Controllability of the time discrete heat equation

Article type: Research Article

Authors: Zheng, Chuang

Affiliations: Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, 100875 Beijing, China | and | Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain. E-mail: chuang.zheng@uam.es

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.

Keywords: null controllability, heat equation, time discretization, time iteration method, filtering

DOI: 10.3233/ASY-2008-0888

Journal: Asymptotic Analysis, vol. 59, no. 3-4, pp. 139-177, 2008