The Space-Time Partial Derivatives of Lyapunov Function for Non-Autonomous Fractional-Order Systems

1. Introduction

The notions of Newton-Leibniz derivatives of integer orders have wide implications in diverse areas of sciences and engineering. The classical continuous-time ordinary differential systems offer many different equations to use. Yet we think these systems seem very complicated to apply to predict the motion of rotating stars in our galaxies. In scientific studies and demonstrations, simple pendulum systems, tunnel-diode circuit systems, mass-spring mechanical systems, etc. [2] have equations of ordinary differential systems that certainly describe various observable phenomena that were previously not available. Beyond that, Sprott [5,6] constructed several algebraically simple ordinary differential systems in ${\mathbb{R}}^3$ that produce chaotic attractors and showed applications to electrical circuits. Chen and Dong [7] used an engineering feedback control method to control the chaotic trajectory of a continuous-time nonlinear system to converge to its equilibrium points and multiperiodic orbits. Farmer et al. [8] introduced various definitions to compute the dimension of complicated attractors arising in continuous-time nonlinear differential systems. Slotine and Li [3] demonstrated that stability analysis of nonlinear systems remains crucial in the studies of theory and engineering control applications.

The continuous-time ordinary differential systems have been growing but still have wide limitations. The first limitation may be that these systems do not allow long-term memory characterization in their state evolution. The second limitation is that the future state of these systems can depend only on the recent past, initial positions, making the first posed initial position redundant, etc. One might pose a reasonable mathematical question: Can we express the equations of such systems by means of derivatives of real order? Though it sounds strange, the so-called continuous-time fractional-order systems use the knowledge of real numbers and describe adequate fractional differential equations that have been of greater interest in scientific literature to many mathematicians. One can go back to the past century, 1695; a question of correspondence from Hôpital to Leibniz has appeared [4]: What could be the meaning of ${\displaystyle \frac{d^n}{d{t}^n}}f$ if n were a fraction? Since then, fractional calculus has drawn the attention of many mathematicians in diverse, independent studies. In short, Podlubny [9] studied basics on Grünwald-Letnikov, Riemann-Liouville, and Caputo fractional derivative concepts and used the Laplace integral transform method to derive explicit solutions to many such fractional differential equations. Kilbas et al. [10] were rigorous about theories of fractional differential equations and their solutions in light of Riemann-Liouville and Caputo fractional derivatives. Many fractional-order systems play a crucial role in the design of RLC electrical circuits [11], the discovery of memory chaos [12,13], strengthening memory complexity of Hopfield-type neural networks [14], etc., and have shown some guidance for understanding in theory and applications.

The goal of this paper is to investigate the stability problem of nonlinear fractional order systems in the sense of the Caputo fractional derivative. Throughout this paper, we consider the standard version of the Caputo-type non-autonomous fractional order system

$$\begin{array}{c}\hfill C_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)=f(t,x\left(t\right)),x\left(t^{\star }\right)=x_{t^{\star }},\end{array}$$

(1)

where state variable $x={\left(x_1,x_2,\dots ,x_n\right)}^T\in {\mathbb{R}}^n$, Caputo fractional derivative operator $C_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)={\left(C_{{\mathfrak{D}}_{t^{\star },t}^{{\gamma }_1}}{x}_1\left(t\right),C_{{\mathfrak{D}}_{t^{\star },t}^{{\gamma }_2}}{x}_2\left(t\right),\dots ,C_{{\mathfrak{D}}_{t^{\star },t}^{{\gamma }_n}}{x}_n\left(t\right)\right)}^T\in {\mathbb{R}}^n$, initial time $t^{\star }\in \mathbb{R}$, order-index $\widehat{\gamma }=\left({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n\right)\in (0,1]\times (0,1]\times \dots \times (0,1]$, and the function $f:[t^{\star },\infty )\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuous with respect to its arguments.

There has been some interest in the current literature that concerns the stability problem of Caputo fractional-order systems, which remains an open issue to our knowledge. A typical important problem dealing with such a system is as follows: Can we predict local and global asymptotic stability (stability) of equilibria of such a system (1)? The ideas of Lyapunov methods [1] are well-known, and Li et al. [15,16] gave a foundational stability analysis of equilibria of fractional-order systems by using Lyapunov functions and introduced several theorems. Their theorems were called the fractional Lyapunov direct method. In [17], Lenka and Banerjee discussed a linearization approach for autonomous fractional order systems and introduced some stabilization results for the fractional order Lorenz system. In [18], Tuan and Trinh proved a Lyapunov theorem by using a convex and differentiable Lyapunov function for autonomus fractional order systems. In [23], Wang et al. proposed comparison inequalities and developed asymptotic stability conditions for equilibrium of a class of nonlinear fractional-order systems. In [25], Lenka and Bora proposed a linearization method by using Metzler asymptotic stability for non-autonomous fractional order systems.

As we see in the literature, a typical Lyapunov problem that concerns our mind is as follows: Can we develop new mathematical insights on Lyapunov’s indirect method to predict the stability of fractional-order systems? To our knowledge, this question has not yet been investigated for fractional order systems (1). The main reason could be that it is still not known what the connection is between the Lyapunov function and the conditions needed for linearization for such nonlinear systems. We are interested in this Lyapunov problem and wish to put forward some new stability results.

The rest of this paper, including contributions, is as follows:

•

In the Section 2, we recall the Riemann-Liouville fractional integral and Caputo fractional derivative operators. We also give definitions of asymptotic stability and stability suggested in the sense of Lyapunov.

•

In Section 3, we pose Conjectures 3.1, 3.2, 3.3, 3.4, and 3.5 for the non-autonomous system (1) and its autonomous counterpart. Our conjectures address sufficient conditions for stability analysis of equilibrium without needing fractional derivatives of Lyapunov functions.

•

In Section 4, we discuss the Lyapunov linearization method for autonomous fractional-order system (4) and introduce Lyapunov theorems Theorems 1 and 2. Our theorems establish sufficient conditions by using a Jacobian matrix that arises in the linearization of nonlinear system (4) around zero equilibrium.

•

In Section 5, we discuss the Lyapunov linearization method for the non-autonomous fractional order system (1) and introduce a Lyapunov theorem (Theorem 3). This theorem establishes a sufficient condition by using a time-dependent Jacobian matrix and satisfies a matrix Lyapunov differential equation.

•

Section 6 establishes fractional Krasovskii’s method that concerns theorems Theorem 4 and Theorem 5 for the autonomous fractional system (4). Both these theorems provide a way to construct a Lyapunov function by using a nonlinear function associated with such a system.

•

The Section 7 considers three typical fractional order systems and discusses novel implications of some applicable results. In particular, we demonstrate that the trivial zero equilibrium point of the non-autonomous fractional-order Lorenz system is globally Lyapunov asymptotically stable.

2. Notations and Definitions

The goal of this section is to give some basic mathematical notions that are absolutely essential to symbolic meanings and definitions that concern this work.

2.1. Notations

Let $\mathbb{N}$ be the set of natural numbers, ${\mathbb{R}}_{+}$ be the set of positive real numbers, $\mathbb{R}$ be the set of real numbers, $\mathbb{C}$ be the set of complex numbers, $X^T$ be the transpose of $X\in {\mathbb{R}}^{m\times n}$, and $\Re \left(\lambda \right)$ be the real part of complex number $\lambda \in \mathbb{C}$. Call ${\lambda }_{min}\left(P\right)$ and ${\lambda }_{max}\left(P\right)$ are the minimum and maximum eigenvalues of symmetric matrix $P\in {\mathbb{R}}^{n\times n}$, respectively, $I_n$ is the identity matrix in ${\mathbb{R}}^{n\times n}$, ${\mathbb{R}}^n$ is the Euclidean space, $\parallel ·\parallel$ is the standard Euclidean norm, ${\mathcal{C}}^0$ is the space of continuous functions, and ${\mathcal{C}}^n$ is the space of $\mathit{n}$-times continuously differentiable functions.

2.2. Definitions

In fractional calculus, the Riemann-Liouville fractional integral and Caputo fractional derivative are quite standard. We put them in definitions as given below.

Definition 1.

[9,10] Let $n\in \mathbb{N}$, $\beta \in {\mathbb{R}}_{+}$, and $I:-\infty \le t^{\star }<t^{•}\le \infty$. Let $\eta \in {\mathcal{C}}^0\left(I\right)$. The left Riemann-Liouville fractional integral of η with order β is defined by

$$\begin{array}{c}\hfill {RL}_{{\mathfrak{I}}_{t^{\star },t}^{-\beta }}\eta \left(t\right):={\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t^{\star }}^t{\left(t-\tau \right)}^{\beta -1}\eta \left(\tau \right)d\tau ,t>t^{\star }.\end{array}$$

(2)

Definition 2.

[9,10] Let $n\in \mathbb{N}$, $\beta \in {\mathbb{R}}_{+}$, and $I:-\infty \le t^{\star }<t^{•}\le \infty$. Let $\eta \in {\mathcal{C}}^0\left(I\right)$, which is ${\mathcal{C}}^n\left(\left(t^{\star },t^{•}\right)\right)$. The left Caputo fractional derivative of η with order β is defined by

$$\begin{array}{c}\hfill C_{{\mathfrak{D}}_{t^{★},t}^{\beta }}\eta \left(t\right):=\{\begin{array}{cc}{RL}_{{\mathfrak{I}}_{t^{★},t}^{-(n-\beta )}}{\eta }^{\left[n\right]}\left(t\right),t>t^{★},\hfill & \mathrm{when}\beta \in (n-1,n),\hfill \\ {\eta }^{\left[n\right]}\left(t\right),\hfill & \mathrm{when}\beta =n,\hfill \end{array}\end{array}$$

(3)

where ${\eta }^{\left[n\right]}\left(t\right):={\left({\displaystyle \frac{d}{dt}}\right)}^n\eta \left(t\right)$.

