A weak method in numerical identification

The parameters identification problem of a sum u(t)=∑\limits^N_i=1a_ie(λ_i,t), \ t∈ [0,T], where N∈ IN, a_i∈ ℜ and λ_i∈ Ω are unknown parameters, and Ω is a bounded open set of ℜⁿ is discussed. For some choices of the function e, this problem is an ill‐posed problem in the classical optimisation methods sense, such as the non‐linear least squares. The identification of parameters N, a_i and λ_i being equivalent to the search of the distribution ℓ=∑\limits^N_i=1a_iδ_λi in the dual space of E=C(¯Omega;), the method developed here consists in finding a weak approximation of ℓ in the sense of the metric of the weak‐* topology on closed spheres of E’. Finally, we will apply this method to the identification problem of a sum of exponentials.