Regional boundary observability for linear time-fractional systems

In the present paper, we investigate the possibility of reconstructing the initial state, of a time-fractional system with Caputo fractional derivative, on a subregion B of the boundary ∂Ω of the evolution domain Ω. In this manuscript, we extend the notion of regional boundary observability from classical systems into fractional ones. We first introduce all the necessary mathematical background, then we show a way to link between regional boundary observability and regional observability by introducing an internal subregion ω, such that B⊂∂ω. This subregion is constructed so that if the considered system is exactly (resp. approximately) regionally observable in ω, then it is also exactly (resp. approximately) regionally boundary observable on B. This allows us to deduce that, in order to reconstruct the initial state on B, it is sufficient to reconstruct it in ω and then extract its value on B. For reconstructing the initial state in ω, we use an extension of the Hilbert Uniqueness Method (HUM). To the best of our knowledge, this is the first time someone has dealt with this concept for time-fractional systems, which are written by means of the Caputo derivative, with this particular approach. The used method leads to an algorithm which also leads to some successful numerical results. Moreover, the obtained reconstruction error of the initial state is very satisfying, which allows us to conclude that the adopted approach in this work is efficient.