Inverse Problem and Trajectory Controllability for the Swift-hohenberg Equation

This article studies an internal type inverse problem that consists of recovering a nonlinear coefficient in the Swift-Hohenberg (SH) equation and establish the controllability to the trajectories of the corresponding control system. First, we establish a Carleman estimates for the linearized problem with Neumann type boundary conditions by internal observation. Applying this Carleman estimate and regularity results, we prove stability estimate of SH equation. Further, by using the fixed point arguments, we prove the null controllability of a linearized SH equation using Carleman estimate and observability inequality derived for adjoint equation. Finally, we extend it for the trajectory controllability of the nonlinear SH equation.