Lyapunov method for nonlinear fractional differential systems with delay

Abstract

This paper deals with the stability of nonlinear fractional differential systems with delay. Based on the Lyapunov functional method and the Lyapunov function method, respectively, several stability criteria including Razumikhin-type stability criteria are derived, which are extensions of some existed results of the Hale and Verduyn Lunel (Introduction to functional differential equations. Springer, Berlin, 1993), Aguila-Camacho et al. (Commun Nonlinear Sci Numer Simul 19, 2951–2957, 2014) and Zhou et al. (Appl Math Lett 28, 25–29, 2014). In addition, some examples are provided to illustrate the applications of these criteria. Numerical simulations show the validity of our results.

1 Introduction

In the past few decades, many researchers have devoted themselves to fractional calculus because many practical systems can be properly described with the help of fractional derivative. For the basic theory of fractional calculus and fractional differential systems (equations), one can see the monographs [1–6] and the references therein.

In the real dynamical systems, delayed phenomenon is unavoidable. Fractional delayed differential systems have received considerable attentions recently because they are more accurate in modeling memory phenomenon and hereditary phenomenon. In [7–14], the authors dealt with the existence of the solution of fractional functional differential systems (equations).

The stability of the equilibrium point of a system is of importance in control theorem. Stability criteria for linear fractional delayed differential systems (equations) can be found in [15–18]. Stability of nonlinear fractional delayed differential systems (equations) was investigated in [19–25].

Lyapunov’s direct method is an effective tool to analyze the stability of nonlinear integer-order differential systems without solving its state equations. It used to be thought that Lyapunov method is invalid to nonlinear fractional differential systems. However, this thought was proved to be wrong. In 2010, professor chen et al. proposed a Lyapunov function to deal with the stability of a class of nonlinear fractional differential systems [26]. In 2014, based on a lemma themselves and a theorem in the Ref. [26], Aguila-Camacho et al. [27] derived a simple stability criterion for a class of nonlinear fractional differential system. Corollary 1 in the Ref. [27] is similar to Theorem 3.1 in the Ref. [28].

Recently, Lyapunov function has been chosen to analyze the stability of nonlinear fractional differential systems with delay. For example, the authors in [20] proposed a Lyapunov function to deal with the Mittag–Leffler stability of the nonlinear fractional neutral differential system

$$\begin{aligned} {{\hbox {}^C}}D_{t_0,t}^{q}(Ex(t)-Ax(t-r))=f(t,x_{t}), \end{aligned}$$

(1)

where $x(t)\in R^n$, $0<q\le 1$, $E, A\in R^{n\times n}$ are constant matrices and E may be singular $(rank E=n_1\le n )$. Further assume that matrix pair (E, A) is regular with index one, $f:R\times (\hbox {bounded sets in C})$ into bounded sets of $R^n$. One of the results in [20] is as follows.

Suppose that the operator D is stable, and $\alpha _1$, $\alpha _2$ and $\alpha _3$ are positive numbers. If there exists a continuous function $V:[t_0, \infty )\times C\rightarrow R^n$ such that

• (a) $\alpha _1\Vert x\Vert \le V(t,x) \le \alpha _2\Vert x\Vert$;

• (b) ${\hbox {}^C}_{t_0}D_{t}^{\beta }V(t, Dx_t)\le -\alpha _3 \Vert Dx_t\Vert$, where $0<\beta \le 1$, then the solution of system (1) is Mittag–Leffler stable.

By proposing a Lyapunov function and taking its integer derivative, the authors in [25] derived a stability criterion for the fractional delayed differential system

$$\begin{aligned} {\hbox {}{\hbox {}^C}}D_{t_0,t}^{\alpha }x(t)=f(x(t),x(t-\tau )), \quad t\ge t_0 \end{aligned}$$

(2)

where $x\in R^n$ is the state vector, $f\in R^n$ is nonlinear vector function satisfying Lipschitz condition and the delay time $\tau =[\tau _1,\tau _2, \ldots , \tau _n]\in R^n$ and $t_0\in R^+$. The stability criterion of the Ref. [25] is as follows:

Suppose that fractional order $0<\alpha \le 1$. If there is a positive definite matrix P and a semi-positive definite matrix Q, for any $x(t)\in R^n$, fractional system (2) satisfies

$$\begin{aligned}&x^\mathrm{T}(t) P\cdot {{{\hbox {}^C}}D}_{a,t}^{\alpha }x(t)+x^\mathrm{T}(t)Qx(t)\nonumber \\&\quad -\,x^\mathrm{T} (t-\tau )Qx(t-\tau )\le 0, \end{aligned}$$

(3)

the fractional delayed system (2) is Lyapunov stable.

Obviously, the stability criterion for the system (2) is inconvenient for application.

Lyapunov functional method is usually used to analyze the stability and designs of switching controller of nonlinear integer-order functional differential equations (systems) [29–32]. However, at present we have not retrieved any related references in which Lyapunov functional was used to investigate the stability of fractional differential systems with delay. Motivated by this fact, in this paper we mainly apply Lyapunov functional method to investigate the stability of nonlinear fractional delayed differential system of the form

$$\begin{aligned} {{\hbox {}^C}}D_{t_0,t}^{\alpha }x(t)=f(t,x_t), \quad t\ge t_0, \end{aligned}$$

(4)

with the initial state

$$\begin{aligned} x_{t_0}(\theta ) =\phi (\theta ), \quad \theta \in [-r, 0], \end{aligned}$$

(5)

where $t_0\in R^+$, ${{\hbox {}^C}}D_{t_0,t}^{\alpha }$ is the Caputo’s fractional derivative of order $\alpha$ with $0<\alpha <1$, r is the delay satisfying $r>0$, $f:[t_0, +\infty )\times C([-r,0], R^n ) \rightarrow R^n$ is a given functional with $f(t,0)=0$, $\phi \in C([-r,0], R^n )$, $f(t,\varphi )$ is continuous and satisfies Lipschitz condition in the second variable $\varphi$ for $\varphi \in C([-r,0], R^n )$. If $x\in C([t_0-r, +\infty ), {R}^n)$, then for any $t\in [t_0, +\infty )$ define $x_t$ by $x_t(\theta )=x(t+\theta )$, $\theta \in [-r,0]$.

