Stability of fractional-order systems with Prabhakar derivatives

TL;DR

This paper investigates the stability of systems with Prabhakar fractional derivatives, providing a precise stability region, asymptotic solutions, and numerical methods validated through experiments, advancing understanding of anomalous relaxation phenomena.

Contribution

It offers the first exact characterization of stability regions for systems with Prabhakar derivatives and develops asymptotic solutions and numerical methods for analysis.

Findings

Exact stability region characterized

Asymptotic expansions derived

Numerical validation confirms theoretical results

Abstract

Fractional derivatives of Prabhakar type are capturing an increasing interest since their ability to describe anomalous relaxation phenomena (in dielectrics and other fields) showing a simultaneous nonlocal and nonlinear behaviour. In this paper we study the asymptotic stability of systems of differential equations with the Prabhakar derivative, providing an exact characterization of the corresponding stability region. Asymptotic expansions (for small and large arguments) of the solution of linear differential equations of Prabhakar type and a numerical method for nonlinear systems are derived. Numerical experiments are hence presented to validate theoretical findings.