The null boundary controllability for the Mullins equation with periodic boundary conditions

In this paper, we study the null controllability of Mullins equation with control acting on periodic boundary. Firstly, using duality relation between and observability, express condition in terms solution backward adjoint system. After showing existence uniqueness system, determine admissible initial data class since system is not always controllable under these boundary conditions. Finally, spectral analysis, reduce problem to moment solve class.