Lack of null controllability of one dimensional linear coupled transport-parabolic system with variable coefficients

Abstract

In this article, we study the null controllability of linear coupled transport-parabolic systems with variable coefficients in one space dimension. We consider coupled systems with coupling of order zero, one and two. The systems are considered with homogeneous Dirichlet boundary conditions and with localized interior controls acting on both transport and parabolic equations. We show that coupled systems are not null controllable at small time. This time depends on the transport velocity and the support of the controls. When the transport velocity is identically zero, the systems are not null controllable at any time. To achieve these results, we construct highly localized solutions, known as Gaussian beams, corresponding to the adjoint systems, and using them, we show that the corresponding observability inequalities fail. However, these systems are null controllable at any time by controls acting everywhere in the parabolic equation, under suitable assumptions on the initial data and the coefficients.