Real-time Control Of The Free Boundary In A Class Of Variable Coefficient Two-phase Stefan Problem

In this paper, we investigate the real time control of a class of two phase Stefan problems with variable coefficients. The free boundary can be observed by fully automatic real-time controllers. The control problem is that two controllers C1, C2 with the free boundaries transmit the signals to heaters Hi, H2 (with certain time lags), which can change the heat fluxes/1, f2 at both ends. We hope that the free boundary can be controlled in the given region or in the given location.The corresponding mathematical model leads to a two-phase Stefan problem as follows and a control equation satisfying(1) a< b< c, ?1(x), ?2(x) is positive lower bound, twice continuously differentiable:(2) ?1?C2[a,b], ?1(x)? 0, x?[a,b],?2? C2[b,c], ?2(x)? 0, x?[b,c],?i(j)b) 0, i=1,2, j=0,1,2;(3) The historical data of s(t) is knownThe main method of this thesis is to transform the problem into a set-valued fixed point equation, and by constructing a mapping, we prove the existence of solutions by using fixed point theorem of set value.The main result are as follows:(1) The solution exists in a finite time interval the minimum length of which can be explicitly described.(2) the free boundary can be controlled in one area or in a given location by controlling the change of heat flux at both ends through s(t).