Florida Institute of Technology
Department of Mathematical Sciences
Research Experience for Undergraduates

Research Program

On the Optimal Control of the Free Boundary Problem for the Second Order Parabolic PDEs

Faculty Mentor: Dr. Ugur Abdulla

Development and investigation of the nonlinear models of the mathematical physics is one of the most important problems of the modern science. This project will shed light into modeling and control of important class of nonlinear processes with phase transition, so called free boundary problems, arising in thermophysics and mechanics of continuous media, bioengineering and materials science. Free boundary problems involve the solution of the partial differential equations in domains that are unknown apriori: not only the solutions of the equations but also the domain of definition of the equations must be determined. The main goal of this project is to gain insight into inverse free boundary problems. These problems arise when not only the solution of the PDEs and the free boundaries are unknown, but also some other characteristics of the media are not available. For example, temperature or heat flux on the fixed boundaries, intensity of the source term or phase transition temperature may not be available. Another important motivation for the analysis of the inverse Stefan problem arises in optimal control of processes with phase transitions.

The prelude for this project are the following two papers, where a new variational method is developed for the solution of the inverse Stefan problem:

  • U. G. Abdulla, 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, Inverse Problems and Imaging, Volume 7, Number 2(2013), 307-340.
  • U. G. Abdulla, On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations. II. Convergence of the Method of Finite Differences Inverse Problems and Imaging, Volume 10, Number 4, 2016, 869-898
  • The following papers reflect recent advances made by Dr. Abdulla's research team:

    • U.G. Abdulla and B.Poggi Optimal Control of the Multiphase Stefan Problem, European Journal of Applied Mathematics (submitted), 2017, arxiv:1508.00290.
    • U.G. Abdulla and J. Goldfarb Frechet Differentiability in Besov Spaces in the Optimal Control of Parabolic Free Boundary Problems, Inverse and Ill-posed Problems (submitted), 2016, arxiv:1604.00057.
    • U.G. Abdulla, E. Cosgrove, J. Goldfarb On the Frechet Differentiability in Optimal Control of Coefficients in Parabolic Free Boundary Problems, Evolution Equations and Control Theory, to appear in 2017.
    In summer session of 2017 students will pursue research on the open problems on the optimal control of the free boundary problems motivated with the recent advance in the field. The project will address the well-posedness of the optimal control problem, discretization and proof of the convergence of discrete optimal control problems to the original problem, Frechet differentiability in Besov-Sobolev spaces, necessary conditions for optimality, Pontryagin's maximum principle and gradient type methods for the numerical implementation.

    Introductory video lectures on this topic are available online here.