Regional Gradient Observability for Fractional Differential Equations with Caputo Time-Fractional Derivatives

TL;DR

This paper studies how to recover the initial gradient in specific regions for fractional diffusion equations with Caputo derivatives, introducing new methods and characterizations for regional gradient observability.

Contribution

It provides new characterizations of regional gradient observability for fractional systems and develops an effective Hilbert Uniqueness Method-based algorithm for initial gradient recovery.

Findings

The method successfully reconstructs initial gradients with low error.

Regional observability can be achieved even when global observability fails.

Numerical simulations validate the effectiveness of the proposed approach.

Abstract

We investigate the regional gradient observability of fractional sub-diffusion equations involving the Caputo derivative. The problem consists of describing a method to find and recover the initial gradient vector in the desired region, which is contained in the spacial domain. After giving necessary notions and definitions, we prove some useful characterizations for exact and approximate regional gradient observability. An example of a fractional system that is not (globally) gradient observable but it is regionally gradient observable is given, showing the importance of regional analysis. Our characterization of the notion of regional gradient observability is given for two types of strategic sensors. The recovery of the initial gradient is carried out using an expansion of the Hilbert Uniqueness Method. Two illustrative examples are given to show the application of the developed approach. The numerical simulations confirm that the proposed algorithm is effective in terms of the reconstruction error.