The simplest of the system (4) is fractional differential-difference equations of the type

$$\begin{aligned} {{\hbox {}^C}}D_{t_0,t}^{\alpha }x(t)=f(t, x(t), x(t-\tau )), \end{aligned}$$

(6)

where delay $\tau >0$ is a constant. It is easy to see that Eq. (2) is a special case of Eq. (4). If A and E in Eq. (1) satisfy that $E=I$(unit matrix) and $A=0$, then Eq. (1) is turned into Eq. (4). In other words, Eq. (4) is a special case of Eq. (1). However, the results of this article are different from ones in [20]. Furthermore, the main method used in this article is Lyapunov functional not the Laplace transform used in [20].

A precondition to study the stability of the equilibrium point $x=0$ of the system (4) is that the system (4)–(5) must has a unique global solution defined on $[t_0-r, \infty )$. In one-dimensional space, the existence and uniqueness of the state solution defined on $[t_0-r,\infty )$ of the system (4)–(5) can be guaranteed by Corollary 5.1 of the Ref. [13]. For the case $R^n$ ($n\ge 2$), there is not any related result to guarantee that the system (4)–(5) has a unique solution defined on $[t_0-r, \infty )$. Throughout this paper, we always assume that the system (4)–(5) has a unique solution defined on $[t_0-r,\infty )$.

The contribution of this paper is that by choosing Lyapunov functionals and Lyapunov functions, respectively, we derived three stability criteria for the system (4)–(5) which are extensions of ones of the Ref. [29]. In addition, Corollary 1 of this article is simple and convenient for application, which is an extension of the results of the Refs. [27, 28].

The rest of this paper is organized as follows. Section 2 is devoted to some preliminaries. In Sect. 3, we present our main results of this article. Some illustrative examples and numerical simulations form the contents of Sect. 4.

2 Preliminaries

In this section, we recall some definitions and lemmas which will be used later.

Throughout this paper, notation ${R}^n$ denotes the n-dimensional Euclidean space, $|\cdot |$ denotes the Euclidean vector norm, $(\cdot )^\mathrm{T}$ denotes the transpose of the vector $(\cdot )$, ${\mathcal {K}}$ denotes the set of class-K functions. Denote ${\mathcal {C}}=C([-r,0], {R}^n )$ the space of continuous functions on $[-r,0]$. For any element $\phi \in {\mathcal {C}}$, define the norm $\Vert \phi \Vert =\max \limits _{-r\le \theta \le 0}|\phi (\theta )|_{{R}^n }$. For more details, readers may refer to the monograph [29].

Definition 1

A continuous function $u(t):[0, +\infty )\rightarrow [0, +\infty )$ is said to be a class-K function if the function u(t) is strictly increasing and $u(0)=0$.

Definition 2

[1] Given an interval [a, b] of R. The fractional order integral of a function $f\in L^{1}([a,b], {R})$ of order $\alpha \in {R}^+$ is defined by

$$\begin{aligned} I_{a,t}^\alpha f(t)= & {} \frac{1}{\varGamma (\alpha )}\int _{a}^{t} (t-s)^{\alpha -1}f(s)\text {d}s,\nonumber \\&t\in [a, b], \quad \alpha >0, \end{aligned}$$

(7)

where $\varGamma$ is the Gamma function.

Property 1

Let $0<\alpha <1$ and $f(t)\ge 0$ on [a, b]. Then it holds

$$\begin{aligned} I_{a,t}^{\alpha }f(t)\ge 0, \quad t\in [a, b]. \end{aligned}$$

(8)

Definition 3

[1] Suppose that a function f is defined on [a, b] and $f^{(n)}(t)\in L^1[a, b]$. The Caputo’s derivative of order $\alpha$ with the lower limit a for the function f is defined as

$$\begin{aligned} \hbox {}^CD_{a,t}^\alpha f(t)= & {} \frac{1}{\varGamma (n-\alpha )} \int _{a}^{t}(t-s)^{n-\alpha -1}f^{(n)}(s)\text {d}s, \nonumber \\&t\in [a,b], \end{aligned}$$

(9)

where $0<n-1<\alpha \le n$.

Particularly, when $0<\alpha \le 1$, it holds

$$\begin{aligned} \hbox {}^CD_{a,t}^\alpha f(t)= & {} \frac{1}{\varGamma (1-\alpha )} \int _{a}^{t}(t-s)^{-\alpha }f^{\prime }(s)\text {d}s\nonumber \\= & {} I_{a,t}^{1-\alpha }f^{\prime }(t),\quad t\in [a, b]. \end{aligned}$$

(10)

Lemma 1

[27] Let $x(t)\in {R}^n$ be a continuous and derivable function. Then, for any instant $t_0\in {R}$, it holds

$$\begin{aligned} \frac{1}{2} {\hbox {}^CD}_{t_0,t}^\alpha \left[ x^\mathrm{T} (t)x(t)\right]\le & {} x^\mathrm{T}(t) {^C D_{t_0,t}^\alpha } x(t),\nonumber \\&\forall \alpha \in ({0,1}), t\ge t_0. \end{aligned}$$

(11)

Lemma 2

(Theorem 5.1 in [13]) Let $\alpha \in {R}^{+}$, $m=[\alpha ]+1$, and $y(t)\in AC^m[0, b]$. Define the set $G:=[0, +\infty )\times {\mathcal {C}}$ and let the function $f:G\rightarrow {R}$ be continuous and fulfill a Lipschitz condition with respect to the second variable with a Lipschitz constant $L\in {R}^+$ that is independent of t, $y_1$ and $y_2$. Then there exists a uniquely defined function $y\in C[-r, b)$ solving the initial value problem

$$\begin{aligned} \hbox {}^CD_{0,t}^{\alpha }y(t)=f(t,y_t) \end{aligned}$$

(12)

with initial conditions

$$\begin{aligned}&D^ky(\theta )=\phi ^{(k)}(\theta ), \quad \theta \in [-r, 0],\nonumber \\&k=0,1,2,\ldots , m-1, \quad D^k=\frac{\text {d}^k}{\text {d}t^k}, \end{aligned}$$

(13)

where $[\alpha ]$ denotes the integer part of $\alpha$.

Lemma 3

(Corollary 5.1 in [13]) Let $\alpha \in {R}^{+}$, $m=[\alpha ]+1$, and $y(t)\in AC^m[0, b]$. Define the set $G:=[0, +\infty )\times {\mathcal {C}}$ and let the function $f:G\rightarrow {R}$ be continuous and fulfill a Lipschitz condition with respect to the second variable with a Lipschitz constant $L\in {R}^+$. Then there exists a uniquely defined function $y\in C[-r, \infty )$ solving the initial value problem (12)–(13).

