Finite-time stability of multiterm fractional nonlinear systems with multistate time delay

Abstract

This work is mainly concentrated on finite-time stability of multiterm fractional system for $0 < \alpha _{2} \leq 1 < \alpha _{1} \leq 2$ with multistate time delay. Considering the Caputo derivative and generalized Gronwall inequality, we formulate the novel sufficient conditions such that the multiterm nonlinear fractional system is finite time stable. Further, we extend the result of stability in the finite range of time to the multiterm fractional integro-differential system with multistate time delay for the same order by obtaining some inequality using the Gronwall approach. Finally, from the examples, the advantage of presented scheme can guarantee the stability in the finite range of time of considered systems.

1 Introduction

Fractional calculus has been utilized as a key to the description of discontinuity and singularity formation. After several years of development, it has gained a lot of attention from physicists and mathematicians. We notice that fractional derivatives can be composite in perspective of pure mathematics and attract increasing interest in establishing the theoretical results and numerical approaches. Since the analysis and synthesis of fractional derivatives have been recognized in a wide-ranging field of practical applications in various applied sciences and have produced tremendous results. The core advantage of fractional derivatives is that numerous interdisciplinary practical applications can be easily formulated [1, 16, 25, 31].

Finite-time stability (FTS) is a more practical idea which is valuable to analyze the nature of a system within a finite interval of time and it is an essential part in the study of transient behavior of systems. Thus, it was extensively studied in both integer and fractional differential systems. Time delay can occur in input, output, or the state variable. The delay of state has appeared several times in physical systems and control problems [15, 24, 29, 32, 34, 35, 40]. On the other hand, in a multistate system the conversion between the behaviors in each state will depend on the passage of time and on inputs of the system. So it is valuable to investigate the FTS concept for a multi-delayed fractional nonlinear system.

Deng et al. [6] investigated the stability analysis of a multiple delay fractional linear system $(0< q<1)$. The FTS of the fractional linear time-invariant system was examined by utilizing a generalized Gronwall inequality in [23]. Liu and Zhong [22] discussed the FTS of a fractional multitime delayed system. Mittag-Leffler stability of a nonlinear fractional system was studied by introducing the Lyapunov method in [20, 41] for order $0<\alpha <1$. The robust stability concept was discussed for the system of fractional order in [5, 19] and for a fractional-order system, various concepts were discussed in [4, 12, 26, 28, 37, 39]. FTS analysis for various types of fractional system was explored in [14, 18, 27, 36]. Zhang and Niu [42] discussed the exponential stability of a class of nonlinear delay-integro-differential equations. In [43], the analysis of FTS of fractional systems with variable coefficients with $\alpha \in (0,1)$ was examined using certain sufficient inequalities which were obtained by applying the Hölder and generalized Gronwall inequalities. Zhang et al. [44] discussed the stability concept for fractional nonlinear systems with order from $(0,2)$. In [8], FTS analysis of delayed nonlinear fractional difference system was investigated by using Gronwall and Jensen inequalities, and the same concept was discussed for a Hopfield neural network with time delay in [11]. In [10], the authors studied FTS of delayed fractional neutral systems by using Gronwall inequality.

By the above deliberations, we were inspired to study the FTS of a multiterm fractional system with multistate time delay. The main idea of this work is made precise as follows:

1. In the literature, the results of FTS for fractional nonlinear systems have been reported. However, there have been no works for the FTS of multiterm fractional nonlinear systems. It is more essential to study the FTS of fractional-order systems with damping behavior. Thus, we consider the multiterm nonlinear fractional system with $0 < \alpha _{2} \leq 1 < \alpha _{1} \leq 2$.

2. Many of the previous results on fractional-order systems are often for a single-delay in state. So it is crucial to pay attention to the study of multiterm nonlinear fractional systems in which multiple delays occur in their states.

