Stability of Nonlinear Switched Fractional Differential Equations with Short Memory

Abstract

Nonlinear switched systems, which combine multiple subsystems with a switching rule, have garnered significant research interest due to their complex stability properties. In this paper we consider the case where the switching times, the switching rule and the family of functions defining the subsystems are given initially. Note that the switching rule could be such that it is not activated at any initially given switching time. When the switching rule is activated, then a subsystem from the given family is chosen. We study the case where the nonlinear subsystems consist of fractional differential equations. To be more concise we apply generalized Caputo fractional derivatives with respect to other functions. Lyapunov functions are used to analyze stability, and several sufficient conditions are obtained. The influence of the switching rule on the stability property of the solutions is discussed and illustrated with examples. It is shown that, in spite of the solutions of some of the subsystems being unstable, the zero solution of the switched system could be stable.

1. Introduction

Switched systems for integer order differential equations are studied by many authors (see, for example, [1,2,3,4]). Several qualitative properties for linear ordinary differential systems are studied, including optimal control [5], common linear co-positive Lyapunov functions and their existence [6], stability [7,8,9] and finite time boundedness [10]. Also, recently, some discrete nonlinear switched systems were studied (see, for example, [11,12]). In theoretical analytical study it is important to develop symbolic computation (see, for example, [13,14,15]).

Recently, fractional derivatives and fractional differential equations have also been applied in modeling switched processes (see, for example, [11,16,17]). Fractional calculus has given us the opportunity to model the complexity of the processes more adequately since fractional derivatives are nonlocal and have weakly singular kernels. At the same time the application of fractional derivatives to dynamical models is more complicated than the ordinary derivative, and this is mainly connected with the memory property of the fractional derivative and the presence of a lower limit in them (see, for example, the classical books [18,19,20]).

In some papers the lower limit of the applied Caputo fractional derivative in the switched system is fixed at the initial time. In [21,22], switched systems with a fixed lower limit of the fractional derivative are considered, but unfortunately, the integral presentation used is incorrect for the case of a changeable lower limit. In [23] the sequence of switching times is initially given, and a Caputo fractional derivative is applied with a changeable lower limit at any interval between two consecutive switching times (so-called short memory). The role of the switching rule in this paper is to determine the used subsystem between two consecutive switching times, and stability is studied by quadratic Lyapunov functions. In [24] a short memory nonlinear switched system and a Markow switching rule are considered, and some sufficient conditions are obtained for asymptotic stability with Lyapunov functions and the fractional Gronwall inequality.

The analysis of the stability of fractional differential equations is more complex than that of classical differential equations since fractional derivatives are nonlocal and have weakly singular kernels. Some applications to various models in Financial Economics, Physics, neural networks and Engineering can be seen in [25,26,27,28], and several results were obtained such as the stability of systems with a Caputo fractional derivative [29], stability in the sense of Lyapunov by using Gronwall’s inequality and Schwartz’s inequality [30], Mittag–Leffler stability [31] and local asymptotic stability [32]. A good survey on stability results as well as applied approaches is given in [33].

In our paper we define and study the stability of a switched system by the Lyapunov method. By a switched system we mean a dynamical system consisting of a family of continuous-time subsystems of fractional differential equations and a rule that orchestrates the switching among them. The switching rule is given initially, and when it is activated, an appropriate subsystem from the given family is chosen. To be more concise we consider a generalized Caputo fractional derivative with respect to another time, and we assume that one and only one subsystem is active at each time instant. We consider the case of short memory, i.e., the lower limit of the considered fractional derivative is changed and equal to the time when the switching rule is activated. The concept of short memory was proposed and discussed in [34] and subsequently adopted by other authors. Note that the short memory in fractional derivatives is very useful in describing some phenomena in physics and engineering, and it is applied successfully to impulsive fractional models (see, for example [35,36,37,38]). The idea of changing the lower limit of the fractional derivative, the so-called short memory, is adapted and applied to the considered switched systems, and it also helps us correct some misunderstandings in the literature.

First, we describe in detail the presence of the switching rule in the studied system and we illustrate it with examples. We focus on the stability of the state variables, and general Lyapunov functions are applied (different from [23]). Comparison results using scalar fractional differential equations with changing nonlinear functions at switching times (without any activated switching rule) are obtained, and several sufficient conditions for stability are proved. These sufficient conditions depend significantly on the switching rule. For example, even if some of the fractional subsystems are unstable, with an appropriate switching rule, the fractional switched system can still be stable. Most of the results are illustrated with examples.

2. Fractional Integrals and Derivatives with Respect to Another Function

Recently, fractional calculus has attracted much attention since it plays an important role in many fields of science and engineering. Note that the behavior of many systems, such as physical phenomena having memory and genetic characteristics, can be adequately modeled with fractional differential systems (see, for example, [39,40,41]).

Let $a,T:0\le a<T\le \infty$. We will use the following sets of functions

$$C^1((a,T],{\mathbb{R}}^n)=\{\upsilon \in C((a,T],{\mathbb{R}}^n):{\upsilon }^{\prime }\text{exists and it is continuous on}(a,T]\},$$

and

$$\mathcal{P}(a,T)=\{\psi \in C^1([a,T],[0,\infty )):{\psi }^{\prime }\left(t\right)>0,t\in (a,T]\}.$$

Definition 1  

(Definition 1 [42]). Let $q>0,\rho >0,$ and the function $\psi \in \mathcal{P}(a,T)$. The generalized fractional integral with respect to another function (FIF) of the function $\upsilon :[a,T]\to \mathbb{R}$ is defined by (where the integral exists)

$${}^{\rho }I_a^{q,\psi }\upsilon \left(t\right)={\displaystyle \frac{{\rho }^{1-q}}{\Gamma \left(q\right)}}{\int }_a^t{\left(\psi \left(s\right)\right)}^{\rho -1}{\psi }^{\prime }\left(s\right){\displaystyle \frac{\upsilon \left(s\right)}{{\left(\psi {\left(t\right)}^{\rho }-\psi {\left(s\right)}^{\rho }\right)}^{1-q}}}ds,t\in (a,T].$$

Definition 2  

([42]). Let $q\in (0,1),\rho >0$ and the function $\psi \in \mathcal{P}(a,T)$. The generalized Caputo fractional derivative with respect to another function (CFDF1) of the function $\upsilon :[a,T]\to \mathbb{R}$ is defined by (where the integral exists)

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_a^{q,\psi }\upsilon \left(t\right)={\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}{\int }_a^t{\displaystyle \frac{{\upsilon }^{\prime }\left(s\right)}{{\left({\left(\psi \left(t\right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho }\right)}^q}}ds,t\in (a,T].\hfill \end{array}$$

(1)

In the case of vector functions the integral IFF and the derivative CFDF are defined component-wise.

In connection with Definition 2 we will introduce the following set of functions:

$$\begin{array}{cc}& C^q((a,T],{\mathbb{R}}^n,\psi ,\rho )=\{\upsilon \in C((a,T],{\mathbb{R}}^n):{\upsilon }^{\prime }\text{exists almost everywhere in}(a,T]\hfill \\ & \text{and there exists CFDF}{}_{\rho }{}^{C}D_a^{q,\psi }\upsilon \left(t\right)\text{for}t\in (a,T]\}.\hfill \end{array}$$

Remark 1. 

The above defined CFDF generalizes the following fractional derivatives.

-

The Caputo fractional derivative if $\rho =1,\psi \left(t\right)\equiv t$;

-

The Caputo fractional derivative with respect to another function if $\rho =1$;

-

The Katugampola-type generalized fractional derivative if $\psi \left(t\right)\equiv t$.

We will use the following results for CFDF.

Lemma 1  

(Example 17 [42]). Let $\rho >0,q\in (0,1),\psi \in \mathcal{P}(a,T)$. Then

$${}_{\rho }{}^{C}D_a^{q,\psi }{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }}{\rho }}\right)}^{\beta -1}={\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma (\beta -q)}}{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }{-(\psi \left(a\right)}^{\rho }}{\rho }}\right)}^{\beta -q-1},t\in (a,T],$$

$${}_{\rho }{}^{C}D_a^{q,\psi }{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }}{\rho }}\right)}^k={\displaystyle \frac{k!}{\Gamma (k-q+1)}}{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }{-(\psi \left(a\right)}^{\rho }}{\rho }}\right)}^{k-q},t\in (a,T],$$

and

$${}_{\rho }{}^{C}D_a^{q,\psi }C=0,$$

where $\beta >1$, k is a natural number, C is a constant.

Corollary 1.  

Let $\rho >0,q\in (0,1)$ and $\psi \in \mathcal{P}(a,T)$. Then ${}_{\rho }{}^{C}D_a^{q,\psi }{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }}{\rho }}\right)}^q=\Gamma (q+1),t\in (a,T].$

From Corollary 1 we have the following existence result and explicit formula for the solution of the scalar fractional differential equation:

Corollary 2.  

Let $\rho >0,q\in (0,1)$ and $\psi \in \mathcal{P}(a,T)$. The solution of the initial value problem for the scalar fractional differential equation

$${}_{\rho }{}^{C}D_a^{q,\psi }\omega \left(t\right)=K,\omega \left(a\right)=B$$

is $\omega \left(t\right)=B+{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }}{\rho }}\right)}^q{\displaystyle \frac{K}{\Gamma (1+q)}},$ where $K,B$ are constants.

From Lemma 1 we obtain the following result:

Corollary 3.  

Let $\rho >0,q\in (0,1),\psi \in \mathcal{P}(a,T)$ and $\lambda \in \mathbb{R}$ be a constant. Then

$${}_{\rho }{}^{C}D_a^{q,\psi }{E}_q\left(\lambda {\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }}{\rho }}\right)}^q\right)=\lambda E_q\left(\lambda {\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }}{\rho }}\right)}^q\right),t\in (a,T],$$

where $E_q$ is the Mittag–Leffler function with one parameter.

Proof. 

