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. 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.
  • 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.