Controllability of quasilinear parabolic equations and application

This paper addresses mainly the controllability of multi-dimensional quasilinear parabolic equations and some related control problems. Notice that the nonlinearity appearing in the principal operator brings new difficulty, which is different from the semilinear case essentially. Also, due to the time-irreversibility, previous approaches solving successfully the controllability of quasilinear hyperbolic equations do not work for this system. We hereby need to develop a delicate Carleman estimate-based approach and solve the controllability problem in the frame of classical solutions. The point is to establish the desired regularity on the control function for the linearized system. Our approach turns out to be quite general, and therefore, it can be applied to establish the local null controllability of both quasilinear parabolic equations and quasilinear Ginzburg-Landau equations. Moreover, as its application, we also show the existence of insensitizing controls for quasilinear parabolic equations.