3 Main results

In this section, three stability criteria which are parallel to ones of the Ref. [29] are derived by applying Lyapunov functional method and Lyapunov function method, respectively.

Theorem 1

Suppose that $f:{R}\times {\mathcal {C}}\rightarrow {R}^n$ takes ${R}\times (\hbox {bounded sets of } {\mathcal {C}})$ into bounded sets of ${R}^n$, $f(t, 0)=0$. Assume further that $u,\upsilon ,\omega : {R}^+\rightarrow {R}^+$ are continuous nondecreasing functions, u(s) and $\upsilon (s)$ are positive for $s>0$, and $u(0)=\upsilon (0)=0$. If there exists a continuous differential functional $V:{R}\times {\mathcal {C}}\rightarrow {R}$ such that

$$\begin{aligned}&u(|\varphi (0)|)\le V(t, \varphi )\le \upsilon (\Vert \varphi \Vert ), \end{aligned}$$

(14)

$$\begin{aligned}&{\hbox {}^CD}_{t_0,t}^{\alpha }{V} (t, \varphi )|_{(4)} \le -\omega (|\varphi (0)|), t\ge t_0. \end{aligned}$$

(15)

then the equilibrium point $x=0$ of the system (4) is stable. If $\omega (s)>0$ for $s>0$, then the equilibrium point $x=0$ of the system (4) is asymptotically stable.

Proof

For any $\varepsilon >0$, there exists a $\delta =\delta (\varepsilon )$, $0<\delta <\varepsilon$, such that $\upsilon (\delta )< u(\varepsilon )$. Let $x(t;t_0,\phi )$ be the state solution of state equation (4) satisfying the initial condition $x_{t_0}=\phi$. The inequality (15) and Property 1 imply

$$\begin{aligned}&I_{t_0,t}^{\alpha }\left( \hbox {}^CD_{t_0,t}^{\alpha } V(t;x_{t}(t_0,\phi ))+\omega (|x(t;t_0,\phi )|) \right) \le 0,\nonumber \\&\quad t\ge t_0. \end{aligned}$$

(16)

That is

$$\begin{aligned} V(t,x_{t}(t_0,\phi ))\le & {} V(t_0,\phi )-I_{t_0,t}^{\alpha }\omega (|x(t;t_0,\phi )|),\nonumber \\&t\ge t_0. \end{aligned}$$

(17)

By the property of the function $\omega$, it follows

$$\begin{aligned} V(t;x_{t}(t_0,\phi ))\le V(t_0,\phi ), \quad t\ge t_0. \end{aligned}$$

(18)

If $\Vert \phi \Vert < \delta$, then the inequalities (14), (15), (18) and the nondecreasing property of the function $\upsilon$ imply

$$\begin{aligned} u(|x(t; t_0, \phi ) | )\le & {} V(t; x_t(t_0, \phi ))\le V(t_0, \phi )\nonumber \\\le & {} \upsilon (\Vert \phi \Vert )\le \upsilon (\delta )< u(\varepsilon ), \quad t\ge t_0. \nonumber \\ \end{aligned}$$

(19)

The nondecreasing property of the function u(s) and the inequality (19) imply $|x(t; t_0, \phi )|<\varepsilon$, $t\ge t_0$. The continuity of the solution $x(t;t_0,\phi )$ on $[t_0-r, \infty )$ implies the stability of the equilibrium point $x=0$ of the system (4).

To prove that the equilibrium point $x=0$ of the system (4) is asymptotically stable, it is sufficient to prove the state solution $x(t; t_0, \phi )$ of the system (4)–(5) satisfies: $\lim \limits _{t\rightarrow +\infty }x(t; t_0, \phi )=0$. Otherwise, if the equilibrium point $x=0$ of the system (4) is not asymptotically stable, then there exist constants $\eta >0$, $t_i\in (t_0,+\infty )$, $i=1,2,\ldots$, such that

$$\begin{aligned} |x(t_i)|\ge \eta , \quad i=1,2,\ldots , \end{aligned}$$

(20)

where $t_i\rightarrow \infty$ as $i\rightarrow +\infty$. It follows from the inequality (15) that

$$\begin{aligned} I_{t_0,t}^{\alpha }\left[ \hbox {}^CD_{t_0,t}^{\alpha } V(t,x_t)+\omega (|x(t)|) \right] \le 0, \quad t\ge t_0. \end{aligned}$$

(21)

That is

$$\begin{aligned} V(t,x_t)\le V(t_0, \phi )-I_{t_0,t}^{\alpha }\omega (|x(t)|). \end{aligned}$$

(22)

Since the function $\omega$ is nondecreasing, we get by the inequality (20) that

$$\begin{aligned} I_{t_0,t}^{\alpha }\omega (x(t_i))\ge & {} I_{t_0,t}^{\alpha } \omega (\eta )\nonumber \\= & {} \frac{\omega (\eta ) }{\varGamma (1+\alpha )}(t-t_0)^{\alpha }, \quad i=1,2,\ldots . \nonumber \\ \end{aligned}$$

(23)

Combining (22) with (23) yields

$$\begin{aligned} V(t_i,x_{t_i})\le & {} V(t_0,\phi )-\frac{\omega (\eta ) }{\varGamma (1+\alpha )}( t_i-t_0)^{\alpha },\nonumber \\&i=1,2,\ldots . \end{aligned}$$

(24)

When $t_i$ is large enough, it follows

$$\begin{aligned} V(t_i, x_{t_i})\le V(t_0,\phi )-\frac{\omega (\eta ) }{ \varGamma (1+\alpha )}(t_i-t_0)^{\alpha }<0. \end{aligned}$$

(25)

This is a contradiction. This proves the asymptotical stability. The proof is completed.

Remark 1

Theorem 1 is an extension of Theorem 2.1 of the Ref. [29, Page 132].

Suppose that f: $R\times {\mathcal {C}}\rightarrow R^n$ is continuous and $f(t,0)=0$. Consider the RFDE(f)

$$\begin{aligned} \dot{x}(t)=f(t,x_t), \quad t\ge \sigma \end{aligned}$$

(26)

subject to the initial condition

$$\begin{aligned} x_{\sigma }(\theta )=\phi (\theta ) , \quad \theta \in [-r,0], \end{aligned}$$

(27)

for any $\sigma \in R$.