3. Further, we extend the result for multiterm fractional-order integro-differential systems with multistate time delay.

The organization of this work is given as follows: In Sect. 2, we have included some useful lemmas and definitions which are helpful to reach our results. In Sect. 3, the FTS concept is discussed for multiterm nonlinear fractional system with multistate time delay and also the same concept is analyzed for multiterm fractional order integro-differential system with a multistate time delay. The main results of this paper are verified through examples in Sect. 4. Finally, the paper is concluded in Sect. 5.

2 Preliminaries

The following notations are used in this paper: $\mathbb{R}^{n}$ denotes the n-dimensional Euclidean space of the reals with maximum norm; $\mathbb{R}^{n\times m}$ consist of all matrices of dimension $n\times m$; $\sigma _{\max }(\mathcal{A})$ denotes the largest singular value of matrix $\mathcal{A}$. Explicitly, $\sigma _{\max }(\mathcal{A})=\sqrt{\lambda _{\max }(\mathcal{A}^{T} \mathcal{A})}$. Now, we present some lemmas and definitions which are needed to obtain our results.

Definition 2.1

([1])

Caputo fractional derivative of $y(t)$ of order $\alpha _{1} \in \mathbb{R}^{+}$ is given by

$$\begin{aligned} {}^{C}_{0}D^{\alpha _{1}}_{t_{0},t}y(t)= \frac{1}{\Gamma (n-\alpha _{1})} \int _{t_{0}}^{t}(t-\vartheta )^{n-\alpha _{1}-1}y^{(n)}( \vartheta )\,d \vartheta, \end{aligned}$$

with $n-1<\alpha _{1}<n \in \mathbb{Z}^{+}$.

Definition 2.2

([31])

The Mittag-Leffler function with two parameters is defined as

$$\begin{aligned} E_{\alpha _{1},\alpha _{2}}(z)=\sum_{j=0}^{\infty } \frac{z^{j}}{\Gamma (\alpha _{1} j+\alpha _{2})}, \quad z \in \mathbb{C}, \alpha _{1}>0, \alpha _{2}>0. \end{aligned}$$

(1)

If $\alpha _{2}=1$ then (1) becomes

$$\begin{aligned} E_{\alpha _{1},1}(z)=\sum_{j=0}^{\infty } \frac{z^{j}}{\Gamma (\alpha _{1} j+1)}\equiv E_{\alpha _{1}}(z). \end{aligned}$$

(2)

Lemma 2.3

([16])

For the fractional integrals and Caputo fractional derivatives, we have

$$\begin{aligned} I^{\alpha }_{t} \bigl({}^{C}_{0}D^{\alpha }y(t) \bigr)=y(t)-\sum_{k=0}^{n-1} \frac{t^{k}}{k }y^{k}(0),\quad t>0, n-1< \alpha < n. \end{aligned}$$

Further, when $1<\alpha <2$,

$$\begin{aligned} I^{\alpha }_{t} \bigl({}^{C}_{0}D^{\alpha }y(t) \bigr)=y(t)-y(0)-ty'(0). \end{aligned}$$

Lemma 2.4

([33])

Assume $0<\alpha _{2}<1<\alpha _{1}<2$, then

$$\begin{aligned} I^{\alpha _{1}}_{t} \bigl({}^{C}_{0}D^{\alpha _{2}}y(t) \bigr)=I^{ \alpha _{1}-\alpha _{2}}_{t}y(t)- \frac{y(0)t^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)}. \end{aligned}$$

Lemma 2.5

(Generalized Gronwall inequality (GGI), [7, 38])

Assume $y(t)>0$, $\omega (t)>0$ are locally integrable and consider a continuous function $\nu (t)>0$ for $t\in [0,T )$. Suppose $\nu (t)\leq M$, $\alpha _{1}>0$ with

$$\begin{aligned} y(t)\leq \omega (t)+\nu (t) \int _{0}^{t} (t-\mu )^{\alpha _{1}-1}y( \mu )\,{d} \mu,\quad 0\leq t< T. \end{aligned}$$

(3)

Then

$$\begin{aligned} y(t)\leq \omega (t)+ \int _{0}^{t} \Biggl[\sum _{n=1}^{\infty } \frac{ (\nu (t)\Gamma (\alpha _{1}) )^{n}}{\Gamma (n\alpha _{1})}(t- \mu )^{n\alpha _{1}-1} \omega (\mu ) \Biggr]\,d\mu,\quad 0\leq t< T. \end{aligned}$$

