A Novel Delay-Dependent Asymptotic Stability Conditions for Differential and Riemann-Liouville Fractional Differential Neutral Systems with Constant Delays and Nonlinear Perturbation

: The novel delay-dependent asymptotic stability of a differential and Riemann-Liouville fractional differential neutral system with constant delays and nonlinear perturbation is studied. We describe the new asymptotic stability criterion in the form of linear matrix inequalities (LMIs), using the application of zero equations, model transformation and other inequalities. Then we show the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with constant delays. Furthermore, we not only present the improved delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with single constant delay but also the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral equation with constant delays. Numerical examples are exploited to represent the improvement and capability of results over another research as compared with the least upper bounds of delay and nonlinear perturbation.


Introduction
Differential systems, or more generally functional differential systems, have been studied rather extensively for at least 200 years and are used as models to describe transportation systems, communication networks, teleportation systems, physical systems and biological systems, and so forth. Parts of fractional-order systems have not received much attention by reason of absence of appropriate utilization circumstances over the past 300 years. However, during the last 10 years fractional-order systems have been widely investigated as they have the qualification to explain various phenomena more precisely in many fields, for example, biological models, material science, finance, cardiac tissues, quantum mechanics, viscoelastic systems, medicine and fluid mechanics [1][2][3][4][5][6][7][8]. Caputo fractional differential systems have been studied in many types of stability such as uniform stability [9], Mittag-Leffler stability [10][11][12][13], Ulam stability [14], finite time stability [15,16] and asymptotic stability [17,18]. Nevertheless, the stability of Riemann-Liouville fractional differential systems is seldom considered, see References [19,20].
The neutral systems with time delays have already been applied in many fields, such as heartbeat, memorization, locomotion, mastication and respiration, see References [21][22][23][24]. Accordingly, the issue

Problem Formulation and Preliminaries
We introduce a differential and fractional differential neutral system with constant delays and nonlinear perturbation for 0 < q ≤ 1, the state vector x(t) ∈ R n , A, B, C are symmetric positive definite matrices with C < 1, τ, σ are positive real constants and ∈ C([−κ, 0]; R n ) with κ = max{τ, σ}. The uncertainty f (.) represents the nonlinear parameter perturbation satisfying where δ, η are given constants. Next, the Riemann-Liouville fractional integral and derivative [36] are defined as, respectively Lemma 2. [17] For a vector of differentiable function x(t) ∈ R n , positive semi-definite matrix K ∈ R n×n and 0 < q < 1, then for all t ≥ t 0 .

Main Results
Consider the asymptotic stability for system (1) with constant delays and nonlinear perturbation. We define a new variable Rewrite the Equation (1) in the following equation Theorem 1. Let δ and η be positive scalars, if there are any appropriate dimensions matrices Q j (j = 1, 2, 3) and symmetric positive definite matrices K i (i = 1, 2, 3, 4, 5) such that satisfy where Then the system (1) is asymptotically stable.
Proof of Theorem 1. For symmetric positive definite matrices K i (i = 1, 2, 3, 4, 5) and any appropriate dimensions matrices Q j (j = 1, 2, 3). Consider the Lyapunov-Krasovskii functional for Computing the differential of V(t) on the solution of system (1) The differential of V 1 (t) is computed by Lemma 2 Taking the differential of V 2 (t), we obtaiṅ Next, from (3), we obtain According to (13), (14) and (15), we can conclude thaṫ where Since linear matrix inequality (10) holds, then the system (1) is asymptotic stability.

Corollary 2.
If there are any appropriate dimensions matrices Q j (j = 1, 2, 3) and symmetric positive definite matrices K i (i = 1, 2, 3) such that satisfy Then the Equation (20) is asymptotically stable.

Numerical Examples
Example 1. The fractional neutral system : Solving the LMI ( Moreover, the least upper bound of the parameter σ that ensures the asymptotic stability of system (27) is 1.3227 when η = 5 × 10 3 and δ = 1. Table 1 represents the least upper bound σ of this example for various values of η, δ.
Solving the LMI (18)  −56.7801 , Moreover, the least upper bound of the parameter τ that ensures the asymptotic stability of system (28) is 3.7 × 10 22 . Example 3. The fractional neutral system : Solving the LMI (21) when A = 3 −1 0 1 , B = 0.2 0.1 0 0.1 , C = 0.1 0 0 0.2 , we obtain the least upper bound of the parameter τ that ensures the asymptotic stability is 2.86 × 10 24 . By the criterion in [35], the least upper bound of the parameter τ is 2.99 × 10 21 . This example represents our result is less conservative than these in [35].
Solving the LMI (25), we have a set of parameters that ensures asymptotic stability of Equation ( Table 4 represents the least upper bound b of this example for various values of a, p.

Conclusions
The aim of this paper is a novel asymptotic stability analysis of differential and Riemann-Liouville fractional differential neutral systems with constant delays and nonlinear perturbation by applying zero equations, model transformation and other inequalities. The new asymptotic stability condition is given in the form of LMIs. Then we show the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with constant delays. Furthermore, we propose the improved delay-dependent asymptotic stability criterion of differential and Riemann-Liouville fractional differential neutral systems with single constant delay and the new delay-dependent asymptotic stability criterion of differential and Riemann-Liouville fractional differential neutral equations with constant delays. Numerical examples illustrate the advantages and applicability of our results.