Abstract
This paper deals with the local null controllability of a free-boundary problem for a class of quasi-linear 1D parabolic equations with an arbitrary located internal controller. Notice that the nonlinearity appearing in the principal operator brings new difficulties, which are different from the semilinear case essentially. Unlike the controllability of a free-boundary problem for the linear and semilinear heat equations, we need to use a delicate Carleman estimate and solve the controllability problem in the frame of classical solutions. We prove that for any initial state that is sufficiently small, there exists a control that derives the state exactly to rest at time t=T.
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