We give the following definitions in the sense of Lyapunov that concern our conjectures and stability theorems.

Definition 3.

The zero equilibrium point of system (1) is said to be Lyapunov asymptotically stable if for any $x\left(t^{\star }\right)\in \mathbb{D}\subset {\mathbb{R}}^n$, the non-trivial solution $\parallel x\left(t\right)\parallel \to 0$ as $t\to \infty$.

Definition 4.

The zero equilibrium point of system (1) is said to be Lyapunov stable if for any $x\left(t^{\star }\right)\in \mathbb{D}\subset {\mathbb{R}}^n$, there exist a $\delta >0$ such that $\parallel x\left(t^{\star }\right)\parallel \le \delta \Rightarrow \parallel x\left(t\right)\parallel \le ϵ$, where $ϵ>0$.

Remark 1.

The notion of Lyapunov asymptotic stability in Definition 3 and Lyapunov stability in Definition 4 can be meaningful upon the existence of a suitable Lyapunov function.

3. Conjectures on Fractional Lyapunov Theory

The stability test of fractional-order system (1) is widely recognised as an important problem, and the fractional Lyapunov direct method provides some mathematical insights for an effective analysis of equilibrium points [18,20]. A typical demerit of this method often lies in the fact that one needs to estimate the fractional derivatives of a scalar Lyapunov function. But the conditions needed to overcome such a difficulty remain unknown in stability theory. The goal of this section is to pose new Conjectures 3.1, 3.2, 3.3, 3.4 and 3.5 (see SubSection 3.1 and SubSection 3.2) for the stability analysis of system (1) to both autonomous and non-autonomous forms. These conjectures (one might call them “Lenka conjectures") were believed to be true and allow a new significance to test stability of equilibrium points of such systems by means of space-time partial derivatives of Lyapunov functions.

Note that when $f(t,x)=f\left(x\right)$, the non-autonomous system (1) becomes an autonomous system. In the case, we consider the autonomous nonlinear fractional order system

$$\begin{array}{c}\hfill C_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)=f\left(x\left(t\right)\right),x\left(t^{\star }\right)=x_{t^{\star }},\end{array}$$

(4)

where $x={\left(x_1,x_2,\dots ,x_n\right)}^T\in {\mathbb{R}}^n$, $C_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)={\left(C_{{\mathfrak{D}}_{t^{\star },t}^{{\gamma }_1}}{x}_1\left(t\right),C_{{\mathfrak{D}}_{t^{\star },t}^{{\gamma }_2}}{x}_2\left(t\right),\dots ,C_{{\mathfrak{D}}_{t^{\star },t}^{{\gamma }_n}}{x}_n\left(t\right)\right)}^T\in {\mathbb{R}}^n$, initial time $t^{\star }\in \mathbb{R}$, order-index $\widehat{\gamma }=\left({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n\right)\in (0,1]\times (0,1]\times \dots \times (0,1]$, and the function $f:\mathbb{D}\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuous.

3.1. Stability conjectures for autonomous systems

Conjecture 3.1.

Let $x=0$ be an equilibrium point of system (4). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:\mathbb{D}\to \mathbb{R}$ be a continuously differentiable and satisfy

$T_1.$

$\mathcal{V}\left(0\right)=0$, and $\mathcal{V}\left(x\right)>0$, $\forall x\in \mathbb{D}-\left\{0\right\}$,

$T_2.$

along system (4) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{\partial \mathcal{V}\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\beta }}}x\left(t\right)<0,\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(5)

where $\widehat{\beta }=\left({\beta }_1,\dots ,{\beta }_n\right)\in (0,1]\times \dots \times (0,1]$.

Then $x=0$ is Lyapunov asymptotically stable on $\mathbb{D}$. If $\mathbb{D}={\mathbb{R}}^n$, and, in addition, $\mathcal{V}\left(x\right)\to \infty$ as $\parallel x\parallel \to \infty$, then $x=0$ is globally Lyapunov asymptotically stable.

Proposition 1.

Let $x=0$ be an equilibrium point of system (4). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:\mathbb{D}\to \mathbb{R}$ be a continuously differentiable and satisfy

$T_1.$

$\mathcal{V}\left(0\right)=0$, and $\mathcal{V}\left(x\right)>0$, $\forall x\in \mathbb{D}-\left\{0\right\}$,

$T_2.$

along system (4) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{\partial \mathcal{V}\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right)}^{T}f\left(x\right(t\left)\right)<0,\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\}.\end{array}$$

(6)

Then $x=0$ is Lyapunov asymptotically stable on $\mathbb{D}$. If $\mathbb{D}={\mathbb{R}}^n$, and, in addition, $\mathcal{V}\left(x\right)\to \infty$ as $\parallel x\parallel \to \infty$, then $x=0$ is globally Lyapunov asymptotically stable.

Proof. 

Set $\widehat{\beta }=\widehat{\gamma }$, that means $\left({\beta }_1,\dots ,{\beta }_n\right)=\left({\gamma }_1,\dots ,{\gamma }_n\right)\in (0,1]\times \dots \times (0,1]$. Then by Conjecture 3.1, one obtains the result. This completes the proof. □

Conjecture 3.2.

Let $x=0$ be an equilibrium point of system (4). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:\mathbb{D}\to \mathbb{R}$ be a continuously differentiable and satisfy

$T_1.$

$\mathcal{V}\left(0\right)=0$, and $\mathcal{V}\left(x\right)>0$, $\forall x\in \mathbb{D}-\left\{0\right\}$,

$T_2.$

along system (4) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{\partial \mathcal{V}\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\beta }}}x\left(t\right)\le 0,\forall t\ge t^{\star },\forall x\in \mathbb{D},\end{array}$$

(7)

where $\widehat{\beta }=\left({\beta }_1,\dots ,{\beta }_n\right)\in (0,1]\times \dots \times (0,1]$.

Then $x=0$ is Lyapunov stable on $\mathbb{D}$. If $\mathbb{D}={\mathbb{R}}^n$, and, in addition, $\mathcal{V}\left(x\right)\to \infty$ as $\parallel x\parallel \to \infty$, then $x=0$ is globally Lyapunov stable.

Proposition 2.

Let $x=0$ be an equilibrium point of system (4). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:\mathbb{D}\to \mathbb{R}$ be a continuously differentiable and satisfy

$T_1.$

$\mathcal{V}\left(0\right)=0$, and $\mathcal{V}\left(x\right)>0$, $\forall x\in \mathbb{D}-\left\{0\right\}$,

$T_2.$

along system (4) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{\partial \mathcal{V}\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right)}^{T}f\left(x\right(t\left)\right)\le 0,\forall t\ge t^{\star },\forall x\in \mathbb{D}.\end{array}$$

(8)

Then $x=0$ is Lyapunov stable on $\mathbb{D}$. If $\mathbb{D}={\mathbb{R}}^n$, and, in addition, $\mathcal{V}\left(x\right)\to \infty$ as $\parallel x\parallel \to \infty$, then $x=0$ is globally Lyapunov stable.

Proof. 

Set $\widehat{\beta }=\widehat{\gamma }$, that means $\left({\beta }_1,\dots ,{\beta }_n\right)=\left({\gamma }_1,\dots ,{\gamma }_n\right)\in (0,1]\times \dots \times (0,1]$. Then the result is immediate from Conjecture 3.2. This completes the proof. □

3.2. Stability conjectures for non-autonomous systems

Conjecture 3.3.

Let $x=0$ be an equilibrium point of system (1). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{\star },\infty )\times \mathbb{D}\to \mathbb{R}$ be a continuously differentiable and satisfy

$T_1.$

$W_1\left(x\right)\le \mathcal{V}(t,x)\le W_2\left(x\right)$, $\forall t\ge t^{\star }$, $\forall x\in \mathbb{D}$, where $W_1$ and $W_2$ are continuous positive definite functions on $\mathbb{D}$,

$T_2.$

along system (1) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial t}}+{\left({\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\beta }}}x\left(t\right)\le -W_3\left(x\right),\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(9)

where $\widehat{\beta }=\left({\beta }_1,\dots ,{\beta }_n\right)\in (0,1]\times \dots \times (0,1]$, and $W_3$ is continuous positive definite function on $\mathbb{D}$.

Then $x=0$ is Lyapunov asymptotically stable on $\mathbb{D}$. If $\mathbb{D}={\mathbb{R}}^n$, and, in addition, $W_1\left(x\right)\to \infty$ as $\parallel x\parallel \to \infty$, then $x=0$ is globally Lyapunov asymptotically stable.

Conjecture 3.4.

Let $x=0$ be an equilibrium point of system (1). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{\star },\infty )\times \mathbb{D}\to \mathbb{R}$ be a continuously differentiable and satisfy

$T_1.$

$W_1\left(x\right)\le \mathcal{V}(t,x)\le W_2\left(x\right)$, $\forall t\ge t^{\star }$, $\forall x\in \mathbb{D}$, where $W_1$ and $W_2$ are continuous positive definite functions on $\mathbb{D}$,

$T_2.$

along system (1) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial t}}+{\left({\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\beta }}}x\left(t\right)<0,\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(10)

where $\widehat{\beta }=\left({\beta }_1,\dots ,{\beta }_n\right)\in (0,1]\times \dots \times (0,1]$.

Then $x=0$ is Lyapunov asymptotically stable on $\mathbb{D}$. If $\mathbb{D}={\mathbb{R}}^n$, and, in addition, $W_1\left(x\right)\to \infty$ as $\parallel x\parallel \to \infty$, then $x=0$ is globally Lyapunov asymptotically stable.

