\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 62, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/62\hfil
 Fractional-like derivative of Lyapunov-type functions]
{Fractional-like derivative of Lyapunov-type functions and
applications to stability analysis of motion}

\author[A. A. Martynyuk, I. M. Stamova \hfil EJDE-2018/62\hfilneg]
{Anatoliy A. Martynyuk, Ivanka M. Stamova}

\address{Anatoliy A. Martynyuk \newline
 S.P. Timoshenko Institute of Mechanics NAS of Ukraine,
 3 Nesterov str. 03057, Kiev-57, Ukraine}
\email{center@inmech.kiev.ua}

\address{Ivanka M. Stamova \newline
Department of Mathematics,
University of Texas at San Antonio,
One UTSA Circle, San Antonio, TX 78249, USA}
\email{ivanka.stamova@utsa.edu}


\thanks{Submitted February 12, 2018. Published March 6, 2018.}
\subjclass[2010]{34A08, 34D20, 34E20}
\keywords{Fractional-like derivative; Lyapunov method;  stability;
\hfill\break\indent  asymptotic stability; instability}

\begin{abstract}
 This article discusses the application of a fractional-like derivative of
 Lyapunov-type functions in the stability analysis of solutions of perturbed
 motion equations with a fractional-like derivative of the state vector.
 The main theorems of the direct Lyapunov method for this class of motion
 equations are established.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

It is known that the Lyapunov function method (or the direct method of Lyapunov) 
is extended to many classes of equations of perturbed motion, including 
systems with distributed parameters and sets of equations in metric spaces. 
See, for example \cite{9} and the references therein. Recall that the  stability 
of motion theory in the sense of Lyapunov was created by him as a result of his
work in 1889--1892 \cite{8}. The key element of the direct Layapunov method is 
the opportunity to calculate the total derivative of a composition of 
functions (chain rule) corresponding to an auxiliary function under 
consideration and the perturbed motion equations.

The great interest in equations with fractional derivatives
 over the last two decades  (see \cite{4, 7, 11, 12, 14} and the bibliography therein) 
has prompted many researchers to generalize the direct Lyapunov method 
to this class of equations. However, the lack of a simple formula for 
calculating the fractional derivative of a composition of functions does 
not allow us to  get results similar to those obtained for many types equations
 for which the total derivative of the Lyapunov function is calculated as 
in the classic analysis. Along with the most common definitions of 
 Riemann-Liouville, Hadamard, Gr\"unwald-Letnikov, 
in 1969 Caputo (see \cite{5}) proposed his definition of a fractional derivative.

In contrast to the classical definitions of fractional derivatives, 
the Caputo definition allows ones to choose the initial values of the solutions 
of fractional differential equations in the same way as for a system of 
ordinary differential equations. This result made it possible to simplify 
somewhat the analysis of the equations of motion with a Caputo's fractional 
derivative. But, as for classical definitions the problem for the evaluation
of the Caputo-type fractional  derivative of a composition of functions
remains open.

It should be noted that some estimates of the Caputo derivative for simple 
functions of Lyapunov (see \cite{3, 10, 11} and the bibliography there) have 
expanded the possibilities of the direct Lyapunov method when analyzing 
the equations of perturbed motion with Caputo derivatives of the state vector. 
The monograph \cite{7} contains the results for equations of this type, 
obtained up to 2009. Many of these results are generalized later for functional 
fractional differential equations and impulsive fractional differential 
equations  \cite{14}.

Recently, in \cite{1}, a definition of a fractional derivative named  
``conformable fractional derivative'' has been proposed by the authors. 
In the opinion of the authors, it is natural to named the new derivative 
as a  ``fractional-like derivative'' (FLD).  In this article the same expression 
is used, since it reflects the essence of the  new definition of a fractional 
derivative.

This article is organized as follows. 
Section 2 provides definitions of a fractional-like derivative and 
some rules for computing it for simple functions. 
In Section 3  our concept of a fractional-like derivative of a Lyapunov-type 
function is introduced, and a Yoshizawa-type relation is established for
such derivatives. In addition, it is shown here that for some 
simple Lyapunov functions of the type of quadratic forms, the fractional-like 
derivative is an upper bound to  the Caputo derivative of these functions. 
In Section 4  sufficient conditions for stability, asymptotic stability and 
instability of the trivial solution of equations of perturbed motion with 
a fractional-like derivative of the state vector are presented. 
In Section 5 we prove the main theorems of the comparison principle on 
the basis of the Lyapunov scalar and vector functions for fractional-like equations. 
In Section 6 sufficient conditions for the stability of motion on a finite 
interval are given. Finally, in Section 7 concluding remarks are presented.

\section{Fractional-like derivatives}

Let $q \in (0,1]$, $\mathbb{R}_+=[0, \infty)$, $t_0 \in \mathbb{R_+}$ 
and given a continuous function  $x(t):[t_0, \infty) \to \mathbb{R}$.

\begin{definition}[\cite{1, 6}] \label{def1}  \rm
  For any $q \in (0,1]$ the fractional-like derivative $\mathcal{D}_{t_0}^q(x(t))$  
of the function $x(t)$ of order $0 < q \leq 1$ is defined by
$$
\mathcal{D}_{t_0}^q(x(t))
= \lim \big\{\frac{x(t+\theta(t-t_0)^{1-q})-x(t)}{\theta},  \theta \to 0 \big\}.
$$
If $t_0=0$, then $\mathcal{D}_{t_0}^q(x(t))$ has the form \cite{6}
$$
\mathcal{D}_{0}^q(x(t))
= \lim \big\{\frac{x(t+\theta t^{1-q})-x(t)}{\theta}, \theta \to 0 \big\}.
$$
In the case $t_0=0$, we will denote $\mathcal{D}_0^q(x(t))=\mathcal{D}^q(x(t))$.

