The heat flux at both fixed
General One-Phase Stefan Problems and Free Boundary Problems for the Heat Equation with Cauchy Data Prescribed on the Free Boundary
where u(x, t) and the free boundary s(t) are to be determined. Here f, g, h, A, p, and a are the data of problem (1.1), with f, g, h, A defined for t > 0 and (p(x) defined for 0 ? x 0 and a > 0 and
Numerical approximation of a Cauchy problem for a parabolic partial differential equation
Abstract : In many physical problems in heat conduction, it is impossible to obtain an initial temperature distribution within a material. In many of these cases, in order to obtain approximations of
The Occurrence of Pathologies in Some Stefan-Like Problems
A one-phase Stefan-like problem for the one-dimensional heat equation is considered and cases are discussed in which the problem does not possess global solution. They occur when the width of the
The Noncharacteristic Cauchy Problem for a Class of Equations with Time Dependence. I. Problems in One Space Dimension
The noncharacteristic Cauchy problem is considered for a general class of operators in one space dimension which are second order in space and first order in time. A weighted energy technique is used
A note on an ill-posed problem for the heat equation
In this paper an ill-posed problem for the heat equation is investigated. Solutions u to the equation u t – u xx = 0, which are approximately known on the positive half-axis t = 0 and on some
The Cauchy problem for a linear parabolic partial differential equation
Numerical approximation of the solution of the Cauchy problem for the linear parabolic partial differential equation is considered. The problem: (p(x)ux)x − q(x)u = p(x)ut, 0 < x < 1,0 < t⩽ T; u(0,
Determining Surface Temperatures from Interior Observations
We consider the inverse heat transfer problem for the case of the one-dimensional heat equation in a quarter plane, and present a new solution algorithm together with error bounds of logarithmic
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