We note that

$$E_q\left(\lambda {\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }}{\rho }}\right)}^q\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{1}{\Gamma (qk+1)}}{\lambda }^k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }}{\rho }}\right)}^{kq}.$$

(2)

Take the CFDF of both sides of (2), use Lemma 1 for $\beta =qk+1$ and obtain

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_a^{q,\psi }{E}_q\left(\lambda {\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }}{\rho }}\right)}^q\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{\lambda }^k}{\Gamma (qk+1)}}{}_{\rho }{}^{C}D_a^{q,\psi }{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }}{\rho }}\right)}^{qk}\hfill \\ & =\sum _{k=1}^{\infty }{\displaystyle \frac{{\lambda }^k}{\Gamma (qk+1)}}{\displaystyle \frac{\Gamma (qk+1)}{\Gamma (qk+1-q)}}{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }{-(\psi \left(a\right)}^{\rho }}{\rho }}\right)}^{qk-q}\hfill \\ & =\sum _{k=0}^{\infty }{\displaystyle \frac{{\lambda }^{k+1}}{\Gamma (qk+1)}}{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }{-(\psi \left(a\right)}^{\rho }}{\rho }}\right)}^{qk}.\hfill \end{array}$$

(3)

□

From Corollary 3 we have the following result:

Lemma 2.  

Let $\rho >0,q\in (0,1),\psi \in \mathcal{P}(a,T)$ and $\lambda \in \mathbb{R}$ be a constant. The solution of the scalar linear fractional initial value problem with CFDF

$${}_{\rho }{}^{C}D_a^{q,\psi }\upsilon \left(t\right)=\lambda \upsilon \left(t\right),fort\in (a,T],and\upsilon \left(a\right)={\upsilon }_0,$$

is the function

$$\upsilon \left(t\right)={\upsilon }_0{E}_q\left(\lambda {\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }}{\rho }}\right)}^q\right),t\in [a,T].$$

(4)

Lemma 3.  

Let $\rho >0,q\in (0,1),\psi \in \mathcal{P}(a,T)$, $\upsilon :[a,T]\to \mathbb{R}$, $a<T\le \infty$ and there exists a point $\xi \in (a,T]$ such that $\upsilon \left(\xi \right)=0$, and $\upsilon \left(t\right)<0$, for $t\in [a,\xi )$ and the CFDF ${}_{\rho }{}^{C}D_a^{q,\psi }{\upsilon \left(t\right)|}_{t=\xi }$ exists. Then if ${lim}_{s\to \xi -}\left({\displaystyle \frac{v\left(s\right)}{{({\left(\psi \left(\xi \right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho })}^q}}\right)=0$ we have ${}_{\rho }{}^{C}D_a^{q,\psi }{\upsilon \left(t\right)|}_{t=\xi }\ge 0$.

Proof. 

Apply ${\displaystyle \frac{{\upsilon }^{\prime }\left(s\right)}{{\left({\left(\psi \left(\xi \right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho }\right)}^q}}={\displaystyle \frac{d}{ds}}\left({\displaystyle \frac{\upsilon \left(s\right)}{{\left({\left(\psi \left(\xi \right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho }\right)}^q}}\right)-{\displaystyle \frac{q\rho {\psi }^{\prime }\left(s\right){\left(\psi \left(s\right)\right)}^{\rho -1}\upsilon \left(s\right)}{{\left({\left(\psi \left(\xi \right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho }\right)}^{1+q}}},\mathrm{a}.\mathrm{e}.s\in (a,\xi )$ and integrating by parts we obtain

$$\begin{array}{cc}\hfill {}_{\rho }{}^{C}D_a^{q,\psi }{\upsilon \left(t\right)|}_{t=\xi }& ={\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}{\int }_a^{\xi }{\left({\left(\psi \left(\xi \right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho }\right)}^{-q}{\upsilon }^{\prime }\left(s\right)ds\hfill \\ & ={\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}{\int }_a^{\xi }{\displaystyle \frac{d}{ds}}\left({\displaystyle \frac{\upsilon \left(s\right)}{{\left({\left(\psi \left(\xi \right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho }\right)}^q}}\right)ds\hfill \\ & -{\displaystyle \frac{{\rho }^{1+q}q}{\Gamma (1-q)}}{\int }_a^{\xi }{\displaystyle \frac{{\psi }^{\prime }\left(s\right){\left(\psi \left(s\right)\right)}^{\rho -1}\upsilon \left(s\right)}{{\left({\left(\psi \left(\xi \right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho }\right)}^{1+q}}}ds\hfill \\ & =-{\displaystyle \frac{{\rho }^q}{\Gamma (1-q)}}\left({\displaystyle \frac{\upsilon \left(a\right)}{{\left({\left(\psi \left(\xi \right)\right)}^{\rho }-{\left(\psi \left(a\right)\right)}^{\rho }\right)}^q}}\right)\hfill \\ & -{\displaystyle \frac{{\rho }^{q+1}q}{\Gamma (1-q)}}{\int }_a^{\xi }{\displaystyle \frac{{\psi }^{\prime }\left(s\right){\left(\psi \left(s\right)\right)}^{\rho -1}\upsilon \left(s\right)}{{\left({\left(\psi \left(\xi \right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho }\right)}^{1+q}}}ds.\hfill \end{array}$$

(5)

From (5), and $\upsilon \left(s\right)\le 0,\psi \left(s\right)\ge 0,{\psi }^{\prime }\left(s\right)>0$ for $s\in [a,\xi ],$ we have the claim in Lemma 3. □

Remark 2.  

In Lemma 3 if $\upsilon \in C^1([a,T],\mathbb{R})$ then L’Hopital’s rule guarantees that

$$\underset{s\to \xi -}{lim}\left({\displaystyle \frac{\upsilon \left(s\right)}{{\left({\left(\psi \left(\xi \right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho }\right)}^q}}\right)=\underset{s\to \xi -}{lim}{\upsilon }^{\prime }\left(s\right){\displaystyle \frac{{\left({\left(\psi \left(\xi \right)\right)}^{\rho }-{\left(\psi \left(s\right)\right)}^{\rho }\right)}^{1-q}}{-q\rho {\left(\psi \left(s\right)\right)}^{\rho -1}{\psi }^{\prime }\left(s\right)}}=0.$$

Note that from $\xi >a$ and ${\psi }^{\prime }\left(s\right)>0,s\in [a.\xi ]$ it follows that $\psi \left(\xi \right)>0$.

Of course one could also put other conditions (other than $\upsilon \in C^1([a,T],\mathbb{R})$), for example, one could instead assume ${lim}_{s\to \xi -}{\upsilon }^{\prime }\left(s\right)$ exists and is a real number) to guarantee that this limit is zero.

3. Description of the Switched Fractional Differential Equations with CFDF

By a switched system we mean a dynamical system consisting of a family of continuous-time subsystems of fractional differential equations with CFDF, initially given switching times (times at which the systems will be changed eventually from one to another) and a given switching rule that orchestrates the switching among them. In this paper, the dynamical system consists of a family of fractional differential subsystems with CFDF. The switching rule plays an exceedingly crucial role in determining

-

The switching time of the system;

-

The subsystem becoming active after each switching;

-

The lower limit of the applied CFDF is equal to the switching time.

Under the influence of the switching rule, the system will exhibit more intriguing dynamic behavior.

The switching rule (sometimes called a switching signal) orchestrates the switching among the subsystems and will be given by a time-dependent function $\sigma :[0,\infty )\to [0,\infty )$, which is piecewise constant and right continuous.

In this paper we will consider the case where the switching rule is $\sigma \left(t\right)=C_k$ for $t\in [{\tau }_k,{\tau }_{k+1})$ where $C_k,k=1,2,\dots ,$ is a non-negative integer.

Let the increasing sequence of switching times ${\left\{{\tau }_k\right\}}_{k=1}^{\infty }$, ${\tau }_1>0$ with ${lim}_{k\to \infty }{\tau }_k=\infty$, the sequence of non-negative integers ${\left\{C_k\right\}}_{k=1}^{\infty }$, and the family of functions ${\left\{f_k(t,\upsilon )\right\}}_{k=0}^{\infty }:f_k\in C([{\tau }_k,\infty )\times {\mathbb{R}}^n,{\mathbb{R}}^n),k=0,1,2,\dots$ be given where ${\tau }_0=0$.

Consider the initial value problem (IVP) for the system of switched fractional differential equations with CFDF (SFDE)

$${}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }\upsilon \left(t\right)=f_{\sigma \left(t\right)}(t,\upsilon \left(t\right)),\text{for}t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,\text{and}\upsilon \left(0\right)={\upsilon }_0,$$

(6)

where ${\upsilon }_0\in {\mathbb{R}}^n$ and $\sigma \left(t\right)=0$ for $t\in [0,{\tau }_1)$.

In this paper we will use the following assumptions concerning the right-hand side parts of the considered system:

Assumption 1.  

The functions $f_k\in C([{\tau }_k,\infty )\times {\mathbb{R}}^n,{\mathbb{R}}^n),k=0,1,2,3,\dots ,$ and $f_k(t,0)\equiv 0,t\ge {\tau }_k$.

Assumption 2.  

For any initial value ${\upsilon }_0^k\in {\mathbb{R}}^n$ and $k=0,1,2,\dots$ the IVP for the system of fractional differential equations with CFDF ${}_{\rho }{}^{C}D_{{\tau }_k}^{q,\psi }\upsilon \left(t\right)=f_k(t,\upsilon \left(t\right)),t>{\tau }_k$ and $\upsilon \left({\tau }_k\right)={\upsilon }_0^k$ has a solution $\upsilon \in C^q([{\tau }_k,\infty ),{\mathbb{R}}^n,\psi ,\rho )$.

Remark 3.  

If Assumptions 1 and 2 are satisfied, the switched system (6) with a zero initial value has a zero solution, and this allows us to define the stability of the zero solution.

Remark 4.  

We will consider the case where any solution $\upsilon (.)$ of SFDE (6) is continuous at any ${\tau }_k,k=1,2,\dots ,$ i.e., $\upsilon ({\tau }_k-0)=\upsilon ({\tau }_k+0)=\upsilon \left({\tau }_k\right),k=1,2,\dots .$

The switching rule has a huge influence on the times at which the system of equations is changed.