If $\mathcal{D}^q(x(t))$ exists on an open  interval of the type $(0, b)$, 
then 
\[
\mathcal{D}^q(x(0))=\lim_{t \to 0^+} \mathcal{D}^q (x(t)).
\]
If the fractional-like derivative of $x(t)$ of order $q$ exists on 
$(t_0, \infty)$, then the function $x(t)$ is said to be $q$-differentiable 
on the interval $(t_0, \infty)$.
\end{definition}

\begin{proposition}[\cite{6}] \label{prop1}
 Let $q \in (0, 1]$ and $x(t), y(t)$ be $q$-differentiable at a point $t > 0$.
Then:
\begin{itemize}
\item[(a)] $\mathcal{D}_{t_0}^q(a x(t)+ b y(t))=a \mathcal{D}_{t_0}^q (x(t))
 + b \mathcal{D}_{t_0}^q (y(t))$  for all  $a, b \in \mathbb{R}$;

\item[(b)] $\mathcal{D}_{t_0}^q(t^p)=p t^{p-q}$  for any  $p \in \mathbb{R}$;

\item[(c)] $\mathcal{D}_{t_0}^q (x(t) y(t))=x(t) 
 \mathcal{D}_{t_0}^q(y(t))+y(t)\mathcal{D}_{t_0}^q (x(t))$;

\item[(d)] 
\[
\mathcal{D}_{t_0}^q \Big( \frac{x(t)}{y(t)}\Big)
=\frac{ y(t)\mathcal{D}_{t_0}^q(x(t))-x(t)\mathcal{D}_{t_0}^q (y(t))}{y^2(t)};
\]

\item[(e)] $\mathcal{D}_{t_0}^q (x(t))=0$  for any  $x(t)=\lambda$, where 
 $\lambda$ is an arbitrary constant.
\end{itemize}
\end{proposition}

\begin{proposition}[\cite{1, 13}] \label{prop2}
Let $h(y(t)):(t_0, \infty)\to \mathbb{R}$. If  $h(\cdot)$ is differentiable 
with respect to $y(t)$
and $y(t)$ is $q$-differentiable, where $0 < q \leq 1$, then for any 
$t \in \mathbb{R}_+$, $t \neq t_0$ and $y(t)\neq 0$
$$
\mathcal{D}_{t_0}^q h(y(t))=h'(y(t))\mathcal{D}_{t_0}^q(y(t)),
$$
where $h'(t)$ is a partial derivative of $h$.
\end{proposition}

The fractional-like integral of order $0 < q \leq 1$ with a lower limit $t_0$ 
is defined by  (see \cite{6})
$$
I_{t_0}^q x(t)= \int_{t_0}^{t}(s-t_0)^{q-1}x(s)ds.
$$

\begin{proposition}[\cite{6}] \label{prop3} 
Let the function $x(t):(t_0, \infty)\to \mathbb{R}$  be $q$-differentiable 
for $0 < q \leq 1$. Then for all $t > t_0$,
$$
I_{t_0}^q (\mathcal{D}_{t_0}^q x(t))=x(t)-x(t_0).
$$
\end{proposition}

\section{Fractional-like derivatives of Lyapunov-type functions}

Consider a system of differential equations with fractional-like derivative 
of the state vector
\begin{gather}
\mathcal{D}_{t_0}^q x(t)=f(t, x(t)),    \label{e1}\\
x(t_0)=x_0, \label{e2}
\end{gather}
where $x \in \mathbb{R}^n$, $f \in C(\mathbb{R}_+ \times \mathbb{R}^n, \mathbb{R}^n)$, 
$t_0\geq0$. It is further assumed that for 
$(t_0, x_0)\in int(\mathbb{R}_+ \times \mathbb{R}^n)$ the initial value problem 
(IVP) \eqref{e1}--\eqref{e2} has a solution 
$x(t, t_0, x_0)\in C^q([t_0,\infty),\mathbb{R}^n) $ for all 
$t \geq t_0$. In addition, it is assumed that $f(t, 0)=0$ for all $t\geq t_0$.

Let for equation \eqref{e1} a Lyapunov-type function  
$V(t, x)\in C^q(\mathbb{R}_+ \times \mathbb{R}^n, \mathbb{R}_+)$
 be constructed in some way such that $V(t, 0)=0$ for all 
 $t \in \mathbb{R}^n$. Introduce the notation 
$B_r=\{x \in \mathbb{R}^n: \|x\|< r\}$,  $r > 0$.

\begin{definition} \label{def2} \rm
  Let $V$  be a continuous and $q$-differentiable function (scalar or vector), 
$V:\mathbb{R}_+ \times B_r \to \mathbb{R}^s$ ($s=1$ or $s=m$, respectively), 
and $x(t, t_0, x_0)$ be the solution of the IVP \eqref{e1}--\eqref{e2}, 
which exists and is defined on $\mathbb{R}_+ \times B_r$. 
Then for $(t, x)\in \mathbb{R}_+ \times B_r$ the expression:
\begin{itemize}
\item[(1)]
\begin{equation}
\begin{aligned}
& ^{+}\mathcal{D}_{t_0}^q  V(t, x) \\
& =\limsup \big\{\frac{V(t+\theta (t-t_0)^{1-q},  x(t+\theta (t-t_0)^{1-q},
t, x))-V(t,x)}{\theta} , \theta \to 0^+ \big\},
\end{aligned} \label{e3}
\end{equation}
is the upper right fractional-like derivative of the Lyapunov function,

\item[(2)]
\begin{align*}
&_{+}\mathcal{D}_{t_0}^q  V(t, x) \\
&=\liminf \big\{\frac{V(t+\theta (t-t_0)^{1-q},  x(t+\theta (t-t_0)^{1-q}, t, x))
-V(t,x)}{\theta} ,  \theta \to 0^+\big\},
\end{align*}
is the lower right fractional-like derivative of the Lyapunov function,

