# Uniform stability of fractional neutral systems: a Lyapunov-Krasovskii functional approach

- KeWei Liu
^{1, 2}Email author and - Wei Jiang
^{1}

**2013**:379

**DOI: **10.1186/1687-1847-2013-379

© Liu and Jiang; licensee Springer. 2013

**Received: **6 September 2013

**Accepted: **2 December 2013

**Published: **27 December 2013

## Abstract

In this paper, we study the stability of nonlinear fractional neutral systems equipped with the Caputo derivative. We extend the Lyapunov-Krasovskii approach for the nonlinear fractional neutral systems. Conditions of uniform stability are obtained for the nonlinear fractional neutral systems.

**MSC:**34K20, 34K37, 34K40.

### Keywords

fractional neutral systems uniform stability Lyapunov-Krasovskii approach## 1 Introduction

In recent decades, fractional calculus and fractional differential equations have attracted great attention. It has been proved that fractional calculus and fractional differential equations are valuable tools in the modeling of many phenomena in various fields of engineering, physics and economics. For details and examples, see [1–5] and the references therein.

Stability analysis is always one of the most important issues in the theory of differential equations and their applications for both deterministic and stochastic cases. The analysis on stability of fractional differential equations is more complex than that of classical differential equations, since fractional derivatives are nonlocal and have weakly singular kernels. Recently, stability of fractional differential equations has attracted increasing interest. The earliest study on stability of fractional differential equations started in [6], the author studied the case of linear fractional differential equations with Caputo derivative and the same fractional order *α*, where $0<\alpha \le 1$. The stability problem comes down to the eigenvalue problem of system matrix. Since then, many researchers have done further studies on the stability of linear fractional differential systems [7–11]. For the nonlinear fractional differential systems, the stability analysis is much more difficult and a few results are available in [12–18]. For more details about the stability results and the methods available to analyze the stability of fractional differential equations, the reader may refer to the recent survey papers [19, 20] and the references therein.

As we all know, Lyapunov’s second method provides a way to analyze the stability of a system without explicitly solving the differential equations. It is necessary to extend Lyapunov’s second method to fractional systems. In [12, 13], the fractional Lyapunov’s second method was proposed, and the authors extended the exponential stability of integer order differential system to the Mittag-Leffler stability of fractional differential system. In [14], by using Bihari’s and Bellman-Gronwall’s inequality, an extension of Lyapunov’s second method for fractional-order systems was proposed. In [15–17], Baleanu *et al.* extended Lyapunov’s method to fractional functional differential systems and developed the Lyapunov-Krasovskii stability theorem, Lyapunov-Razumikhin stability theorem and Mittag-Leffler stability theorem for fractional functional differential systems. As far as we know, there are few papers with respect to the stability of fractional neutral systems. In this paper, we consider the stability of a class of nonlinear fractional neutral functional differential equations with the Caputo derivative. Motivated by Li *et al.* [12, 13], Baleanu *et al.* [15] and Cruz and Hale [21], we aim in this paper to extend the Lyapunov-Krasovskii method for the nonlinear fractional neutral systems.

The rest of the paper is organized as follows. In Section 2, we give some notations and recall some concepts and preparation results. In Section 3, we extend the Lyapunov-Krasovskii approach for the nonlinear fractional neutral systems, results of uniform stability for the nonlinear fractional neutral systems are presented. Conclusions are presented in Section 4.

## 2 Preliminaries

In this section, we introduce notations, definitions, and preliminary facts needed here. Throughout this paper, let ${\mathbb{R}}^{n}$ be a real *n*-dimensional linear vector space with the norm $|\cdot |$, let $\mathcal{C}=C([-r,0],{\mathbb{R}}^{n})$ be the space of continuous functions taking $[-r,0]$ into ${\mathbb{R}}^{n}$ with $\parallel \varphi \parallel $, $\varphi \in \mathcal{C}$ defined by $\parallel \varphi \parallel ={sup}_{s\in [-r,0]}|\varphi (s)|$, $r>0$ be a real constant. If $\sigma \in \mathbb{R}$, $A>0$ and $x\in C([\sigma -r,\sigma +A],{\mathbb{R}}^{n})$, then for any $t\in [\sigma ,\sigma +A]$, we let ${x}_{t}\in \mathcal{C}$ be defined by ${x}_{t}(\theta )=x(t+\theta )$, $\theta \in [-r,0]$.

