arXiv subjects

Hanaa Zitane

Publications and source records attributed to Hanaa Zitane.

8 recordsLinked to original sources

arXiv2024

Controllability and observability of tempered fractional differential systems

We study controllability and observability concepts of tempered fractional linear systems in the Caputo sense. First, we formulate a solution for the class of tempered systems under investigation by means of the Laplace transform method. Then, we derive necessary and sufficient conditions for the controllability, as well as for the observability, in terms of the Gramian controllability matrix and the Gramian observability matrix, respectively. Moreover, we establish the Kalman criteria that allows one to check easily the controllability and the observability for tempered fractional systems. Applications to the fractional Chua's circuit and Chua--Hartley's oscillator models are provided to illustrate the theoretical results developed in this manuscript.

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arXiv2024

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

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.

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arXiv2024

Stability criteria of nonlinear generalized proportional fractional delayed systems

This work deals with the finite time stability of generalized proportional fractional systems with time delay. First, based on the generalized proportional Grönwall inequality, we derive an explicit criterion that enables the system trajectories to stay within a priori given sets during a pre-specified time interval, in terms of the Mittag-Leffler function. Then, we investigate the finite time stability of nonlinear nonhomogeneous delayed systems by means of an approach based on Hölder's and Jensen's inequalities. Numerical applications are presented to illustrate the validity and feasibility of the developed results.

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arXiv2023

Finite time stability of tempered fractional systems with time delays

We investigate the notion of finite time stability for tempered fractional systems (TFSs) with time delays and variable coefficients. Then, we examine some sufficient conditions that allow concluding the TFSs stability in a finite time interval, which include the nonhomogeneous and the homogeneous delayed cases. We present two different approaches. The first one is based on Hölder's and Jensen's inequalities, while the second one concerns the Bellman--Grönwall method using the tempered Grönwall inequality. Finally, we provide two numerical examples to show the practicability of the developed procedures.

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arXiv2023

Pharmacokinetic/Pharmacodynamic Anesthesia Model Incorporating psi-Caputo Fractional Derivatives

We present a novel Pharmacokinetic/Pharmacodynamic (PK/PD) model for the induction phase of anesthesia, incorporating the $ψ$-Caputo fractional derivative. By employing the Picard iterative process, we derive a solution for a nonhomogeneous $ψ$-Caputo fractional system to characterize the dynamical behavior of the drugs distribution within a patient's body during the anesthesia process. To explore the dynamics of the fractional anesthesia model, we perform numerical analysis on solutions involving various functions of $ψ$ and fractional orders. All numerical simulations are conducted using the MATLAB computing environment. Our results suggest that the $ψ$ functions and the fractional order of differentiation have an important role in the modeling of individual-specific characteristics, taking into account the complex interplay between drug concentration and its effect on the human body. This innovative model serves to advance the understanding of personalized drug responses during anesthesia, paving the way for more precise and tailored approaches to anesthetic drug administration.

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arXiv2023

A class of fractional differential equations via power non-local and non-singular kernels: existence, uniqueness and numerical approximations

We prove a useful formula and new properties for the recently introduced power fractional calculus with non-local and non-singular kernels. In particular, we prove a new version of Gronwall's inequality involving the power fractional integral; and we establish existence and uniqueness results for nonlinear power fractional differential equations using fixed point techniques. Moreover, based on Lagrange polynomial interpolation, we develop a new explicit numerical method in order to approximate the solutions of a rich class of fractional differential equations. The approximation error of the proposed numerical scheme is analyzed. For illustrative purposes, we apply our method to a fractional differential equation for which the exact solution is computed, as well as to a nonlinear problem for which no exact solution is known. The numerical simulations show that the proposed method is very efficient, highly accurate and converges quickly.

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arXiv2023

Generalized Taylor's formula for power fractional derivatives

We establish a new generalized Taylor's formula for power fractional derivatives with nonsingular and nonlocal kernels, which includes many known Taylor's formulas in the literature. Moreover, as a consequence, we obtain a general version of the classical mean value theorem. We apply our main result to approximate functions in Taylor's expansions at a given point. The explicit interpolation error is also obtained. The new results are illustrated through examples and numerical simulations.

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arXiv2020

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

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 is symmetric and uniformly elliptic 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.

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