\item[(3)]
\begin{align*}
&^{-}\mathcal{D}_{t_0}^q  V(t, x) \\
&=\limsup \big\{\frac{V(t+\theta (t-t_0)^{1-q},  x(t+\theta (t-t_0)^{1-q}, t, x))
-V(t,x)}{\theta} , \ \theta \to 0^-\big\},
\end{align*}
is the upper left fractional-like derivative of the Lyapunov function,
\item[(4)]
\begin{align*}
&_{-}\mathcal{D}_{t_0}^q V(t, x) \\
&=\liminf \big\{\frac{V(t+\theta (t-t_0)^{1-q},  x(t+\theta (t-t_0)^{1-q}, t, x))
-V(t,x)}{\theta} ,  \theta \to 0^-\big\},
\end{align*}
is the lower left fractional-like derivative of the Lyapunov function.
\end{itemize}
\end{definition}

An efficient application of the upper right fractional-like derivatives of 
Lyapunov functions in the construction of his direct method is based on 
the following result (cf. \cite{15}).

\begin{lemma} \label{lem1} 
 Let $V(t, x)$ be continuous, $q$-differentiable and locally Lipschitz 
with respect to its second variable  $x$ on $\mathbb{R}_+ \times B_r$.
Then the fractional-like derivative of the function $V(t, x)$ with respect 
to the solution $x(t, t_0, x_0)$ is defined by
\begin{equation}
\begin{aligned}
&^{+}\mathcal{D}_{t_0}^q  V(t, x)\\
&=\limsup \big\{\frac {V(t+\theta (t-t_0)^{1-q},  x+ \theta (t-t_0)^{1-q} f (t, x))
-V(t,x)}{\theta} ,  \theta \to 0^+\big\},  
\end{aligned}\label{e3*}
\end{equation}
where $(t, x)\in \mathbb{R}_+ \times B_r$.
\end{lemma}

If $V(t, x(t))=V(x(t))$, $0 < q \leq 1$, the function $V$ is differentiable on
 $x$, and the function $x(t)$ is $q$-differentiable on $t$ for $t > t_0$, then
$$
^{+}\mathcal{D}_{t_0}^q  \ V(t, x)=V'(x(t))  \mathcal{D}_{t_0}^q x(t),
$$
where  $V'$ is a partial derivative of the function $V$.

Taking relations \eqref{e3} and \eqref{e3*} into account, we obtain the 
result by Yoshizawa \cite{15} for a
fractional-like derivative of the function $V(t, x)$ in the form
$$
^{+}\mathcal{D}_{t_0}^q  V(t, x(t, t_0, x_0))
={}^{+}\mathcal{D}_{t_0}^q \ V(t, x)\big|_{\eqref{e1}}.
$$

\begin{definition} \label{def3} \rm
If the function $V(t, x)$ together with one of its fractional-like derivatives 
resolves the problem of stability (instability) of the solutions of \eqref{e1}, 
we will call $V(t, x)$ a Lyapunov function for the fractional-like 
system \eqref{e1}.
\end{definition}

\begin{example} \label{examp1}\rm
 Let $t > t_0$, $V(t, x)=V_1(x)=x^2(t),  x \in \mathbb{R}$. Then, according 
to (c) in Proposition \ref{prop1}, we have
\begin{gather}
{}^{+}\mathcal{D}_{t_0}^q \ V(x(t))=^{+}\mathcal{D}_{t_0}^q (x(t) x(t))
=x(t) ^{+}\mathcal{D}_{t_0}^q (x(t)), \nonumber \\
+^{+}\mathcal{D}_{t_0}^q (x(t))x(t)=2x(t) ^{+}\mathcal{D}_{t_0}^q (x(t))
\label{e4}
\end{gather}
for all $t \geq t_0$.
Consider the following scalar fractional-like equation for $0 < q \leq 1$,
\begin{equation}
\mathcal{D}_{t_0}^q x(t)=f(t, x(t)), \quad  t \geq t_0,
\label{e5}
\end{equation}
where $f: \mathbb{R}\times\mathbb{R}^n \to \mathbb{R},  f(t, 0)=0$ for $t \geq t_0$.
For the function $V(x)=\frac{1}{2}x^2(t)$, considering \eqref{e4}, we obtain
\begin{equation}
^{+}\mathcal{D}_{t_0}^q   V_1(x(t))\big|_{\eqref{e5}}
=x(t) ^{+}\mathcal{D}_{t_0}^q x(t)=x(t)f(t, x(t)) \label{e6}
\end{equation}
in the domain of the function $f(t,x)$.


Let $V(t, x)=V_2(x)=x^T x,  x \in \mathbb{R}^n$.  Then, according to 
(c) in Proposition \ref{prop1}, we have
\begin{equation}
^{+}\mathcal{D}_{t_0}^q (V_2(x(t))=^{+}\mathcal{D}_{t_0}^q(x^T(t)x(t))
=2x^T(t) ^{+}\mathcal{D}_{t_0}^qx(t)). \label{e7}
\end{equation}
\end{example}

\begin{example} \label{examp2}\rm
 Let  $x_1, x_2:[t_0, \infty) \to \mathbb{R}$ and $x_1, x_2$ be $q$-differentiable. 
Consider the equations of perturbed motion with  fractional-like derivatives 
in the form
\begin{equation}
\begin{gathered}
\mathcal{D}_{t_0}^q x_1(t)=-\mu(t)x_2-\nu(t)x_1, \\
\mathcal{D}_{t_0}^q x_2(t)=\mu(t)x_1-\nu(t)x_2.
\end{gathered}\label{e8}
\end{equation}
where  $\mu(t)$ and  $\nu(t)$ are continuous single-valued functions defined on
 $t \geq t_0$.