Theorem 2.1 of the Ref. [29], which is a stability criterion for the integer-order retarded functional differential equation (RFDE) (26), is as follows:

Suppose that f: $R\times {\mathcal {C}}\rightarrow R^n$ takes $R\times$ (bounded set of ${\mathcal {C}})$ into bounded sets of $R^n$, and u, v, w: $R^+\rightarrow R^+$ are continuous nondecreasing functions, u(s) and v(s) are positive for $s>0$, and $u(0)=v(0)=0$. If there is a continuous function V: $R\times {\mathcal {C}}\rightarrow R$ such that

$$\begin{aligned} \begin{aligned} u(|\varphi (0)|)&\le V(t,\varphi )\le v(|\varphi |),\\ {\dot{V}}(t, \varphi )&\le -\omega (|\varphi (0)|), \end{aligned} \end{aligned}$$

(28)

then the solution $x=0$ of Eq. (26) is uniformly stable. If $u(s)\rightarrow \infty$ as $s\rightarrow \infty$, the solutions of Eq. (26) are uniformly bounded. If $\omega (s)>0$ for $s>0$, then the solution $x=0$ of Eq. (26) is uniformly asymptotically stable.

It is worth to mention that the change of $t_0$ yields the change of the system (4)–(5). Therefore, the uniform stability of the equilibrium point $x=0$ of any fractional differential system with Caputo’s derivative including the system (4) does not hold.

By similar arguments used in the proof of Theorem 1, one can get the following theorem.

Theorem 2

Suppose that $f:{R}\times {\mathcal {C}}\rightarrow {R}^n$ takes ${R}\times (\hbox {bounded sets of } {\mathcal {C}})$ into bounded sets of ${R}^n$ and $f(t,0)=0$. Assume further that $u,\upsilon ,\omega : {R}^+\rightarrow {R}^+$ are continuous nondecreasing functions, u(s) and $\upsilon (s)$ are positive for $s>0$, and $u(0)=\upsilon (0)=0$. If there exists a continuous differential function $V:{R}\times {R}^n\rightarrow {R}$ such that for $t\in {R}$, $x\in {R}^n$ and $\varphi \in {\mathcal {C}}$,

$$\begin{aligned}&u(|x|)\le V(t,x)\le \upsilon (|x|), \end{aligned}$$

(29)

$$\begin{aligned}&{\hbox {}^CD}_{t_0,t}^{\alpha }{V}(t, \varphi (0))|_{(4)} \le -\,\omega (|\varphi (0)|), t\ge t_0, \end{aligned}$$

(30)

then the equilibrium point $x=0$ of the system (4) is stable. If $\omega (s)>0$ for $s>0$, then the equilibrium point $x=0$ of the system (4) is asymptotically stable.

Remark 2

Theorem 2 also holds for $\theta =0$. That is, Theorem 2 is valid for nonlinear fractional differential system

$$\begin{aligned} {}^C D_{t_0,t}^{a}x(t)=f(t,x(t)), \quad t\ge t_0 \end{aligned}$$

(31)

for $t_0\in R^+$. If $t_0=0$ and $\theta =0$, then Theorem 2 is similar to theorems of the Ref. [26].

By Theorem 2 and Lemma 1, we can get the following corollary.

Corollary 1

Suppose that $f:{R}\times {\mathcal {C}}\rightarrow {R}^n$ takes ${R}\times (\hbox {bounded sets of} \ {\mathcal {C}})$ into bounded sets of ${R}^n$ and $f(t, 0)=0$. If for any $x\in {R}^n$, $x^\mathrm{T}(t)f(t, x_t)\le 0$ for $t\ge t_0$, then the system (4) is stable. Furthermore, if for any $x\ne 0$, it holds $x^\mathrm{T}(t)f(t,x_t)<0$, then the system (4) is asymptotically stable.

Proof

Take a Lyapunov function $V({t,x(t)}) = \frac{1}{2}x^\mathrm{T} (t)x(t)$. Lemma 1 implies

$$\begin{aligned} {{\hbox {}^C}} D_{t_0,t}^\alpha V({t,x(t)}) \le x^\mathrm{T} (t){}^C D_{t_0t}^\alpha x(t)=x^\mathrm{T}(t) f(t,x_t). \end{aligned}$$

(32)

If $x^\mathrm{T} (t)f({t,x_t }) \le 0$, then ${}^C D_{t_0,t}^\alpha V(t, x(t)) \le 0$. Theorem 2 implies the system (4) is stable.

If $x^\mathrm{T} (t)f(t,x_t)< 0, \quad \forall x\ne 0$, then ${\hbox {}^C} D_{t_0,t}^\alpha V(t, x(t)) < 0$. Theorem 2 implies that the system (4) is asymptotically stable.

Remark 3

For the case $\theta =0$, Corollary 1 also holds. Therefore, Corollary 1 here is an extension of Corollary 1 of the Ref. [27] and Theorem 3.1 of the Ref. [28]. In order to compare the Corollary 1 of this article with Corollary 1 of the Ref. [27] and Theorem 3.1 of the Ref. [28], we list them in the following.

Corollary 1 of the Ref. [27] is as follows:

For the fractional order system

$$\begin{aligned} \hbox {}^{C}D^{\alpha }_{t_0,t}x(t)=f(x(t)) \end{aligned}$$

(33)

where $\alpha \in (0,1)$, $x=0$ is the equilibrium point and $x(t)\in R$, if the following condition is satisfied

$$\begin{aligned} x(t)f(x(t))\le 0, \quad \forall x \end{aligned}$$

(34)

then the origin of the system (33) is stable. And if

$$\begin{aligned} x(t)f(x(t))<0, \quad \forall x\ne 0 \end{aligned}$$

(35)

then the origin of the system (33) is asymptotically stable.

Theorem 3.1 of the Ref. [28] is as follows.

Consider the system

$$\begin{aligned} \hbox {}^CD_{0,t}^{\alpha }x(t)=f(x), \quad t\ge 0 \end{aligned}$$

(36)

where $0<\alpha <1$. $\varOmega \subset R$ is a domain that contains the origin $x=0$. Suppose further that $f(x)\in C^1(\varOmega )$ with $f(0)=0$. If $x\cdot f(x)\le 0$, then the equilibrium point $x=0$ is stable. Further, if $x\ne 0$ implies $x\cdot f(x)< 0$, then the equilibrium point $x=0$ is asymptotically stable.

