关键词: 数据同化, 热方程, 零能控

Abstract:

We consider an one dimensional heat conduction model ∂_ty-Δy=0, in  (0, T)×Ω, which we would like to "predict" on a time interval (T0, T₀) but for which the initial value of the state variable is unknown. However, "measures"of the solutions are known only at one point of Ω on a time interval (0, T₀) where 0<T₀<T. 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 T₀. The approximation needs a sharp estimation of the cost of the null controllability of the heat equation.

Key words: Data assimilation, Heat equation, Null controllability, Stability