(4)

Lemma 2.6

([38])

Under the assumptions of Lemma 2.5, if $\omega (t)>0$ is a nondecreasing function on $[0,T )$ then

$$\begin{aligned} y(t)\leq \omega (t)E_{\alpha _{1}} \bigl(\nu (t) \bigl(\Gamma (\alpha _{1})\bigr)t^{ \alpha _{1}} \bigr). \end{aligned}$$

(5)

Lemma 2.7

(Extended form of Gronwall inequality, [33])

Suppose both fractional orders $\alpha _{1}$ and $\alpha _{2}$ are nonzero and positive, $\omega (t)>0$ is locally integrable, the continuous functions $\nu _{1}(t)>0$ and $\nu _{2}(t)>0$ are nondecreasing on $[0,T)$, and $\nu _{1}(t)\leq M_{1}$, $\nu _{2}(t)\leq M_{2}$. Assume $y(t)>0$ is locally integrable on [0,T) and

$$\begin{aligned} y(t)\leq \omega (t)+\nu _{1}(t) \int _{0}^{t} (t-\mu )^{\alpha _{1}-1}y( \mu )\,d\mu +\nu _{2}(t) \int _{0}^{t} (t-\mu )^{\alpha _{2}-1}y( \mu )\,d\mu. \end{aligned}$$

(6)

Then for $t\in [0,T)$,

$$\begin{aligned} y(t)\leq {}&\omega (t)+ \int _{0}^{t} \sum_{n=1}^{\infty } \bigl[\nu (t) \bigr]^{n} \\ &{}\times \sum_{k=0}^{n} \frac{c_{n}^{k} [\Gamma (\alpha _{1}) ]^{n-k} [\Gamma (\alpha _{2}) ]^{k}}{\Gamma ((n-k)\alpha _{1}+k\alpha _{2})}(t- \mu )^{(n-k)\alpha _{1}+k\alpha _{2}-1}\omega (\mu )\,d\mu, \end{aligned}$$

(7)

where $\nu (t)=\nu _{1}(t)+\nu _{2}(t)$ and $c_{n}^{k}=\frac{n(n-1)(n-2)\cdots (n-k+1)}{k!}$.

Lemma 2.8

([33])

Under the assumptions of Lemma 2.7, if $\omega (t)>0$ is a nondecreasing function on $[0,T )$ then

$$\begin{aligned} y(t)\leq \omega (t)E_{\gamma } \bigl[\nu (t) \bigl(\Gamma (\alpha _{1})t^{ \alpha _{1}}+\Gamma (\alpha _{2})t^{\alpha _{2}} \bigr) \bigr], \end{aligned}$$

(8)

where $\gamma =\min \{ \alpha _{1},\alpha _{2} \}$.

At this instant, we impose the following conditions for deriving the results:

${{(\mathrm{H}_{1})}}$ The function $f(t,y(t))$ satisfies Lipschitz condition on $[0, T)$ and there exists $K>0$ such that

$$\begin{aligned} \bigl\Vert f\bigl(t,y(t)\bigr) \bigr\Vert \leq K \bigl\Vert y(t) \bigr\Vert ,\quad \forall t \in L, y \in \mathbb{R}^{n}; \end{aligned}$$

${{(\mathrm{H}_{2})}}$ The function $f(t,x,y)$ is Lipschitz continuous and there exist $D_{1}>0$ and $D_{2}>0$ such that

$$\begin{aligned} \bigl\Vert f(t,x,y) \bigr\Vert \leq D_{1} \Vert x \Vert +D_{2} \Vert y \Vert ,\quad x,y \in \mathbb{R}^{n}. \end{aligned}$$

