Identifying the diffusion coefficient by optimization from the final observation

This paper deals with the inverse problem of determining a pair (a,u)(a,u) in the reaction–diffusion equationut-(aux)x=f(x,t),ut-(aux)x=f(x,t),with initial and boundary conditionsu(x,0)=ϕ(x),ux|x=0=ux|x=1=0,from the final measurement data u(x,T)=z(x)u(x,T)=z(x), which has important application in a large fields of applied science. Based on the optimal control framework, the existence, uniqueness and stability of the minimizer for the cost functional are established. A necessary condition which is a couple system of a parabolic equation and a parabolic variational inequality is deduced. The gradient iteration algorithm is applied to the inverse problem and some numerical results are presented for various typical test examples.