The Stability and Stabilization of Infinite Dimensional Caputo-Time Fractional Differential Linear Systems

Hanaa Zitane,Ali Boutoulout,Delfim F M Torres

doi:10.3390/math8030353

Hanaa Zitane, Ali Boutoulout + Show 1 more

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https://doi.org/10.3390/math8030353

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Abstract

We investigate the stability and stabilization concepts for infinite dimensional time fractional differential linear systems in Hilbert spaces with Caputo derivatives. Firstly, based on a family of operators generated by strongly continuous semigroups and on a probability density function, we provide sufficient and necessary conditions for the exponential stability of the considered class of systems. Then, by assuming that the system dynamics are symmetric and uniformly elliptical and by using the properties of the Mittag–Leffler function, we provide sufficient conditions that ensure strong stability. Finally, we characterize an explicit feedback control that guarantees the strong stabilization of a controlled Caputo time fractional linear system through a decomposition approach. Some examples are presented that illustrate the effectiveness of our results.

Highlights

Fractional order calculus is a natural generalization of classical integer order calculus
We study the stabilization of system (16) using the decomposition method, which consists of decomposing the state space and the system using the spectral properties of operator A
We investigated the stability problem of infinite dimensional time fractional differential linear systems under Caputo derivatives of order α ∈ (0, 1), where the state space is the Hillbert space L2 (Ω)

Summary

Introduction

Fractional order calculus is a natural generalization of classical integer order calculus. Fractional order calculus has become very popular in recent years, due to its demonstrated applications in many fields of applied sciences and engineering, such as the spread of contaminants in underground water, charge transport in amorphous semiconductors, and diffusion of pollution in the atmosphere [1,2,3]. Because it generalizes and includes in the limit the integer order calculus, fractional calculus has the potential to accomplish much more than what integer order calculus achieves [4]. The operator introduced by Caputo in 1967, and used by us in the present work, represents an observed

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