Let us recall the following known definitions. For more details, we refer the reader to [1, 2, 4, 5].

**Definition 2.1**The fractional order integral of a function $f:[{t}_{0},\mathrm{\infty})\to \mathbb{R}$ of order $\alpha \in {\mathbb{R}}^{+}=[0,+\mathrm{\infty})$ is defined by

where $\mathrm{\Gamma}(\cdot )$ is the gamma function.

**Definition 2.2**For a function

*f*given on the interval $[{t}_{0},\mathrm{\infty})$, the

*α*order Riemann-Liouville fractional derivative of

*f*is defined by

where $\mathbb{N}=\{1,2,3,\dots \}$.

**Definition 2.3**For a function

*f*given on the interval $[{t}_{0},\mathrm{\infty})$, the

*α*order Caputo fractional derivative of

*f*is defined by

Some properties of the aforementioned operators are recalled below [1].

**Property 2.1**

*The following results are especially interesting*:

- (i)
*For*$\nu >-1$,*we have*${D}_{{t}_{0}}^{\alpha}{(t-{t}_{0})}^{\nu}=\frac{\mathrm{\Gamma}(1+\nu )}{\mathrm{\Gamma}(1+\nu -\alpha )}{(t-{t}_{0})}^{\nu -\alpha}$. - (ii)
*When*$n-1<\alpha <n$, $n\in \mathbb{N}$,*we have*${}^{c}D_{{t}_{0}}^{\alpha}f(t)={D}_{{t}_{0}}^{\alpha}[f(t)-\sum _{i=0}^{n-1}\frac{{f}^{(i)}({t}_{0}){(t-{t}_{0})}^{i}}{i!}].$ - (iii)
*For*$\alpha \in (0,1)$, $T\ge {t}_{0}$*and*$f\in C([{t}_{0},T],{\mathbb{R}}^{n})$,*we have*${}^{c}D_{{t}_{0}}^{\alpha}{I}_{{t}_{0}}^{\alpha}f(t)=f(t)$, ${{I}_{{t}_{0}}^{\alpha}}^{c}{D}_{{t}_{0}}^{\alpha}f(t)=f(t)-f({t}_{0})$.

**Remark 2.1**From Property 2.1, if ${}^{c}D_{{t}_{0}}^{\alpha}f(t)\ge 0$, $\alpha \in (0,1)$, then for $t\ge {t}_{0}$, we have

- (i)
$f(t)\ge f({t}_{0})$.

- (ii)
In general, it is not true that $f(t)$ is nondecreasing in

*t*.

In [21], Cruz and Hale studied a class of functional difference operators which are very useful in stability theory and the asymptotic behavior of solutions of functional differential equations of neutral type. In monograph [22], Hale *et al.* presented the following definitions and results of the difference operators.

*X*and

*Y*, $\mathcal{L}(X,Y)$ is the Banach space of bounded linear mappings from

*X*to

*Y*with the operator topology. If $L\in \mathcal{L}(\mathcal{C},{\mathbb{R}}^{n})$, then the Riesz representation theorem implies that there is an $n\times n$ matrix function

*μ*on $[-r,0]$ of bounded variation such that

**Definition 2.4**Let Ω be an open subset of a metric space. We say $L:\mathrm{\Omega}\to \mathcal{L}(\mathcal{C},{\mathbb{R}}^{n})$ has smoothness on the measure if, for any $B\in \mathbb{R}$, there is a scalar function $\gamma (\lambda ,t)$ continuous for $\lambda \in \mathrm{\Omega}$, $t\in \mathbb{R}$, $\gamma (\lambda ,0)=0$, such that if $L(\lambda )\varphi ={\int}_{-r}^{0}[d\mu (\lambda ,\theta )]\varphi (\theta )$, $\lambda \in \mathrm{\Omega}$, $s>0$, then

If $B\in \mathbb{R}$ and the matrix $A(\lambda ;B,L)=\mu (\lambda ,{B}^{+})-\mu (\lambda ,{B}^{-})$ is nonsingular at $\lambda ={\lambda}_{0}$, we say $L(\lambda )$ is atomic at *B* at ${\lambda}_{0}$. If $A(\lambda ;B,L)$ is nonsingular on a set $K\subseteq \mathrm{\Lambda}$, we say $L(\lambda )$ is atomic at *B* on *K*.