Proposition 3.

Let $x=0$ be an equilibrium point of system (1). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{\star },\infty )\times \mathbb{D}\to \mathbb{R}$ be a continuously differentiable and satisfy

$T_1.$

$W_1\left(x\right)\le \mathcal{V}(t,x)\le W_2\left(x\right)$, $\forall t\ge t^{\star }$, $\forall x\in \mathbb{D}$, where $W_1$ and $W_2$ are continuous positive definite functions on $\mathbb{D}$,

$T_2.$

along system (1) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial t}}+{\left({\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial x\left(t\right)}}\right)}^{T}f(t,x(t\left)\right)\le -W_3\left(x\right),\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(11)

where $W_3$ is continuous positive definite function on $\mathbb{D}$.

Then $x=0$ is Lyapunov asymptotically stable on $\mathbb{D}$. If $\mathbb{D}={\mathbb{R}}^n$, and, in addition, $W_1\left(x\right)\to \infty$ as $\parallel x\parallel \to \infty$, then $x=0$ is globally Lyapunov asymptotically stable.

Proof. 

We set $\widehat{\beta }=\widehat{\gamma }$, that means $\left({\beta }_1,\dots ,{\beta }_n\right)=\left({\gamma }_1,\dots ,{\gamma }_n\right)\in (0,1]\times \dots \times (0,1]$. By Conjecture 3.3, one obtains the result. This completes the proof. □

Conjecture 3.5.

Let $x=0$ be an equilibrium point of system (1). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{\star },\infty )\times \mathbb{D}\to \mathbb{R}$ be a continuously differentiable and satisfy

$T_1.$

$W_1\left(x\right)\le \mathcal{V}(t,x)\le W_2\left(x\right)$, $\forall t\ge t^{\star }$, $\forall x\in \mathbb{D}$, where $W_1$ and $W_2$ are continuous positive definite functions on $\mathbb{D}$,

$T_2.$

along system (1) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial t}}+{\left({\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\beta }}}x\left(t\right)\le 0,\forall t\ge t^{\star },\forall x\in \mathbb{D},\end{array}$$

(12)

where $\widehat{\beta }=\left({\beta }_1,\dots ,{\beta }_n\right)\in (0,1]\times \dots \times (0,1]$.

Then $x=0$ is Lyapunov stable on $\mathbb{D}$. If $\mathbb{D}={\mathbb{R}}^n$, and, in addition, $W_1\left(x\right)\to \infty$ as $\parallel x\parallel \to \infty$, then $x=0$ is globally Lyapunov stable.

Proposition 4.

Let $x=0$ be an equilibrium point of system (1). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{\star },\infty )\times \mathbb{D}\to \mathbb{R}$ be a continuously differentiable and satisfy

$T_1.$

$W_1\left(x\right)\le \mathcal{V}(t,x)\le W_2\left(x\right)$, $\forall t\ge t^{\star }$, $\forall x\in \mathbb{D}$, where $W_1$ and $W_2$ are continuous positive definite functions on $\mathbb{D}$,

$T_2.$

along system (1) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial t}}+{\left({\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial x\left(t\right)}}\right)}^{T}f(t,x(t\left)\right)\le 0,\forall t\ge t^{\star },\forall x\in \mathbb{D},\end{array}$$

(13)

where $W_3$ is continuous positive definite function on $\mathbb{D}$.

Then $x=0$ is Lyapunov stable on $\mathbb{D}$. If $\mathbb{D}={\mathbb{R}}^n$, and, in addition, $W_1\left(x\right)\to \infty$ as $\parallel x\parallel \to \infty$, then $x=0$ is globally Lyapunov stable.

Proof. 

We set $\widehat{\beta }=\widehat{\gamma }$, that means $\left({\beta }_1,\dots ,{\beta }_n\right)=\left({\gamma }_1,\dots ,{\gamma }_n\right)\in (0,1]\times \dots \times (0,1]$. By Conjecture 3.5, one obtains the result. This completes the proof. □

Remark 2.

In contrast to literature works [15,16,18,19,20,21,22], Conjectures 3.1, 3.2, 3.3, 3.4, and 3.5 offer space-time partial derivatives of Lyapunov functions, which sharpen understanding by new intuitive mathematical tools. The proofs of these conjectures remain unknown and are left as open exercise problems to the readers.

4. Linearization of Autonomous Systems

This section introduces a linearization procedure for the autonomous nonlinear fractional-order system and proposes two new Lyapunov theorems. These two theorems can be called the fractional Lyapunov indirect method (fractional Lyapunov linearization method).

The Lyapunov linearization method for continuous-time ordinary differential systems is well-known in the work of Khalil [2]. Going beyond such a method to fractional-order systems (4) remains crucial in the basic studies to stability theory. We thus proceed as follows.

Suppose that origin $x=0$ is an equilibrium point of (4); that means $f\left(0\right)=0$, and let $x=0\in \mathbb{D}\subset {\mathbb{R}}^n$. Assume that $f:\mathbb{D}\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuously differentiable. By mean value theorem, we have

$$\begin{array}{c}\hfill f_i\left(x\right)=f_i\left(0\right)+{\displaystyle \frac{\partial f_i}{\partial x}}\left(z_i\right)x,\end{array}$$

(14)

where $z_i$ is a point on the line segment connecting x to origin 0. Note that the mentioned equality is valid for any point $x\in \mathbb{D}$ such that the line segment connecting x to origin 0 lies entirely in $\mathbb{D}$. By noticing $f\left(0\right)=0$, we can write

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f_i\left(x\right)& ={\displaystyle \frac{\partial f_i}{\partial x}}\left(z_i\right)x\hfill \\ \hfill & ={\displaystyle \frac{\partial f_i}{\partial x}}\left(0\right)x+\left[{\displaystyle \frac{\partial f_i}{\partial x}}\left(z_i\right)x-{\displaystyle \frac{\partial f_i}{\partial x}}\left(0\right)x\right].\hfill \end{array}\end{array}$$

(15)

Hence, we express the function $f\left(x\right)$ as given below

$$\begin{array}{c}\hfill f\left(x\right)=Ax+g\left(x\right),\end{array}$$

(16)

where $A={\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]}_{x=0}\in {\mathbb{R}}^{n\times n}$ and $g_i\left(x\right)=\left[{\displaystyle \frac{\partial f_i}{\partial x}}\left(z_i\right)-{\displaystyle \frac{\partial f_i}{\partial x}}\left(0\right)\right]x$. The function $g_i\left(x\right)$ satisfies

$$\begin{array}{c}\hfill |g_i\left(x\right)|\le \parallel {\displaystyle \frac{\partial f_i}{\partial x}}\left(z_i\right)-{\displaystyle \frac{\partial f_i}{\partial x}}\left(0\right)\parallel \parallel x\parallel .\end{array}$$

(17)

Thus, we have

$$\begin{array}{c}\hfill \parallel g\left(x\right)\parallel \le {\left({\displaystyle \sum _{i=1}^n}\parallel {\displaystyle \frac{\partial f_i}{\partial x}}\left(z_i\right)-{\displaystyle \frac{\partial f_i}{\partial x}}\left(0\right){\parallel }^2\right)}^{1/2}\parallel x\parallel .\end{array}$$

(18)

Since $\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]$ is continuous on domain $\mathbb{D}$, for any $L_i>0$ there exists $r>0$ such that

$$\begin{array}{c}\hfill \parallel x\parallel <r\Rightarrow \parallel {\displaystyle \frac{\partial f_i}{\partial x}}\left(z_i\right)-{\displaystyle \frac{\partial f_i}{\partial x}}\left(0\right)\parallel <L_i,i=1,\dots ,n.\end{array}$$

(19)

Set ${\tilde{L}}^2={\displaystyle \sum _{i=1}^n}{L}_i^2$. Then, it follows from (18) that

$$\begin{array}{c}\hfill \parallel g\left(x\right)\parallel <\tilde{L}\parallel x\parallel ,\forall \parallel x\parallel <r.\end{array}$$

(20)

This suggests that in a small neighbourhood of origin, we can approximate the nonlinear system (4) with a linearized fractional order system

$$\begin{array}{c}\hfill C_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)=Ax\left(t\right).\end{array}$$

(21)

Remark 3.

For the nonlinear system (4), the linear system (21) is called a linearized system around equilibrium point $x=0$.

Lemma 1.

Consider the linear invariant system (21). If there exist a constant, symmetric and positive definite matrix $P=\left[p_{ij}\right]\in {\mathbb{R}}^{n\times n}$ such that

$$\begin{array}{c}\hfill PA+A^{T}P\le -\lambda I_n,\end{array}$$

(22)

for some constant $\lambda >0$ and identity matrix $I_n\in {\mathbb{R}}^{n\times n}$, then the equilibrium point $x=0$ of system (21) is globally Lyapunov asymptotically stable on ${\mathbb{R}}^n$.

Proof. 

