Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 135, pp. 1–17. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu DATA ASSIMILATION AND NULL CONTROLLABILITY OF DEGENERATE/SINGULAR PARABOLIC PROBLEMS KHALID ATIFI, EL-HASSAN ESSOUFI Communicated by Jerome A. Goldstein Abstract. In this article, we use the variational method in data assimilation to study numerically the null controllability of degenerate/singular parabolic problem ∂t ψ − ∂x(x α ∂xψ(x)) − λ x β ψ = f, (x,t) ∈]0, 1[×]0,T [, ψ(x, 0) = ψ 0 ,ψ ˛ ˛ x=0 = ψ ˛ ˛ x=1 =0. To do this, we determine the source term f with the aim of obtaining ψ(·,T )= 0, for all ψ 0 ∈ L 2 (]0, 1[). This problem can be formulated in a least-squares framework, which leads to a non-convex minimization problem that is solved using a regularization approach. Also we present some numerical experiments. 1. Introduction In this article, we study an inverse problem of identifying the source term in degenerate/singular parabolic equation. This in the aim to study the null control- lability, which has important applications in various areas of applied science and engineering. Controllability properties of degenerate/singular parabolic equations has been widely studied (see [1, 4, 13, 12, 26]) using Carleman estimates. Our main con- tribution is to study numerically the null controllability of problem (1.1), below, using the variational method in data assimilation. The problem can be stated as follows: Estimate the source term in the degenerate parabolic equation with singular potential ∂ t ψ − ∂ x (x α ∂ x ψ(x)) − λ x β ψ = f, (x,t) ∈ Ω×]0,T [ (1.1) where Ω =]0, 1[, α ∈]0, 1[, β ∈]0, 2 − α[, λ ≤ 0, and f ∈ L 2 (Ω×]0,T [). The mathematical model leads to a non-convex minimization problem find ˆ f ∈ A ad such that E( ˆ f )= min f ∈A ad E(f ), (1.2) 2010 Mathematics Subject Classification. 15A29, 47A52, 93C20, 35K05, 35K65, 35K65, 93B05. Key words and phrases. Data assimilation; null controllability; regularization; heat equation; inverse problem; degenerate equations; optimization. c 2017 Texas State University. Submitted February 24, 2017. Published May 18, 2017. 1