First, we will describe the sequence of the switching times in detail.

At the time ${\tau }_1$ there could be an activation of the switching rule $\sigma \left(t\right)=C_1,t\in [{\tau }_1,{\tau }_2)$.

If $C_1\ne 0$, then the switching rule is activated at time ${\tau }_1$, and the system of equations is changed to a system with a lower limit of CFDF ${\tau }_1$ and the function $f_1(t,\upsilon \left(t\right))$.

If $C_1=0$, then the switching rule is not activated at time ${\tau }_1$. Let $k_1=min\{j>1:C_j\ne 0\}$. Then $C_1=C_2=\cdots =C_{k_1-1}=0$, $C_{k_1}\ne 0$, and the switching rule is not activated at the times ${\tau }_1,{\tau }_2,\dots ,{\tau }_{k_1-1}$. It is activated at time ${\tau }_{k_1}$ and the system of equations is changed to a system with a lower limit of CFDF ${\tau }_{k_1}$ and the function $f_{C_{k_1}}(t,\upsilon \left(t\right))$.

To combine both cases ($C_1=0$ and $C_1\ne 0$) we denote ${\xi }_1={\tau }_{m_1}$ where

$$\begin{array}{c}\hfill m_1=\left\{\begin{array}{c}1,if{C}_1\ne 0,\hfill \\ k_1,if{C}_1=0.\hfill \end{array}\right.\end{array}$$

Therefore, the switching rule is activated at time ${\xi }_1={\tau }_{m_1}$ and the system of equations is changed to the system with lower limit of CFDF ${\xi }_1$ and the function $f_{C_{m_1}}(.,.)$ from the given family of functions.

At the time ${\tau }_{m_1+1}$ there could be an activation of the switching rule $\sigma \left(t\right)=C_{m_1+1},t\in [{\tau }_{m_1+1},{\tau }_{m_1+2})$.

If $C_{m_1+1}\ne C_{m_1}$, then the switching rule is activated at time ${\tau }_{m_1+1}$ and the system of equations is changed.

If $C_{m_1+1}=C_{m_1}$ then the switching rule is not activated at time ${\tau }_{m_1+1}$. Let $k_2=min\{j>m_1+1:C_j\ne C_{m_1}\}$. Then $C_{{\tau }_{m_1+1}}=C_{{\tau }_{m_1+2}}=\cdots =C_{k_2-1}=C_{m_1}$, $C_{k_2}\ne C_{m_1}$, and the switching rule is not activated at the times ${\tau }_{m_1+1},{\tau }_{m_1+2},{\tau }_{m_1+3},\dots ,{\tau }_{k_2-1}$. It is activated at time ${\tau }_{k_2}$ and the system of equations is changed.

Denote ${\xi }_2={\tau }_{m_2}$ where

$$\begin{array}{c}\hfill m_2=\left\{\begin{array}{c}{m}_1+1,\text{if}{C}_{m_1+1}\ne C_{m_1},\hfill \\ k_2,\text{if}{C}_{m_1+1}=C_{m_1}.\hfill \end{array}\right.\end{array}$$

Therefore, the switching rule is activated at time ${\xi }_2={\tau }_{m_2}$ and the system of equations is changed to a system with a lower limit of CFDF ${\xi }_2$ and the function $f_{C_{m_2}}(.,.)$ from the given family of functions.

Continue this procedure and we obtain a sequence of times ${\left\{{\xi }_j\right\}}_{j=1}^{\infty }$ at which the switching rule is activated and the system of equations is changed at time ${\xi }_j={\tau }_{m_j}$ with a lower limit of CFDF ${\xi }_j$ and a function $f_{C_{m_j}}(.,.)$ from the given family of functions where the sequence of subscripts ${\left\{m_j\right\}}_{j=1}^{\infty }$ is recursively defined by

$$\begin{array}{c}\hfill m_j=\left\{\begin{array}{c}{m}_{j-1}+1,\text{if}{C}_{m_{j-1}+1}\ne C_{m_{j-1}},\hfill \\ k_j,\text{if}{C}_{m_{j-1}+1}\ne C_{m_{j-1}},\hfill \end{array}\right.\end{array}$$

(7)

and $k_j=min\{l>m_{j-1}+1:C_j=C_{m_j-1}\}$.

From the procedure described above, we obtain the sequence ${\left\{{\xi }_k\right\}}_{k=1}^{\infty }$ and a sequence of subscripts ${\left\{m_k\right\}}_{k=1}^{\infty }$, which depends only on the given switching rule, i.e., on the sequence of non-negative integers ${\left\{C_k\right\}}_{k=1}^{\infty }$.

Now, we will give a detailed description of the SFDE (6).

Let ${\xi }_0={\tau }_0=0$.

The switched system SFDE (6) reduces to

$${}_{\rho }{}^{C}D_{{\xi }_k}^{q,\psi }\upsilon \left(t\right)=f_{C_{m_k}}(t,\upsilon \left(t\right)),\text{for}t\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots ,\text{and}\upsilon \left(0\right)={\upsilon }_0.$$

(8)

We will describe the solution of SFDE (6) and its equivalent (8).

The initial IVP for SFDE is

$${}_{\rho }{}^{C}D_0^{q,\psi }\upsilon \left(t\right)=f_0(t,\upsilon \left(t\right)),\text{for}t\in (0,{\xi }_1],\text{and}\upsilon \left(0\right)={\upsilon }_0.$$

(9)

According to Assumption 2, the IVP (9) has a solution ${\upsilon }_0\left(t\right),t\in [0,{\xi }_1]$.

At time ${\xi }_1$ the switching rule $\sigma \left(t\right)=C_{m_1}$ is activated and the system of equations is changed, i.e., consider the IVP for the system of fractional differential equations with CFDF

$${}_{\rho }{}^{C}D_{{\xi }_1}^{q,\psi }\upsilon \left(t\right)=f_{C_{m_1}}(t,\upsilon \left(t\right)),\text{for}t\in ({\xi }_1,{\xi }_2],\text{and}\upsilon \left({\xi }_1\right)={\upsilon }_0\left({\xi }_1\right).$$

(10)

According to Assumption 2, the IVP (10) has a solution ${\upsilon }_1\left(t\right),t\in [{\xi }_1,{\xi }_2]$.

At time ${\xi }_2$ the switching rule $\sigma \left(t\right)=C_{m_2}$ is activated and the system of equations is changed, i.e., consider the IVP for the system of fractional differential equations with CFDF

$${}_{\rho }{}^{C}D_{{\xi }_2}^{q,\psi }\upsilon \left(t\right)=f_{C_{m_2}}(t,\upsilon \left(t\right)),\text{for}t\in ({\xi }_2,{\xi }_3],\text{and}\upsilon \left({\xi }_2\right)={\upsilon }_1\left({\xi }_2\right).$$

(11)

According to Assumption 2, the IVP (11) has a solution ${\upsilon }_2\left(t\right),t\in [{\xi }_2,{\xi }_3]$.

Continue this process for any $k=1,2,3,\dots$ at time ${\xi }_k$ the switching rule $\sigma \left(t\right)=C_{m_k}$ is activated and the system of equations is changed, i.e., consider the IVP for the system of fractional differential equations with CFDF

$${}_{\rho }{}^{C}D_{{\xi }_k}^{q,\psi }\upsilon \left(t\right)=f_{C_{m_k}}(t,\upsilon \left(t\right)),\text{for}t\in ({\xi }_k,{\xi }_{k+1}],\text{and}\upsilon \left({\xi }_k\right)={\upsilon }_{k-1}\left({\xi }_k\right).$$

(12)

According to Assumption 2, the IVP (12) has a solution ${\upsilon }_k\left(t\right),t\in [{\xi }_k,{\xi }_{k+1}]$.

Then, the solution of (6) and its equivalent (8) is given by

$$\begin{array}{c}\hfill \upsilon \left(t\right)=\left\{\begin{array}{c}{\upsilon }_0\left(t\right),\text{for}t\in [0,{\xi }_1],\hfill \\ {\upsilon }_1\left(t\right),\text{for}t\in ({\xi }_1,{\xi }_2]\hfill \\ {\upsilon }_2\left(t\right),\text{for}t\in ({\xi }_2,{\xi }_3]\hfill \\ \dots \hfill \\ {\upsilon }_m\left(t\right),\text{for}t\in ({\xi }_m,{\xi }_{m+1}]\hfill \\ \dots \dots \dots \dots \dots .\hfill \end{array}\right.\end{array}$$

(13)

Remark 5.  

Note the solution $\upsilon \left(t\right)$ of SFDE (6) is continuous at any switching point ${\xi }_k,k=1,2,3,\dots$ because of the initial condition ${\upsilon }_k\left({\xi }_k\right)={\upsilon }_{k-1}\left({\xi }_k\right)$ (compare with Remark 4).

Remark 6.  

From the description above it can be seen that not all initially given switching times ${\tau }_k$ can be applied.

Example 1.  

Let $\rho =1,q=0.7,\psi \left(t\right)\equiv t,$ and ${\tau }_k=k,k=0,1,2,3,\dots$.

Consider the initial value problem (IVP) for the scalar switched fractional differential equation with CFDF

$${}_1{}^{C}D_4^{0.7,t}\upsilon \left(t\right)={(-1)}^{\sigma \left(t\right)}0.5\upsilon \left(t\right),fort\in (k,k+1],k=0,1,2,\dots ,and\upsilon \left(0\right)={\upsilon }_0,$$

(14)

where ${\upsilon }_0\in \mathbb{R}$.

We will solve the IVP (14) for various initial values and various switching rules.

Case 1. 

Let $\sigma \left(t\right)=\left[0.3k\right],t\in (k,k+1]$, where $\left[z\right]$ is the integer part of $z\in \mathbb{R}$ and ${\upsilon }_0=20$.