For the function $V_2(x_1, x_2)=\frac{1}{2}(x_1^2+x_2^2)$ according to (c) 
in Proposition \ref{prop1}, we obtain
\begin{equation}
\begin{aligned}
&^{+}\mathcal{D}_{t_0}^q \Big(\frac{1}{2}(x_1^2 + x_2^2) \Big)
=x_1(t)^{+}\mathcal{D}_{t_0}^q x_1(t)+ x_2(t) ^{+}\mathcal{D}_{t_0}^q x_2(t) \\
&=-2\nu(t)(x_1^2(t)+x_2^2(t)).
\end{aligned} \label{e9}
\end{equation}
\end{example}

\begin{remark} \label{rmk1}\rm
 In  \cite{2} the authors obtain the following estimate for a fractional derivative 
in the Caputo sense of the Lyapunov function $V(t,x_1, x_2)=(x_1^2+x_2^2)$
 with respect to the system \eqref{e8}
$$
^{c}_{t_0}\!D_{t}^q (V(t,x_1, x_2)) \leq -2(x_1^2(t)+x_2^2(t))
$$
for  $x\in \mathbb{R}^2$. Comparing this estimate with the estimate \eqref{e9}, 
we see that when estimating a fractional Caputo derivative we ``lose''
the effect of the function $\nu (t)$ on the properties of the zero
solution of the system of equations \eqref{e8}.
\end{remark}

\begin{lemma} \label{lem2} 
Let $x \in \mathbb{R}$, $y \in \mathbb{R}^n$ and  $P$ is an $n \times n$ 
constant matrix. Then for the functions  $V_1=x^2(t)$, $V_2=y^T(t)y(t)$, and
 $V_3=y^T(t)Py(t)$ the following estimates hold:
\begin{itemize}
\item[(a)] $^{c}_{t_0}\!D_{t}^q (x^2(t)) \leq ^{+}\mathcal{D}_{t_0}^q (x^2(t))$ 
 for $x \in \mathbb{R}$;

\item[(b)] $^{c}_{t_0}\!D_{t}^q (y^T(t)y(t)) \leq ^{+}
 \mathcal{D}_{t_0}^q (y^T(t)y(t))$  for  $y \in \mathbb{R}^n$;

\item[(c)] $^{c}_{t_0}\!D_{t}^q (y^T(t)P y(t)) 
\leq ^{+}\mathcal{D}_{t_0}^q (y^T(t)P y(t))$,
  for $y \in \mathbb{R}^n$.
\end{itemize}
\end{lemma}

\begin{proof} 
We apply \cite[Lemma 1]{3} to the Caputo fractional derivative of the function 
$V_1$ and obtain
$$
^{c}_{t_0}\!D_{t}^q  \ (x^2(t)) \leq 2x(t) \ ^{c}_{t_0}\!D_{t}^q \ (x(t)).
$$
Similar estimates we can obtain for the functions $V_2$ and $V_3$. 
Taking this into account the equalities \eqref{e4} and \eqref{e7}, we obtain
assertions (a)--(c) of Lemma \ref{lem2}.
\end{proof}

From Lemma \ref{lem2} it follows that the fractional-like derivative of a  
Lyapunov-type function is an upper bound of the Caputo fractional 
derivatives of this Lyapunov function. 

\section{Direct Lyapunov's method and main results}

The Lyapunov-type stability definitions  for a fractional-like system \eqref{e1} 
remain the same as for ordinary differential equations and differential 
equations with Caputo's fractional derivatives. 
See, for example, \cite{7, 14} and the references therein.

In our main theorems we will  use the Hahn class of functions
 $K=\{ a\in C[\mathbb{R}_+,\mathbb{R}_+]: a(u)$ is strictly increasing and 
$a(0)=0 \}$.

\begin{theorem} \label{thm1} 
Assume that for the fractional-like system \eqref{e1} there exist a 
$q$-differentiable function $V(t, x)$, $V(t, 0)=0$ for $t \geq t_0$ and 
functions  $a, b \in K$ such that
\begin{itemize}
\item[(i)] $V(t, x)\geq a (\|x\|)$,  $(t, x)\in \mathbb{R}_+ \times B_r$,
\item[(ii)]   $V(t, x)\leq b (\|x\|)$, $(t, x)\in \mathbb{R}_+ \times B_r$,
\item[(iii)] 
\begin{equation}
{}^{+}\mathcal{D}_{t_0}^q (V(t, x(t))) \leq 0 \quad\text{for }
 (t, x)\in \mathbb{R}_+ \times B_r. \label{e10}
\end{equation}
\end{itemize}
Then the state $x=0$ of \eqref{e1} is uniformly stable.
\end{theorem}

\begin{proof}  Let $x(t)=x(t, t_0, x_0)$ be the solution of \eqref{e1} for  
$(t_0, x_0)\in (\mathbb{R}_+ \times B_r)$ defined for all $t \geq t_0$. 
Let $t_0 \in \mathbb{R}_+$ and $0 < \varepsilon < r$  be given. By 
conditions (i), (ii) of Theorem \ref{thm1} we can choose 
$\delta=\delta(\varepsilon)> 0$ so that
\begin{equation}
b(\delta)< a(\varepsilon). \label{e11}
\end{equation}

