Regional gradient observability for fractional differential equations with Caputo time-fractional derivatives

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 spatial 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.

The Regional Enlarged Observability for Hilfer Fractional Differential Equations

“…Zguaid et al [20] studied ReOb for a class of linear time-fractional systems using the HUM approach and proved that the considered approach allows to transform the ReOb problem into a solvability one. Regional gradient observability for Caputo fractional diffusion systems is considered in [21]. In [22], Ge et al presented the notion of the regional gradient observability for Riemann-Liouville (R-L) diffusion systems for the first time.…”

On the regional boundary observability of semilinear time-fractional systems with Caputo derivative

“…Recently, FC started to penetrate the domain of control theory [10,13,14]; in particular, it is used to investigate the notion of regional observability; see [15][16][17][18] for linear fractional systems and [19,20] for semilinear ones. In this paper, we investigate the notion of regional boundary observability, which is basically regional observability where the desired subregion is a part of the boundary of the evolution domain [21,22].…”