3 Main results

3.1 Multiterm nonlinear fractional system

The multiterm fractional nonlinear system with multistate time delay is described as

$$\begin{aligned} \textstyle\begin{cases} {}^{C}_{0}D^{\alpha _{1}}_{t}y(t)-\mathcal{A}^{C}_{0}D^{\alpha _{2}}_{t}y(t)= \mathcal{B}_{0}y(t)+\sum_{i=1}^{n}\mathcal{B}_{i}y(t-\rho _{i})+f(t,y(t))+ \mathcal{C}u(t), \\ \quad t\in L=[t_{0},t_{0}+T], \\ y(t)=y_{0},\qquad y^{\prime }(t)=y_{1},\quad -\rho \leq t \leq 0, \end{cases}\displaystyle \end{aligned}$$

(9)

with $0<\alpha _{2}\leq 1<\alpha _{1}\leq 2$. Here, the state vector $y(t)$ is in $\mathbb{R}^{n}$, the matrices $\mathcal{A},\mathcal{B}_{i}$ in $\mathbb{R}^{n\times n}$ and matrix $\mathcal{C}$ in $\mathbb{R}^{n\times m}$, $u(t)\in \mathbb{R}^{m}$ denotes the control vector, $\rho =\max (\rho _{1},\rho _{2},\ldots,\rho _{n})$, $\rho _{i}$ is a constant with $\rho _{i}>0$, and T is either positive or +∞. Also, here $\Vert \cdot \Vert$ indicates the max norm.

Definition 3.1

([17, 21])

The system (9) is said to be finite-time stable with respect to $\{ t_{0},L, \delta,\epsilon,\alpha _{1u},\rho \}$ iff $\kappa <\delta$ and $\forall t\in L, \Vert u(t) \Vert <\alpha _{1u}$ implies $\Vert y(t) \Vert <\epsilon, \forall t\in L$, where $\kappa =\max \{ \Vert y_{0} \Vert , \Vert y_{1} \Vert \}$ and $\delta, \alpha _{1u}, \epsilon$ are positive constants.

Definition 3.2

([17, 21])

The system (9) is said to be finite-time stable with respect to $\{ t_{0},L,\delta,\epsilon,\rho \}$ at $(u(t)\equiv 0, \forall t)$ iff $\kappa <\delta$, $\forall t\in L$ implies $\Vert y(t) \Vert <\epsilon$, $\forall t\in L$, where $\kappa =\max \{ \Vert y_{0} \Vert , \Vert y_{1} \Vert \}$ and $\delta, \epsilon$ are positive constants.

Theorem 3.3

Assume that $t_{0}=0$. The multiterm fractional-order nonlinear system (9) is finite-time stable with respect to $\{ \delta,\epsilon,L_{0},\alpha _{1u} \}$, $\delta <\epsilon$, if it satisfies

$$\begin{aligned} &\biggl\{ 1+t+ \frac{\sigma _{\mathrm{max}}(\mathcal{A})t^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)} \biggr\} E_{\gamma } \bigl( \nu (t) \bigl(\Gamma (\alpha _{1}-\alpha _{2})t^{ \alpha _{1}-\alpha _{2}}+ \Gamma (\alpha _{1})t^{\alpha _{1}} \bigr) \bigr) \\ &\quad{}+\frac{\eta _{u}}{\Gamma (\alpha _{1}+1)}t^{\alpha _{1}}\leq \frac{\epsilon }{\delta },\quad \forall t\in L_{0}=[0,T], \end{aligned}$$

(10)

where $\eta _{u}=\frac{c\alpha _{1u}}{\delta }$, $\nu (t)=\nu _{1}(t)+\nu _{2}(t)$; $\nu _{1}(t)= \frac{\sigma _{\max }(\mathcal{A})}{\Gamma (\alpha _{1}-\alpha _{2})}$, $\nu _{2}(t)=\frac{K+\sigma (n+1)}{\Gamma (\alpha _{1})}$ and $\sigma _{\max }(\cdot )$ denotes the highest singular value of a given matrix $(\cdot )$.

