In this study, we introduce the concept of observability for a specific category of time-fractional systems characterized by the Hilfer fractional derivative (HFD) of order \(\alpha \in ]0,1[,\) with type \(0 \le \beta \le 1.\) Initially, we provide definitions and discuss key properties of this concept. Subsequently, we outline a technique for determining the system’s state at \(t=0,\) which builds upon the principles of the Hilbert uniqueness method (HUM). This methodological extension culminates in an algorithm that yields compelling numerical outcomes. Notably, the reconstruction error associated with the initial state is remarkably low, affirming the efficacy of the approach employed in this investigation.
