# Stability Concepts of Riemann-Liouville Fractional-Order Delay Nonlinear Systems

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## Abstract

**:**

## 1. Introduction

- -
- the initial conditions connected with the RL fractional derivative are set up in an appropriate way;
- -
- new types of stability connected with the type of initial conditions are defined;
- -
- the RL fractional modification of the Razumikhin method is presented;
- -
- new sufficient conditions for the defined stability are obtained;
- -
- two types of fractional derivatives of the Lyapunov functions are used.

## 2. Preliminary Notes

- -
- Riemann–Liouville fractional integral of order $q>0$$${\phantom{\rule{4pt}{0ex}}}_{0}{I}_{t}^{q}m\left(t\right)=\frac{1}{\Gamma \left(q\right)}\underset{0}{\overset{t}{\int}}\frac{m\left(s\right)}{{(t-s)}^{1-q}}ds,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t\ge 0,$$Note that the notation ${}_{0}{D}_{t}^{-\alpha}m\left(t\right)={\phantom{\rule{4pt}{0ex}}}_{0}{I}_{t}^{\alpha}m\left(t\right)$ is sometimes used.
- -
- Riemann–Liouville fractional derivative of order $q\in (0,1)$$${}_{0}^{RL}{D}_{t}^{q}m\left(t\right)=\frac{1}{\Gamma \left(1-q\right)}\frac{d}{dt}\underset{{t}_{0}}{\overset{t}{\int}}{\left(t-s\right)}^{-q}m\left(s\right)ds,\phantom{\rule{4pt}{0ex}}t\ge 0$$
- -
- Grünwald–Letnikov fractional derivative of order $q\in (0,1)$$${}_{0}^{GL}{D}_{t}^{q}m\left(t\right)=\underset{h\to 0+}{\mathrm{lim}}\frac{1}{{h}^{q}}\sum _{j=0}^{\frac{\left[t\right]}{h}}{(-1)}^{j}{\phantom{\rule{4pt}{0ex}}}_{q}{C}_{j}\phantom{\rule{4pt}{0ex}}m(t-jh),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t\ge 0,$$$${\phantom{\rule{4pt}{0ex}}}_{q}{C}_{j}=\frac{q(q-1)(q-2)\cdots (q-j-1)}{j!}=\frac{\Gamma (q+1)}{j!\Gamma (q-j+1)}.$$

**Remark**

**1.**

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3**

**Remark**

**2.**

**Proposition**

**4**

**Proposition**

**5.**

**Remark**

**3.**

**Lemma**

**1**

- (a)
- If there exists a.e. a limit ${\mathrm{lim}}_{t\to 0+}\left[{t}^{1-q}m\left(t\right)\right]=c\in \mathbb{R}$, then there also exists a limit$${\phantom{\rule{4pt}{0ex}}}_{0}{I}_{t}^{1-q}m\left(t\right){{|}_{t=0}={\phantom{\rule{4pt}{0ex}}}_{0}{D}_{t}^{q-1}m\left(t\right)|}_{t=0}:=\underset{t\to 0+}{\mathrm{lim}}\frac{1}{\Gamma \left(1-q\right)}\underset{0}{\overset{t}{\int}}\frac{m\left(s\right)}{{(t-s)}^{q}}ds=c\Gamma \left(q\right).$$
- (b)
- If there exists a.e. a limit ${\mathrm{lim}}_{t\to 0+}{\phantom{\rule{4pt}{0ex}}}_{0}{I}_{t}^{1-q}m\left(t\right)=c\in \mathbb{R},$ and if there exists the limit ${\mathrm{lim}}_{t\to 0+}\left[{t}^{1-q}m\left(t\right)\right]$, then$$\underset{t\to 0+}{\mathrm{lim}}\left[{t}^{1-q}m\left(t\right)\right]=\frac{c}{\Gamma \left(q\right)}.$$

