CONTROLLABILITY AND OBSERVABILITY OF SOME STOCHASTIC COMPLEX GINZBURG-LANDAU EQUATIONS

This paper addresses a study of the null controllability and observability of some forward linear stochastic complex Ginzburg-Landau equations. By the standard duality technique, it suffices to establish suitable observability inequalities for backward and forward linear stochastic complex Ginzburg-Landau equations, respectively. For this purpose, two different methods are adopted. First, by a new pointwise weighted identity for a backward stochastic complex Ginzburg-Landau operator itself, we derive a global Carleman estimate of it. Meanwhile, a global Carleman estimate for forward linear stochastic complex Ginzburg-Landau operators is established directly by the known Carleman estimate for deterministic complex Ginzburg-Landau operators, by means of the duality argument. Based on the latter method, the requirement for regularity of some coefficients may be relaxed.