On the State and Free Boundary Estimation for Stochastic Parabolic Systems

The purpose of this paper is to construct the estimator system for the stochastic parabolic system with the free boundary. The free boundary problem is one of important nonlinear problems in engineering. The intrinsic characteristic of the free boundary problem is that the domain of the solution of the considered equation is unknown and must be determined with the solution of the considered equation. In ordinary initial and boundary value problems, we can seek the solution from the initial and boundary conditions, however, in the free boundary problems, an additional condition besides initial and boundary data is required to determine the solution and the domain of the solution. Generally, such a condition is called Stefan condition, which denotes physically the energy balance on the free boundary. Stefan problems [1] are often cited as a typical example of the free boundary problem. In this paper, focusing our attention on two-phase Stefan problem, stochastic modeling of the two-phase Stefan problems is firstly considered. In real situations, various kinds of disturbances exist, in this paper, we consider the disturbance in a heat input caused by impurities and obstacles in fuel. Secondly, the state and free boundary estimation problem to the stochastic two-phase Stefan system under noisy observations is studied. Finally, since the estimator dynamics derived here is nonlinear, simple approximation technique is proposed and the efficiency of the approximation is examined through the numerical simulations.