NULL CONTROLLABILITY OF DEGENERATE NONAUTONOMOUS PARABOLIC EQUATIONS

摘要

In this paper we are interested in the study of the null controllability for the one dimensional degenerate non autonomous parabolic equation $$ u_{t}-M(t)(a(x)u_{x})_{x}=hchi_{omega},qquad? (x,t)in Q=(0,1)imes(0,T),$$ where $omega=(x_{1},x_{2})$ is a small nonempty open subset in $(0,1)$, $hin L^{2}(omegaimes(0,T))$, 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.