Proof

Applying fractional integral on both sides of system (9), we get

$$\begin{aligned} I^{\alpha _{1}} \bigl({}^{C}_{0}D^{\alpha _{1}}_{t}y(t) \bigr)- \mathcal{A}I^{\alpha _{1}} \bigl({}^{C}_{0}D^{\alpha _{2}}_{t}y(t) \bigr)=I^{\alpha _{1}} \Biggl(\mathcal{B}_{0}y(t) +\sum _{i=1}^{n} \mathcal{B}_{i}y(t-\rho _{i})+f\bigl(t,y(t)\bigr)+\mathcal{C}u(t) \Biggr). \end{aligned}$$

Now utilizing Lemmas 2.3 and 2.4, we obtain the solution of system (9) as

$$\begin{aligned} y(t)={}&y_{0}+ty_{1}- \frac{\mathcal{A}t^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)}y_{0}+ \frac{\mathcal{A}}{\Gamma (\alpha _{1}-\alpha _{2})} \int _{0}^{t}(t- \mu )^{\alpha _{1}-\alpha _{2}-1}y(\mu )\,d\mu \\ &{}+\frac{1}{\Gamma (\alpha _{1})} \int _{0}^{t}(t-\mu )^{\alpha _{1}-1} \Biggl[ \mathcal{B}_{0}y(\mu )+\sum_{i=1}^{n} \mathcal{B}_{i}y( \mu -\rho _{i}) \\ &{}+f\bigl(\mu,y(\mu )\bigr)+\mathcal{C}u(\mu ) \Biggr]\,d\mu. \end{aligned}$$

The above equation implies that

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert \leq{} & \Vert y_{0} \Vert +t \Vert y_{1} \Vert + \frac{ \Vert \mathcal{A} \Vert (t)^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)} \Vert y_{0} \Vert \\ &{}+ \frac{ \Vert \mathcal{A} \Vert }{\Gamma (\alpha _{1}-\alpha _{2})} \int _{0}^{t}(t-\mu )^{\alpha _{1}-\alpha _{2}-1} \bigl\Vert y( \mu ) \bigr\Vert \,d\mu \\ &{}+\frac{1}{\Gamma (\alpha _{1})} \int _{0}^{t}(t-\mu )^{\alpha _{1}-1} \Biggl\Vert \mathcal{B}_{0}y(\mu )+\sum_{i=1}^{n} \mathcal{B}_{i}y( \mu -\rho _{i}) \\ &{}+f\bigl(\mu,y(\mu )\bigr)+\mathcal{C}u(\mu ) \Biggr\Vert \,d \mu. \end{aligned}$$

(11)

Now,

$$\begin{aligned} \Biggl\Vert \mathcal{B}_{0}y(t)+\sum _{i=1}^{n}\mathcal{B}_{i}y(t- \rho _{i})+f\bigl(t,y(t)\bigr)+\mathcal{C}u(t) \Biggr\Vert \leq{} & \Vert \mathcal{B}_{0} \Vert \bigl\Vert y(t) \bigr\Vert +\sum _{i=1}^{n} \Vert \mathcal{B}_{i} \Vert \bigl\Vert y(t-\rho _{i}) \bigr\Vert \\ &{}+ \bigl\Vert f\bigl(t,y(t)\bigr) \bigr\Vert + \Vert \mathcal{C} \Vert \bigl\Vert u(t) \bigr\Vert . \end{aligned}$$

(12)

Consider $\sigma _{1}=\max_{1\leq i\leq n}\sigma _{\max }(\mathcal{B}_{i})$ and $\sigma =\max \{ \sigma _{\max }(\mathcal{B}_{0}),\sigma _{1} \}$. From this consideration we get

$$\begin{aligned} \Vert \mathcal{B}_{i} \Vert \leq \sigma;\quad \forall i=0,1,2, \ldots,n. \end{aligned}$$

(13)

