Controllability of the time discrete heat equation

Abstract

In this paper we study the controllability of an Euler Implicit time discrete heat equation in a bounded domain with a local internal controller. We prove that, based on Lebeau–Robbiano's time iteration method, the projection in appropriate filtered space is null controllable with uniformly bounded control. In this way, the well-known null-controllability property of the heat equation can be proven as the limit, as ▵t→0, of the controllability of projections of the time-discrete one. Consequently we prove the uniform approximate controllability after filtering with bounded control. A further study is made and analogous results are obtained for other discrete schemes, i.e. Euler Explicit schemes, θ-method schemes. We also discuss the null controllability of the Euler Implicit time discrete parabolic equation of fractional order.