Regional observability of time-fractional systems with a differentiation order 1 &lt; <i>α</i> &lt; 2

Abstract

Abstract In this article, we explore the concept of regional observability for a category of time-fractional systems characterized with the Caputo derivative of order belonging to the open interval ]1, 2[. To do so, we develop the conventional understanding of regional observability, generally applied to classical systems, in particular hyperbolic systems, and extend it to fractional systems. After establishing the essential concepts and definitions, we establish valid criteria for exact and approximate regional observability. We illustrate the importance of regional analysis by providing an example of a fractional system that exhibits regional observability despite the absence of global observability. Next, we look at the methodology for determining the initial state in a specified sub-region ω which is a subset of the larger domain Ω. Our approach involves the use of Hilbert’s uniqueness method, which effectively transforms the initial state reconstruction problem into a solvability problem. Then, we present an algorithm to reconstruct the initial state in the desired sub-region ω . At the end, we provide two successful numerical results to support our theoretical framework. Each result is obtained using different types of sensors and retains a reasonable margin of error, confirming the effectiveness of our approach.