The Regional Enlarged Observability for Hilfer Fractional Differential Equations

Abstract

In this paper, we investigate the concept of regional enlarged observability (ReEnOb) for fractional differential equations (FDEs) with the Hilfer derivative. To proceed this, we develop an approach based on the Hilbert uniqueness method (HUM). We mainly reconstruct the initial state ν01 on an internal subregion ω from the whole domain Ω with knowledge of the initial information of the system and some given measurements. This approach shows that it is possible to obtain the desired state between two profiles in some selective internal subregions. Our findings develop and generalize some known results. Finally, we give two examples to support our theoretical results.

Some results regarding observability and initial state reconstruction for time-fractional systems

“…Furthermore, [14] explored regional observability for Hadamard-Caputo time fractional distributed parameter systems, and in [15], the pseudo-state representation was used to construct Luenberger-like observers for estimating various variables. For a deeper understanding of observability for classical and fractional systems we refer the reader to the literature [16][17][18][19][20][21][22][23][24][25][26][27][28]. According to the academic literature, the observability of linear systems has been widely studied, with multiple methods proposed for calculation.…”