We use the quadratic Lyapunov function $\mathcal{V}\left(x\right)=x^{T}Px$, where $P=\left[p_{ij}\right]\in {\mathbb{R}}^{n\times n}$ is a constant, symmetric and positive definite matrix. Then along system (21) non-trivial solution $x\left(t\right)$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left({\displaystyle \frac{\partial \mathcal{V}\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)& ={\left(2Px\left(t\right)\right)}^{T}Ax\left(t\right)\hfill \\ \hfill & =x^T\left(t\right)\left(PA+A^{T}P\right)x\left(t\right)\hfill \\ \hfill & \le -\lambda x^T\left(t\right)I_{n}x\left(t\right)\hfill \\ \hfill & <0,\forall t>t^{\star },\forall x\ne {\mathbb{R}}^n-\left\{0\right\},\hfill \end{array}\end{array}$$

(23)

where the inequality (22) was used. Since $\mathcal{V}\left(0\right)=0$ and the inequality

$$\begin{array}{c}\hfill {\lambda }_{min}\left(P\right){\parallel x\parallel }^2\le \mathcal{V}\left(x\right)\le {\lambda }_{max}\left(P\right){\parallel x\parallel }^2,\end{array}$$

(24)

holds, where ${\lambda }_{min}\left(P\right)>0$ and ${\lambda }_{max}\left(P\right)>0$ are the minimum and maximum eigenvalues of matrix P, respectively. One has $\mathcal{V}\left(x\right)\to \infty$ as $\parallel x\parallel \to \infty$. Set $\widehat{\beta }=\widehat{\gamma }$. Thus, by Conjecture 3.1, we conclude that the equilibrium point $x=0$ of system (21) should be globally Lyapunov asymptotically stable. □

Theorem 1.

Let $x=0$ be an equilibrium point of autonomous nonlinear system (4) where $f:\mathbb{D}\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuously differentiable and $\mathbb{D}\subset {\mathbb{R}}^n$ is a neighbourhood of $x=0\in {\mathbb{R}}^n$. Let $A={\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]}_{x=0}\in {\mathbb{R}}^{n\times n}$. If there exist a constant, symmetric and positive definite matrix $P=\left[p_{ij}\right]\in {\mathbb{R}}^{n\times n}$ such that

$$\begin{array}{c}\hfill PA+A^{T}P\le -\lambda I_n,\end{array}$$

(25)

for some constant $\lambda >0$ and identity matrix $I_n\in {\mathbb{R}}^{n\times n}$, then the equilibrium point $x=0$ of nonlinear system (4) is Lyapunov asymptotically stable on $\mathbb{D}$.

Proof. 

Since f is continuously differentiable on $\mathbb{D}$, we have

$$\begin{array}{c}\hfill f\left(x\right)=Ax+g\left(x\right)\end{array}$$

(26)

where $A={\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]}_{x=0}\in {\mathbb{R}}^{n\times n}$ and the function $g:\mathbb{D}\to {\mathbb{R}}^n$ satisfies ${\displaystyle \frac{\parallel g\left(x\right)\parallel }{\parallel x\parallel }}\to 0$ as $\parallel x\parallel \to 0$. Take the domain $\mathbb{D}=\{x\in {\mathbb{R}}^n:\parallel x\parallel <r,r>0\}$. Therefore, for any $ϵ>0$, there exists $r>0$ such that

$$\begin{array}{c}\hfill \parallel g\left(x\right)\parallel <ϵ\parallel x\parallel ,\forall x\in \mathbb{D}.\end{array}$$

(27)

Now we consider the linearized system (21), where $A={\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]}_{x=0}\in {\mathbb{R}}^{n\times n}$. Observe that when the condition (25) holds, by Lemma 1 one assures that origin of linear system (21) is Lyapunov asymptotically stable on $\mathbb{D}$. We use the same Lyapunov function $\mathcal{V}\left(x\right)=x^{T}Px$ for nonlinear system (4), where $P=\left[p_{ij}\right]\in {\mathbb{R}}^{n\times n}$ is a constant, symmetric and positive definite matrix. Then along system (4) non-trivial solution $x\left(t\right)$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left({\displaystyle \frac{\partial \mathcal{V}\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)& =2x^T\left(t\right)Pf\left(x\left(t\right)\right)\hfill \\ \hfill & =x^T\left(t\right)Pf\left(x\left(t\right)\right)+f^T\left(x\left(t\right)\right)Px\left(t\right),\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\}.\hfill \end{array}\end{array}$$

(28)

Consequently, it follows from (28) that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left({\displaystyle \frac{\partial \mathcal{V}\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)& =x^T\left(t\right)Pf\left(x\left(t\right)\right)+f^T\left(x\left(t\right)\right)Px\left(t\right)\hfill \\ \hfill & =x^T\left(t\right)P\left(Ax\left(t\right)+g\left(x\right(t\left)\right)\right)+{\left(Ax\left(t\right)+g\left(x\right(t\left)\right)\right)}^{T}Px\left(t\right)\hfill \\ \hfill & =x^T\left(t\right)\left(PA+A^{T}P\right)x\left(t\right)+2x^T\left(t\right)Pg\left(x\left(t\right)\right)\hfill \\ \hfill & \le -\lambda x^T\left(t\right)x\left(t\right)+2\parallel x\left(t\right)\parallel \parallel P\parallel \parallel g\left(x\right(t\left)\right)\parallel \hfill \\ \hfill & <-{\lambda \parallel x\left(t\right)\parallel }^2+{2ϵ\parallel P\parallel \parallel x\left(t\right)\parallel }^2,\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\},\hfill \end{array}\end{array}$$

(29)

where (25), (26), and (27) were utilized. Set $ϵ<{\displaystyle \frac{\lambda }{2\parallel P\parallel }}$. Then, the inequality (29) becomes

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\left({\displaystyle \frac{\partial \mathcal{V}\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)<0,\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\}.\end{array}\end{array}$$

(30)

Set $\widehat{\beta }=\widehat{\gamma }$. By Conjecture 3.1, the equilibrium point $x=0$ of system (4) should be Lyapunov asymptotically stable on $\mathbb{D}$. This completes the proof. □

Lemma 2.

Let $A\in {\mathbb{R}}^{n\times n}$ be a matrix and all eigenvalues of A satisfy $\Re \left({\lambda }_i\right)<0$. Then for any given symmetric, positive definite matrix $Q\in {\mathbb{R}}^{n\times n}$ there exists a symmetric, positive definite matrix $P\in {\mathbb{R}}^{n\times n}$ that satisfies the matrix Lyapunov equation

$$\begin{array}{c}\hfill PA+A^{T}P=-Q.\end{array}$$

(31)

Moreover, the matrix P is a unique solution of (31).

Proof. 

Suppose that all eigenvalues of A satisfy $\Re \left({\lambda }_i\right)<0$. Define the matrix P by

$$\begin{array}{c}\hfill P={\int }_{t_{\star }}^{\infty }{e}^{A^T(t-t_{\star })}Q{e}^{A(t-t_{\star })}dt,\end{array}$$

(32)

where $t^{\star }\in \mathbb{R}$. This integral actually exists if we write $A=UJ{U}^{-1}$, where $U\in {\mathbb{R}}^{n\times n}$ is an invertible matrix and $J\in {\mathbb{R}}^{n\times n}$ is Jordan canonical form. In this case, one obtains $e^{A(t-t^{\star })}=U{e}^{J(t-t^{\star })}{U}^{-1}$. Thus, every entry of $e^{J(t-t^{\star })}$ of the form $e^{k-1}{e}^{{\lambda }_i(t-t^{\star })}$, where $\Re \left({\lambda }_i\right)<0$. Consequently, one has $e^{A(t-t^{\star })}\to 0$ as $t\to \infty$. Further, we have

$$\begin{array}{c}\hfill \parallel e^{A(t-t^{\star })}x\parallel \le C^{\#}\parallel x\parallel e^{\Re \left({\lambda }_i\right)(t-t^{\star })},\end{array}$$

(33)

for some $C^{\#}>0$ and $\Re \left({\lambda }_i\right)<0$. Now by using (33), one gets

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill x^{T}Px& ={\int }_{t_{\star }}^{\infty }{x}^T{e}^{A^T(t-t_{\star })}Q{e}^{A(t-t_{\star })}xdt\hfill \\ \hfill & \le {\lambda }_{max}\left(Q\right){\int }_{t_{\star }}^{\infty }{\parallel e^{A(t-t_{\star })}x\parallel }^{2}dt\hfill \\ \hfill & \le {\lambda }_{max}\left(Q\right){\left(C^{\#}\right)}^2{\parallel x\parallel }^2{\int }_{t_{\star }}^{\infty }{e}^{2\Re \left({\lambda }_i\right)(t-t^{\star })}dt\hfill \\ \hfill & <\infty .\hfill \end{array}\end{array}$$

(34)

Suppose that the matrix P is not positive definite. Then, there is a vector $x\ne 0$ such that $x^{T}Px=0$. It implies that

$$\begin{array}{c}\hfill {\int }_{t_{\star }}^{\infty }{x}^T{e}^{A^T(t-t_{\star })}Q{e}^{A(t-t_{\star })}xdt=0.\end{array}$$

(35)

Since

$$\begin{array}{c}\hfill {\lambda }_{min}\left(Q\right){\int }_{t_{\star }}^{\infty }\parallel e^{A(t-t_{\star })}x{\parallel }^{2}dt\le {\int }_{t_{\star }}^{\infty }{x}^T{e}^{A^T(t-t_{\star })}Q{e}^{A(t-t_{\star })}xdt\le {\lambda }_{max}\left(Q\right){\int }_{t_{\star }}^{\infty }{\parallel e^{A(t-t_{\star })}x\parallel }^{2}dt,\end{array}$$

(36)

by using (35) in (36), we have

$$\begin{array}{c}\hfill e^{A(t-t_{\star })}x=0,\forall t\ge t^{\star }.\end{array}$$

(37)

Note that $e^{A(t-t_{\star })}$ is non-singular for all $t\ge t^{\star }$. Thus, we see that $x=0$ solves the equation (37). This contradiction proves that P is positive definite. We use (32) in the left-hand side of (31) and obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill PA+A^{T}P& ={\int }_{t_{\star }}^{\infty }{e}^{A^T(t-t_{\star })}Q{e}^{A(t-t_{\star })}Adt+{\int }_{t_{\star }}^{\infty }{A}^T{e}^{A^T(t-t_{\star })}Q{e}^{A(t-t_{\star })}dt\hfill \\ \hfill & ={\int }_{t_{\star }}^{\infty }{\displaystyle \frac{d}{dt}}\left(e^{A^T(t-t_{\star })}Q{e}^{A(t-t_{\star })}\right)dt\hfill \\ \hfill & ={\left[e^{A^T(t-t_{\star })}Q{e}^{A(t-t_{\star })}\right]}_{t=t^{\star }}^{t=\infty }\hfill \\ \hfill & =-Q.\hfill \end{array}\end{array}$$