The switching rule $\sigma \left(t\right)=0$ for $[1,3]$ does not act at the switching times $\tau =1,2,3$, i.e., we consider the IVP for the scalar equation

$${}_{\rho }{}^{C}D_0^{q,\psi }\upsilon \left(t\right)=f_0(t,\upsilon \left(t\right))=0.5\upsilon \left(t\right),t\in (0,4],\upsilon \left(0\right)=20.$$

(15)

According to Lemma 2 with $a=0,T=4$ and $\lambda =0.5,$ IVP (15) has a solution ${\upsilon }_0\left(t\right)=20E_{0.7}\left(0.5t^{0.7}\right)$ for $t\in [0,4].$

The switching rule is activated at time ${\xi }_1=4$ and $C_{m_1}=1$ because $\sigma \left(t\right)=[0.3\ast 4]=1\ne 0$ and it changes the scalar equation to an equation with CFDF with lower limit ${\xi }_1=4$ and the function $f_1(t,\upsilon \left(t\right))=-0.5\upsilon \left(t\right)$. Also, the switching rule $\sigma \left(t\right)=1$ for $[4,6]$ and does not act at the times ${\tau }_k=5,6$, it is activated at time ${\xi }_2=7$ because $\sigma \left(t\right)=[0.3\ast 7]=2\ne 1$, i.e., we consider the following IVP for the scalar equation

$${}_1{}^{C}D_4^{0.7,t}\upsilon \left(t\right)=-0.5\upsilon \left(t\right),fort\in (4,7],and\upsilon \left(4\right)=20E_{0.7}\left(0.54^{0.7}\right).$$

(16)

According to Lemma 2 with $a=4,T=7,\lambda =-0.5$, the IVP (16) has a solution ${\upsilon }_1\left(t\right)=20E_{0.7}\left(0.54^{0.7}\right)E_{0.7}\left(-0.5{(t-4)}^{0.7}\right),t\in [4,7]$.

The switching rule is activated at time ${\xi }_2=7$ and $C_{m_2}=2$. Also, the switching rule $\sigma \left(t\right)=2$ for $[7,9]$ does not act at the times ${\tau }_k=8,9$; it is activated at time ${\xi }_3=10$ because $\sigma \left(10\right)=[0.3\ast 10]=3\ne 2$, i.e., we consider the following IVP for the scalar equation

$$\begin{array}{cc}& {}_1{}^{C}D_7^{0.7,t}\upsilon \left(t\right)=0.5\upsilon \left(t\right),fort\in (7,10],\hfill \\ & \upsilon \left(7\right)=20E_{0.7}\left(0.54^{0.7}\right)E_{0.7}\left(-0.5{(7-4)}^{0.7}\right).\hfill \end{array}$$

(17)

According to Lemma 2 with $a=7,T=10,\lambda =0.5$ the IVP (17) has a solution

$${\upsilon }_2\left(t\right)=20E_{0.7}\left(0.54^{0.7}\right)E_{0.7}\left(-0.53^{0.7}\right)E_{0.7}\left(0.5{(t-7)}^{0.7}\right),t\in [7,10].$$

The switching rule is activated at time ${\xi }_3=10$ and $C_{m_3}=3$. Also, the switching rule $\sigma \left(t\right)=3$ for $[10,13]$ does not act at the times ${\tau }_k=11,12,13$, it is activated at time ${\xi }_4=14$ because $\sigma \left(14\right)=[0.3\ast 14]=4\ne 3$, i.e., we consider the following IVP for the scalar equation

$$\begin{array}{cc}& {}_1{}^{C}D_{10}^{0.7,t}\upsilon \left(t\right)=-0.5\upsilon \left(t\right),fort\in (10,14],\hfill \\ & \upsilon \left(10\right)=20E_{0.7}\left(0.54^{0.7}\right)E_{0.7}\left(-0.53^{0.7}\right)E_{0.7}\left(0.5{(10-7)}^{0.7}\right).\hfill \end{array}$$

(18)

According to Lemma 2 with $a=10,T=14,\lambda =-0.5$ the IVP (18) has a solution

$${\upsilon }_3\left(t\right)=20E_{0.7}\left(0.54^{0.7}\right)E_{0.7}\left(-0.53^{0.7}\right)E_{0.7}\left(0.53^{0.7}\right)E_{0.7}\left(-0.5{(t-10)}^{0.7}\right),t\in [10,14].$$

The switching rule is activated at time ${\xi }_4=14$ and $C_{m_4}=4$. Also, the switching rule $\sigma \left(t\right)=4$ for $[14,16]$ does not act at the times ${\tau }_k=15,16$, it is activated at time ${\xi }_5=17$ because $\sigma \left(17\right)=[0.3\ast 17]=5\ne 4$, i.e., we consider the following IVP for the scalar equation

$$\begin{array}{cc}& {}_1{}^{C}D_{14}^{0.7,t}\upsilon \left(t\right)=0.5\upsilon \left(t\right),fort\in (14,17],\hfill \\ & \upsilon \left(14\right)=20E_{0.7}\left(0.54^{0.7}\right)E_{0.7}\left(-0.53^{0.7}\right)E_{0.7}\left(0.53^{0.7}\right)E_{0.7}\left(-0.5{(14-10)}^{0.7}\right).\hfill \end{array}$$

(19)

According to Lemma 2 with $a=14,T=17,\lambda =0.5$ the IVP (19) has a solution

$$\begin{array}{cc}& {\upsilon }_4\left(t\right)=20E_{0.7}\left(0.54^{0.7}\right)E_{0.7}\left(-0.53^{0.7}\right)E_{0.7}\left(0.53^{0.7}\right)E_{0.7}\left(-0.54^{0.7}\right)E_{0.7}\left(0.5{(t-14)}^{0.7}\right),\hfill \\ & fort\in [14,17].\hfill \end{array}$$

Continue this process we obtain the solution of IVP (26) (see Figure 1):

$$\begin{array}{c}\hfill \upsilon \left(t\right)=\left\{\begin{array}{c}20E_{0.7}\left(0.5t^{0.7}\right),t\in [0,4],\hfill \\ 20E_{0.7}\left(0.54^{0.7}\right)E_{0.7}\left(-0.5{(t-4)}^{0.7}\right),t\in (4,7],\hfill \\ 20E_{0.7}\left(0.54^{0.7}\right)E_{0.7}\left(-0.53^{0.7}\right)E_{0.7}\left(0.5{(t-7)}^{0.7}\right),t\in (7,10],\hfill \\ 20E_{0.7}\left(0.54^{0.7}\right)E_{0.7}\left(-0.53^{0.7}\right)E_{0.7}\left(0.53^{0.7}\right)E_{0.7}\left(-0.5{(t-10)}^{0.7}\right),\hfill \\ t\in (10,14],\hfill \\ 20E_{0.7}\left(0.54^{0.7}\right)E_{0.7}\left(-0.53^{0.7}\right)E_{0.7}\left(0.53^{0.7}\right)E_{0.7}\left(-0.54^{0.7}\right)E_{0.7}\left(0.5{(t-14)}^{0.7}\right),\hfill \\ t\in (14,17],\hfill \\ \dots \dots \dots \dots .\hfill \end{array}\right.\end{array}$$

Case 2. 

Let $\sigma \left(t\right)=\left[0.3k^2\right],t\in (k,k+1]$, where $\left[z\right]$ is the integer part of $z\in \mathbb{R}$ and ${\upsilon }_0=20$.

The switching rule $\sigma \left(t\right)=0$ for $t\in [1,2)$ does not act at the switching time ${\tau }_k=1$, i.e., we consider the IVP for the scalar equation

$${}_1{}^{C}D_0^{0.7,t}\upsilon \left(t\right)=0.5\upsilon \left(t\right),t\in (0,2],\upsilon \left(0\right)=20.$$

(20)

IVP (20) has a solution ${\upsilon }_0\left(t\right)=20E_{0.7}\left(05.5t^{0.7}\right)$ for $t\in [0,2].$

The switching rule is activated at time ${\xi }_1=2$ and $C_{m_1}=1$ because $\sigma \left(t\right)=[0.3\ast 2^2]=1\ne 0$, i.e., we consider the following IVP for the scalar equation

$${}_1{}^{C}D_2^{0.7,t}\upsilon \left(t\right)=-0.5\upsilon \left(t\right),fort\in (2,3],and\upsilon \left(2\right)=20E_{0.7}\left(0.52^{0.7}\right).$$

(21)

The IVP (21) has a solution ${\upsilon }_1\left(t\right)=20E_{0.7}\left(0.52^{0.7}\right)E_{0.7}\left(-0.5{(t-2)}^{0.7}\right),t\in [2,3]$.

