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 frame of classical solutions. order to minimize number control forces, prove Carleman inequality associated linear system. By constructing a sequence optimal problems iteration method based on parabolic regularity, find qualified Hölder space Based theory Kakutani’s fixed point theorem, that system is null-controllable when initial datum small smooth enough.