We will prove that  $\|x_0\|< \delta$ implies $\|x(t)\|< \varepsilon$ for all 
$t \geq t_0$. If this is not true there exists a solution 
$x(t, t_0, x_0)=x(t)$ of \eqref{e1} such that for  $\|x_0\|< \delta$ there is 
$t_1> t_0$ for which
$$
\|x(t_1)\|=\varepsilon, \quad \|x(t)\|< \varepsilon  \quad \text{for all } 
 t \in [t_0, t_1).
$$
By Proposition \ref{prop3} and condition \eqref{e10}, the Lyapunov relation
$$
V(t, x(t))-V(t_0, x_0)=I_{t_0}^q (^{+}\mathcal{D}_{t_0}^q (V(t, x(t))))
$$
becomes
\begin{equation}
V(t, x(t))-V(t_0, x_0) \leq 0.   \label{e12}
\end{equation}
For  $t=t_1$ we have from \eqref{e12},
\begin{equation}
a(\varepsilon)\leq V(t_1, x(t_1)) \leq V(t_0, x_0)
\leq b(\|x_0\|) < a(\varepsilon). \label{e13}
\end{equation}
This inequality contradicts condition \eqref{e11}.
This completes the proof.
\end{proof}

\noindent\textbf{Example \ref{examp1} continued.}
 From \eqref{e6} and Theorem \ref{thm1} it follows that the state $x=0$ of the 
fractional-like equation \eqref{e5} is uniformly stable if
$$
x(t)f(t, x(t)) \leq 0
$$
for $(t, x) \in \mathbb{R}_+ \times B_r$.


\begin{theorem} \label{thm2} 
 Let the condition of Theorem \ref{thm1} be satisfied and instead of \eqref{e10} 
the following estimate hold
\begin{equation}
^{+}\mathcal{D}_{t_0}^q (V(t, x(t))) \leq -d(\|x\|)   \label{e14}
\end{equation}
for $(t, x)\in \mathbb{R}_+ \times B_r$, where $d \in K$.
Then the state $x=0$ of system \eqref{e1} is uniformly asymptotically stable.
\end{theorem}

\begin{proof}
Since all conditions of Theorem \ref{thm1} are satisfied the state $x=0$ is uniformly 
stable. We will prove that it is uniformly asymptotically stable.

Let $0 < \varepsilon < r$ and $\delta=\delta(\varepsilon)> 0$  be the same
 as in in Theorem \ref{thm1}. For $\varepsilon_0 \leq r$ we choose 
$\delta_0=\delta_0(\varepsilon_0)> 0$ and consider the solution 
$x(t, t_0, x_0)$ with initial data $t_0 \in \mathbb{R}_+$ and 
$\|x_0\|< \delta_0$. Let for $t_0 < t \leq t_0 + T(\varepsilon)$, where  
$T(\varepsilon)\geq \big( qb(\delta_0)/ d(\delta(\varepsilon))\big)^{1/q}$ 
 for $x(t)$ we have $\|x(t)\|\geq \delta(\varepsilon)$.
 We will show that this is not possible under the conditions of Theorem \ref{thm2}. 
From the Lyapunov relation we obtain
\begin{equation}
\begin{aligned}
V(t, x(t))-V(t_0, x_0)
&=I_{t_0}^q (^{+}\mathcal{D}_{t_0}^q(V(t, x(t)) \\
&\leq - I_{t_0}^q (d(\|x(t)\|))=- \int_{t_0}^{t}(s-t_0)^{q-1}d(\|x(s)\|)ds.
\end{aligned} \label{e15}
\end{equation}
From \eqref{e15} we obtain
\begin{equation}
\begin{aligned}
V(t, x(t))
&\leq V(t_0, x_0)- \int_{t_0}^{t}(s-t_0)^{q-1} d (\|x(s)\|)ds \\
&\leq b(\delta_0)-d(\delta(\varepsilon))\frac{(t-t_0)^q}{q}. \label{e16}
\end{aligned}
\end{equation}
For  $t=t_0+T(\varepsilon)$ by \eqref{e16} we have
\begin{align*}
0&< a(\delta(\varepsilon))\leq V(t_0+T(\varepsilon), x(t_0+T(\varepsilon))\\
&\leq b(\delta_0)-d(\delta(\varepsilon))\frac{T(\varepsilon)^q}{q} \leq 0,
\end{align*}
which is a contradiction.

The above contradiction shows that there exists $t_1 \in [t_0, t_0 + T(\varepsilon)]$ 
such that $\|x(t_1)\|< \delta(\varepsilon)$. Hence $\|x(t)  \|< \varepsilon$ 
for all $t \geq t_0 + T(\varepsilon)$  as far as $\|x_0\|< \delta_0$ and 
$\lim \|x(t)\|=0$  as $t \to \infty$ uniformly on $t_0 \in \mathbb{R}_+$. 
This completes the proof.
\end{proof}

\noindent\textbf{Example \ref{examp2} continued.} 
It follows from \eqref{e4} and conditions of Theorem \ref{thm2}
  that the state $x_1=x_2=0$  
of \eqref{e8} will be uniformly asymptotically stable if the function 
$\nu(t)$ satisfies the condition  $\nu(t)\geq \nu_0 > 0$, 
since in this case we have
$$
^{+}\mathcal{D}_{t_0}^q(V(x_1(t), x_2(t)))\leq-2\nu_0(x_1^2(t)+x_2^2(t))< 0
$$
for all $t \geq t_0$ and $0 < q \leq 1$.

