Regional observability for linear time fractional systems

Zguaid, Khalid; Alaoui, Fatima-Zahrae El; Boutoulout, Ali

doi:10.1016/j.matcom.2020.12.013

Cited by 16 publications

(15 citation statements)

References 12 publications

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“…Proof. We shall prove that Λ is coercive, that is, there exists σ > 0 that verifies Λ𝑣, 𝑣 K ≥ σ 𝑣 K for all 𝑣 ∈ K. We take φ0 to be in K, [48], we have that…”

Section: The Regional Gradient Reconstruction Methodsmentioning

confidence: 99%

“…The Hilbert space O is called the observation space. The operator S α (𝑡) defined in ( 4) is a linear bounded operator, see [48], which describes the evolution of the considered time-fractional system in function of its initial state. Moreover, if the operator C is bounded, then the admissibility condition (6) is always satisfied, which means that any bounded observation operator is an admissible observation operator.…”

Section: 𝜕 𝜕𝑥mentioning

confidence: 99%

“…where N α is a linear and bounded operator defined in terms of a probability density function [48]. From Proposition 3.3 of [48], we have that:…”

Section: Remark 12 the Problem Of This Article Is Formulated With Cap...mentioning

confidence: 99%

See 2 more Smart Citations

Regional Gradient Observability for Fractional Differential Equations with Caputo Time-Fractional Derivatives

Zguaid¹,

Alaoui²,

Torres³

2023

Preprint

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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 spacial 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.

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“…Proof. We shall prove that Λ is coercive, that is, there exists σ > 0 that verifies Λ𝑣, 𝑣 K ≥ σ 𝑣 K for all 𝑣 ∈ K. We take φ0 to be in K, [48], we have that…”

Section: The Regional Gradient Reconstruction Methodsmentioning

confidence: 99%

Section: 𝜕 𝜕𝑥mentioning

confidence: 99%

See 1 more Smart Citation

Regional Gradient Observability for Fractional Differential Equations with Caputo Time-Fractional Derivatives

Zguaid¹,

Alaoui²,

Torres³

2023

Preprint

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show abstract

“…The Hilbert space O is called the observation space. The operator S α (t) defined in ( 4) is a linear bounded operator, see [44], which describes the evolution of the considered time-fractional system in function of its initial state.…”

Section: Problem Statement and Regional Gradient Observabilitymentioning

confidence: 99%

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

Zguaid

Alaoui

Torres

2023

Int. J. Dynam. Control

Self Cite

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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.

show abstract

“…More information on fractional calculus can be found in [1,12,13,14,22,24,32,35,36,37,39,40] Control theory is an important and very active branch of mathematics which serves as a link between theoretical mathematics and its applications in the real world where most processes are modeled by nonlinear distributed parameter systems, which explains the big interest of researchers in the study (Controllability, Observability, Stability...) of nonlinear and semilinear systems. As for the study of linear systems, there exists a very wide literature for integer order system, see [9] - [34] and the references therein, whereas for fractional order systems there is a much less literature, see [17,26,29,30,38] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Regional Reconstruction of Semilinear Caputo Type Time-Fractional Systems Using the Analytical Approach

Alaoui

Boutoulout

Zguaid

2021

Advances in the Theory of Nonlinear Analysis and Its Application

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Regional observability for linear time fractional systems

Cited by 16 publications

References 12 publications

Regional Gradient Observability for Fractional Differential Equations with Caputo Time-Fractional Derivatives

Regional Gradient Observability for Fractional Differential Equations with Caputo Time-Fractional Derivatives

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

Regional Reconstruction of Semilinear Caputo Type Time-Fractional Systems Using the Analytical Approach

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