Optimal control for one-phase Stefan problem with random emission

We study the optimal control for the stochastic one-phase Stefan problem where the cost functional is quadratic with respect to the state and control. A one-phase Stefan problem with random disturbance is formulated as the stochastic variational inequality. After studying the existence and uniqueness of the solution of the stochastic variational inequality, the necessary condition of optimality for the control problem is derived.The first author would like to express his thanks to the Department of Applied Mathematics, Twente University of Technology, Enschede, The Netherlands, for the financial support that it provided during the completion of the final two sections of this paper.