In the next theorem, we will establish conditions for the instability 
of the state $x=0$  of system \eqref{e1}.

\begin{theorem} \label{thm3} 
 Let for the system \eqref{e1}  there exists a  $q$-differentiable 
function $V(t, x): \mathbb{R}_+ \times B_{\varepsilon}\to \mathbb{R}$, 
such that on $[t_0, \infty)\times G(h)$, where $G(h)\subset B_{\varepsilon}$, 
$t_0 \geq 0$, the following conditions are satisfied:
\begin{itemize}\itemsep=0cm\parskip=0cm
\item[(1)] $0 < V(t, x)\leq c < \infty$ for some constant $c$;
\item[(2)] $^{+}\mathcal{D}_{t_0}^q  \ V(t, x)\big|_{\eqref{e1}} \geq a(V(t, x))$, 
where $a \in K$,  $0 < q \leq 1$;
\item[(3)] the state $x=0$ belongs to $\partial G(h)$;
\item[(4)] $V(t, x)$=0 for $[t_0, \infty)\times(\partial G(h)
\cap B_{\varepsilon})$.
\end{itemize}
Then the state $x=0$ of system \eqref{e1} is unstable.
\end{theorem}

\begin{proof}
It follows from condition (3) of Theorem \ref{thm3} that for any  $\delta > 0$ 
there exists a $x_0 \in G(h)\cap B_{\delta}$ such that $V(t_0, x_0)> 0$. 
For the solution $x(t)=x(t, t_0, x_0)$ while $x(t)\in G(h)$ from 
conditions \eqref{e1}, (2) we have
\begin{equation}
\begin{aligned}
c &\geq V(t, x(t)) - V(t_0, x_0)\geq I_{t_0}^q a(V(t, x(t)) \\
&\geq V(t_0, x_0) + a (V(t_0, x_0)) \frac{(t-t_0)^q}{q}.
\end{aligned}\label{e17}
\end{equation}
From this inequality it follows that the solution $x(t)$ must leave the
domain $G(h)$ at some moment $t_1 > t_0$.

Since condition (4) of Theorem \ref{thm3} is satisfied, then $x(t)$ can not leave the 
domain $G(h)$ across the boundary $\partial G(h)$, because 
$G(h)\subset B_{\varepsilon}$. Therefore $x(t)$ will leave $B_{\varepsilon}$,
 i.e. $\|x(t_1)\|\geq \varepsilon$. This completes the proof.
\end{proof}

From Theorem \ref{thm3}, we have the following corollary.


\begin{corollary} \label{coro1} 
Suppose that all conditions of Theorem \ref{thm3} hold and conditions
 (1)  and (2)  are replaced by the following conditions, respectively:
\begin{itemize}
\item[(1*)] $0 < V(t, x) \leq b(\|x\|)$,

\item[(2*)] $ ^{+}\mathcal{D}_{t_0}^q  V(t, x) \geq a(\|x\|)$, where 
 $a, b \in K$.

\end{itemize}
Then the state $x=0$ of system \eqref{e1} is unstable.
\end{corollary}

\begin{corollary} \label{coro2}
 Suppose that all conditions of Theorem \ref{thm3} hold and condition  (2) 
 is replaced by
\begin{equation}
{}^{+}\mathcal{D}_{t_0}^q  V(t, x)
= \lambda V(t, x)+ W(x(t)), t \in[t_0, \infty),\quad
 x \in G(h),\; \lambda > 0,  \label{e18}
\end{equation}
where the function $W$ is continuous and  $W(x)\geq 0$.
Then the state $x=0$ of system \eqref{e1} is unstable.
\end{corollary}

\begin{proof} 
Relation \eqref{e18} can be represented in the integral form
\begin{align*}
V(t,x(t))&=V(t_0,x(t_0))\exp\Big(\lambda\frac{(s-t_0)^q}{q}\Big)
+ \int_{t_0}^{t}\exp\Big(\lambda\frac{(s-t_0)^q}{q}\Big)\\
&\quad \times \exp \Big(-\lambda \frac{(s-t_0)^q}{q}\Big)(s-t_0)^{q-1}W(x(s))ds.
\end{align*}
From the above relation, since the second term is positive by the
 conditions of Corollary \ref{coro2}, for any $0 < q \leq 1$ we have
\begin{equation}
V(t,x(t))\geq V(t_0,x(t_0))\exp \Big(\lambda\frac{(t-t_0)^q}{q}\Big), \quad
 t \geq t_0,
\label{e19}
\end{equation}

Let the initial state of the solution $x(t)=x(t, t_0, x_0)$ be $x_0 \in U$, 
where $U$ is a neighborhood of the origin $x=0$. Since for any $t \geq t_0$ 
the estimate \eqref{e19} is satisfied with respect to the solution $x(t)$, 
then for $t \to \infty$  the function $V(t,x(t))$ increases while, 
by the conditions of Theorem \ref{thm3} it is bounded. Hence for $x(t)$ there exists  
$t^*$ such that $x(t^*)$ will leave $B_r$. This proves the instability of 
the state $x=0$ of system \eqref{e1}.
\end{proof}

\begin{example} \label{examp3} \rm
Consider the  fractional-like system for $0 < q \leq 1$,
\begin{equation}
\begin{gathered}
\mathcal{D}_{t_0}^q  x(t)=n(t)y-xg(t, x, y), \quad x(t_0)=x_0; \\
\mathcal{D}_{t_0}^q  y(t)=-n(t)x-yg(t, x, y), \quad y(t_0)=y_0,
\end{gathered}\label{e20}
\end{equation}
where $n(t)$ is a continuous function for all $t \geq t_0$,  $g(t, x, y)$
is a sum of a convergent power series,  $g(t, 0, 0)=0$ for  $t \geq t_0$.
Applying the function $2V(x, y)=x^2 +y^2$ to system \eqref{e20} we have
\begin{equation}
{}^{+}\mathcal{D}_{t_0}^q \ V(x(t)), y (t))=-(x^2+y^2)g(t, x, y).
\label{e21}
\end{equation}
Performing a $q$-integration of \eqref{e21}, we obtain the Lyapunov relation
\begin{equation}
V(x(t), y (t))- V(x_0, y_0)
\leq -r^2 \int_{t_0}^{t}\frac{g(s,x(s), y(s))}{(s-t_0)^{1-q}}ds
\label{e22}
\end{equation}
on the domain $x^2+y^2\leq r^2$ of the equilibrium state $x=y=0$.
From the relation \eqref{e21} and inequality \eqref{e22}  it follows that:
\begin{itemize}
\item[(a)] By Theorem \ref{thm1} the state $x=y=0$ of \eqref{e20} is uniformly
 stable provided the function is such that  $g(t, x, y)\geq 0$ for $t \geq t_0$;

