Frechet Differentiability in the Optimal Control of Parabolic Free Boundary Problems - An Alternative Approach

Abstract

We consider an optimal control of the Stefan type free boundary problem for the following general second order linear parabolic PDE: (a(x; t)ux)x + b(x; t)ux + c(x; t)u - ut = f(x; t) where u(x; t) is the temperature function. The density of heat sources f, unknown free boundary s, and boundary heat ux g are components of the control vector, and the cost functional consists of the L2-declination of the trace of the temperature at the final moment, the temperature at the free boundary, and the final position of the free boundary from available measurements. We follow a new variational formulation developed in U. G. Abdulla, Inverse Problems and Imaging, 7,2(2013),307- 340. Freechet differentiability of the optimal control problem has been proven. In this project, we consider an alternative approach in proving Frechet differentiability. The idea of the alternative approach is the following: changing the space variable as y = x=s(t) and keeping the time variable as it is, one can transform the problem to a new optimal control problem of the system described by the parabolic PDE in a fixed region, with the control parameters distributed in the coefficients of the PDE. In this paper, we derive the expression for the Frechet differential of the transformed cost functional.