**Definition 2.5** Suppose that $\mathrm{\Omega}\subseteq \mathbb{R}\times \mathcal{C}$ is open with elements $(t,\varphi )$. A function $\mathcal{D}:\mathrm{\Omega}\to {\mathbb{R}}^{n}$ (not necessarily linear) is said to be atomic at *B* on Ω if D is continuous together with its first and second Fréchet derivatives with respect to *ϕ*; and ${\mathcal{D}}_{\varphi}$, the derivative with respect to *ϕ*, is atomic at *B* on Ω.

**Remark 2.2**If $\mathcal{D}(t,\varphi )$ is linear in

*ϕ*and continuous in $(t,\varphi )\in \mathbb{R}\times \mathcal{C}$,

*ϕ*and

Thus, $\mathcal{D}(t,\varphi )$ is atomic at *B* on $\mathbb{R}\times \mathcal{C}$ for all $t\in \mathbb{R}$. In particular, if $B\ne 0$, $B\in [-r,0]$, $\mathcal{D}(t,\varphi )=\varphi (0)+F(t)\varphi (B)$, then $A(t,B)=F(t)$ and $\mathcal{D}(t,\varphi )$ is atomic at *B* on $\mathbb{R}\times \mathcal{C}$ if $detF(t)\ne 0$ for all $t\in \mathbb{R}$.

where $0<\alpha <1$, ${t}_{0}\in \mathbb{R}$ is a constant, $\mathcal{D},f:\mathbb{R}\times \mathcal{C}\to {\mathbb{R}}^{n}$ are given continuous functions, nonlinear difference operator $\mathcal{D}$ atomic at zero. For more details about the operator $\mathcal{D}$, the reader may refer to [21, 22, 24] and the references therein. In the sequel, we always assume that, for any given ${t}_{0}\in \mathbb{R}$ and a given function $\phi \in \mathcal{C}$, there exists a unique continuous solution of (2.1), denoted by $x(t)=x(t,{t}_{0},\phi )$, such that it satisfies (2.1) for all $t>{t}_{0}$ and (2.2). To deal with stability, as usual, we assume that $\mathcal{D}(t,0)=f(t,0)=0$ so that (2.1) has the zero solution.

**Definition 2.6** [23]

The zero solution $x=0$ of (2.1) is stable if for any ${t}_{0}\in \mathbb{R}$ and any $\epsilon >0$, there exists $\delta =\delta ({t}_{0},\epsilon )$ such that any solution $x(t)=x(t,{t}_{0},\phi )$ of (2.1) with initial value *φ* at ${t}_{0}$, $\parallel \phi \parallel <\delta $ satisfies $|x(t)|<\epsilon $ for $t\ge {t}_{0}$. It is asymptotically stable if it is stable and for any ${t}_{0}\in \mathbb{R}$ and any $\epsilon >0$, there exists ${\delta}_{0}=\delta ({t}_{0},\epsilon )$, $T({t}_{0},\epsilon )>0$ such that $\parallel \phi \parallel <{\delta}_{0}$ implies $|x(t)|\le \epsilon $ for $t\ge {t}_{0}+T({t}_{0},\epsilon )$, *i.e.*, ${lim}_{t\to +\mathrm{\infty}}x(t)=0$. It is uniformly stable if it is stable and $\delta =\delta (\epsilon )>0$ can be chosen independently of ${t}_{0}$. It is uniformly asymptotically stable if it is uniformly stable and there exists ${\delta}_{0}>0$ for any $\eta >0$, there exists $T=T(\eta )>0$ such that $\parallel \phi \parallel <{\delta}_{0}$ implies $|x(t)|<\eta $ for $t>{t}_{0}+T$. It is globally (uniformly) asymptotically stable if it is (uniformly) asymptotically stable and ${\delta}_{0}$ can be an arbitrary large, finite number.