The switching rule is activated at time ${\xi }_2=3$ and $C_{m_3}=2$ because $\sigma \left(t\right)=[0.3\ast 3^2]=2\ne 1$. Also, the switching rule $\sigma \left(t\right)=2$ for $t\in [3,4)$, i.e., we consider the following IVP for the scalar equation

$$\begin{array}{cc}& {}_1{}^{C}D_3^{0.7,t}\upsilon \left(t\right)=0.5\upsilon \left(t\right),fort\in (3,4],\hfill \\ & \upsilon \left(3\right)=20E_{0.7}\left(0.52^{0.7}\right)E_{0.7}\left(-0.5{(3-2)}^{0.7}\right).\hfill \end{array}$$

(22)

The IVP (22) has a solution

$${\upsilon }_2\left(t\right)=20E_{0.7}\left(0.52^{0.7}\right)E_{0.7}\left(-0.5\right)E_{0.7}\left(0.5{(t-3)}^{0.7}\right),t\in [3,4].$$

The switching rule is activated at time ${\xi }_3=4$ and $C_{m_3}=4$ because $\sigma \left(t\right)=[0.3\ast 4^2]=4\ne 3$. Also, the switching rule $\sigma \left(t\right)=4$ for $t\in [4,5)$, i.e., we consider the following IVP for the scalar equation

$$\begin{array}{cc}& {}_1{}^{C}D_4^{0.7,t}\upsilon \left(t\right)=0.5\upsilon \left(t\right),fort\in (4,5],\hfill \\ & \upsilon \left(4\right)=20E_{0.7}\left(0.52^{0.7}\right)E_{0.7}\left(-0.5\right)E_{0.7}\left(0.5{(4-3)}^{0.7}\right).\hfill \end{array}$$

(23)

The IVP (23) has a solution

$${\upsilon }_3\left(t\right)=20E_{0.7}\left(0.52^{0.7}\right)E_{0.7}\left(-0.5\right)E_{0.7}\left(0.5\right)E_{0.7}\left(0.5{(t-4)}^{0.7}\right),t\in [4,5].$$

The switching rule is activated at time ${\xi }_4=5$ and $C_{m_4}=7$ because $\sigma \left(t\right)=[0.3\ast 5^2]=7\ne 4$. Also, the switching rule $\sigma \left(t\right)=7$ for $t\in [5,6)$, i.e., we consider the following IVP for the scalar equation

$$\begin{array}{cc}& {}_1{}^{C}D_5^{0.7,t}\upsilon \left(t\right)=-0.5\upsilon \left(t\right),fort\in (5,6],\hfill \\ & \upsilon \left(5\right)=20E_{0.7}\left(0.52^{0.7}\right)E_{0.7}\left(-0.5\right)E_{0.7}\left(0.5\right)E_{0.7}\left(0.5{(5-4)}^{0.7}\right).\hfill \end{array}$$

(24)

The IVP (24) has a solution

$${\upsilon }_4\left(t\right)=20E_{0.7}\left(0.52^{0.7}\right)E_{0.7}\left(-0.5\right){\left(E_{0.7}\left(0.5\right)\right)}^2{E}_{0.7}\left(-0.5{(t-5)}^{0.7}\right),t\in [5,6].$$

Continue this process we obtain the solution of IVP (26) (see Figure 2)

$$\begin{array}{c}\hfill \upsilon \left(t\right)=\left\{\begin{array}{c}20E_{0.7}\left(0.5t^{0.7}\right),t\in [0,2],\hfill \\ 20E_{0.7}\left(0.52^{0.7}\right)E_{0.7}\left(-0.5{(t-2)}^{0.7}\right),t\in (2,3],\hfill \\ 20E_{0.7}\left(0.52^{0.7}\right)E_{0.7}\left(-0.5\right)E_{0.7}\left(0.5{(t-3)}^{0.7}\right),t\in (3,4],\hfill \\ 20E_{0.7}\left(0.52^{0.7}\right)E_{0.7}\left(-0.5\right)E_{0.7}\left(0.5\right)E_{0.7}\left(0.5{(t-4)}^{0.7}\right),t\in (4,5],\hfill \\ 20E_{0.7}\left(0.52^{0.7}\right)E_{0.7}\left(-0.5\right){\left(E_{0.7}\left(0.5\right)\right)}^2{E}_{0.7}\left(-0.5{(t-5)}^{0.7}\right),t\in (5,6],\hfill \\ \dots \dots \dots \dots .\hfill \end{array}\right.\end{array}$$

Case 3. 

Let $\sigma \left(t\right)=k,t\in (k,k+1]$. Then the switching rule is activated at any time k and the solution of (26) is given by

$$\begin{array}{c}\hfill \upsilon \left(t\right)=\left\{\begin{array}{c}{\upsilon }_0{E}_{0.7}\left(0.5t^{0.7}\right),t\in [0,1],\hfill \\ {\upsilon }_0{E}_{0.7}\left(0.5\right)E_{0.7}\left(-0.5{(t-1)}^{0.7}\right),t\in (2,3],\hfill \\ {\upsilon }_0{E}_{0.7}\left(0.5\right)E_{0.7}\left(-0.5\right)E_{0.7}\left(0.5{(t-3)}^{0.7}\right),t\in (3,4],\hfill \\ 20E_{0.7}\left(0.5\right)E_{0.7}\left(-0.5\right)E_{0.7}\left(0.5\right)E_{0.7}\left(-0.5{(t-4)}^{0.7}\right),t\in (4,5]\hfill \\ {\upsilon }_0{E}_{0.7}\left(0.5\right)E_{0.7}\left(-0.5\right)E_{0.7}\left(0.5\right)E_{0.7}\left(-0.5\right)E_{0.7}\left(0.5{(t-5)}^{0.7}\right),t\in (5,6],\hfill \\ \dots \dots \dots \dots .\hfill \end{array}\right.\end{array}$$

Case 4. 

Let $\sigma \left(t\right)=3,t\in (k,k+1],k=1,2,\dots .$ Then the switching rule will be activated only at time ${\xi }_1=1$ and the SFDE will be reduced to

$$\begin{array}{cc}& {}_1{}^{C}D_0^{0.7,t}\upsilon \left(t\right)=0.5\upsilon \left(t\right),fort\in (0,1],\hfill \\ & {}_1{}^{C}D_1^{0.7,t}\upsilon \left(t\right)=-0.5\upsilon \left(t\right),fort>1,\hfill \\ & \upsilon \left(0\right)=20,\upsilon (1-)=\upsilon (1+).\hfill \end{array}$$

(25)

Remark 7.  

From the example as well as the above description of SFDE, it follows that the times ${\xi }_k$ at which the switching rule is activated and the applied function from the family of functions depend only on the switching rule and do not depend on any solution of SFDE.

4. Comparison Results for Scalar Switched Fractional Differential Equations with CFDF

Let the switching rule $\sigma \left(t\right)=C_k,t\in [{\tau }_k.{\tau }_{k+1}),k=1,2,3,\dots$ be given. According to Remark 7, we could find the sequence of times ${\left\{{\xi }_k\right\}}_{k=1}^{\infty }:{\xi }_j={\tau }_{m_j},$ at which the switching rule is activated and the sequence of integers ${\left\{m_k\right\}}_{k=1}^{\infty }$ defined by (7).

Let ${\xi }_0=0$, $m_0=0,C_0=0$.

Let a sequence of scalar nonlinear functions ${\left\{G_{C_{m_k}}(,x)\right\}}_{k=0}^{\infty }:G_{C_{m_k}}\in C([{\xi }_k,\infty )\times \mathbb{R},\mathbb{R}),k=0,1,2,\dots$, be given.

Consider the IVP nonlinear scalar fractional differential equation with changeable right-hand side (FDEC)

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\xi }_k}^{q,\psi }\omega \left(t\right)=G_{C_{m_k}}(t,\omega ),\mathrm{for}t\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots ,\hfill \\ & \omega ({\xi }_k-0)=\omega ({\xi }_k+0),k=1,2,3,\dots ,\hfill \\ & \omega \left(0\right)={\omega }_0\hfill \end{array}$$

(26)

where ${\omega }_0\in \mathbb{R}$.

We will use the following assumption:

Assumption 3.  

For any point ${\omega }_0\in \mathbb{R}$ the IVP for the FDEC (26) has an unique solution $\omega (.)\in {\bigcup }_{k=0}^{\infty }{C}^q(({\xi }_k,{\xi }_{k+1}],\mathbb{R},\psi ,\rho )$.

Definition 3.  

The zero solution of the FDEC (26) is said to be stable if for every $ϵ>0$ there exist $\delta =\delta \left(ϵ\right)>0$ such that for any ${\omega }_0\in \mathbb{R}$ the inequality $|{\omega }_0|<\delta$ implies $|\omega (t)|<ϵ$ for $t\ge 0$ where $\omega \left(t\right)$ is a solution of FDEC (26) with an initial value ${\omega }_0$.

We will now obtain a comparison result for FDEC (26).

Lemma 4. 

Assume the following conditions are satisfied:

1. 

The function ${\omega }^{*}(.)\in {\bigcup }_{k=0}^{\infty }{C}^q(({\xi }_k,{\xi }_{k+1}],\mathbb{R},\psi ,\rho )$ is a solution of FDEC (26).

2. 

The function $\varpi \in {\bigcup }_{k=0}^{\infty }{C}^q(({\xi }_k,{\xi }_{k+1}],\mathbb{R},\psi ,\rho )$ satisfies the nonlinear scalar fractional differential inequalities with changeable right-hand side

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\xi }_k}^{q,\psi }\varpi \left(t\right)<G_{C_{m_k}}(t,\varpi ),\mathit{for}t\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots ,\hfill \\ & \varpi ({\xi }_k-0)=\varpi ({\xi }_k+0),k=1,2,\dots ,\hfill \\ & \varpi \left(0\right)<{\omega }^{*}\left(0\right).\hfill \end{array}$$

(27)

Then $\varpi \left(t\right)<{\omega }^{*}\left(t\right)$ for $t\ge 0.$

Proof. 

We will use induction to prove the claim.

Case 1.

Consider the interval $[0,{\xi }_1]$.

Note $\varpi \left(0\right)<{\omega }^{*}\left(0\right)$.

We will prove the inequality

$$\varpi \left(t\right)<{\omega }^{*}\left(t\right)\text{for}t\in [0,{\xi }_1].$$

(28)

Assume the contrary, i.e., there exists a point $\eta \in (0,{\xi }_1]$ such that $\varpi \left(t\right)<{\omega }^{*}\left(t\right)$ on $[0,\eta )$ and $\varpi \left(\eta \right)={\omega }^{*}\left(\eta \right)$.

All the conditions in Lemma 3 are satisfied for the points $\eta ,a=0$ and the function $\varpi \left(t\right)-{\omega }^{*}\left(t\right),t\in [0,\eta ]$. According to Lemma 3 applied to the function $\varpi (.)-{\omega }^{*}(.)$ with $a=0,T={\xi }_1$, and Remark 2 we have ${}_{\rho }{}^{C}D_0^{q,\psi }\left(\varpi \left(t\right)-{\omega }^{*}\left(t\right)\right){|}_{t=\eta }\ge 0$.

From the first inequality of (27) for $t=\eta$ and $k=0$ we have

$${}_{\rho }{}^{C}D_0^{q,{\psi }_0}\left(\varpi \left(t\right)-{\omega }^{*}\left(t\right)\right){|}_{t=\eta }<G_0(\eta ,\varpi \left(\eta \right))-G_0(\eta ,{\omega }^{*}\left(\eta \right))=0.$$

The obtained contradiction proves inequality (28) on $[0,{\xi }_1]$.

