Synthesis of robust boundary control for systems governed by semi-discrete differential equations

Boundary control for systems governed by partial differential equations (PDEs) is an important field with many practical and theoretical issues. The topic of boundary control of PDEs has been the subject of a considerable literature since the seminal works of J.-L. Lions in the 90s. In this paper, we consider the boundary control of systems represented by spatial discretizations of PDEs (i.e. semi-discrete equations). We focus on control laws which are sampled and piecewise constant: periodically, at every sampling time, a fixed control amplitude is applied to the system until the next sampling instant. We show that, under some conditions, sampled piecewise-constant boundary control allows to achieve approximate controllability: Given a time T > 0, the controlled system evolves to a neighborhood of a given final state. The result is illustrated on the boundary control of the semi-discrete version of the heat equation.