For a nonlinear operator $\mathcal{D}$, in [24], Zhang gave the following definition.

**Definition 2.7**$\mathcal{D}(t,\varphi )$ is said to be uniformly stable if there exist positive constants

*a*,

*b*,

*c*such that for any $h:\mathbb{R}\to {\mathbb{R}}^{n}$ and any $\sigma \in \mathbb{R}$, $\phi \in \mathbb{C}$, with $\mathcal{D}(\sigma ,\phi )=h(\sigma )$, the solution $x(t,\sigma ,\phi )$ of

The following lemma plays a major role in our analysis.

**Lemma 2.1**

*Let*$\mathcal{D}(t,\varphi )$

*be uniformly stable*, ${x}_{t}={x}_{t}(\sigma ,\phi )$

*be the solution of equation*(2.3).

*Suppose that*$\alpha (t)$

*is any continuous and nondecreasing function with*$\alpha (0)=0$,

*and*$\alpha (t)>0$

*for*$t>0$.

*Then*,

*for small*$s>0$,

*there is a continuous and strictly increasing function*$\beta (s)\ge bs+c\alpha (s)$, $\beta (0)=0$, $\beta (t)>0$

*for*$t>0$

*such that*

- (i)
*for each small*$\delta >0$, $\parallel \phi \parallel <\delta $, $|\mathcal{D}(t,{x}_{t})|<\alpha (\delta )$, $t\ge \sigma $,*then*$\parallel {x}_{t}\parallel \le \beta (\delta ),\phantom{\rule{1em}{0ex}}t\ge \sigma .$(2.5) - (ii)
*for each small*$\delta >0$, $\mu >0$*and a nonnegative constant**L*,*there exists*$T(\mu ,\delta ,L)>0$*such that*$\parallel \phi \parallel <\delta $*and*$|\mathcal{D}(t,{x}_{t})|\le \alpha (\mu )$, $t\ge \sigma $*imply*$\parallel {x}_{t}\parallel \le \beta (\mu )+L,\phantom{\rule{1em}{0ex}}t\ge \sigma +T(\mu ,\delta ,L).$(2.6)

*Proof*(i) From Definition 2.5, for sufficiently small $\delta >0$ and $t\ge \sigma $, we have

Trivially, we can choose a continuous and increasing function $\beta (\delta )\ge b\delta +c\alpha (\delta )$, $\beta (0)=0$, $\beta (t)>0$ for $t>0$, so that (2.5) holds.

Therefore, if we take $T(\mu ,\delta ,L)=max\{0,\frac{1}{a}ln\frac{b\delta}{b\mu +L}\}$, then (2.6) holds. □

## 3 Main results

*φ*. By Property 2.1(iii), we can obtain that initial value problem (2.1)-(2.2) is equivalent to the integral equation

Now, we give the following Lyapunov-Krasovskii methods for nonlinear fractional neutral systems as counterpart to Lyapunov-Krasovskii methods for classical neutral systems proposed in [21].

**Theorem 3.1**

*Suppose that*$\mathcal{D}(t,\varphi )$

*is uniformly stable*,

*f*

*takes closed bounded sets into bounded sets*,

*and suppose that*$u(s)$, $v(s)$

*are strictly increasing functions with*$u(0)=v(0)=0$,

*and*$w(s)$

*is a continuous*,

*nonnegative*,

*nondecreasing function*.

*If there exists a continuously differentiable functional*$V:\mathbb{R}\times \mathcal{C}\to \mathbb{R}$

*such that*

*where* $\gamma \in (0,1]$. *Then the zero solution of* (2.1) *is uniformly stable*. *If*, *in addition*, $w(s)>0$ *for* $s>0$, *then it is uniformly asymptotically stable*.