## 3. Statement of the Problem

**Remark**

**4.**

**Hypothesis**

**1**

**Hypothesis**

**2**

**Remark**

**5.**

**Definition**

**1.**

- -
- stable in time if for any number $\epsilon >0$ there exist numbers $\delta >0$ and ${T}_{\epsilon}>0$ depending on ε such that for any initial functions $\varphi \in {C}_{0}{:\phantom{\rule{4pt}{0ex}}\left|\right|\varphi \left|\right|}_{0}<\delta $, the corresponding solution $x(t;\varphi )$ of IVP (2) and (3) satisfies $\left|\right|x(t;\varphi )\left|\right|<\epsilon $ for $t\ge {T}_{\epsilon}$;
- -
- asymptotically stable if it is stable in time and additionally $x(t;\varphi )\to 0$ as $t\to +\infty $;
- -
- generalized Mittag-Leffler stable in time if there exist positive numbers $\lambda ,b$ and $\gamma \in (0,1)$ and a locally Lipschitz function $h\in C\left(\right[0,\infty ),[0,\infty \left)\right):\phantom{\rule{4pt}{0ex}}h\left(0\right)=0$ such that for any $\epsilon >0$, there exists ${T}_{\epsilon}>0$ the solution of IVP (2) and (3) satisfies$$\left|\right|x(t;\varphi )\left|\right|\le {\epsilon \left\{h\right(\left|\right|\varphi \left|\right|}_{0}{\left){t}^{-\gamma}{E}_{q,1-\gamma}(-\lambda {t}^{q})\right\}}^{b},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t\ge {T}_{\epsilon}.$$

**Example**

**1.**

## 4. Stability of Nonlinear RL Fractional Differential Equations

#### 4.1. Lyapunov Functions and Their Derivatives

**Definition**

**2**

- -
- RL fractional derivative—Let $x\left(t\right):[0,T)\to \Delta $ be a solution of the IVP for the RLFrDDE (2) and (3). Then, we consider$${\phantom{\rule{4.pt}{0ex}}}_{0}^{RL}{D}^{q}V(t,x\left(t\right))=\frac{1}{\Gamma \left(1-q\right)}\frac{d}{dt}\underset{0}{\overset{t}{\int}}{\left(t-s\right)}^{-q}V(s,x\left(s\right))ds,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t\in [0,T).$$
- -
- Dini fractional derivative—Let $\varphi :[-\tau ,0]\to \Delta $. Then, consider (see [8])$$\begin{array}{cc}& {D}_{\left(2\right)}^{+}V(t,\varphi \left(0\right),\varphi )\hfill \\ & =\underset{h\to 0}{\mathrm{lim}\mathrm{sup}}\frac{1}{{h}^{q}}\left[V(t,\varphi \left(0\right))-\sum _{r=1}^{\left[\frac{t}{h}\right]}{(-1)}^{r+1}{\phantom{\rule{4pt}{0ex}}}_{q}{C}_{r}V(t-rh,\varphi \left(0\right)-{h}^{q}f(t,{\varphi}_{0}))\right],\hfill \end{array}$$The Dini fractional derivative is applicable for continuous Lyapunov functions.

**Example**

**2.**

**Remark**

**6.**

#### 4.2. Stability by the RL Fractional Derivative of Lyapunov Functions

**Theorem**

**1.**

- (i)
- for any $\epsilon >0$ there exists ${T}_{\epsilon}>0$ such that$$\epsilon a\left(\right|\left|x\right|\left|\right)\le V(t,x)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}t>{T}_{\epsilon},\phantom{\rule{4pt}{0ex}}x\in {\mathbb{R}}^{n},$$
- (ii)
- there exists an increasing function $g\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ such that for any function $y\in {C}_{1-q}([0,\infty ),{\mathbb{R}}^{n}):\phantom{\rule{4pt}{0ex}}{t}^{1-q}{y\left(t\right)|}_{t=0+}={y}_{0}$ the inequality$${t}^{1-q}{V(t,y\left(t\right))|}_{t=0+}=\underset{t\to 0+}{\mathrm{lim}}{t}^{1-q}V(t,y\left(t\right))\le g\left(\right||{y}_{0}\left|\right|)$$holds;
- (iii)
- for any point $t>0$ such that ${(t+\Theta )}^{1-q}V(t+\Theta ,x(t+\Theta ))<{t}^{1-q}V(t,x\left(t\right))$ for $\Theta \in (-\mathrm{min}\{t,\tau \},0)$, the RL fractional derivative ${\phantom{\rule{4.pt}{0ex}}}_{0}^{RL}{D}_{t}^{q}V(t,x\left(t\right))$ exists and the inequality$${\phantom{\rule{4.pt}{0ex}}}_{0}^{RL}{D}_{t}^{q}V(t,x\left(t\right))=\frac{1}{\Gamma \left(1-q\right)}\frac{d}{dt}\underset{0}{\overset{t}{\int}}{\left(t-s\right)}^{-q}V(s,x\left(s\right))ds<0,$$

