Null controllability for the parabolic equation with a complex principal part

This paper is addressed to a study of the null controllability for the semilinear parabolic equation with a complex principal part. For this purpose, we establish a key weighted identity for partial differential operators $(\text{a}+i\text{b}){\text{pa}}_t+{\sum }_{j,k=1}^n{\text{pa}}_k(a^{jk}{\text{pa}}_j)$ (with real functions $\text{a}$ and $\text{b}$), by which we develop a universal approach, based on global Carleman estimate, to deduce not only the desired explicit observability estimate for the linearized complex Ginzburg-Landau equation, but also all the known controllability/observability results for the parabolic, hyperbolic, Schr"odinger and plate equations that are derived via Carleman estimates.