热方程数据同化问题的非典型方法

Data Assimilation:a Nonstandard Approach to a Heat Conduction Model

考虑一维热方程■_ty-Δy=0,在(0,T)×Ω中的数据同化问题.未知初始条件,已知(0,T_0)时间内可以测量方程的解在Ω中某一点的值,经典的数据同化是要求解方程的初值,但此问题在热方程中是不适定的,该文借助控制转换的方法给出了在时刻T(T>T_0)方程解的估计.

We consider an one dimensional heat conduction model■y -Δy = 0,in（0,T）×Ω, which we would like to ＂predict＂ on a time interval（T_0,T） but for which the initial value of the state variable is unknown.However,＂measures＂ of the solutions are known only at one point ofΩon a time interval（0,T_0） where 0T_0T.The classical approach in data assimilation is to look for the initial value at time 0 and this is known to be an ill-posed problem for heat equations.In this paper,by the property of null controllability of heat equations we give a result of approximate reconstruction of the value at T_0.The approximation needs a sharp estimation of the cost of the null controllability of the heat equation.