Geodesic
Null controllability of degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain
Eurasian journal of mathematical and computer applications, Tome 8 (2020) no. 1, pp. 4-29.

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The main purpose of this work is to study numerically a null controllability for a class of one-dimensional degenerate/singular parabolic equations, in divergence and nondivergence form. For this, we resolve an inverse source problem reformulated in a least-squares framework, which leads to a non-convex minimization problem of a cost function $J$, that is solved using a Tikhonov regularization. Firstly, we prove the well-posedness of the minimization problem and the direct problem. Secondly we prove the differentiability of the functional $J$, which gives the existence of the gradient of $J$, that is computed using the adjoint state method. Finally, to show the convergence of the descent method, we prove the Lipschitz continuity of the gradient of $J$. Also we present some numerical experiments.
Keywords: Null controllability, singular equations, inverse source problem, adjoint-state method. 
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     author = {Kh. Atifi and E.-H. Essoufi and B. Khouiti},
     title = {Null controllability of degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJMCA_2020_8_1_a0/}
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Kh. Atifi; E.-H. Essoufi; B. Khouiti. Null controllability of degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Eurasian journal of mathematical and computer applications, Tome 8 (2020) no. 1, pp. 4-29. http://geodesic.mathdoc.fr/item/EJMCA_2020_8_1_a0/