Applying (13) and Hypothesis ${{(\mathrm{H}_{1})}}$ in (12), we get

$$\begin{aligned} \Biggl\Vert \mathcal{B}_{0}y(t)+\sum _{i=1}^{n}\mathcal{B}_{i}y(t- \rho _{i})+f\bigl(t,y(t)\bigr)+\mathcal{C}u(t) \Biggr\Vert \leq {}& \sigma \bigl\Vert y(t) \bigr\Vert +\sum_{i=1}^{n} \sigma \bigl\Vert y(t-\rho _{i}) \bigr\Vert \\ &{}+K \bigl\Vert y(t) \bigr\Vert +c \bigl\Vert u(t) \bigr\Vert , \end{aligned}$$

(14)

where $\Vert \mathcal{C} \Vert \leq c$. Substituting inequality (14) into (11), we get

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert \leq {}& \Vert y_{0} \Vert +t \Vert y_{1} \Vert + \frac{\sigma _{\max }(\mathcal{A})t^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)} \Vert y_{0} \Vert + \frac{\sigma _{\max } (\mathcal{A})}{\Gamma (\alpha _{1}-\alpha _{2})} \\ &{}\times \int _{0}^{t}(t-\mu )^{\alpha _{1}-\alpha _{2}-1} \bigl\Vert y( \mu ) \bigr\Vert \,d\mu +\frac{1}{\Gamma (\alpha _{1})} \int _{0}^{t}(t- \mu )^{\alpha _{1}-1} \Biggl\{ \sigma \bigl\Vert y(\mu ) \bigr\Vert \\ &{}+\sum_{i=1}^{n}\sigma \bigl\Vert y(\mu -\rho _{i}) \bigr\Vert +K \bigl\Vert y(\mu ) \bigr\Vert +c \bigl\Vert u(\mu ) \bigr\Vert \Biggr\} \,d \mu. \end{aligned}$$

(15)

Now let

$$\begin{aligned} x(t)=\sup_{\beta \in [-\rho,t]} \bigl\Vert y(\beta ) \bigr\Vert ,\quad \forall t\in L_{0} \end{aligned}$$

and

$$\begin{aligned} \bigl\Vert y(\mu ) \bigr\Vert \leq x(\mu ),\qquad \bigl\Vert y(\mu -\rho _{i}) \bigr\Vert \leq x(\mu ), \quad\forall i=1,2,\ldots,n, \mu \in [0,t]. \end{aligned}$$

From (15) it follows that

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert \leq {}& \Vert y_{0} \Vert +t \Vert y_{1} \Vert + \frac{\sigma _{\max }(\mathcal{A})t^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)} \Vert y_{0} \Vert + \frac{\sigma _{\max }(\mathcal{A})}{\Gamma (\alpha _{1}-\alpha _{2})} \\ &{}\times \int _{0}^{t}(t-\mu )^{\alpha _{1}-\alpha _{2}-1}x(\mu )\,d \mu + \biggl(\frac{\sigma (n+1)+K}{\Gamma (\alpha _{1})} \biggr) \int _{0}^{t}(t- \mu )^{\alpha _{1}-1}x(\mu )\,d\mu \\ &{} +\frac{c\alpha _{1u}}{\Gamma (\alpha _{1}+1)}t^{\alpha _{1}} \\ ={}& \Vert y_{0} \Vert +t \Vert y_{1} \Vert + \frac{\sigma _{\max }(\mathcal{A})t^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)} \Vert y_{0} \Vert + \frac{\sigma _{\max }(\mathcal{A})}{\Gamma (\alpha _{1}-\alpha _{2})} \\ &{}\times \int _{0}^{t}\mu ^{\alpha _{1}-\alpha _{2}-1}x(t-\mu )\,d \mu + \biggl(\frac{\sigma (n+1)+K}{\Gamma (\alpha _{1})} \biggr) \int _{0}^{t} \mu ^{\alpha _{1}-1}x(t-\mu )\,d\mu \\ &{} +\frac{c\alpha _{1u}}{\Gamma (\alpha _{1}+1)}t^{\alpha _{1}}. \end{aligned}$$

(16)

Here $\Vert u(\mu ) \Vert \leq \alpha _{1u}$ and $\sigma _{\max }(\mathcal{A})$ indicates the highest singular value for the given matrix $\mathcal{A}$.