Corollary 2

Assume $\alpha \in (0,1)$. Consider the fractional delayed differential system

$$\begin{aligned} \hbox {}^CD_{t_0,t}^{\alpha }x(t)=f(t,x_t)+g(t, x_t), \quad t\ge t_0, \end{aligned}$$

(37)

where f, g is continuous and satisfy Lipschitz conditions in the second variable $\varphi \in C([-r,0], {R}^n)$, $f(t,0)=0$, $\lim \limits _{x\rightarrow 0} \frac{|g(t,x_t)|}{|f(t,x_t)|}=0$. If $x^\mathrm{T} f<0$ for $x\ne 0$, then the system (37) is locally asymptotically stable.

Proof

Obviously, there exists a positive real number $\delta >0$ ($\delta$ may be small enough) such that if $|x(t)|<\delta$, it holds $\left| \frac{f(t,x_t)}{g(t,x_t)}\right| <\frac{1}{2}$. In consequence, $|x(t)|<\delta$ and $x^\mathrm{T} f<0$ imply

$$\begin{aligned}&x(t)^\mathrm{T}[f(t,x_t)+g(t, x_t)]\nonumber \\&\quad = x(t)^\mathrm{T} f(t,x_t)\left[ 1+\frac{g(t,x_t)}{f(t, x_t)} \right] <0, \quad t\ge t_0.\nonumber \\ \end{aligned}$$

(38)

By Corollary 1, the system (37) is locally asymptotically stable.

The following theorems are Razumikhin-type stability criteria for the fractional delayed differential system (4).

Theorem 3

Suppose that $f:{R}\times {\mathcal {C}}\rightarrow {R}^n$ takes ${R}\times (\hbox {bounded sets of}\ {\mathcal {C}})$ into bounded sets of ${R}^n$, $f(t, 0)=0$. Assume further that $u,\upsilon ,\omega : {R}^+\rightarrow {R}^+$ are continuous nondecreasing functions, u(s) and $\upsilon (s)$ are positive for $s>0$, and $u(0)=\upsilon (0)=0$. If there exists a continuous function $V:{R} \times {R}^n \rightarrow {R}$ such that for $t\in {R}$, $x\in {R}^n$ and $\varphi \in {\mathcal {C}}$,

$$\begin{aligned} u(|x|) \le V(t, x)\le v(|x|) \end{aligned}$$

(39)

and $V(t+\theta , \varphi (\theta ))\le V(t, \varphi (0))$ implies

$$\begin{aligned} \hbox {}^CD_{t_0,t}^{\alpha }V(t, \varphi (0))|_{(4)}\le -w(|\varphi (0)|), \quad t\ge t_0. \end{aligned}$$

(40)

for $\theta \in [-r,0]$, then the equilibrium point $x=0$ of the system (4) is stable.

Proof

For $t\in [t_0, \infty )$, $\varphi \in {\mathcal {C}}$, define

$$\begin{aligned} {\overline{V}}(t, \varphi )=\sup \limits _{\theta \in [-r,0]} V(t+\theta , \varphi (\theta )). \end{aligned}$$

(41)

Then there is a $\theta _0(t)\in [-r, 0]$ such that ${\overline{V}}(t, \varphi ) = V(t+\theta _0, \varphi (\theta _0))$. It follows $V(t+\theta , \varphi (\theta ))\le V(t+\theta _0, \varphi (\theta _0))$ for $\theta \in [-r, 0]$. If $\theta _0<0$, then for $h>0$ sufficiently small

$$\begin{aligned} {\overline{V}}(t+h, x_{t+h}(t,\phi ))={\overline{V}}(t,\phi ) \end{aligned}$$

(42)

and therefore $\dot{{\overline{V}}}(t,\varphi )=0$. Thus it follows that

$$\begin{aligned} \hbox {}^CD_{t_0,t}^{\alpha }\overline{V}(t, \varphi )=0. \end{aligned}$$

(43)

If $\theta _0=0$, then ${\overline{V}}(t, \varphi )=V(t, \varphi (0))$. We have from the condition (40) that

$$\begin{aligned} \hbox {}^CD_{t_0,t}^{\alpha }{\overline{V}}(t,\varphi )|_{(4){\text {--}}(5)}\le -\omega (|\varphi (0)|). \end{aligned}$$

(44)

By similar arguments used in the proof of Theorem 1, it follows that the equilibrium point $x=0$ of the system (4) is stable. This completes the proof.

Remark 4

Theorem 3 is parallel to Theorem 4.1 of the Ref. [29, P152], which is as follows:

Suppose f:$R\times {\mathcal {C}}\rightarrow R^n$ takes $R\times$ (bounded sets of ${\mathcal {C}}$) into bounded sets of $R^n$ and suppose further that $u,v, w:R^+\rightarrow R^+$ are continuous, nondecreasing functions, u(s), v(s) positive for $s>0$, $u(0)=v(0)=0$, v strictly increasing. If there is a continuous function $V:R\times R^n \rightarrow R$ such that

$$\begin{aligned} u(|x|)\le V(t,x)\le v(|x|), t\in R, x\in R^n \end{aligned}$$

(45)

and

$$\begin{aligned} {\dot{V}}(t,\varphi (0))\le & {} -\omega (|\varphi (0)| ) \quad \hbox {if }\nonumber \\&V(t+\theta , \varphi (\theta ))\le V(t, \varphi (0)) \end{aligned}$$

(46)

for $\theta \in [-r,0]$, then the solution $x=0$ of Eq. (26) is uniformly stable.

Theorem 4

Suppose that all the conditions of Theorem 3 are satisfied. Assume further that the function $\upsilon$ is strictly increasing. Namely, $\upsilon \in {\mathcal {K}}$. If there is a continuous nondecreasing function $p(s)>s$ for $s>0$ such that the condition (40) is strengthened to

$$\begin{aligned} \hbox {}^CD_{t_0,t}^{\alpha }V(t, \varphi (0))|_{(4)}\le & {} -\omega (\varphi (0)) \hbox { if}\nonumber \\ V(t+\theta , \varphi (\theta ))< & {} p(V(t, \varphi (0))) \end{aligned}$$

(47)

for $t\ge t_0$ and $\theta \in [-r,0]$, then the equilibrium point $x=0$ of the system (4) is asymptotically stable.

Proof

By Theorem 3, the equilibrium point $x=0$ of the system (4) is stable.

The fact that the solution $x=0$ of the system (4) is stable shows that for the stability constant $\varepsilon _0=1$, we can choose a constant $\delta _0>0$ such that for $\Vert \phi \Vert \le \delta _0$, the state solution $|x(t; t_0,\phi )|$ of the system (4)–(5) satisfying $|x(t;t_0,\phi )|<\varepsilon _0=1$. In the remainder of the proof, we denote the state solution $x(t;t_0,\phi )$ of the system (4)–(5) as x(t) for convenience.