(38)

To prove the solution P is unique, we proceed as given below. Let $\tilde{P}\ne P$ be another solution of (31). Then

$$\begin{array}{c}\hfill \left(P-\tilde{P}\right)A+A^T\left(P-\tilde{P}\right)=0.\end{array}$$

(39)

Pre-multiplying by $e^{A^T(t-t^{\star })}$ and post-multiplying by $e^{A(t-t^{\star })}$ in (39), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 0& =e^{A^T(t-t^{\star })}\left[\left(P-\tilde{P}\right)A+A^T\left(P-\tilde{P}\right)\right]e^{A(t-t^{\star })}\hfill \\ \hfill & ={\displaystyle \frac{d}{dt}}\left[e^{A^T(t-t^{\star })}\left(P-\tilde{P}\right)e^{A(t-t^{\star })}\right].\hfill \end{array}\end{array}$$

(40)

By integrating (40) from $t^{\star }$ to ∞, we obtain

$$\begin{array}{c}\hfill -\left[P-\tilde{P}\right]=0.\end{array}$$

(41)

Therefore,

$$\begin{array}{c}\hfill P=\tilde{P}.\end{array}$$

(42)

This completes the proof. □

Lemma 3.

Consider the linear invariant system (21). If all eigenvalues of A satisfy $\Re \left({\lambda }_i\right)<0$, then the equilibrium point $x=0$ of system (21) is globally Lyapunov asymptotically stable on ${\mathbb{R}}^n$.

Proof. 

If all eigenvalues of A satisfy $\Re \left({\lambda }_i\right)<0$, by Lemma 2, we see that for any given symmetric, positive-definite matrix $Q\in {\mathbb{R}}^{n\times n}$, there exists a symmetric, positive definite matrix $P\in {\mathbb{R}}^{n\times n}$ that satisfies the so-called matrix Lyapunov equation

$$\begin{array}{c}\hfill PA+A^{T}P=-Q.\end{array}$$

(43)

We let the Lyapunov function $\mathcal{V}\left(x\right)=x^{T}Px$, where $P=\left[p_{ij}\right]\in {\mathbb{R}}^{n\times n}$ is a constant, symmetric and positive definite matrix. Then along system (21) non-trivial solution $x\left(t\right)$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left({\displaystyle \frac{\partial \mathcal{V}\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)& ={\left(2Px\left(t\right)\right)}^{T}Ax\left(t\right)\hfill \\ \hfill & =x^T\left(t\right)\left(PA+A^{T}P\right)x\left(t\right)\hfill \\ \hfill & \le -x^T\left(t\right)Qx\left(t\right)\hfill \\ \hfill & <0,\forall t>t^{\star },\forall x\ne {\mathbb{R}}^n-\left\{0\right\},\hfill \end{array}\end{array}$$

(44)

where (43) was used. Since $\mathcal{V}\left(0\right)=0$ and the inequality

$$\begin{array}{c}\hfill {\lambda }_{min}\left(P\right){\parallel x\parallel }^2\le \mathcal{V}\left(x\right)\le {\lambda }_{max}\left(P\right){\parallel x\parallel }^2,\end{array}$$

(45)

holds, where ${\lambda }_{min}\left(P\right)>0$ and ${\lambda }_{max}\left(P\right)>0$ are the minimum and maximum eigenvalues of matrix P, respectively. One has $\mathcal{V}\left(x\right)\to \infty$ as $\parallel x\parallel \to \infty$. Set $\widehat{\beta }=\widehat{\gamma }$. Thus, by Conjecture 3.1, we conclude that the equilibrium point $x=0$ of system (21) should be globally Lyapunov asymptotically stable. This completes the proof. □

Theorem 2.

Let $x=0$ be an equilibrium point of autonomous nonlinear system (4) where $f:\mathbb{D}\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuously differentiable and $\mathbb{D}\subset {\mathbb{R}}^n$ is a neighbourhood of $x=0\in {\mathbb{R}}^n$. Let $A={\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]}_{x=0}\in {\mathbb{R}}^{n\times n}$. If all eigenvalues of A satisfy $\Re \left({\lambda }_i\right)<0$, then the equilibrium point $x=0$ of nonlinear system (4) is Lyapunov asymptotically stable on $\mathbb{D}$.

Proof. 

Since f is continuously differentiable on $\mathbb{D}$, we have the expression

$$\begin{array}{c}\hfill f\left(x\right)=Ax+g\left(x\right)\end{array}$$

(46)

where $A={\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]}_{x=0}\in {\mathbb{R}}^{n\times n}$ and the function $g:\mathbb{D}\to {\mathbb{R}}^n$ satisfies ${\displaystyle \frac{\parallel g\left(x\right)\parallel }{\parallel x\parallel }}\to 0$ as $\parallel x\parallel \to 0$. Take the domain $\mathbb{D}=\{x\in {\mathbb{R}}^n:\parallel x\parallel <r,r>0\}$. Therefore, for any $ϵ>0$, there exists $r>0$ such that

$$\begin{array}{c}\hfill \parallel g\left(x\right)\parallel <ϵ\parallel x\parallel ,\forall x\in \mathbb{D}.\end{array}$$

(47)

Now we consider the linearized system (21), where $A={\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]}_{x=0}\in {\mathbb{R}}^{n\times n}$. Observe that if all eigenvalues of A satisfy $\Re \left({\lambda }_i\right)<0$, by Lemma 2, we see that for any given symmetric, positive definite matrix $Q\in {\mathbb{R}}^{n\times n}$, there exists a symmetric, positive definite matrix $P\in {\mathbb{R}}^{n\times n}$ that satisfies the matrix Lyapunov equation

$$\begin{array}{c}\hfill PA+A^{T}P=-Q.\end{array}$$

(48)

As a result, if we take the Lyapunov function $\mathcal{V}\left(x\right)=x^{T}Px$, by Lemma 3 and using (48), the origin of linear system (21) is Lyapunov asymptotically stable on $\mathbb{D}$. We use the same Lyapunov function $\mathcal{V}\left(x\right)=x^{T}Px$ for nonlinear system (4), where $P=\left[p_{ij}\right]\in {\mathbb{R}}^{n\times n}$ is a constant, symmetric, and positive definite matrix. Then along the system (4) non-trivial solution $x\left(t\right)$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left({\displaystyle \frac{\partial \mathcal{V}\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)& =2x^T\left(t\right)Pf\left(x\left(t\right)\right)\hfill \\ \hfill & =x^T\left(t\right)Pf\left(x\left(t\right)\right)+f^T\left(x\left(t\right)\right)Px\left(t\right),\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\}.\hfill \end{array}\end{array}$$

(49)

Consequently, it follows from (49) that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left({\displaystyle \frac{\partial \mathcal{V}\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)& =x^T\left(t\right)Pf\left(x\left(t\right)\right)+f^T\left(x\left(t\right)\right)Px\left(t\right)\hfill \\ \hfill & =x^T\left(t\right)P\left(Ax\left(t\right)+g\left(x\right(t\left)\right)\right)+{\left(Ax\left(t\right)+g\left(x\right(t\left)\right)\right)}^{T}Px\left(t\right)\hfill \\ \hfill & =x^T\left(t\right)\left(PA+A^{T}P\right)x\left(t\right)+2x^T\left(t\right)Pg\left(x\left(t\right)\right)\hfill \\ \hfill & \le -x^T\left(t\right)Qx\left(t\right)+2\parallel x\left(t\right)\parallel \parallel P\parallel \parallel g\left(x\right(t\left)\right)\parallel \hfill \\ \hfill & <-x^T\left(t\right)Qx\left(t\right)+{2ϵ\parallel P\parallel \parallel x\left(t\right)\parallel }^2,\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\},\hfill \end{array}\end{array}$$

(50)

where (46), (47) and (48) were utilized. Set $ϵ<{\displaystyle \frac{{\lambda }_{min}\left(Q\right)}{2\parallel P\parallel }}$. Then, the inequality (50) becomes

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\left({\displaystyle \frac{\partial \mathcal{V}\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)<0,\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\}.\end{array}\end{array}$$

(51)

Set $\widehat{\beta }=\widehat{\gamma }$. By Conjecture 3.1, the equilibrium point $x=0$ of system (4) should be Lyapunov asymptotically stable on $\mathbb{D}$. This completes the proof. □

Remark 4.

The result in Theorem 2 develops an equivalent analogue of Theorem 1 by using eigenvalues of the linearized matrix of the nonlinear system (4).

Remark 5.

Note that Tavazoei and Haeri [24] considered Taylor’s series approximation-based linearization and proposed order-dependent stability conditions for autonomous fractional-order systems. In contrast, Theorems 1 and 2 in Section 4 develop order-independent conditions for an effective stability analysis by means of Lyapunov-based linearization.

5. Linearization of Nonautonomous Systems

The goal of this section is to give an extension of the previous section’s linearization procedure and generalize Theorem 1 to a non-autonomous fractional-order system (1).

