  The aim of this paper is to study the null controllability of a class of
quasilinear parabolic equations. In a first step we prove that the associated
linear parabolic equations with non-constant diffusion coefficients are
approximately null controllable by the means of regular controls and that these
controls depend continuously to the diffusion coefficient. A fixed-point
strategy is employed in order to prove the null approximate controllability for
the considered quasilinear parabolic equations. We also show the exact null
controllability in arbitrary small time for a class of parabolic equations
including the parabolic $p$-Laplacian with $\frac{3}{2} < p < 2$. The
theoretical results are numerically illustrated combining a fixed point
algorithm and a reformulation of the controllability problem for linear
parabolic equation as a mixed-formulation which is numerically solved using a
finite elements method.
