Research and Recent Publications
2. Optimal Control and Inverse Problems for PDEs
I developed 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 boundary. This research project is motivated by the bioengineering problem on the
laser ablation of biological tissues. I employed optimal control framework, where boundary heat flux and
free boundary are components of the control vector, and optimality criteria consists of the minimization of
the sum of $L_2$norm declinations from the available measurement of the temperature on the fixed boundary
and available information on the phase transition temperature on the free boundary. This approach allows one
to tackle situations when the phase transition temperature is not known explicitly, and is available through
measurement with possible error. It also allows for the development of iterative numerical methods of least
computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed
region instead of full free boundary problem. Here are recent papers of my research group on Optimal Control and Inverse Problems :
 U.G. Abdulla On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations. I. Wellposedness and
Convergence of the Method of Lines,
Inverse Problems and Imaging, Volume 7, Number 2(2013), 307340.
 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), 869898.
 U.G. Abdulla and B.Poggi Optimal Control of the Multiphase Stefan Problem,
Applied Mathematics & Optimization, Volume 77, 2018,
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 Illposed Problems, Volume 26, Issue 2, 2018,
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, Volume 6, Issue 3, 2017, 319344.