Case 2.

Consider the interval $({\xi }_1,{\xi }_2]$.

We will prove the inequality

$$\varpi \left(t\right)<{\omega }^{*}\left(t\right)\text{for}t\in [{\xi }_1,{\xi }_2].$$

(29)

According to Case 1 the inequality (29) holds for $t={\xi }_1$.

Assume the contrary, i.e., there exists a point $\eta \in ({\xi }_1,{\xi }_2]$ such that $\varpi \left(t\right)<{\omega }^{*}\left(t\right)$ on $[{\xi }_1,\eta )$ and $\varpi \left(\eta \right)={\omega }^{*}\left(\eta \right)$.

From Lemma 3 applied to the function $\varpi (.)-{\omega }^{*}(.)$ with $a={\xi }_1,T={\xi }_2$, and Remark 2 we have ${}_{\rho }{}^{C}D_{{\xi }_1}^{q,\psi }\left(\varpi \left(t\right)-{\omega }^{*}\left(t\right)\right){|}_{t=\eta }\ge 0$.

According to the first inequality of (27) for $t=\eta$ and $k=1$ we have

$${}_{\rho }{}^{C}D_{{\xi }_1}^{q,\psi }\left(\varpi \left(t\right)-{\omega }^{*}\left(t\right)\right){|}_{t=\eta }<G_{C_{m_1}}(\eta ,\varpi \left(\eta \right))-G_{C_{m_1}}(\eta ,{\omega }^{*}\left(\eta \right))=0.$$

The obtained contradiction proves inequality (29) on $[{\xi }_1,{\xi }_2]$.

Case 3.

Assume that there exist an integer $j\ge 2$ such that inequality

$$\varpi \left(t\right)<{\omega }^{*}\left(t\right)\text{for}t\in [0,{\xi }_j]$$

(30)

holds.

We will prove the inequality

$$\varpi \left(t\right)<{\omega }^{*}\left(t\right)\text{for}t\in [{\xi }_j,{\xi }_{j+1}].$$

(31)

Assume the contrary, i.e., there exists a point $\eta \in ({\xi }_j,{\xi }_{j+1}]$ such that $\varpi \left(t\right)<{\omega }^{*}\left(t\right)$ on $[{\xi }_j,\eta )$ and $\varpi \left(\eta \right)={\omega }^{*}\left(\eta \right)$.

From Lemma 3 applied to the function $\varpi (.)-{\omega }^{*}(.)$ with $a={\xi }_j,T={\xi }_{j+1}$ and Remark 2 we have ${}_{\rho }{}^{C}D_{{\xi }_j}^{q,\psi }\left(\varpi \left(t\right)-{\omega }^{*}\left(t\right)\right){|}_{t=\eta }\ge 0.$

According to the first inequality of (27) for $t=\eta$ and $k=j$ we have

$${}_{\rho }{}^{C}D_{{\xi }_j}^{q,\psi }\left(\varpi \left(t\right)-{\omega }^{*}\left(t\right)\right){|}_{t=\eta }<G_{C_{m_j}}(\eta ,\varpi \left(\eta \right))-G_{C_{m_j}}(\eta ,{\omega }^{*}\left(\eta \right))=0.$$

The obtained contradiction proves inequality (31) on $[{\xi }_j,{\xi }_{j+1}]$.

□

Consider the special case of FDEC (26) with $G_{C_{m_k}}(t,x)={\mu }_{k}x,k=0,1,2,\dots$, i.e.,

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\xi }_k}^{q,\psi }\omega \left(t\right)={\mu }_k\omega \left(t\right),\text{for}t\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots ,\hfill \\ & \omega ({\xi }_k-0)=\omega ({\xi }_k+0),k=1,2,3,\dots ,\hfill \\ & \omega \left(0\right)={\omega }_0,\hfill \end{array}$$

(32)

where ${\mu }_k\in \mathbb{R},k=0,1,2,\dots$ are constants.

Applying Lemma 2 on each interval $({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots ,$ with $a={\xi }_k,T={\xi }_{k+1}$ and $\lambda ={\mu }_k$ we obtain the solution of IVP (32),

$$\begin{array}{cc}& \omega \left(t\right)={\omega }_0\prod _{j=0}^{k-1}{E}_q\left({\mu }_j{\left({\displaystyle \frac{\psi {\left({\xi }_{j+1}\right)}^{\rho }-{\left(\psi \left({\xi }_j\right)\right)}^{\rho }}{\rho }}\right)}^q\right)E_q\left({\mu }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\xi }_k\right)\right)}^{\rho }}{\rho }}\right)}^q\right),\hfill \\ & \text{for}t\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots .\hfill \end{array}$$

(33)

Example 2.  

Let ${\xi }_k=k,k=0,1,2,\dots ,$ $\rho =1$, $\psi \left(t\right)=t$, $q=0.7$, and consider the scalar Equation (32).

We will consider the solutions for various values of the coefficients ${\mu }_k$.

Case 1. 

Let ${\mu }_k=1$ if k is even, and ${\mu }_k=-2$ if k is odd. The solution of (32) is

$$\begin{array}{cc}& \omega \left(t\right)={\omega }_0\prod _{j=0}^{k-1}{E}_{0.7}\left({\mu }_j\right)E_q\left({\mu }_k{\left(t-k\right)}^{0.7}\right),fort\in (k,k+1],k=0,1,2,\dots .\hfill \end{array}$$

(34)

From Figure 3 it can be seen that the solution is stable.

Case 2. 

Let ${\mu }_k=-1$ if k is even, and ${\mu }_k=-2$ if k is odd. The solution of (32), given by (34) is graphed on Figure 4. It can be seen that the solution is stable.

Case 3. 

Let ${\mu }_k=-1$ if k is even, and ${\mu }_k=2$ if k is odd. The solution of (32), given by (34) is graphed on Figure 5. It can be seen that the solution is not stable.

Case 4. 

Let ${\mu }_k=1$ for all $k=0,1,2,\dots$. The solution of (32) is given by (34) and since $E_{0.7}\left(z\right)>1$ for $z>0$, the solution is not stable.

From the cases above it can be seen that if all ${\mu }_k<0$, then the solution is stable. Also, it is stable if ${\mu }_{2j}<0<{\mu }_{2j+1},|{\mu }_{2j}|>{\mu }_{2j+1},j=0,1,2,\dots$.

As a special case of Lemma 4 we obtain the following comparison result:

Corollary 4.  

Assume the function $\varpi \in {\bigcup }_{k=0}^{\infty }{C}^q(({\xi }_k,{\xi }_{k+1}],\mathbb{R},\psi ,\rho )$ satisfies the linear scalar fractional differential inequalities with changeable right-hand side

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\xi }_k}^{q,\psi }\varpi \left(t\right)<{\mu }_k\varpi \left(t\right),fort\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots ,\hfill \\ & \varpi ({\xi }_k-0)=\varpi ({\xi }_k+0),k=1,2,\dots ,\hfill \\ & \varpi \left(0\right)<{\omega }_0.\hfill \end{array}$$

(35)

Then

$$\varpi \left(t\right)<{\omega }_0\prod _{j=0}^{k-1}{E}_q\left({\mu }_j{\left({\displaystyle \frac{\psi {\left({\xi }_{j+1}\right)}^{\rho }-{\left(\psi \left({\xi }_j\right)\right)}^{\rho }}{\rho }}\right)}^q\right)E_q\left({\mu }_k{\left({\displaystyle \frac{\psi {\left(t\right)}^{\rho }-{\left(\psi \left({\xi }_k\right)\right)}^{\rho }}{\rho }}\right)}^q\right)$$

for $t\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots .$

In the special case of zero function in (26) we obtain another comparison result. First, consider the IVP for FDEC

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\xi }_k}^{q,\psi }\omega \left(t\right)=\epsilon ,\text{for}t\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots ,\hfill \\ & \omega ({\xi }_k-0)=\omega ({\xi }_k+0),k=1,2,3,\dots ,\hfill \\ & \omega \left(0\right)=\epsilon .\hfill \end{array}$$

(36)

Corollary 5.  

The solution of IVP for FDEC (36) is

$$\begin{array}{cc}\hfill \omega \left(t\right)& =\epsilon \left(1+{\displaystyle \frac{1}{\Gamma (1+q)}}\left(\sum _{j=0}^{k-1}{\left({\displaystyle \frac{\psi \left({\xi }_{j+1}\right)-\psi \left({\xi }_j\right)}{\rho }}\right)}^q+{\left({\displaystyle \frac{\psi \left(t\right)-\psi \left({\xi }_k\right)}{\rho }}\right)}^q\right)\right),\hfill \\ & t\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots .\hfill \end{array}$$

(37)

Proof. 

Let $t\in [0,{\xi }_1].$ According to Corollary 2 with $a=0,T={\xi }_1,B=\epsilon =K$ the solution of (36) is

$$\omega \left(t\right)=\epsilon \left(1+{\displaystyle \frac{1}{\Gamma (1+q)}}{\left({\displaystyle \frac{\psi \left(t\right)-\psi \left({\xi }_0\right)}{\rho }}\right)}^q\right),t\in [0,{\xi }_1].$$

Let $t\in ({\xi }_1,{\xi }_2].$ According to Corollary 2 with $a={\xi }_1,T={\xi }_2,B=\omega ({\xi }_1-0)=\epsilon \left(1+{\displaystyle \frac{1}{\Gamma (1+q)}}{\left({\displaystyle \frac{\psi \left({\xi }_1\right)-\psi \left({\xi }_0\right)}{\rho }}\right)}^q\right),K=\epsilon$ the solution of (36) is

$$\begin{array}{cc}\hfill \omega \left(t\right)& =\epsilon \left(1+{\displaystyle \frac{1}{\Gamma (1+q)}}{\left({\displaystyle \frac{\psi \left({\xi }_1\right)-\psi \left({\xi }_0\right)}{\rho }}\right)}^q\right)+{\displaystyle \frac{\epsilon }{\Gamma (1+q)}}{\left({\displaystyle \frac{\psi \left(t\right)-\psi \left({\xi }_1\right)}{\rho }}\right)}^q\hfill \\ & =\epsilon \left(1+{\displaystyle \frac{1}{\Gamma (1+q)}}{\left({\displaystyle \frac{\psi \left({\xi }_1\right)-\psi \left({\xi }_0\right)}{\rho }}\right)}^q+{\displaystyle \frac{1}{\Gamma (1+q)}}{\left({\displaystyle \frac{\psi \left(t\right)-\psi \left({\xi }_1\right)}{\rho }}\right)}^q\right).\hfill \end{array}$$

Continue this process by induction and we have (37). □

Corollary 6.  

