On finite-time stability for stochastic parabolic systems with free boundary-application to micro-tunneling machine

We study a stability property for stochastic parabolic equation with free boundary. This model is motivated by a micro-tunneling machine which is used for the conduit installation for communication cables. By using this tunneling machine, a flexible pilot pipe is thrust into the ground and hence the total length of the inserted pipe becomes a function of time. Furthermore, the thrust force generated by a hydraulic jack is stochastically perturbed. The transverse displacement of the inserted pipe is firstly shown to be governed by a parabolic equation with free boundary. Secondly, the existence of a solution of this equation is proved. For checking the buckling phenomena the concept of finite-time stability is introduced and its stability property is studied with a numerical example