Suppose that origin $x=0$ is an equilibrium point of system (1); that means $f(t,0)=0$, $\forall t\ge t^{\star }$. Assume that $f:[t^{\star },\infty )\times \mathbb{D}\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuously differentiable, $\mathbb{D}=\{x\in {\mathbb{R}}^n:\parallel x\parallel <r\}$. Furthermore, suppose that the Jacobian matrix $\left[{\displaystyle \frac{\partial f(t,x)}{\partial x}}\right]$ is bounded and Lipschitz on $\mathbb{D}$, uniformly in t. Then we have

$$\begin{array}{c}\hfill \parallel {\displaystyle \frac{\partial f_j(t,u)}{\partial x}}-{\displaystyle \frac{\partial f_j(t,w)}{\partial x}}\parallel \le L_1\parallel u-w\parallel ,\forall u,w\in \mathbb{D},\forall t\ge t^{\star },\end{array}$$

(52)

for all $j=1,2,\dots ,n$. By mean value theorem,

$$\begin{array}{c}\hfill f_j(t,x)=f_j(t,0)+{\displaystyle \frac{\partial f_j(t,z_j)}{\partial x}}x,\end{array}$$

(53)

where $z_j$ is the point on the line segment connecting x to origin 0. Since $f(t,0)=0$, we can write (53):

$$\begin{array}{c}\hfill f_j(t,x)={\displaystyle \frac{\partial f_j(t,0)}{\partial x}}x+\left[{\displaystyle \frac{\partial f_j(t,z_j)}{\partial x}}x-{\displaystyle \frac{\partial f_j(t,0)}{\partial x}}x\right].\end{array}$$

(54)

Thus, we have

$$\begin{array}{c}\hfill f(t,x)=A\left(t\right)x+g(t,x)\end{array}$$

(55)

where $A\left(t\right)={\displaystyle \frac{\partial f(t,0)}{\partial x}}$ and $g_j(t,x)=\left[{\displaystyle \frac{\partial f_j(t,z_j)}{\partial x}}-{\displaystyle \frac{\partial f_j(t,0)}{\partial x}}\right]x$. Observe that the function g satisfies

$$\begin{array}{c}\hfill \parallel g(t,x)\parallel \le {\left[{\displaystyle \sum _{j=1}^n}\parallel {\displaystyle \frac{\partial f_j(t,z_j)}{\partial x}}-{\displaystyle \frac{\partial f_j(t,0)}{\partial x}}{\parallel }^2\right]}^{1/2}\parallel x\parallel .\end{array}$$

(56)

By using (52) in (56), one obtains

$$\begin{array}{c}\hfill \parallel g(t,x)\parallel \le {L\parallel x\parallel }^2,\end{array}$$

(57)

where $L=n^{1/2}{L}_1$. Therefore, in the small neighbourhood of origin, we may approximate the nonlinear system (1) by a linear fractional order system

$$\begin{array}{c}\hfill C_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)=A\left(t\right)x\left(t\right),\end{array}$$

(58)

where $A\left(t\right)={\displaystyle \frac{\partial f(t,0)}{\partial x}}$.

Lemma 4.

Let $x=0$ be an equilibrium point of non-autonomous linear system (58). If there exist a continuously differentiable, bounded, symmetric and positive definite matrix $P\left(t\right)=\left[p_{ij}\left(t\right)\right]\in {\mathbb{R}}^{n\times n}$ and satisfy

$i)$

$0<c_1{I}_n\le P\left(t\right)\le c_2{I}_n$,$\forall t\ge t^{\star }$,

$ii)$

the matrix Lyapunov equation

$$\begin{array}{c}\hfill {\displaystyle \frac{dP\left(t\right)}{dt}}+P\left(t\right)A\left(t\right)+A^T\left(t\right)P\left(t\right)=-Q\left(t\right),\end{array}$$

(59)

where $Q\left(t\right)=\left[q_{ij}\left(t\right)\right]\in {\mathbb{R}}^{n\times n}$ is continuous, symmetric and positive definite; that means $Q\left(t\right)\ge c_3{I}_n>0$, $\forall t\ge t^{\star }$, and identity matrix $I_n\in {\mathbb{R}}^{n\times n}$,

then the equilibrium point $x=0$ of system (58) is globally Lyapunov asymptotically stable on ${\mathbb{R}}^n$.

Proof. 

We take the Lyapunov function $\mathcal{V}(t,x)=x^{T}P\left(t\right)x$, where $P\left(t\right)=\left[p_{ij}\left(t\right)\right]\in {\mathbb{R}}^{n\times n}$ is a continuously differentiable, bounded, symmetric, and positive definite matrix that satisfies condition $i)$. Then along the system (58) non-trivial solution $x\left(t\right)$, one has

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial t}}+{\left({\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)& =x^T\left(t\right){\displaystyle \frac{dP\left(t\right)}{dt}}x\left(t\right)+{\left(2P\left(t\right)x\left(t\right)\right)}^{T}A\left(t\right)x\left(t\right)\hfill \\ \hfill & =x^T\left(t\right)\left({\displaystyle \frac{dP\left(t\right)}{dt}}+P\left(t\right)A\left(t\right)+A^T\left(t\right)P\left(t\right)\right)x\left(t\right)\hfill \\ \hfill & =-x^T\left(t\right)Q\left(t\right)x\left(t\right)\hfill \\ \hfill & \le -c_3{\parallel x\left(t\right)\parallel }^2\hfill \\ \hfill & <0,\forall t>t^{\star },\forall x\ne {\mathbb{R}}^n-\left\{0\right\},\hfill \end{array}\end{array}$$

(60)

where (59) of condition $ii)$ was used. Since

$$\begin{array}{c}\hfill W_1\left(x\right)=c_1{\parallel x\parallel }^2\le \mathcal{V}(t,x)\le c_2{\parallel x\parallel }^2=W_2\left(x\right),\end{array}$$

(61)

it follows from Conjecture 3.4 that $x=0$ should be globally asymptotically stable. This completes the proof. □

Theorem 3.

Let $x=0$ be an equilibrium point of autonomous nonlinear system (1) where $f:[t^{\star },\infty )\times \mathbb{D}\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuously differentiable, $\mathbb{D}=\{x\in {\mathbb{R}}^n:\parallel x\parallel <r\}$ and the Jacobian matrix $\left[{\displaystyle \frac{\partial f(t,x)}{\partial x}}\right]$ is bounded and Lipschitz on $\mathbb{D}$, uniformly in t. Let $A={\left[{\displaystyle \frac{\partial f(t,x)}{\partial x}}\right]}_{x=0}\in {\mathbb{R}}^{n\times n}$. If there exist a continuously differentiable, bounded, symmetric and positive definite matrix $P\left(t\right)=\left[p_{ij}\left(t\right)\right]\in {\mathbb{R}}^{n\times n}$ and satisfy

$i)$

$0<c_1{I}_n\le P\left(t\right)\le c_2{I}_n$,$\forall t\ge t^{\star }$,

$ii)$

the Lyapunov equation

$$\begin{array}{c}\hfill {\displaystyle \frac{dP\left(t\right)}{dt}}+P\left(t\right)A\left(t\right)+A^T\left(t\right)P\left(t\right)=-Q\left(t\right),\end{array}$$

(62)

where $Q\left(t\right)=\left[q_{ij}\left(t\right)\right]\in {\mathbb{R}}^{n\times n}$ is continuous, symmetric and positive definite; that means $Q\left(t\right)\ge c_3{I}_n>0$, $\forall t\ge t^{\star }$, and identity matrix $I_n\in {\mathbb{R}}^{n\times n}$,

then the equilibrium point $x=0$ of nonlinear system (1) is Lyapunov asymptotically stable on $\mathbb{D}$.

Proof. 

Since f is continuously differentiable on $\mathbb{D}$, we have

$$\begin{array}{c}\hfill f(t,x)=A\left(t\right)x+g(t,x)\end{array}$$

(63)

where $A={\left[{\displaystyle \frac{\partial f(t,x)}{\partial x}}\right]}_{x=0}\in {\mathbb{R}}^{n\times n}$ and the function $g:[t^{\star },\infty )\times \mathbb{D}\to {\mathbb{R}}^n$ satisfies

$$\begin{array}{c}\hfill \parallel g(t,x)\parallel \le {L\parallel x\parallel }^2,\forall x\in \mathbb{D}.\end{array}$$

(64)

Note that when the conditions $i)$ and $ii)$ hold, by Lemma 4, one can assure that the origin of linear system (58) is Lyapunov asymptotically stable on $\mathbb{D}$. We take the same Lyapunov function $\mathcal{V}(t,x)=x^{T}P\left(t\right)x$, where $P\left(t\right)=\left[p_{ij}\left(t\right)\right]\in {\mathbb{R}}^{n\times n}$ is a continuously differentiable, bounded, symmetric, and positive definite matrix that satisfies condition $i)$. Then along the system (1) non-trivial solution $x\left(t\right)$, one has

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial t}}+{\left({\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)& =x^T\left(t\right){\displaystyle \frac{dP\left(t\right)}{dt}}x\left(t\right)+{\left(2P\left(t\right)x\left(t\right)\right)}^{T}f(t,x\left(t\right))\hfill \\ \hfill & =x^T\left(t\right){\displaystyle \frac{dP\left(t\right)}{dt}}x\left(t\right)+x^T\left(t\right)P\left(t\right)f(t,x\left(t\right))\hfill \\ \hfill & +f^T(t,x\left(t\right))P\left(t\right)x\left(t\right)\hfill \\ \hfill & =x^T\left(t\right)\left[{\displaystyle \frac{dP\left(t\right)}{dt}}+P\left(t\right)A\left(t\right)+A^T\left(t\right)P\left(t\right)\right]x\left(t\right)\hfill \\ \hfill & +2x^T\left(t\right)P\left(t\right)g(t,x\left(t\right))\hfill \\ \hfill & =-x^T\left(t\right)Q\left(t\right)x\left(t\right)+2x^T\left(t\right)P\left(t\right)g(t,x\left(t\right))\hfill \\ \hfill & \le -c_3{\parallel x\left(t\right)\parallel }^2+2c_{2}L{\parallel x\left(t\right)\parallel }^3\hfill \\ \hfill & <-\left(c_3-2c_{2}Lr\right){\parallel x\left(t\right)\parallel }^2,\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\},\hfill \end{array}\end{array}$$

(65)

where (63), (64) and $ii)$ were used. Set $L<{\displaystyle \frac{c_3}{2c_{2}r}}$. Then

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial t}}+{\left({\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)<0,\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\}.\end{array}$$

(66)

Set $\widehat{\beta }=\widehat{\gamma }$. By Conjecture 3.4, the equilibrium point $x=0$ of system (1) should be Lyapunov asymptotically stable on $\mathbb{D}$. This completes the proof. □

Remark 6.