Note that for all $\beta \in [0,t]$, we have

$$\begin{aligned} \bigl\Vert y(\beta ) \bigr\Vert \leq {}& \Vert y_{0} \Vert +t \Vert y_{1} \Vert + \frac{\sigma _{\max }(\mathcal{A})t^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)} \Vert y_{0} \Vert + \frac{\sigma _{\max }(\mathcal{A})}{\Gamma (\alpha _{1}-\alpha _{2})} \\ &{}\times \int _{0}^{\beta }\mu ^{\alpha _{1}-\alpha _{2}-1}x(\beta -\mu ) \,d \mu + \biggl(\frac{\sigma (n+1)+K}{\Gamma (\alpha _{1})} \biggr) \int _{0}^{\beta }\mu ^{\alpha _{1}-1}x(\beta -\mu )\,d \mu \\ &{} +\frac{c\alpha _{1u}}{\Gamma (\alpha _{1}+1)}t^{\alpha _{1}}. \end{aligned}$$

(17)

Since the functions $\int _{0}^{t}\mu ^{\alpha _{1}-\alpha _{2}-1}x(t-\mu )\,d\mu$ and $\int _{0}^{t}\mu ^{\alpha _{1}-1}x(t-\mu )\,d\mu$ are increasing with respect to $t\geq 0$, because of the increasing of the nonnegative function $x(t)$, we get

$$\begin{aligned} \begin{aligned} &\int _{0}^{\beta }\mu ^{\alpha _{1}-\alpha _{2}-1}x(\beta -\mu ) \,d \mu \leq \int _{0}^{t}\mu ^{\alpha _{1}-\alpha _{2}-1}x(t- \mu )\,d\mu, \\ &\int _{0}^{\beta }\mu ^{\alpha _{1}-1}x(\beta -\mu )\,d \mu \leq \int _{0}^{t}\mu ^{\alpha _{1}-1}x(t-\mu )\,d\mu. \end{aligned} \end{aligned}$$

(18)

Therefore

$$\begin{aligned} \bigl\Vert y(\beta ) \bigr\Vert \leq {}& \Vert y_{0} \Vert +t \Vert y_{1} \Vert + \frac{\sigma _{\max }(\mathcal{A})t^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)} \Vert y_{0} \Vert + \frac{\sigma _{\max }(\mathcal{A})}{\Gamma (\alpha _{1}-\alpha _{2})} \\ &{}\times \int _{0}^{t}\mu ^{\alpha _{1}-\alpha _{2}-1}x(t-\mu )\,d \mu + \biggl(\frac{\sigma (n+1)+K}{\Gamma (\alpha _{1})} \biggr) \int _{0}^{t} \mu ^{\alpha _{1}-1}x(t-\mu )\,d\mu \\ &{} +\frac{c\alpha _{1u}}{\Gamma (\alpha _{1}+1)}t^{\alpha _{1}}, \quad\forall \beta \in [0,t]. \end{aligned}$$

(19)

Hence, we have

$$\begin{aligned} x(t)={}&\sup_{\beta \in [-\rho,t]} \bigl\Vert y(\beta ) \bigr\Vert \leq \max \Bigl\{ \sup_{\beta \in [-\rho,0]} \bigl\Vert y(\beta ) \bigr\Vert , \sup_{\beta \in [0,t]} \bigl\Vert y(\beta ) \bigr\Vert \Bigr\} \\ \leq {}& \max \biggl\{ \Vert y_{0} \Vert , \biggl( \Vert y_{0} \Vert +t \Vert y_{1} \Vert + \frac{\sigma _{\max }(\mathcal{A})t^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)} \Vert y_{0} \Vert + \frac{\sigma _{\max }(\mathcal{A})}{\Gamma (\alpha _{1}-\alpha _{2})} \\ &{}\times \int _{0}^{t}\mu ^{\alpha _{1}-\alpha _{2}-1}x(t-\mu )\,d \mu + \biggl(\frac{\sigma (n+1)+K}{\Gamma (\alpha _{1})} \biggr) \int _{0}^{t} \mu ^{\alpha _{1}-1}x(t-\mu )\,d\mu \\ &{} +\frac{c\alpha _{1u}}{\Gamma (\alpha _{1}+1)}t^{\alpha _{1}} \biggr) \biggr\} \\ ={}& \Vert y_{0} \Vert +t \Vert y_{1} \Vert + \frac{\sigma _{\max }(\mathcal{A})t^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)} \Vert y_{0} \Vert + \frac{\sigma _{\max }(\mathcal{A})}{\Gamma (\alpha _{1}-\alpha _{2})} \\ &{}\times \int _{0}^{t}(t-\mu )^{\alpha _{1}-\alpha _{2}-1}x(\mu )\,d \mu + \biggl(\frac{\sigma (n+1)+K}{\Gamma (\alpha _{1})} \biggr) \int _{0}^{t}(t- \mu )^{\alpha _{1}-1}x(\mu )\,d\mu \\ &{} +\frac{c\alpha _{1u}}{\Gamma (\alpha _{1}+1)}t^{\alpha _{1}}. \end{aligned}$$

