Finite Time Stability of 2D Fractional Hyperbolic System with Time Delay

In this work, a class of two-dimensional fractional hyperbolic di ﬀ erential linear system (2D-FHDLS) with time delay is investigated. By using generalized Gronwall ’ s inequality, su ﬃ cient conditions for the ﬁ nite time stability (FTS) of two-dimensional fractional hyperbolic di ﬀ erential system with time delay are given. Numerical examples are also given to illustrate the stability result.


Introduction
Since twenty years, the area of fractional calculus has gained much attentions by the researchers, and numerous works have been published in this context. In fact, in [1], for a magnetic resonance imaging, a robust corner detection is developed. Authors have made a comparative experiment between the proposed methods and integer-order one. Furthermore, for the Hilfer stochastic delay fractional differential equations with the Poisson jumps, authors in [2] have analyzed the averaging principle. The author in [3] introduced a new approach for solving diffusive systems governed by the Caputo operator. Also, in [4], a hyperchaotic economic system was studied using fractional differential operator. Bayrak et al. in [5] established a novel approach for solving diffusive problems with conformable derivative. A new extension of the Hermite-Hadamard inequalities via generalized fractional integral has been given in [6]. Nagy and Ben Makhlouf in [7] studied the finite time stability of the linear Caputo-Katugampola fractional time delay systems.
Fractional differential equations have recently proved to be valuable tools in the modeling of many phenomena in different domain applications, whether in control theory, diffusion [5], viscoelasticity [8], or biology [9][10][11]. For example, in regard to the biology field, the pandemic transmission model of fractional-order COVID-19 type has been studied numerically by Higazy et al. in [9]. Regarding control theory field, a new adaptive surface control method based on fractional calculus is developed by Zouari et al. in [12]. It was found that all the variables, errors, and signals are practical finite time stability with an asymptotic convergence to zero of the tracking errors. Also, a regional observability for linear fractional systems has been studied in [13]. In [14], Xu et al. proved a global asymptotic stability for fractional neural networks with multiple time varying delay. A finite time stability for a class of fractional fuzzy neural networks with delay has been described and studied in [15]. In addition, authors in [16] have studied the FTS for fractional-order time delay systems.
For some basic results in the theory of fractional partial differential equations, the reader is referred to many various works. For example, for a perturbed partial fractional-order differential equations with finite delay, the Darboux problem is proposed by Abbas and Benchohra in [17]. A nonlinear fractional optimal control problem has been solved by generalized Bernoulli polynomials [18]. Wang and Zhang in [19] studied a Lyapunov inequality for PDE with mixed Caputo derivative. Also, Benchohra and Hellal in [20] proved a global uniqueness results for fractional partial hyperbolic differential equations with delay.
Motivated by the above interpretations, the main objective of this paper is to study the FTS for the linear Darboux fractional partial differential equations with delay or simply, as mentioned above, the 2D-FHDLS with delay. In fact, we were able to establish a new result for the FTS of Caputo 2D-FHDLS with delay. Indeed, thanks to the generalized Gronwall's inequality, we have determined sufficient conditions for the FTS of the 2D-FHDLS with delay. Recall that in [21], we have proved a similar result, but using a fixed point approach. By comparing the two methods, we have shown by numerical tests that the generalized Gronwall's inequality method gives a wider stability interval than that given by fixed point method which proves that generalized Gronwall's inequality method gives very satisfactory stability results.
The paper is organized as follows. In Section 2, some preliminaries are given. In Section 3, FTS results are presented. In Section 4, some numerical examples which show the efficiency of the results were presented.