Assume the function $\varpi \in {\bigcup }_{k=0}^{\infty }{C}^q(({\xi }_k,{\xi }_{k+1}],\mathbb{R},\psi ,\rho )$ satisfies the nonlinear scalar fractional differential inequalities with changeable right-hand side

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\xi }_k}^{q,\psi }\varpi \left(t\right)\le 0,fort\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots ,\hfill \\ & \varpi ({\xi }_k-0)=\varpi ({\xi }_k+0),k=1,2,\dots ,\hfill \\ & \varpi \left(0\right)\le 0.\hfill \end{array}$$

(38)

Then $\varpi \left(t\right)\le 0$ for $t\ge 0.$

Proof. 

Let $\epsilon >0$ be an arbitrary number and consider the IVP for FDEC (36) whose solution by Corollary 5 is given by (37).

Denote $\widehat{\varpi }\left(t\right)=\varpi \left(t\right)+\epsilon ,t\ge 0.$ The function $\widehat{\varpi }\in {\bigcup }_{k=0}^{\infty }{C}^q(({\xi }_k,{\xi }_{k+1}],\mathbb{R},\psi ,\rho )$ and from inequalities (38) it follows that

$$\begin{array}{cc}& {}_{\rho }{}^{C}D_{{\xi }_k}^{q,\psi }\widehat{\varpi }<\epsilon ,\text{for}t\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots ,\hfill \\ & \widehat{\varpi }({\xi }_k-0)=\widehat{\varpi }({\xi }_k+0),k=1,2,3,\dots ,\hfill \\ & \widehat{\varpi }\left(0\right)<\epsilon .\hfill \end{array}$$

(39)

According to Lemma 4 the inequality

$$\widehat{\varpi }\left(t\right)<\epsilon \left(1+{\displaystyle \frac{1}{\Gamma (1+q)}}\left(\sum _{j=0}^{k-1}{\left({\displaystyle \frac{\psi \left({\xi }_{j+1}\right)-\psi \left({\xi }_j\right)}{\rho }}\right)}^q+{\left({\displaystyle \frac{\psi \left(t\right)-\psi \left({\xi }_k\right)}{\rho }}\right)}^q\right)\right)$$

(40)

holds for any $t\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots$. From inequality (40) after taking a limit as $\epsilon$ approaches zero, we obtain $\varpi \left(t\right)\le 0,t\ge 0$. □

5. Stability of Nonlinear Switched Fractional Differential Equations with CFDF

Now we will study the stability properties of nonlinear switched systems with fractional derivatives. Note that stability properties of nonlinear discrete switched systems are studied in [43,44]. In [45] the stability properties of a nonlinear system with ordinary differential subsystems and piecewise constant switching rule are investigated. Detailed stability analysis for switched systems with ordinary derivatives is given in the book [46].

Definition 4.  

The zero solution of the switched fractional differential equations with CFDF (6) is said to be stable if for every $ϵ>0$ there exist $\delta =\delta \left(ϵ\right)>0$ such that for any ${\upsilon }_0\in {\mathbb{R}}^n$ the inequality $||{\upsilon }_0||<\delta$ implies $||\upsilon (t;{\upsilon }_0)||<ϵ$ for $t\ge 0$ where $\upsilon (t;{\upsilon }_0)$ is a solution of (6) with the initial value ${\upsilon }_0$.

Consider the following set:

$$\begin{array}{ccc}\hfill \mathcal{K}& =& \{a\in C[{\mathbb{R}}_{+},{\mathbb{R}}_{+}]:a\mathrm{is}\mathrm{strictly}\mathrm{increasing}\mathrm{and}a\left(0\right)=0\}.\hfill \end{array}$$

Theorem 1.  

Let the following conditions be satisfied:

1. 

The Assumptions 1, 2 and 3 are satisfied.

2. 

There exists a function $V(t,x)\in C([0,\infty )\times {\mathbb{R}}^n,{\mathbb{R}}_{+}),V(t,0)\equiv 0$ which is locally Lipschitzian with respect to its second argument such that

(i) 

For any solution $v\left(t\right)$ of the IVP for SFDE (6) the inequality

$${}_{\rho }{}^{C}D_{{\xi }_k}^{q,\psi }V(t,v\left(t\right))<G_{C_{{\tau }_k}}(t,V(t,v\left(t\right)),t\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots ,$$

(41)

holds;

(ii) 

$b(||x||)\le V(t,x)\le a(||x||)$ for $t\ge 0,x\in {\mathbb{R}}^n,$ where $a,b\in \mathcal{K}$.

3. 

The zero solution of FDEC (26) is stable.

Then the zero solution of the switched system of fractional differential equations with CFDF (6) is stable.

Proof. 

Let $ϵ>0$ be an arbitrary number.

According to condition 3 there exists $0<\delta \le b\left(ϵ\right)$ such that for any initial value ${\omega }_0^{*}\in \mathbb{R},|{\omega }_0^{*}|<\delta$ the inequality

$$|{\omega }^{*}\left(t\right)|<b\left(ϵ\right)$$

(42)

holds for $t\ge 0$ where ${\omega }^{*}\left(t\right)$ is a solution of FDEC (26) with initial value ${\omega }_0={\omega }_0^{*}$.

There exists ${\delta }_1>0$ such that $a\left({\delta }_1\right)<0.5\delta$.

Let ${\delta }_2<min\{\epsilon ,{\delta }_1\}$. Choose ${\upsilon }_0\in {\mathbb{R}}^n,||{\upsilon }_0||<{\delta }_2$. Consider the solution $\upsilon \left(t\right)$ of SFDE (6) with the chosen initial value ${\upsilon }_0$. We will prove

$$||\upsilon \left(t\right)||<\epsilon ,t\ge 0.$$

(43)

Note the inequality $||\upsilon \left(0\right)||=||{\upsilon }_0||<{\delta }_2\le \epsilon$ holds. Assume inequality (43) is not true. Therefore, there exists $\eta >0$ such that

$$||\upsilon \left(\eta \right)||=\epsilon \mathrm{and}||\upsilon \left(t\right)||<\epsilon ,t\in [0,\eta ).$$

(44)

Consider the solution $\widehat{\omega }\left(t\right)$ of FDEC (26) with initial value ${\widehat{\omega }}_0=2V(0,{\upsilon }_0)\in (0,\infty )$. Then $|{\widehat{\omega }}_0|=2V(0,{\upsilon }_0)\le 2a(||{\upsilon }_0||)<2a\left({\delta }_2\right)\le 2a\left({\delta }_1\right)<\delta$. Therefore the inequality (42) holds for the solution $\widehat{\omega }\left(t\right)$, i.e., $|\widehat{\omega }\left(t\right)|<b\left(\epsilon \right),t\ge 0$.

From condition 2(i) and inequality $V(0,{\upsilon }_0)=0.5{\widehat{\omega }}_0<{\widehat{\omega }}_0$ according to Lemma 4 for the functions $\widehat{\omega }\left(t\right)$ and $V(t,\upsilon (t\left)\right)$ we obtain $V(t,\upsilon \left(t\right))<\widehat{\omega }\left(t\right),t\ge 0$. Apply $||\upsilon \left(\eta \right)||=\epsilon$ and condition 2(ii) and obtain $b\left(\epsilon \right)=b(||\upsilon \left(\eta \right)||)\le V(\eta ,\upsilon \left(\eta \right))<\widehat{\omega }\left(\eta \right)<b\left(\epsilon \right)$.

The contradiction proves inequality (43). □

As a special case of Theorem 1, we obtain the following result:

Theorem 2.  

Let the following conditions be satisfied:

1. 

The Assumptions 1 and 2 are satisfied.

2. 

There exists a function $V(t,x)\in C([0,\infty )\times {\mathbb{R}}^n,{\mathbb{R}}_{+}),V(t,0)\equiv 0$ which is locally Lipschitzian with respect to its second argument such that

(i) 

For any solution $v\left(t\right)$ of the IVP for SFDE (6), the inequality

$${}_{\rho }{}^{C}D_{{\xi }_k}^{q,\psi }v\left(t\right)<-L_{k}V(t,v\left(t\right)),t\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots ,$$

(45)

holds, where $L_k>0,k=0,1,2,\dots ,$ are constants;

(ii) 

$b(||x||)\le V(t,x)\le a(||x||)$ for $t\ge 0,x\in {\mathbb{R}}^n,$ where $a,b\in \mathcal{K}$.

Then, the zero solution of SFDE (6) is stable.

Proof. 

Consider the scalar Equation (32) with ${\mu }_k=-L_k$ which solution is given by equality (33) replacing ${\mu }_k$ by $(-L_k)$. Since $E_q\left(z\right)\le 1$ for $z\in \mathbb{R}:z\le 0$ the condition 3 of Theorem 1 holds for the scalar Equation (32) with ${\mu }_k=L_k$ (compare with Example 2 Case 2), and this proves the claim. □

Theorem 3.  

Let the following conditions be satisfied:

1. 

The Assumptions 1 and 2 are satisfied.

2. 

