In this article, we investigate the concept of regional observability for a class of time-fractional systems characterized by the Hilfer fractional derivative with parameters \((\alpha , \beta ) \in ]1,2[\times [0,1]\) . We extend the conventional understanding of regional observability, typically applied to classical systems, especially hyperbolic systems, to fractional-order systems. After introducing the essential concepts and definitions, we establish rigorous criteria for both exact and approximate regional observability. To demonstrate the significance of regional analysis, we provide an example of a fractional system that achieves regional observability despite lacking global observability. Afterward, we introduce a methodology for reconstructing the initial state within a specified subregion \(\omega\) , of the larger domain \(\Omega\) , which leverages the Hilbert Uniqueness Method (HUM). The key idea of this methodology is to reformulate the initial state reconstruction problem as a solvability problem. An algorithm based upon the used approach for reconstructing the initial state in the desired subregion \(\omega\) is presented. Finally, we validate our theoretical framework with two successful numerical examples, each employing different sensor types and achieving a reasonable error margin, thus confirming the effectiveness of our approach.
