# Generalized Mittag-Leffler Input Stability of the Fractional Differential Equations

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## Abstract

**:**

## 1. Introduction

- the converging-input converging-state
- the bounded-input bounded-state
- the uniform global asymptotic stability of the trivial solution of the unforced fractional differential equation (fractional differential equation without exogenous input).

## 2. Background on Fractional Derivatives

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

**Definition**

**11.**

## 3. New Stability Notion of the Fractional Differential Equations

**Definition**

**12.**

**Definition**

**13.**

## 4. Lyapunov Characterizations of the Generalized Mittag-Leffler input Stability

**Theorem**

**1.**

- 1.
- ${\u2225x\u2225}^{a}\le V(t,x)\le {\chi}_{1}\left(\u2225x\u2225\right).$
- 2.
- If for any$\u2225x\u2225\ge {\chi}_{2}\left(\left(\left|u\right|\right)\right)\u27f9{D}_{c}^{\alpha ,\rho}V(t,x)\le -{\chi}_{3}\left(\left(\u2225x\u2225\right)\right)$.

**Proof.**

**Theorem**

**2.**

- 1.
- ${\u2225x\u2225}^{a}\le V(t,x)\le {\chi}_{1}\left(\u2225x\u2225\right).$
- 2.
- ${D}_{c}^{\alpha ,\rho}V(t,x)\le -kV(x,t)+\gamma \left(\u2225u\u2225\right)$.

**Proof.**

## 5. Practical Applications

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Sene, N.; Srivastava, G.
Generalized Mittag-Leffler Input Stability of the Fractional Differential Equations. *Symmetry* **2019**, *11*, 608.
https://doi.org/10.3390/sym11050608

**AMA Style**

Sene N, Srivastava G.
Generalized Mittag-Leffler Input Stability of the Fractional Differential Equations. *Symmetry*. 2019; 11(5):608.
https://doi.org/10.3390/sym11050608

**Chicago/Turabian Style**

Sene, Ndolane, and Gautam Srivastava.
2019. "Generalized Mittag-Leffler Input Stability of the Fractional Differential Equations" *Symmetry* 11, no. 5: 608.
https://doi.org/10.3390/sym11050608