There exists a function $V(t,x)\in C([0,\infty )\times {\mathbb{R}}^n,{\mathbb{R}}_{+}),V(t,0)\equiv 0$ which is locally Lipschitzian with respect to its second argument such that

(i) 

For any solution $v\left(t\right)$ of the IVP for SFDE (6) the inequality

$${}_{\rho }{}^{C}D_{{\xi }_k}^{q,\psi }v\left(t\right)\le 0,t\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots ,$$

(46)

holds;

(ii)

$b(||x||)\le V(t,x)\le a(||x||)$ for $t\ge 0,x\in {\mathbb{R}}^n,$ where $a,b\in \mathcal{K}$.

Then the zero solution of SFDE (6) is stable.

Proof. 

The proof is similar to that in Theorem 1 where Corollary 6 is applied instead of Lemma 4 and using the solution $\omega \left(t\right)=\omega \left(0\right),t\ge 0$ of (26) with $G_k\equiv 0,k=0,1,2,\dots ,$ (see Corollary 2). □

As a special case of Theorem 1 it follows the result:

Theorem 4.  

Let the following conditions be satisfied:

1. 

The Assumptions 1 and 2 are satisfied.

2. 

There exists a function $V(t,x)\in C([0,\infty )\times {\mathbb{R}}^n,{\mathbb{R}}_{+}),V(t,0)\equiv 0$ which is locally Lipschitzian with respect to its second argument such that

(i) 

For any solution $v\left(t\right)$ of the IVP for SFDE (6) the inequality

$${}_{\rho }{}^{C}D_{{\xi }_k}^{q,\psi }v\left(t\right)<L_{k}V(t,v\left(t\right)),t\in ({\xi }_k,{\xi }_{k+1}],k=0,1,2,\dots ,$$

(47)

holds where $L_k\in \mathbb{R},k=0,1,2,\dots ,$ are constants such that for any $j=0,1,2,3,\dots$ the inequalities $L_{2j}<0<L_{2j+1}$ and $\left|L_{2j}{\displaystyle \frac{\psi (2j+1)-\psi \left(2j\right)}{\rho }}\right|>L_{2j+1}{\displaystyle \frac{\psi (2j+2)-\psi (2j+1)}{\rho }}$ hold;

(ii) 

$b(||x||)\le V(t,x)\le a(||x||)$ for $t\ge 0,x\in {\mathbb{R}}^n,$ where $a,b\in \mathcal{K}$.

Then the zero solution of SFDE (6) is stable.

Proof. 

Consider the scalar Equation (32) with ${\mu }_k=L_k$. The solution of the scalar Equation (32) is stable because of the conditions for $L_k$ (compare with Example 2 Case 1) and this proves the claim. □

Remark 8.  

Note that if we consider a system of the family of subsystems of fractional differential equations with CFDF, then its zero solution is not necessarily stable. At the same time, because the switching rule might be applied to the system, the zero solution of SFDE (6) could be stable.

Example 3.  

Let $\rho =1,\psi \left(t\right)\equiv t,q=0.7$, ${\tau }_k=2+k,k=1,2,\dots$ and consider the family of functions $\{f_k{(t,x\}}_{k=0}^{\infty }:f_k(t,x)={(-1)}^{k+1}x,t\in [k,\infty ),x\in \mathbb{R}$.

Let $\sigma \left(t\right)=2k,t\in [k,k+1],k=1,2,\dots$. Then ${\xi }_0=0,{\xi }_k=k,$ and $m_k=2k,k=1,2,\dots .$

Consider the IVP for the scalar SFDE (14) which reduces in our case to

$${}_1{}^{C}D_k^{0.7,t}\upsilon \left(t\right)={(-1)}^{2k+1}\upsilon \left(t\right),fort\in (k,k+1],k=0,1,2,\dots ,and\upsilon \left(0\right)={\upsilon }_0.$$

(48)

The IVP (48) has a solution $\upsilon \left(t\right)={\upsilon }_0{\left(E_{0.7}\left(-1\right)\right)}^k{E}_{0.7}\left(-{(t-k)}^{0.7}\right),t\in (k,k+1],k=0,1,2,\dots ,$ then its zero solution is stable (see Figure 6).

At the same time, if we consider the function $f_1(t,x)={(-1)}^{2}x=x$ from the given family of functions and the corresponding IVP

$${}_1{}^{C}D_1^{0.7,t}\upsilon \left(t\right)=\upsilon \left(t\right)fort>1,and\upsilon \left(1\right)={\upsilon }_0$$

(49)

with a solution $\upsilon \left(t\right)={\upsilon }_0{E}_{0.7}\left({(t-1)}^{0.7}\right),t\ge 1$, then its zero solution is not stable (see Figure 7).

Therefore, because of the appropriately chosen switched rule, the zero solution of SFDE could be stable in spite of the behavior of the zero solution of any of the subsystems.

We will now illustrate some of the obtained sufficient conditions.

Example 4.  

Let $q=0.7$, $\rho =1,\psi \left(t\right)=t$, ${\tau }_k=k,k=0,1,2,\dots$.

Consider the following IVP for SFDE

$$\begin{array}{cc}& {}_1{}^{C}D_k^{0.7,t}x\left(t\right)=F_{\sigma \left(t\right)}(t,x\left(t\right),y\left(t\right))={(-1)}^{C_k+1}{t}^{2}x\left(t\right)-x^3\left(t\right)si{n}^2\left(y\left(t\right)\right),\hfill \\ & {}_1{}^{C}D_k^{0.7,t}y\left(t\right)=f_{\sigma \left(t\right)}(t,x\left(t\right),y\left(t\right))={(-1)}^{C_k+1}y\left(t\right),t\in (k,k+1],k=0,1,2,\dots ,\hfill \\ & x\left(0\right)=x_0,y\left(0\right)=y_0.\hfill \end{array}$$

(50)

Case 1. 

Let $\sigma \left(t\right)=2k,t\in [k,k+1)$. Then ${\xi }_k=k,m_k=k,C_{m_k}=2k$ and system (50) reduces to

$$\begin{array}{cc}& {}_1{}^{C}D_k^{0.7,t}x\left(t\right)=F_{\sigma \left(t\right)}(t,x\left(t\right),y\left(t\right))=-t^{2}x\left(t\right)-x^3\left(t\right)si{n}^2\left(y\left(t\right)\right),\hfill \\ & {}_1{}^{C}D_k^{0.7,t}y\left(t\right)=f_{\sigma \left(t\right)}(t,x\left(t\right),y\left(t\right))=-y\left(t\right),t\in (k,k+1],k=0,1,2,\dots ,\hfill \\ & x\left(0\right)=x_0,y\left(0\right)=y_0.\hfill \end{array}$$

(51)

Consider the quadratic Lyapunov function $V(t,x,y)=x^2+y^2$ for $x,y\in \mathbb{R}$.

Let $\left(x\right(t),y(t\left)\right)$ be any solution of (51). Then we have

$$\begin{array}{cc}& {}_1{}^{C}D_k^{0.7,t}V(t,x\left(t\right),y\left(t\right))\le 2x\left(t\right)\left(-t^{2}x\left(t\right)-x^3\left(t\right)si{n}^2\left(y\left(t\right)\right)\right)+2y\left(t\right)\left(-y\left(t\right)\right)\hfill \\ & =-2t^2{x}^2\left(t\right)-2x^4\left(t\right)si{n}^2\left(y\left(t\right)\right)-2y^2\left(t\right)\le 0,t\in (k,k+1],k=0,1,2,\dots .\hfill \end{array}$$

(52)

According to Theorem 3 the zero solution of SFDE (50) is stable.

Case 2. 

Let $\sigma \left(t\right)=2k+1,t\in [k,k+1)$. Then ${\xi }_k=k,m_k=k,C_{m_k}=2k+1$ and system (50) reduces to

$$\begin{array}{cc}& {}_1{}^{C}D_k^{0.7,t}x\left(t\right)=F_{\sigma \left(t\right)}(t,x\left(t\right),y\left(t\right))=t^{2}x\left(t\right)-x^3\left(t\right)si{n}^2\left(y\left(t\right)\right),\hfill \\ & {}_1{}^{C}D_k^{0.7,t}y\left(t\right)=f_{\sigma \left(t\right)}(t,x\left(t\right),y\left(t\right))=y\left(t\right),t\in (k,k+1],k=0,1,2,\dots ,\hfill \\ & x\left(0\right)=x_0,y\left(0\right)=y_0.\hfill \end{array}$$

(53)

Consider the quadratic Lyapunov function $V(t,x,y)=x^2+y^2$ for $x,y\in \mathbb{R}$.

Let $\left(x\right(t),y(t\left)\right)$ be any solution of (51). Then we have

$$\begin{array}{cc}& {}_1{}^{C}D_k^{0.7,t}V(t,x\left(t\right),y\left(t\right))\le 2x\left(t\right)\left(t^{2}x\left(t\right)-x^3\left(t\right)si{n}^2\left(y\left(t\right)\right)\right)+2y\left(t\right)\left(y\left(t\right)\right)\hfill \\ & =2t^2{x}^2\left(t\right)-2x^4\left(t\right)si{n}^2\left(y\left(t\right)\right)+2y^2\left(t\right)\le 2({(k+1)}^2+1)V(t,x,y),\hfill \\ & t\in (k,k+1],k=0,1,2,\dots .\hfill \end{array}$$

(54)

None of the obtained sufficient conditions could be applied to conclude the stability. Also, if we consider the corresponding comparison Equation (32) with ${\mu }_k=2({(k+1)}^2+1)>0,k=0,1,2,\dots$, then its zero solution is not stable.

As mentioned above, in this example, the type of the switching rule can totally change the behavior of the switched system.

6. Conclusions

In this paper a nonlinear switched system of fractional differential equations with generalized Caputo fractional derivatives with respect to other functions is studied in the case where the switched rule is given initially. Some comparison results for scalar fractional differential equations with a changeable right-hand side are obtained. These results and Lyapunov functions are applied to study the stability of the given equations. The influence of the type of switched rule on the stability properties of the switched system is studied and illustrated with several examples.

Several future works connected with the studied problem could be considered in the future. One could study various types of stability, and in addition, one could consider equations with delays. Also, one could study stability properties of dynamic models in biology, engineering, and neural networks.