Local Null-Controllability for Some Quasi-Linear Phase-Field Systems with Neumann Boundary Conditions by one Control Force

In this article, we study the local null-controllability for some quasi-linear phase-field systems with homogeneous Neumann boundary conditions and an arbitrary located internal controller in the frame of classical solutions. In order to minimize the number of control forces, we prove the Carleman inequality for the associated linear system. By constructing a sequence of optimal control problems and an iteration method based on the parabolic regularity, we find a qualified control in Hölder space for the linear system. Based on the theory of Kakutani’s fixed point theorem, we prove that the quasi-linear system is local null-controllable when the initial datum is small and smooth enough.