In order to prove that the equilibrium point $x=0$ of the system (4) is asymptotical stability, it is sufficient to prove that, for any $\eta >0$ and $u(\eta )<v(\varepsilon _0)=v(1)$, there exists a $T_0\ge t_0$ such that $V(t,x(t))<u(\eta )$ on $[T_0, \infty )$.

From the properties of the continuous function p(s), there exists a number $a>0$ such that $p(s)-s>a$ on the interval $[u(\eta ), v(1)]$. Let $n_0$ be the first positive integer such that $u(\eta )+n_0 a \ge v(1)$. We first prove that when t is large enough, $V(t,x(t))\le u(\eta )+(n_0-1)a.$ Otherwise, there exist constants $t_i>t_0$, $i=1,2, \ldots$, satisfying $t_i\rightarrow \infty$ as $i\rightarrow \infty$ such that

$$\begin{aligned} V(t,x(t))>u(\eta )+(n_0-1)a, t=t_i, i=1,2,\ldots . \end{aligned}$$

(48)

It follows that

$$\begin{aligned}&p(V(t,x(t)))\nonumber \\&\quad >V(t,x(t))+a >u(\eta )+(n_0-1)a+a\nonumber \\&\quad = u(\eta )+n_0 a>v(1), \quad t=t_i,\quad i=1,2,\ldots .\nonumber \\ \end{aligned}$$

(49)

By the condition (39) and (49), we have

$$\begin{aligned} V(t+\theta , x(t+\theta ))\le & {} v(1)< p(V(t,x(t))),\nonumber \\&t=t_i, \quad i=1,2,\ldots . \end{aligned}$$

(50)

(50) and (47) imply

$$\begin{aligned} \hbox {}^CD_{t_0,t}^{\alpha }V(t,x(t))\le -\omega (|x|),\quad t=t_i, \quad i=1,2,\ldots . \end{aligned}$$

(51)

From (51), we have

$$\begin{aligned} V(t,x(t))\le & {} V(t_0,x(t_0))-I_{t_0,t}^{\alpha }\omega (|x|), \nonumber \\&t= t_i, \quad i=1,2,\ldots . \end{aligned}$$

(52)

By (39) and (48), it follows

$$\begin{aligned} v(|x|)>u(\eta )+(n_0-1)a, \quad t=t_i, \quad i=1,2,\ldots . \end{aligned}$$

(53)

Since v is class-K function, we have

$$\begin{aligned} |x|>v^{-1}(u(\eta )), \quad t=t_i, \quad i=1,2,\ldots . \end{aligned}$$

(54)

By the fact that the function $\omega$ is continuous and nondecreasing, it follows

$$\begin{aligned} \omega (|x|) \ge \omega (v^{-1}(u(\eta )) ), \quad t=t_i, \quad i=1,2,\ldots . \end{aligned}$$

(55)

From (52) and (55), we have

$$\begin{aligned} V(t,x(t))\le & {} V(t_0,x(t_0))-\frac{\omega \left( v^{-1}(u(\eta ))\right) }{\varGamma ( 1+\alpha )}(t-t_0)^{\alpha }, \nonumber \\&t=t_i, \quad i=1,2,\ldots . \end{aligned}$$

(56)

$t_i\rightarrow \infty$ will yield $V(t,x(t))<0$. This is a contraction. In consequence, $V(t,x(t))\le u(\eta )+(n_0-1)a$ when t is large enough. Without loss of generality, we assume that there exists a constant $T_1>t_0$ such that

$$\begin{aligned} V(t,x(t))\le u(\eta )+(n_0-1)a, \quad t\ge T_1. \end{aligned}$$

(57)

We now prove that there exists a constant $T_2>T_1+r$ such that

$$\begin{aligned} V(t, x(t))\le u(\eta )+(n_0-2)a, \quad t>T_2. \end{aligned}$$

(58)

Otherwise, there exist constants $t_{m}>T_1+r$, $m=1,2,\ldots$, $t_{m}\rightarrow \infty$ as $m\rightarrow \infty$ such that

$$\begin{aligned} V(t,x(t))> & {} u(\eta )+(n_0-2)a,\quad t=t_{m},\nonumber \\&m=1,2,\ldots . \end{aligned}$$

(59)

It then follows from (59) that

$$\begin{aligned} p(V(t,x(t)))> & {} V(t,x(t))+a>u(\eta )+(n_0-1)a,\nonumber \\&t=t_{m}, \quad m=1,2,\ldots . \end{aligned}$$

(60)

Combining (57) with (60) yields

$$\begin{aligned} p(V(t,x(t))> & {} V(t+\theta , x(t+\theta )),\quad \theta \in [-r,0],\nonumber \\&t=t_{m}, \quad m=1,2,\ldots . \end{aligned}$$

(61)

(61) and (47) imply

$$\begin{aligned} \hbox {}^CD_{t_0,t}^{\alpha }V(t,x(t))\le & {} -\omega (|x|), \quad t=t_{m},\nonumber \\&m=1,2,\ldots . \end{aligned}$$

(62)

By (59) and (62), we have

$$\begin{aligned} V(t,x(t))\le & {} V(t_0,x(t_0))-\frac{\omega \left( v^{-1}(u(\eta ))\right) }{\varGamma (1+\alpha )}(t-t_0)^{\alpha }, \nonumber \\&t=t_{m}, \quad m=1,2,\ldots . \end{aligned}$$

(63)

By (63), if $t_{m}\rightarrow \infty$, then $V(t,x(t))<0$. This is a contraction. In consequence, we can choose a constant $T_2>T_1+r$ such that

$$\begin{aligned} V(t,x(t))<u(\eta )+(n-2)a, \quad t>T_2. \end{aligned}$$

(64)

Repeat above process $n_0$ times, we can choose a constant $T_{n_0}>t_0$ such that

$$\begin{aligned} V(t,x(t))<u(\eta ), \quad t>T_{n_0}. \end{aligned}$$

(65)

From (65) and the condition (39), it follows

$$\begin{aligned} u(|x|)<u(\eta ), \quad t>T_{n_0}. \end{aligned}$$

(66)

It then follows $|x(t)|<\eta$ on $[T_{n_0}, +\infty )$. This shows that the equilibrium point $x=0$ of the system (4) is asymptotically stable. The proof is completed.