*Proof*It is possible to choose a continuous function $\alpha (t)$ so that $\alpha (0)=0$ and $\alpha (t)>{u}^{-1}(v(t))\ge 0$ for small $t>0$. Then $v(t)<u(\alpha (t))$ for small $t>0$. For the above-chosen $\alpha (t)$, by Lemma 2.1, we can find a corresponding $\beta (t)$ with the desired properties. Now, for any $\epsilon >0$, we can find a sufficiently small

*δ*such that $\beta (\delta )<\epsilon $. Hence, for any initial time ${t}_{0}$ and any initial condition ${x}_{{t}_{0}}=\phi $ with $\parallel \phi \parallel <\delta $, (3.2) implies

Therefore, the zero solution is uniformly stable.

*f*takes bounded sets into bounded sets, there is a constant ${L}_{0}$ such that $|f(t,{x}_{t}({t}_{0},\phi ))|\le {L}_{0}$ for $t\ge {t}_{0}$, $\parallel \phi \parallel \le {\delta}_{0}$. Then

where $\theta =min\{r,{[\frac{\delta \mathrm{\Gamma}(1+\alpha )}{4b{L}_{0}}]}^{\frac{1}{\alpha}}\}$.

This proves the uniform asymptotic stability of the zero solution of (2.1). □

**Remark 3.1** From the proof of inequality (3.8), we can know that the analysis on stability of fractional differential equations is more complex than that of classical differential equations, since fractional derivatives are nonlocal and have weakly singular kernels.

**Remark 3.2** If $\mathcal{D}(t,\varphi )$ is linear in *ϕ* and $\alpha =\gamma =1$, Theorem 3.1 is just the same as Theorem 4.1 in [21].

**Remark 3.3** If $\mathcal{D}(t,\varphi )=\varphi (0)$, the conclusions of Theorem 3.1 are just the same as the corresponding conclusions of Theorem in [15].

**Theorem 3.2** *Suppose that the assumptions in Theorem * 3.1 *are satisfied except replacing* ${}^{c}D_{{t}_{0}}^{\gamma}$ *by* ${D}_{{t}_{0}}^{\gamma}$, *then one has the same result for uniform stability and uniform asymptotic stability*.

*Proof*By using Property 2.1, we have

Since $V({t}_{0},\phi )\ge 0$, then ${}^{c}D_{{t}_{0}}^{\gamma}V(t,{x}_{t})\le {D}_{{t}_{0}}^{\gamma}V(t,{x}_{t})$. Then we can obtain the same result for uniform stability and uniform asymptotic stability. □

## 4 Conclusions

In this paper, we have studied the stability of nonlinear fractional order neutral systems. We introduce the Lyapunov-Krasovskii approach for the nonlinear fractional neutral systems, which enrich the knowledge of both the system theory and the fractional calculus. We partly extend the application of Caputo fractional systems by using Lyapunov-Krasovskii approach. By using Caputo and Riemann-Liouville derivatives and Lyapunov-Krasovskii technique, uniform stability criteria are obtained for the nonlinear fractional neutral systems. The obtained conclusions generalize the corresponding conclusions in [15, 21].

## Declarations

### Acknowledgements

This work is supported by the National Natural Science Foundation of China (11371027), the Fundamental Research Funds for the Central Universities (2013HGXJ0226) and the Fund of Anhui University Graduate Academic Innovation Research (10117700004).

