Optimal Control of Systems with Distributed Parameters
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 was developed. This research project is motivated by the bioengineering problem on the laser ablation of biological tissues. Optimal control framework was employed, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consist of the minimization of the sum of L2-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 the least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of a full free boundary problem. Below are recent papers on Optimal Control and Inverse Problems:
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.
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.
Optimal control of the multiphase Stefan problem, Applied Mathematics & Optimization, 80, 2(2019), 479-513.
Frechet Differentiability in Besov Spaces in the Optimal Control of Parabolic Free Boundary Problems, Inverse and Ill-posed Problems, Volume 26, Issue 2, 2018, 211-228.
On the Frechet Differentiability in Optimal Control of Coefficients in Parabolic Free Boundary Problems, Evolution Equations and Control Theory, Volume 6, Issue 3, 2017, 319-344.
Gradient Method in Hilbert-Besov Spaces for the Optimal Control of Parabolic Free Boundary Problems, Journal of Computational and Applied Mathematics, Volume 346, January 2019, 84-109.
Optimal Stefan Problem, Calculus of Variations and Partial Differential Equations, 59, 61(2020).
Optimal Control of Multiphase Free Boundary Problems for Nonlinear Parabolic Equations, Applied Mathematics and Optimization (2020).
Optimal Control of Coefficients in Parabolic Free Boundary Problems Modeling Laser Ablation, Journal of Computational and Applied Mathematics (2020).