Boundary Feedback Control of Complex Ginzburg-Landau Equation with A Simultaneously Space and Time Dependent Coefficient

Linearized complex Ginzburg-Landau equation models various physical phenomena and the stability controls of them are important. In this paper, we study the control of the LCGLE with a simultaneously space and time dependent coefficient by transforming it into a complex heat equation. It is shown that under certain conditions on the coefficient functions $a_2(\tilde{x},\tilde{t})$, the exponential stability of the system at any rate can be achieved by boundary control based on the state feedback. The kernels are {\it explicitly} calculated as series of approximation and shown to be twice differentiable by using the {\it method of dominant}. Both the exponential stabilities of the systems with Dirichlet and Neumann boundary conditions are strictly proven.