Remark 5

Theorem 4 is parallel to Theorem 4.2 of the Ref. [29, P152] which is as follows:

Suppose that all of the conditions of Theorem 4.1 are satisfied and in addition $\omega (s)>0$ for $s>0$. If there is a continuous nondecreasing function $p(s)>s$ for $s>0$ such that the condition (46) is strengthened to

$$\begin{aligned} {\dot{V}}(t,\varphi (0))\le & {} -\omega (|\varphi (0)|) \quad \hbox {if}\nonumber \\&V(t+\theta , \varphi (\theta ))< p(V(t,\varphi (0))) \end{aligned}$$

(67)

for $\theta \in [-r,0]$, then the solution $x=0$ of the RFDE(f) (26) is uniformly asymptotically stable. If $u(s)\rightarrow \infty$ as $s\rightarrow \infty$, then the solution $x=0$ is also a globally attractor for the RFDE(f).

4 Illustrative examples and numerical simulations

In this section, we present four examples to illustrate the applications of our results. All numerical simulations are based on the predictor–corrector algorithm [33].

Example 1

Consider the system

$$\begin{aligned} \hbox {}^CD_{0, t}^{\frac{1}{2}}x(t)=-x(t)x^2(t-1), \quad t\ge 0. \end{aligned}$$

(68)

We first check the system (68) with a initial state $x_{0}(\theta )=\phi (\theta )$ has only unique solution on $[0, \infty )$. Here the initial state $\phi (\theta )$ is continuous for $\theta \in [-1,0]$. Denote $f(t,x_t)=-x(t)x^2(t-1)$. For $t\in [0,1]$, the system (68) is equivalent to

$$\begin{aligned} \hbox {}^CD_{0,t}^{\frac{1}{2}}x(t)=-x(t)\phi ^2(t-1). \end{aligned}$$

(69)

For any $x_t, y_t\in C([-1,0], R)$, $t\in [0, 1]$, we have

$$\begin{aligned}&\left| f(t,x_t)-f(t,y_t) \right| \nonumber \\&\quad = \left| -x(t)\phi ^2(t-1)-y(t)\phi ^2(t-1) \right| \nonumber \\&\quad \le \max \limits _{t\in [0,1]}|\phi ^2(t-1)||x(t)-y(t)|\nonumber \\&\quad \le \max \limits _{t\in [0,1]}|\phi ^2(t-1)|\sup \limits _ {\theta \in [-1, 0]}|x_t(\theta )-y_{t}(\theta )|\nonumber \\&\quad = L_1 \sup \limits _{\mathop {t\in [0,1] }\limits ^{\theta \in [-1, 0]}}|x_t(\theta )-y_{t}(\theta )|, \end{aligned}$$

(70)

where the Lipschitz constant $L_1=\max \limits _{t\in [0,1]}|\phi ^2(t-1)|$. In consequence, the functional $f(t,x_t)=-x(t)\cdot x^2(t-1)$ satisfies Lipschitz condition with a Lipschitz constant $L_1$. By Lemma 2, the system (68) with the initial state $x_0(\theta )=\phi (\theta )$ has a unique solution on [0, 1]. Without loss of generality, we denote $x(t;0,\phi (\theta ))=\phi _{1}(t)$ defined on [0, 1] the state solution of the system (68) with the initial state $\phi (\theta )$.

For $t\in [1,2]$, the system (68) is equivalent to

$$\begin{aligned} \hbox {}^CD_{0,t}^{\frac{1}{2}}x(t)=-x(t)\phi _{1}^2(t-1). \end{aligned}$$

(71)

For any $x_t,y_t\in C([-1,0], R)$, $t\in [1,2]$, we have

$$\begin{aligned}&|f(t, x_t)-f(t, y_t)|\nonumber \\&\quad = |x(t)\phi _1^2(t-1)-y(t)\phi _1^2(t-1)|\nonumber \\&\quad \le \max \limits _{t\in [1,2]}\phi _1^2(t-1) |x(t)-y(t)|\nonumber \\&\quad \le L_2 \sup \limits _{\mathop {\theta \in [-1,0]}\limits ^{t\in [1,2]}}|x_t(\theta )-y_t(\theta )|, \end{aligned}$$

(72)

where $L_2=\max \limits _{t\in [1,2]}\phi _1^2(t-1)$. Take $L_{1,2}=\max \{L_1,L_2\}$. For any $x_t, y_t\in {\mathcal {C}}$, $t\in [0, 2]$, combining (70) with (72) yields

$$\begin{aligned}&|f(t, x_t)-f(t, y_t)|\nonumber \\&\quad \le L_{1,2} \sup \limits _{\mathop {\theta \in [-1,0]}\limits ^{t\in [0,2]}}|x_t(\theta )-y_t(\theta )|. \end{aligned}$$

(73)

Therefore, the functional $f(t,x_t)=-x(t)\cdot x^2(t-1)$ in the system (68) satisfied Lipschitz condition on [0, 2] with a Lipschitz constant $L_{1,2}$. By Lemma 2, the system (68) with the initial state $\phi (\theta )$ has a unique solution on [0, 2].

Similarly, we can show that the system (68) with the initial state $\phi (\theta )$ has a unique state solution on [0, 3], $[0,4], \ldots$. By mathematical induction, it follows that the system (68) with the initial state $\phi (\theta )$ has a unique solution $x(t;0,\phi (\theta ))$ on $[0, \infty )$.

Obviously, $x(t)f(t,x_t)=-x^2(t)x^2(t-1)\le 0$. By Corollary 1, the system (68) is stable. The numerical simulation of the state solution $x(t;0,\phi (\theta ))$ of the system (68) with the initial state $\phi (\theta )=1+\theta$, $\phi (\theta )=0.5(1+\theta )$, respectively, for $\theta \in [-1, 0]$ is given in Fig. 1.

On the one hand, it can be seen from Fig. 1 that the initial states satisfy $\Vert 0.5(1+\theta )\Vert < \Vert 1+\theta \Vert$ for $\theta \in [-1, 0]$ and the state solutions satisfy $\Vert x(t;0, 0.5(1+\theta ))\Vert < \Vert x(t; 0, 1+\theta )\Vert$ for $t\in [0, \infty )$. This fact coincides with the stability of the system (68).

On the other hand, the state solutions $x(t;0, 1+\theta )$ and $x(t; 0, 0.5(1+\theta ))$ in Fig. 1 do not tend to 0 as $t \rightarrow \infty$. This does not show that the system (68) is asymptotically stable.

