Carleman estimate and controllability of a time-discrete coupled parabolic system

This article investigates the so-called $ \phi( {\Delta t}) $-null controllability of a time-discrete 1D coupled parabolic system using a single Dirichlet boundary control, where the coupling is made through Kirchhoff-type condition. To prove such controllability result, we first derive a Carleman estimate for the associated time-discrete adjoint parabolic system under some condition relating the large Carleman parameter to the time step of the discretization scheme. This estimate then helps us to obtain a relaxed observability inequality, which yields, by duality argument, the $ \phi( {\Delta t}) $-null controllability of our system. Furthermore, we establish that the limit of these controls for the discrete system turns out to be a control responsible for the null controllability of the associated continuous system.