In [25], the authors have used the ideas of Metzler asymptotic stability and suggested order-dependent stability conditions for non-autonomous fractional-order systems. In contrast, Theorem 3 develops an order-independent condition that is linked with the matrix Lyapunov differential equation (62) associated with a uniformly positive definite matrix.

6. Fractional Krasovskii’s Method

This section establishes Krasovskii’s method for stability analysis of autonomous fractional-order system (4). The method provides a way to construct a Lyapunov function by using the non-linearity of such a system.

Theorem 4.

Consider the autonomous nonlinear system (4) with $f\left(0\right)=0$. Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing origin $x=0$. Assume that $f:\mathbb{D}\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuously differentiable and its Jacobian matrix $\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]$ satisfies

$$\begin{array}{c}\hfill \left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]+{\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]}^T\le -k^{+}{I}_n,\forall t\ge t^{\star },\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(67)

where some constant $k^{+}>0$. Then $x=0$ is locally Lyapunov asymptotically stable on $\mathbb{D}$. If $\mathbb{D}={\mathbb{R}}^n$ and ${\parallel f\left(x\right)\parallel }^2\to \infty$ as $\parallel x\parallel \to \infty$, then $x=0$ is globally Lyapunov asymptotically stable.

Proof. 

We take the Lyapunov function $\mathcal{V}\left(x\right)=f^T\left(x\right)f\left(x\right)$. Note that $\mathcal{V}\left(0\right)=0$ and $\mathcal{V}\left(x\right)>0$ for all $x\in \mathbb{D}-\left\{0\right\}$. Then along the system (4) non-trivial solution $x\left(t\right)$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left({\displaystyle \frac{\partial \mathcal{V}\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)& ={\left(2\left[{\displaystyle \frac{\partial f\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right]f\left(x\left(t\right)\right)\right)}^{T}f\left(x\left(t\right)\right)\hfill \\ \hfill & =2f^T\left(x\left(t\right)\right){\left[{\displaystyle \frac{\partial f\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right]}^{T}f\left(x\left(t\right)\right)\hfill \\ \hfill & =f^T\left(x\left(t\right)\right){\left[{\displaystyle \frac{\partial f\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right]}^{T}f\left(x\left(t\right)\right)+f^T\left(x\left(t\right)\right)\left[{\displaystyle \frac{\partial f\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right]f\left(x\left(t\right)\right)\hfill \\ \hfill & =f^T\left(x\left(t\right)\right)\left({\left[{\displaystyle \frac{\partial f\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right]}^T+\left[{\displaystyle \frac{\partial f\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right]\right)f\left(x\left(t\right)\right)\hfill \\ \hfill & \le -k^{+}{f}^T\left(x\left(t\right)\right)I_{n}f\left(x\left(t\right)\right)\hfill \\ \hfill & =-k^{+}\mathcal{V}\left(x\left(t\right)\right)\hfill \\ \hfill & <0,\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\},\hfill \end{array}\end{array}$$

(68)

where (67) was used. Set $\widehat{\beta }=\widehat{\gamma }$. Thus, it follows from Conjecture 3.1 that $x=0$ should be locally Lyapunov asymptotically stable. When $\mathbb{D}={\mathbb{R}}^n$ and ${\parallel f\left(x\right)\parallel }^2\to \infty$ as $\parallel x\parallel \to \infty$, by Conjecture 3.1, $x=0$ should be globally Lyapunov asymptotically stable. This completes the proof. □

Here we give an extension of the previous Theorem 4 by using a constant, symmetric, and positive definite matrix.

Theorem 5.

Consider the autonomous nonlinear system (4) with $f\left(0\right)=0$. Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing origin $x=0$. Assume that $f:\mathbb{D}\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuously differentiable and its Jacobian matrix $\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]$ satisfies

$$\begin{array}{c}\hfill P\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]+{\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]}^{T}P\le -k^{+}{I}_n,\forall t\ge t^{\star },\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(69)

where constant $k^{+}>0$, and $P=\left[p_{ij}\right]\in {\mathbb{R}}^{n\times n}$ is constant, symmetric and positive definite matrix. Then $x=0$ is locally Lyapunov asymptotically stable on $\mathbb{D}$. If $\mathbb{D}={\mathbb{R}}^n$ and $f^T\left(x\right)Pf\left(x\right)\to \infty$ as $\parallel x\parallel \to \infty$, then $x=0$ is globally Lyapunov asymptotically stable.

Proof. 

Take the Lyapunov function $\mathcal{V}\left(x\right)=f^T\left(x\right)Pf\left(x\right)$, where $P=\left[p_{ij}\right]\in {\mathbb{R}}^{n\times n}$ is a constant, symmetric, and positive definite matrix. Note that $\mathcal{V}\left(0\right)=0$ and $\mathcal{V}\left(x\right)>0$ for all $x\in \mathbb{D}-\left\{0\right\}$. Then along the system (4) non-trivial solution $x\left(t\right)$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left({\displaystyle \frac{\partial \mathcal{V}\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)& ={\left(2P\left[{\displaystyle \frac{\partial f\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right]f\left(x\left(t\right)\right)\right)}^{T}f\left(x\left(t\right)\right)\hfill \\ \hfill & =2f^T\left(x\left(t\right)\right){\left[{\displaystyle \frac{\partial f\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right]}^{T}Pf\left(x\left(t\right)\right)\hfill \\ \hfill & =f^T\left(x\left(t\right)\right){\left[{\displaystyle \frac{\partial f\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right]}^{T}Pf\left(x\left(t\right)\right)+f^T\left(x\left(t\right)\right)P^T\left[{\displaystyle \frac{\partial f\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right]f\left(x\left(t\right)\right)\hfill \\ \hfill & =f^T\left(x\left(t\right)\right)\left({\left[{\displaystyle \frac{\partial f\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right]}^{T}P+P\left[{\displaystyle \frac{\partial f\left(x\right(t\left)\right)}{\partial x\left(t\right)}}\right]\right)f\left(x\left(t\right)\right)\hfill \\ \hfill & \le -k^{+}{\parallel f\left(x\right(t\left)\right)\parallel }^2\hfill \\ \hfill & <0,\forall t>t^{\star },\forall x\in \mathbb{D}-\left\{0\right\},\hfill \end{array}\end{array}$$

(70)

where (69) was used. Set $\widehat{\beta }=\widehat{\gamma }$. Thus, by Conjecture 3.1, $x=0$ should be locally Lyapunov asymptotically stable. When $\mathbb{D}={\mathbb{R}}^n$ and $f^T\left(x\right)Pf\left(x\right)\to \infty$ as $\parallel x\parallel \to \infty$, by Conjecture 3.1, $x=0$ should be globally Lyapunov asymptotically stable. This completes the proof. □

Remark 7.

The fractional Krasovskii’s method uses the fractional Lyapunov direct method (see Conjecture 3.1) to achieve the stability conditions of autonomous system (4). The Lyapunov functions $\mathcal{V}\left(x\right)=f^T\left(x\right)f\left(x\right)$ and $\mathcal{V}\left(x\right)=f^T\left(x\right)Pf\left(x\right)$ in Theorem 4 and Theorem 5, respectively, provide a constructive tool for an effective stability analysis of equilibrium points of such class systems.

7. Demonstration

This section demonstrates the novel significance of theoretical results to some complicated fractional-order systems.

The non-autonomous Lorenz-type system was considered by Zhang in a paper [26]. We thought of the fractional-order Lorenz system in Example 1 and investigated the stability of the origin by the consideration of time-varying parameters.

Example 1.

Consider the non-autonomous fractional-order Lorenz system

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_{{\mathfrak{D}}_{t^{\star },t}^{{\gamma }_1}}{x}_1\left(t\right)& =\sigma \left(t\right)\left(x_2\left(t\right)-x_1\left(t\right)\right)\hfill \\ \hfill C_{{\mathfrak{D}}_{t^{\star },t}^{{\gamma }_2}}{x}_2\left(t\right)& =\rho \left(t\right)x_1\left(t\right)-x_2\left(t\right)-x_1\left(t\right)x_3\left(t\right)\hfill \\ \hfill C_{{\mathfrak{D}}_{t^{\star },t}^{{\gamma }_3}}{x}_3\left(t\right)& =x_1\left(t\right)x_2\left(t\right)-\beta \left(t\right)x_3\left(t\right)\hfill \end{array}\end{array}$$

(71)

where initial-time $t^{\star }\in \mathbb{R}$, orders ${\gamma }_1,{\gamma }_2,{\gamma }_3\in (0,1]$, and time-varying parameters $\sigma \left(t\right)$, $\rho \left(t\right)$ and $\beta \left(t\right)$ are non-negative and bounded.

