On the temperature control for stochastic hyperbolic heat systems with free boundaries

This paper is concerned with the temperature control problems for stochastic hyperbolic heat systems with free boundaries. It is well known that the parabolic equation has an infinite thermal propagation speed. In order to avoid this physically unacceptable aspect, the heat conduction model of the hyperbolic type is derived from the physical point of view. First, taking the randomness in the input signal into consideration, the stochastic hyperbolic heat conduction model is proposed in this paper. Secondly, the free boundary problem for the stochastic hyperbolic heat system is studied. It is shown that the free boundary problem for the stochastic hyperbolic heat equation is formulated by the stochastic variational inequality of a new type. The existence and uniqueness theorem of the solution to the stochastic variational inequality is given. Finally, the temperature control problem for the stochastic hyperbolic heat equation with the free boundary is considered and a simple but very useful control method is proposed.