Gradient Mittag-Leffler and strong stabilizability of time fractional diffusion processes

TL;DR

This paper investigates gradient stability and stabilizability of Caputo time fractional diffusion systems, providing spectral-based conditions, feedback control strategies, and an algorithm validated through numerical simulations.

Contribution

It introduces new spectral conditions for gradient Mittag-Leffler stability and develops a feedback control algorithm for stabilizing fractional diffusion processes.

Findings

Spectral conditions ensure gradient Mittag-Leffler stability.

A feedback control algorithm effectively stabilizes the system.

Numerical simulations confirm the algorithm's effectiveness.

Abstract

This paper deals with the gradient stability and the gradient stabilizability of Caputo time fractional diffusion linear systems. First, we give sufficient conditions that allow the gradient Mittag-Leffler and strong stability, where we use a direct method based essentially on the spectral properties of the system dynamic. Moreover, we consider a class of linear and distributed feedback controls that Mittag-Leffler and strongly stabilize the state gradient. The proposed results lead to an algorithm that allows us to gradient stabilize the state of the fractional systems under consideration. Finally, we illustrate the effectiveness of the developed algorithm by a numerical example and simulations.