Main Result
In this paper, we are interested on the study of the initial value fractional-order linear system defined on the bounded domain Q = ½0, T × ½0, T as follows: for all ðξ, ζÞ ∈ Q. The initial condition where C D ν 0 ð·Þ is the CFD of order ν = ðν 1 , ν 2 Þ, 0 < ν 1 , ν 2 < 1. The functions e 1 , e 2 are positive and continuous on ½0, T and ½0, T , respectively. The matrices G, K ∈ ℝ n×n and L ∈ 2 Journal of Function Spaces ℝ n×p and the function ψ ∈ CðU, ℝ n Þ. Here, the domain U is given by where the constants m 1 , m 2 are given by The function δ is a perturbation. We assume that the function δ ∈ Cðℝ 2 + , ℝ p Þ and satisfies Let us introduce the following constants a, b, c which are defined by Definition 6. Let ε > 0 and γ > 0 such that ε < γ. System (9) is robustly FTS with respect to fε, γ, ρ, T , T g, if the following relation is satisfied: for all perturbation δ satisfying equation (9) and condition (13).
Recall that the solution of system (9) is defined by where the functions Ψ, ρ are defined by The main result in this work is as follows. (9) is FTS with respect to fε, γ, ρ, T , T g, ε < γ, if the following inequality holds:
Proof. The solution of system (9) is given by relation (16). Then, we can deduce the following estimation: for all ðξ, ζÞ ∈ Q Let us consider the function yðξ, ζÞ defined on the extended bounded domain J ξζ = ½−m 1 , ξ × ½−m 2 , ζ as follows: We have, for all ðr, μÞ ∈ ½0, ξ × ½0, ζ, the following estimations: Then, for all ðξ, ζÞ ∈ Q, we obtain where the constant M is given by Let us notice that the function is nondecreasing with respect to each of its variables, because yðu, vÞ is nondecreasing with respect to each 3 Journal of Function Spaces of its variables. Then, for all ðr, μÞ ∈ ½0, ξ × ½0, ζ: Then, we get Now, using the generalized Gronwall inequality, we get for all ðξ, ζÞ ∈ Q. The proof is completed.

Remark 8.
Note that a similar result, to that given in Theorem 7, has been proved in ( [21], Theorem 2) by a fixed point method.

Numerical Scheme
From relation (16), we have for all ðξ, ζÞ ∈ ½0, T × ½0, T , where the state ϑ is the solution of system (9), and the functions Ψ, ρ are given by relations (17) and (18). In this section, we study system (9) where ϑ, ρ, Ψ ∈ ℝ 2 . Then, let us assume that the solution ϑ is of the following form: In this section, we use the same techniques of discretization and approximations that we have already used for the numerical resolution of the nonlinear problem in [21]. Thus, we build an uniform grid on the domain ½−m 1 , T × ½−m 2 , T . Let r, h ∈ ℝ and K, J , n, m ∈ ℕ such that Then, we introduce two sequences ðξ i Þ i and ðζ j Þ j defined by ξ i = ir,∀i = −n,−n + 1,−n + 2:⋯,−1, 0, ⋯, K, So, the state ϑ can be expressed at the point ðξ i , ζ j Þ as follows: where ρðξ i , ζ j Þ = ψð0, ζ j Þ + ψðξ i , 0Þ − ψð0, 0Þ. By considering the following approximations, we can rewrite equation (33) as follows: Then, we deduce that ϑ 0j = ρ 0j = ψ 0j , By using the properties of integration, we can rewrite equation (35) as follows: Now, using approximation proposed in [25], we obtain Journal of Function Spaces where we have the approximation Ψðξ k , ζ l Þ ≈ Ψ kl and and the term So, we deduce that Calculating and simplifying the integral in equation (38) gives the following expression: where σ ik , τ lj are given by Remark 9. More details about the convergence, the consistency, and behavior of the error kϑðξ i , ζ j Þ − ϑ ij k with respect to r, h, ν 1 , ν 2 can be deduced from [25].

Numerical Simulation and Interpretation
In this section, we propose some numerical examples. With the help of these examples, we show that the solution of system (9) is consistent with Definition 6. Indeed, for any ε > 0,

Journal of Function Spaces
γ > 0 such that ε < γ, we can verify that System (9) is defined by for all ðξ, ζÞ ∈ ½0, T × ½0, T , ν = ðν 1 , ν 2 Þ. The initial condition is given by Recall that we have denoted the solution ϑ of system (45) as follows: The initial condition