Abbes Benaissa, Abdelatif Kainane Mezadek, Lahcen Maniar

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In this paper we are interested in the study of the null controllability for the one dimensional degenerate non autonomous parabolic equation
u_{t}-M(t)(a(x)u_{x})_{x}=h\chi_{\omega},\qquad  (x,t)\in Q=(0,1)\times(0,T),$$ where $\omega=(x_{1},x_{2})$ is a
small nonempty open subset in $(0,1)$, $h\in L^{2}(\omega\times(0,T))$, the diffusion coefficients $a(\cdot)$ is
degenerate at $x=0$ and $M(\cdot)$ is non degenerate on $[0,T]$. Also the boundary conditions are considered to
be Dirichlet or Neumann type related to the degeneracy rate of $a(\cdot)$. Under some conditions on the functions
$a(\cdot)$ and $M(\cdot)$, we prove some global Carleman estimates which will yield the  observability inequality
of the associated adjoint system and equivalently the null controllability of our parabolic equation.


Null controllability; nonautonomous parabolic equation; Carleman estimates.


Null controllability, Carleman estimates, Hardy inequality, Observability inequality.

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