## Authors’ Affiliations

## References

- Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In
*Theory and Applications of Fractional Differential Equations*. Elsevier, Amsterdam; 2006.Google Scholar - Miller KV, Ross B:
*An Introduction to the Fractional Calculus and Fractional Differential Equations*. Wiley, New York; 1993.MATHGoogle Scholar - Monje CA, Chen YQ, Vinagre BM, Xue D, Feliu V:
*Fractional-Order Systems and Controls: Fundamentals and Applications*. Springer, London; 2010.View ArticleMATHGoogle Scholar - Podlubny I:
*Fractional Differential Equations*. Academic Press, San Diego; 1999.MATHGoogle Scholar - Samko SG, Kilbas AA, Marichev OI:
*Fractional Integrals and Derivatives: Theory and Applications*. Gordon and Breach, Yverdon; 1993.MATHGoogle Scholar - Matignon, D: Stability results for fractional differential equations with applications to control processing. In: IMACS-SMC Proceedings, Lille, France, July 1996
- Chen YQ, Moore KL: Analytical stability bound for a class of delayed fractional-order dynamic systems.
*Nonlinear Dyn.*2002, 29: 191-200. 10.1023/A:1016591006562MathSciNetView ArticleMATHGoogle Scholar - Deng WH, Li CP, Lü JH: Stability analysis of linear fractional differential system with multiple time delays.
*Nonlinear Dyn.*2007, 48(4):409-416. 10.1007/s11071-006-9094-0View ArticleMathSciNetMATHGoogle Scholar - Lazarevic MP, Spasic AM: Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach.
*Math. Comput. Model.*2009, 49: 475-481. 10.1016/j.mcm.2008.09.011MathSciNetView ArticleMATHGoogle Scholar - Liu KW, Jiang W: Finite-time stability of linear fractional order neutral systems.
*Math. Appl.*2011, 24(4):724-730.MathSciNetMATHGoogle Scholar - Sabatier J, Moze M, Farges C: LMI stability conditions for fractional order systems.
*Comput. Math. Appl.*2010, 59: 1594-1609. 10.1016/j.camwa.2009.08.003MathSciNetView ArticleMATHGoogle Scholar - Li Y, Chen YQ, Podlubny I: Mittag-Leffler stability of fractional order nonlinear dynamic systems.
*Automatica*2009, 45(8):1965-1969. 10.1016/j.automatica.2009.04.003MathSciNetView ArticleMATHGoogle Scholar - Li Y, Chen YQ, Podlubny I: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability.
*Comput. Math. Appl.*2010, 59: 1810-1821. 10.1016/j.camwa.2009.08.019MathSciNetView ArticleMATHGoogle Scholar - Delavari H, Baleanu D, Sadati J: Stability analysis of Caputo fractional-order nonlinear systems revisited.
*Nonlinear Dyn.*2012, 67(4):2433-2439. 10.1007/s11071-011-0157-5MathSciNetView ArticleMATHGoogle Scholar - Baleanu D, Ranjbar A, Delavari H, Sadati JR, Abdeljawad T, Gejji V: Lyapunov-Krasovskii stability theorem for fractional systems with delay.
*Rom. J. Phys.*2011, 56: 636-643.MathSciNetMATHGoogle Scholar - Baleanu D, Sadati SJ, Ghaderi R, Ranjbar A, Abdeljawad T, Jarad F: Razumikhin stability theorem for fractional systems with delay.
*Abstr. Appl. Anal.*2010., 2010: Article ID 124812Google Scholar - Sadati SJ, Baleanu D, Ranjbar A, Ghaderi R, Abdeljawad T: Mittag-Leffler stability theorem for fractional nonlinear systems with delay.
*Abstr. Appl. Anal.*2010., 2010: Article ID 108651Google Scholar - Zhang FR, Li CP, Chen YQ: Asymptotical stability of nonlinear fractional differential system with Caputo derivative.
*Int. J. Differ. Equ.*2011., 2011: Article ID 635165Google Scholar - Li CP, Zhang FR: A survey on the stability of fractional differential equations.
*Eur. Phys. J. Spec. Top.*2011, 193: 27-47. 10.1140/epjst/e2011-01379-1View ArticleGoogle Scholar - Rivero M, Rogosin SV, Machado JAT, Trujillo JJ: Stability of fractional order systems.
*Math. Probl. Eng.*2013., 2013: Article ID 356215Google Scholar - Cruz MA, Hale JK: Stability of functional differential equations of neutral type.
*J. Differ. Equ.*1970, 7: 334-355. 10.1016/0022-0396(70)90114-2MathSciNetView ArticleMATHGoogle Scholar - Hale JK, Verduyn Lunel SM:
*Introduction to Functional Differential Equations*. Springer, New York; 1993.View ArticleMATHGoogle Scholar - Gu K, Kharitonov VL, Chen J:
*Stability of Time-Delay Systems*. Birkhäuser, Boston; 2003.View ArticleMATHGoogle Scholar - Zhang SN: Unified stability theorem in RFDE and NFDE.
*J. Math. Anal. Appl.*1990, 150: 397-413. 10.1016/0022-247X(90)90112-SMathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.