Observe that system (71) has an equilibrium point ${\left(0,0,0\right)}^T\in {\mathbb{R}}^3$. Here, we consider the Lyapunov function $\mathcal{V}(t,x)=x_1^2+x_2^2+x_3^2$, where $x={\left(x_1,x_2,x_3\right)}^T\in {\mathbb{R}}^3$. This function obeys bound

$$\begin{array}{c}\hfill W_1\left(x\right)={0.5\parallel x\parallel }^2\le \mathcal{V}(t,x)\le 2{\parallel x\parallel }^2=W_2\left(x\right),\forall t\ge t^{\star },\forall x\in {\mathbb{R}}^n.\end{array}$$

(72)

Then, along system (71) solution $x\left(t\right)$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial t}}& +{\left({\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)\hfill \\ \hfill & ={\displaystyle \sum _{j=1}^3}2x_j\left(t\right)C_{{\mathfrak{D}}_{t^{\star },t}^{{\gamma }_j}}{x}_j\left(t\right)\hfill \\ \hfill & =2x_1\left(t\right)\sigma \left(t\right)\left(x_2\left(t\right)-x_1\left(t\right)\right)+2x_2\left(t\right)\rho \left(t\right)x_1\left(t\right)-2x_2^2\left(t\right)-2x_2\left(t\right)x_1\left(t\right)x_3\left(t\right)\hfill \\ \hfill & +2x_3\left(t\right)x_1\left(t\right)x_2\left(t\right)-2\beta \left(t\right)x_3^2\left(t\right)\hfill \\ \hfill & =-2\sigma \left(t\right)x_1^2\left(t\right)-2x_2^2\left(t\right)-2\beta \left(t\right)x_3^2\left(t\right)+2\left(\sigma \left(t\right)+\rho \left(t\right)\right)x_1\left(t\right)x_2\left(t\right)\hfill \\ \hfill & \le -\left(\sigma \left(t\right)-\rho \left(t\right)\right)x_1^2\left(t\right)-\left(2-\sigma \left(t\right)-\rho \left(t\right)\right)x_2^2\left(t\right)-2\beta \left(t\right)x_3^2\left(t\right),\forall t>t^{\star },\forall x\ne 0.\hfill \end{array}\end{array}$$

(73)

Since the parameters are bounded, we let

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 0<\tilde{\rho }& \le \rho \left(t\right)\le {\rho }^{\#},\forall t\ge t^{\star },\hfill \\ \hfill 0<\tilde{\sigma }& \le \sigma \left(t\right)\le {\sigma }^{\#},\forall t\ge t^{\star },\hfill \\ \hfill 0<\tilde{\beta }& \le \beta \left(t\right)\le {\beta }^{\#},\forall t\ge t^{\star },\hfill \end{array}\end{array}$$

(74)

where $\tilde{\sigma },{\sigma }^{\#},\tilde{\rho },{\rho }^{\#},\tilde{\beta },{\beta }^{\#}$ are some positive constants. By substituting (74) in (73), one obtains

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial t}}& +{\left({\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)\hfill \\ \hfill & \le -\left(\tilde{\sigma }-{\rho }^{\#}\right)x_1^2\left(t\right)-\left(2-{\sigma }^{\#}-{\rho }^{\#}\right)x_2^2\left(t\right)-2\tilde{\beta }{x}_3^2\left(t\right),\forall t>t^{\star },\forall x\ne 0.\hfill \end{array}\end{array}$$

(75)

Here, we assume that the time-varying parameters’ bounds satisfy

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \tilde{\sigma }>{\rho }^{\#},2>{\sigma }^{\#}+{\rho }^{\#},\tilde{\beta }>0.\end{array}\end{array}$$

(76)

Consequently, when (74) and (76) are satisfied, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial t}}& +{\left({\displaystyle \frac{\partial \mathcal{V}(t,x(t\left)\right)}{\partial x\left(t\right)}}\right)}^T{C}_{{\mathfrak{D}}_{t^{\star },t}^{\widehat{\gamma }}}x\left(t\right)\hfill \\ \hfill & <0,\forall t>t^{\star },\forall x\ne 0.\hfill \end{array}\end{array}$$

(77)

Set $\widehat{\beta }=\widehat{\gamma }=\left({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n\right)$. We see that $T_1$ and $T_2$ of Conjecture 3.4 are clearly satisfied. Therefore, by Conjecture 3.4, the equilibrium point ${\left(0,0,0\right)}^T$ should be globally Lyapunov asymptotically stable, provided (74) and (76) hold.

The integer-order pendulum system was discussed by Khalil in his book [2]. We consider a fractional-order version of such a system described in equation (78). By using a linearization tool, we will show that this system equilibrium point is locally Lyapunov asymptotically stable.

Example 2.

Consider the fractional order pendulum system with friction

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_{{\mathfrak{D}}_{t^{\star },t}^{{\gamma }_1}}{x}_1\left(t\right)& =x_2\left(t\right)\hfill \\ \hfill C_{{\mathfrak{D}}_{t^{\star },t}^{{\gamma }_2}}{x}_2\left(t\right)& =-asin\left(x_1\left(t\right)\right)-b{x}_2\left(t\right)\hfill \end{array}\end{array}$$

(78)

where initial-time $t^{\star }\ge 0$, orders $\left({\gamma }_1,{\gamma }_2\right)\in (0,1]\times (0,1]$, and parameters a and b are positive.

It is reasonably clear that the system (78) has an equilibrium point ${\left(0,0\right)}^T\in {\mathbb{R}}^2$. Take the function

$$\begin{array}{c}\hfill f\left(x\left(t\right)\right)={\left(x_2\left(t\right),-asin\left(x_1\left(t\right)\right)-b{x}_2\left(t\right)\right)}^T\in {\mathbb{R}}^2.\end{array}$$

(79)

Here we investigate the asymptotic stability of the origin by the Lyapunov linerization method. The Jacobian matrix is

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial f}{\partial x}}=\left[\begin{array}{cc}{\displaystyle \frac{\partial f_1}{\partial x_1}}& {\displaystyle \frac{\partial f_1}{\partial x_2}}\\ {\displaystyle \frac{\partial f_2}{\partial x_1}}& {\displaystyle \frac{\partial f_2}{\partial x_2}}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -acos\left(x_1\right)& -b\end{array}\right].\end{array}$$

(80)

By evaluating Jacobian at $x=0$, we define

$$\begin{array}{c}\hfill A={\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]}_{x=0}=\left[\begin{array}{cc}0& 1\\ -a& -b\end{array}\right].\end{array}$$

(81)

The eigenvalues of matrix A are

$$\begin{array}{c}\hfill {\lambda }_{1,2}={\displaystyle \frac{1}{2}}\left(-b\pm \sqrt{b^2-4a}\right).\end{array}$$

(82)

We see that for all $a,b>0$, the eigenvalues satisfy $\Re \left({\lambda }_{1,2}\right)<0$. Take the domain $\mathbb{D}=\{x:x_1^2+x_2^2<{\pi }^2\}$. By Theorem 2, the equilibrium point ${\left(0,0\right)}^T\in {\mathbb{R}}^2$ should be Lyapunov asymptotically stable on $\mathbb{D}$.

To end our demonstration, we give the following constructive system (83), where it is shown that the zero equilibrium point is globally Lyapunov asymptotically stable by using fractional Krasovskii’s method.

Example 3.

Consider the autonomous nonlinear fractional-order system

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_{{\mathfrak{D}}_{t^{\star },t}^{{\gamma }_1}}{x}_1\left(t\right)& =-3x_1\left(t\right)+x_2\left(t\right)\hfill \\ \hfill C_{{\mathfrak{D}}_{t^{\star },t}^{{\gamma }_2}}{x}_2\left(t\right)& =x_1\left(t\right)-3x_2\left(t\right)-x_2^3\left(t\right)\hfill \end{array}\end{array}$$

(83)

where initial-time $t^{\star }\in \mathbb{R}$, and orders ${\gamma }_1,{\gamma }_2\in (0,1]$.

Note that the origin ${\left(0,0\right)}^T$ is an equilibrium point of system (83) in ${\mathbb{R}}^2$. Take the function

$$\begin{array}{c}\hfill f\left(x\left(t\right)\right)={\left(-3x_1\left(t\right)+x_2\left(t\right),x_1\left(t\right)-3x_2\left(t\right)-x_2^3\left(t\right)\right)}^T\in {\mathbb{R}}^2.\end{array}$$

(84)

Here we apply fractional Krasovskii’s method to investigate this system. The Jacobian matrix is

$$\begin{array}{c}\hfill \left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]=\left[\begin{array}{cc}-3& 1\\ 1& -3-3x_2^2\end{array}\right].\end{array}$$

(85)

We have

$$\begin{array}{c}\hfill \widehat{K}=\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]+{\left[{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}\right]}^T=\left[\begin{array}{cc}-6& 2\\ 2& -6-6x_2^2\end{array}\right].\end{array}$$

(86)

By Geršgorin theorem [Theorem $6.1.1$, [27]], the eigenvalues of $\widehat{K}$ satisfy

$$\begin{array}{c}\hfill \left|\lambda +6\right|\le 2,|\lambda +6+6x_2^2|\le 2.\end{array}$$

(87)

Consequently, one obtains that all the eigenvalues $\lambda$ have negative real parts for all $x\in {\mathbb{R}}^2$. Thus, the matrix $\widehat{K}$ is negative definite for all $x\in {\mathbb{R}}^2$. Since

$$\begin{array}{c}\hfill {\parallel f\left(x\right)\parallel }^2={\left(-3x_1\left(t\right)+x_2\left(t\right)\right)}^2+{\left(x_1\left(t\right)-3x_2\left(t\right)-x_2^3\left(t\right)\right)}^2\to \infty \end{array}$$

(88)

as $\parallel x\parallel \to \infty$, by Theorem 4 origin ${\left(0,0\right)}^T$ should be globally Lyapunov asymptotically stable. This closes the demonstration.