\item[(b)] By Theorem  \ref{thm2} the state $x=y=0$ of system \eqref{e20} is uniformly
asymptotically stable, if $g(t, x, y)> 0$ on the domain $x^2+y^2 \leq r^2$
for $t \geq t_0$;

\item[(c)] By Theorem \ref{thm3} the state $x=y=0$ of system \eqref{e20} is unstable
 if $g(t, x, y)< 0$ for $t \geq t_0$ on a sufficiently small neighborhood.
\end{itemize}
\end{example}


\section{Comparison principle}

We continue our consideration of system \eqref{e1} together with the 
$q$-differentiable function $V(t, x)\in C (\mathbb{R}_+\times \mathbb{R}^n, 
\mathbb{R}_+)$. Consider the total fractional-like derivative of the function 
$V(t, x)$ of the type  \eqref{e3*}.

As in the general theory of stability of motion, the application of the 
comparison principle allows us to indicate in a general form the structure 
of the stability conditions for fractional-like equations of perturbed motion.
We will show that the following comparison theorem  holds.


\begin{theorem} \label{thm4} 
 Assume that:
\begin{itemize}
\item[(1)] For the system \eqref{e1} there exists a $q$-differentiable
 function $V(t, x)$ with а fractional-like derivative of the type \eqref{e3*};

\item[(2)] There exists a function $g(t, u)\in C (\mathbb{R}_+^2, \mathbb{R})$ 
such that
\begin{equation}
{}^{+}\mathcal{D}_{t_0}^q \ V(t, x) \leq g(t, V(t, x)),\label{e22b}
\end{equation}
for $(t, x)\in \mathbb{R}_+ \times \mathbb{R}^n$ and $0 < q \leq 1$;

\item[(3)] There exists a maximal solution 
$r(t)=r(t, t_0, r_0)\in C^q([t_0,\infty),\mathbb{R})$ 
of the comparison scalar fractional-like equation
\begin{equation}
\mathcal{D}_{t_0}^q \ u(t)=g(t, u), \ u(t_0)=u_0 \geq 0 \label{e23}
\end{equation}
for all $t \geq t_0$.
\end{itemize}
Then along the solutions of system \eqref{e1} the estimate
\begin{equation}
V(t, x(t))\leq r(t), \label{e24}
\end{equation}
is valid for all $t \geq t_0$ whenever $V(t_0, x_0)\leq u_0$.
\end{theorem}

\begin{proof} 
Let the solution $x(t)=x(t, t_0, x_0)$ of the IVP \eqref{e1}--\eqref{e2} 
exists on $t \in [t_0, \infty)$ and $V(t_0, x_0)\leq u_0$. 
Denote by $m(t)=V(t, x(t))$ and evaluate the fractional-like derivative 
of the function $m(t)$ by the formula \eqref{e3*}. 
From  condition (2) of Theorem \ref{thm4} we obtain
\begin{equation}
\mathcal{D}_{t_0}^q \ m(t)\leq g(t, V(t, x))=g(t, m(t)).
 \label{e25}
\end{equation}
Similar to \cite[Theorem 2.8.3]{7} we have
\begin{equation}
\mathcal{D}_{t_0}^q \ u(t)=g(t, u)+\varepsilon,  u(t_0)
=u_0 +\varepsilon \geq 0, \quad \varepsilon > 0. \label{e26}
\end{equation}
From  this equality it follows that
$$
\mathcal{D}_{t_0}^q u(t, \varepsilon)=g(t, u(t, \varepsilon))
+\varepsilon>g(t, u(t, \varepsilon)),
$$
so $m(t)< u(t, \varepsilon)$  and, hence
$\lim u(t, \varepsilon)=r(t)$ as $\varepsilon \to 0$,  uniformly on $t$
for $t_0 \leq t < T < + \infty$. This completes the proof.
\end{proof}

Further, we will represent the fractional-like system \eqref{e1} in the form
\begin{equation}
\mathcal{D}_{t_0}^q \ x_i(t)=f_i(t, x_i)+R_i(t, x_1(t), \ldots, x_m(t)),\label{e27}
\end{equation}
where $x_i \in \mathbb{R}^{n_i}$, $\sum_{i=1}^{m}n_i=n$,
$f_i \in C(\mathbb{R}_+ \times \mathbb{R}^{n_i}, \mathbb{R}^{n_i})$,
$R_i \in C(\mathbb{R}_+ \times \mathbb{R}^{n_1} \times \ldots
\times \mathbb{R}^{n_m},  \mathbb{R}^{n_i})$, and $f_i(t, 0)=0$ for $t \geq t_0$.

Suppose that for the independent subsystems
\begin{equation}
\mathcal{D}_{t_0}^q \ x_i(t)=f_i(t, x_i), \ \ i=1, 2, \ldots, m \label{e28}
\end{equation}
Lyapunov-type functions $V_i(t, x_i)$ exist such that
\begin{equation}
{}^{+}\mathcal{D}_{t_0}^q \ (V_i(t, x_i(t)))\big|_{(27)}
\leq -d_i(\|x_i\|)+w_i(t, x_i, x). \label{e29}
\end{equation}
Here $d_i(\cdot)$ are functions of the Hahn class of functions $K$,
 $w_i(t, \cdot, \cdot)$ are continuous with respect to $t$ functions,
and $w_i(t, 0, 0)=0$ for $t \geq t_0$.

