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 (\a+i\b)\pa_t+\sum\limits_{j,k=1}^n\pa_k(a^{jk}\pa_j) (with real functions \a and \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

An Internal Observability Estimate for Stochastic Hyperbolic Equations

This paper is addressed to establishing an internal observability estimate for some linear stochastic hyperbolic equations. The key is to establish a new global Carleman estimate for forward stochastic hyperbolic equations in the $L^2$-space. Different from the deterministic case, a delicate analysis of the adaptedness for some stochastic processes is required in the stochastic setting