Regional observability in the context of Riemann–Liouville fractional differential systems

Abstract

Abstract In this article, we investigate the notion of regional observability within a specific category of linear time‐fractional systems with a differentiation order ranging between 1 and 2. This extends the conventional understanding of regional observability beyond classical systems, particularly hyperbolic systems, to encompass fractional systems. After presenting essential concepts and definitions, we establish criteria for both exact and approximate regional observability. The significance of regional analysis is exemplified through a fractional system that lacks global observability but displays observability within a specific region. Subsequently, we outline a method for determining the system's state at within the region of interest ( ). This involves employing Hilbert's uniqueness method to transform the state reconstruction problem into a solvability problem. We then introduce a numerical approach tailored for reconstructing the initial state within the specified region. To substantiate our theoretical findings, we conclude with two successful numerical results, each involving different sensor types and demonstrating a reasonable margin of error.