**Proof.**

**Remark**

**7.**

**Remark**

**8.**

**Remark**

**9.**

**Theorem**

**2.**

**Proof.**

#### 4.3. Stability by the Dini Fractional Derivative of Lyapunov Functions

**Lemma**

**2.**

- 1.
- 2.
- The function $V\in \mathsf{\Lambda}\left(\right[-\tau ,T],\Delta )$, $\Delta \subset {\mathbb{R}}^{n}$, is such that:
- (i)
- There exists an increasing function $g\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ such that the inequality ${t}^{1-q}{V(t,x\left(t\right))|}_{t=0+}={\mathrm{lim}}_{t\to 0+}{t}^{1-q}V(t,x\left(t\right))\le g\left(\frac{\left|\right|\varphi \left(0\right)\left|\right|}{\Gamma \left(q\right)}\right)\le {g\left(\right|\left|\varphi \right||}_{0})$ holds;
- (ii)
- for any point $t>0$ such that $V(t+\Theta ,x(t+\Theta \left)\right)<V(t,x(t\left)\right)$ for $\Theta \in (-\mathrm{min}\{t,\tau \},0)$, the inequality$${D}_{\left(2\right)}^{+}V(t,\psi \left(0\right),\psi )<0$$

Then, $V(t,x(t;\varphi ))\le {t}^{q-1}{g\left(\right|\left|\varphi \right||}_{0})$ for $t\in [0,T].$

**Remark**

**10.**

**Remark**

**11.**

**Proof.**

**Theorem**

**3.**

- (i)
- for any $\epsilon >0$, there exists ${T}_{\epsilon}>0$ such that$$\epsilon a\left(\right|\left|x\right|\left|\right)\le V(t,x)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}t>{T}_{\epsilon},\phantom{\rule{4pt}{0ex}}x\in {\mathbb{R}}^{n},$$
- (ii)
- there exists an increasing function $g\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ such that for any function $y\in {C}_{1-q}([0,\infty ),{\mathbb{R}}^{n}):\phantom{\rule{4pt}{0ex}}{t}^{1-q}{y\left(t\right)|}_{t=0+}={y}_{0}$ the inequality$${t}^{1-q}{V(t,y\left(t\right))|}_{t=0+}=\underset{t\to 0+}{\mathrm{lim}}{t}^{1-q}V(t,y\left(t\right))\le g\left(\right||{y}_{0}\left|\right|)$$holds;
- (iii)
- for any function $\psi \in C([-\tau ,0],{\mathbb{R}}^{n}$ such that if for a point t we have $V(t+\Theta ,\psi (\Theta \left)\right)<V(t,\psi (0\left)\right)$ for $\Theta \in [-\tau ,0)$, then the inequality$${D}_{\left(2\right)}^{+}V(t,\psi \left(0\right),\psi )<0$$holds.Then, the zero solution of (2) with the zero initial function is stable.

**Proof.**

**Example**

**3.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Agarwal, R.; Hristova, S.; O’Regan, D.
Stability Concepts of Riemann-Liouville Fractional-Order Delay Nonlinear Systems. *Mathematics* **2021**, *9*, 435.
https://doi.org/10.3390/math9040435

**AMA Style**

Agarwal R, Hristova S, O’Regan D.
Stability Concepts of Riemann-Liouville Fractional-Order Delay Nonlinear Systems. *Mathematics*. 2021; 9(4):435.
https://doi.org/10.3390/math9040435

**Chicago/Turabian Style**

Agarwal, Ravi, Snezhana Hristova, and Donal O’Regan.
2021. "Stability Concepts of Riemann-Liouville Fractional-Order Delay Nonlinear Systems" *Mathematics* 9, no. 4: 435.
https://doi.org/10.3390/math9040435