Regional boundary observability for Riemann–Liouville linear fractional evolution systems

Abstract

This paper deals with regional boundary observability of linear time-fractional systems with Riemann–Liouville derivative. The main purpose is to reconstruct the initial state of the considered system in a subregion B of the boundary ∂Ω of the evolution domain Ω. For that, we show the link between regional boundary observability on B and regional observability, of the considered system, in a suitable subregion (ω⊂Ω) defined such that B⊂∂ω. This will allow us to firstly recover the initial state in the subregion ω by using an extension of Hilbert Uniqueness Method for fractional systems, after that we take the restriction, on B, of its trace, on ∂ω, in order to obtain the initial state on B. We also propose an algorithm that enables us to recover the initial state in ω and eventually on B, with a satisfying reconstruction error, which is illustrated in two numerical simulations, using a zonal sensor and a pointwise one.