If for the right-hand side of \eqref{e29} there is a majorizing function
 $H (t; u)$ which is quasi-monotonic (see \cite{7, 14, 15})  non-decreasing with 
respect to $u$ and such that
\begin{equation}
{}^{+}\mathcal{D}_{t_0}^q  V(t, x(t))\leq H(t, V(t, x(t))), \label{e30}
\end{equation}
where  $V(t, x)=(V_1(t, x_1), \dots, V_m(t, x_m))$, then the following theorem holds.

\begin{theorem} \label{thm5} 
 Assume that $V \in C(\mathbb{R}_+ \times \mathbb{R}^{n}, \mathbb{R}^{m}_+)$ 
is $q$-differentiable,
$$
^{+}\mathcal{D}_{t_0}^q (V(t, x(t)))\leq H(t, V(t, x (t))),
$$
where $H \in C(\mathbb{R}_+ \times \mathbb{R}^{m}_+, \mathbb{R}^{m})$ 
and for all $t \geq t_0$ there exists the maximal solution $u(t)$ of the
 fractional-like equation
$$
\mathcal{D}_{t_0}^q \ u(t)=H(t, u), \quad u(t_0)=u_0,
$$
for values  $0 < q \leq 1$.
Then  $V(t_0, x_0)\leq u_0$ implies
\begin{equation}
V(t, x(t)) \leq u(t), \quad t \geq t_0.  \label{e31}
\end{equation}
\end{theorem}

\begin{proof} 
The proof of Theorem \ref{thm5} is similar to the proof of \cite[Theorem 4.2.1]{7}, 
taking into account that the total fractional-like derivative of the 
Lyapunov is evaluated according to Proposition \ref{prop2}.
\end{proof}

Estimates \eqref{e24} and \eqref{e31} allow us to establish stability criteria 
for the state  $x = 0$ of system \eqref{e1} in the same way as it is done 
 the monograph \cite{7}.

\begin{corollary} \label{coro3}
 If in the estimate \eqref{e22b} the majorizing function
$$
g(t, V(t, x))\leq k V(t, x),\quad  k=const > 0,
$$
then
\begin{equation}
V(t, x(t)) \leq V(t_0, x_0) \exp \Big(k\frac{(t-t_0)^q}{q}\Big) \label{e32}
\end{equation}
for all $t \in [t_0, t_0+T]$ and any values of  $0 < q \leq 1$.
\end{corollary}

\begin{corollary} \label{coro4} 
 If in estimate \eqref{e22} the majorizing function
$$
g(t, V(t, x))\leq k(t) V(t, x),
$$
where $k(t)$ is a $q$-differentiable function, then
\begin{equation}
V(t, x(t))\leq V(t_0, x_0)\exp \Big(\int_{t_0}^{t}k(s)(s-t_0)^{q-1}ds\Big) \label{e33}
\end{equation}
for all $t \in [t_0, t_0 + T]$ and any values of $0 < q \leq 1$.
\end{corollary}


\section{Stability on a finite interval}

For the  fractional-like system \eqref{e1} we give the following definition 
of the stability on a finite interval.

\begin{definition} \label{def4}\rm
 System \eqref{e1} is stable on a finite interval, if for given  
$0 < c_1 < c_2$, $t_0$ and $T > 0$  the solution $x(t)$ satisfies the estimate
$$
V(t, x(t))\leq c_2  \quad \text{for all }  t \in [t_0, t_0 +T]
$$
whenever  $V(t_0, x_0)< c_1$.
\end{definition}

The following theorem follows directly from the estimates \eqref{e32} and \eqref{e33}.

\begin{theorem} \label{thm6} 
 If for system \eqref{e1} there exists a $q$-differentiable function  
$V(t, x)\in C (\mathbb{R}_+ \times \mathbb{R}^{n}, \mathbb{R_+})$, 
such that the conditions of corollaries 3 or 4 are satisfied, then 
system \eqref{e1} is stable on a finite interval if one of the conditions:
\begin{itemize}
\item[(1)] $\exp \big(k\frac{(t-t_0)^q}{q}\big) \leq\frac{c_2}{c_1}$ 
  for all  $t \in [t_0, t_0 +T]$ and any values of $0 < q \leq 1$;

\item[(2)] $\exp \big(\int_{t_0}^{t}k(s)(s-t_0)^{q-1}ds\big)\leq \frac{c_2}{c_1}$
  for all  $t \in [t_0, t_0 +T]$ and any values of  $0 < q \leq 1$ is satisfied, 
respectively.
\end{itemize}
\end{theorem}


\subsection*{Concluding remarks}
For  systems of equations with Caputo fractional  derivatives of the state
 vector there exist several definitions of fractional derivatives of a 
Lyapunov-type function (see, \cite{2, 7}). The actual calculation of the Caputo 
fractional derivative for a Lyapunov-type function is difficult due to 
the absence of a chain rule for this derivative, as for other fractional 
derivatives (Riemann-Liouville,  Gr\"unwald-Letnikov, etc.). 
For this reason, when considering particular examples,  it is necessary 
to estimate the fractional derivative of the Lyapunov function \cite{2, 3, 10, 11}.

In this paper, the direct Lyapunov method is extended to systems of equations 
of perturbed motion with fractional-like derivatives. 
Theorems of the direct Lyapunov method and the comparison principle are 
established for the scalar and vector Lyapunov functions, taking into account 
that for a fractional-like derivative, a chain rule takes place. 
The relationship between a fractional-like derivative and a Caputo fractional 
derivative  (Lemma \ref{lem2}) indicates that the fractional-like  derivative of 
Lyapunov functions under consideration is a majorant for the Caputo fractional 
derivative of these functions. This circumstance
should be taking into account when considering specific problems of the 
stability of motions. 

\subsection*{Acknowledgements}
The authors express their  sincere gratitude to Professor T. A. Burton  
for the careful reading of the manuscript, very interesting historical and
 mathematical comments and an outline of promising directions for the 
development of this approach.

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\end{document}