Fig. 1

The numerical simulation of the system (68) with the initial states $x_{0}(\theta )=\theta +1, 0.5(\theta +1)$, respectively, $\theta \in [-1, 0]$

Example 2

Consider the system

$$\begin{aligned} \hbox {}^CD_{0,t}^{\frac{1}{2}}x(t)=-x(t)\left[ 1+x^2(t-1)\right] , \quad t\ge 0. \end{aligned}$$

(74)

Just as the existence and uniqueness of the global state solution of the system (68) with a initial state is checked, it is easy to check that the system (74) with the initial state $\phi (\theta )$ for $\theta \in [-1, 0]$ has a unique state solution $x(t; 0, \phi (\theta ))$ on $[0,\infty )$.

Denote $f(t,x_t)=-x(t)\left[ 1+x^2(t-1)\right]$. Obviously,

$$\begin{aligned} x(t)f(t,x_t)=-x^2(t)\left[ 1+x^2(t-1)\right] < 0 \end{aligned}$$

(75)

for $x\ne 0$. Corollary 1 implies that the system (74) is asymptotically stable. The numerical simulation of the state solution of the system (74) with the initial state $\phi (\theta )=1+\theta$ for $\theta \in [-1, 0]$ is given in Fig. 2.

From Fig. 2, the state solution $x(t;0, 1+\theta )\rightarrow 0$ as $t\rightarrow \infty$. This shows that the state solution $x(t;0, 1+\theta )$ is attractive, which coincides with the asymptotical stability of the system (74).

Fig. 2

The numerical simulation of the system (74) with the initial state $x_0(\theta )=\theta +1, \theta \in [-1, 0]$

Example 3

Consider the system

$$\begin{aligned} \hbox {}^CD_{0, t}^{\frac{1}{2}}x(t)= & {} - x(t)\left[ 1+x^2(t-1)\right] \nonumber \\&\pm \, 2x(t) x^3(t-1), \quad t\ge 0. \end{aligned}$$

(76)

By Lemma 2, it is easy to check that the system (76) with a initial state $x_0(\theta )=\phi (\theta )$ has a unique solution on $[0, \infty )$.

Denote $f(t,x_t)=-x(t) \left[ 1+x^2(t-1)\right]$, $g(t,x_t)=\pm 2x(t)x^3(t-1)$. Then we can check that $\lim \limits _{x\rightarrow 0}\frac{g(t,x_t)}{f(t,x_t)}=0$. Obviously, $x(t)f(t,x_t)<0$ for $x\ne 0$. Corollary 2 implies that the system (76) is locally asymptotically stable. The numerical simulation of the system (76) with the initial state $x_0(\theta )=0.1(1+\theta )$ is given in Fig. 3.

From Fig. 3, it can be seen that the state solution $x(t; 0, 0.1(1+\theta ))$ of the system (76) tends to be 0. This coincides with the local stability of equilibrium point $x=0$ of the system (76).

Fig. 3

The numerical simulation of the system (76) with the initial state $x_0(\theta )=0.1(\theta +1), \theta \in [-1, 0]$

Remark 6

Denote $F(t,x_t)=-x(t)\left[ 1+x^2(t-1)\right] \pm 2x(t) x^3(t-1)$. If follows

$$\begin{aligned} x(t)F(t,x_t)\!=\!-x^2(t)\left[ 1+x^2(t-1)\pm 2x^3(t-1) \right] . \end{aligned}$$

(77)

Corollary 1 is invalid to the system (76). However, Corollary 2 implies that the system (76) is locally asymptotically stable. This shows that Corollary 2 is sometimes suitable for application.

Example 4

Consider the system

$$\begin{aligned} \hbox {}^CD_{0,t}^{\alpha }x(t)=-2x(t)+x(t-1), \quad t\ge 0, \end{aligned}$$

(78)

where $\alpha \in (0, 1)$. We first check that the functional $f(t,x_t)=-2x(t)+x(t-1)$ satisfies Lipschitz condition. In fact, for any $x_t, y_t\in C([-1,0], R)$ and $t\in R^+$, we have

$$\begin{aligned}&|f(t,x_t)-f(t, y_t)|\nonumber \\&\quad = | -2x(t)+x(t-1)+2y(t)-y(t-1) |\nonumber \\&\quad \le 2|x(t)-y(t)|+|x(t-1)-y(t-1)|\nonumber \\&\quad \le 2\sup \limits _{\theta \in [-1,0]}| x_t(\theta )-y_t(\theta )|\nonumber \\&\qquad +\sup \limits _{\theta \in [-1,0]}|x_t(\theta )-y_t(\theta )|\nonumber \\&\quad = 3\sup \limits _{\theta \in [-1,0]}|x_t(\theta )-y_t(\theta )|, \end{aligned}$$

(79)

which implies that the functional $f(t,x_t)=-2x(t)+x(t-1)$ satisfies Lipschitz condition with a Lipschitz constant $L=3$. By Lemma 3, the system (78) with the initial state $\phi (\theta )\in C([-1,0], R)$ has a unique state solution on $[0, \infty )$.

Choose a constant $q=\frac{3}{2}$. When $q|x(t)|>|x(t-1)|$, we get

$$\begin{aligned} x(t)f(t,x_t)= & {} x(t)(-2x(t)+x(t-1)) \nonumber \\\le & {} -\,2x^2(t)+ |x(t)||x(t-1)| \nonumber \\< & {} -\,\frac{1}{2}x^2(t)<0 \end{aligned}$$

(80)

for $x\ne 0$. By Theorem 4, the equilibrium point $x=0$ of the system (78) is asymptotically stable.

The numerical simulations of the state solution of the system (78) with $\alpha =1, 0.9, 0.5$ and $\phi (\theta )=\theta ^2$ for $\theta \in [-1, 0]$ is given in Fig. 4.

From Fig. 4, it can be seen that no matter $\alpha =1, 0.9$ or $\alpha =0.5$, the state solution $x(t; 0, \theta ^2 )\rightarrow 0$ as $t\rightarrow \infty$. This coincides with the fact that the equilibrium point $x=0$ of the system (78) is asymptotically stable.

Fig. 4

Simulation of the system (78) with the initial state $x_0(\theta )=\theta ^2, \theta \in [-1, 0]$ and $\alpha =1, 0.9, 0.5$, respectively

5 Conclusions

In this paper, Lyapunov functional and Lyapunov function are used, respectively, to investigate the stability of the fractional delayed differential systems. Some stability criteria including the two Razumikhin-type stability theorems are derived. Four examples are given for illustrations. Moreover, numerical simulations are provided to show the validity of the results.