(20)

Let

$$\begin{aligned} \omega (t)= \Vert y_{0} \Vert +t \Vert y_{1} \Vert + \frac{\sigma _{\max }(\mathcal{A})t^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)} \Vert y_{0} \Vert , \end{aligned}$$

(21)

which is a nondecreasing function, and let $\nu _{1}(t)= \frac{\sigma _{\max }(\mathcal{A})}{\Gamma (\alpha _{1}-\alpha _{2})}, \nu _{2}(t)=\frac{\sigma (n+1)+K}{\Gamma (\alpha _{1})}$. Utilizing this consideration, we get

$$\begin{aligned} x(t)\leq{} & \omega (t) +\nu _{1}(t) \int _{0}^{t}(t-\mu )^{\alpha _{1}- \alpha _{2}-1}x(\mu )\,d\mu \\ &{}+\nu _{2}(t) \int _{0}^{t}(t-\mu )^{\alpha _{1}-1}x(\mu )\,d \mu + \frac{c\alpha _{1u}}{\Gamma (\alpha _{1}+1)}t^{\alpha _{1}}. \end{aligned}$$

(22)

Now applying Lemma 2.8, we obtain

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert \leq x(t)\leq \omega (t)E_{\gamma } \bigl\{ \nu (t) \bigl(\Gamma (\alpha _{1}-\alpha _{2})t^{\alpha _{1}-\alpha _{2}}+ \Gamma (\alpha _{1})t^{\alpha _{1}} \bigr) \bigr\} + \frac{c\alpha _{1u}}{\Gamma (\alpha _{1}+1)}t^{\alpha _{1}}, \end{aligned}$$

where $\nu (t)=\nu _{1}(t)+\nu _{2}(t)$. Now applying the conditions of FTS, one can obtain

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert \leq {}& \delta \biggl(1+t+ \frac{\sigma _{\max }(\mathcal{A})t^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)} \biggr) E_{\gamma } \bigl\{ \nu (t) \bigl(\Gamma (\alpha _{1}-\alpha _{2})t^{ \alpha _{1}-\alpha _{2}}+\Gamma (\alpha _{1})t^{\alpha _{1}} \bigr) \bigr\} \\ &{}+\frac{c\alpha _{1u}}{\Gamma (\alpha _{1}+1)}t^{\alpha _{1}}. \end{aligned}$$

(23)

Hence from (10), we have

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert < \epsilon, \quad\forall t\in L_{0}. \end{aligned}$$

(24)

This completes the proof. □

Corollary 3.4

If $\alpha _{1}=2$ and $\alpha _{2}=1$ then system (9) becomes the second-order integer system with multistate time delay which is given by

$$\begin{aligned} \textstyle\begin{cases} \frac{d^{2}y}{dt^{2}}-\mathcal{A}\frac{dy}{dt}=\mathcal{B}_{0}y(t)+ \sum_{i=1}^{n}\mathcal{B}_{i}y(t-\rho _{i})+f(t,y(t))+ \mathcal{C}u(t), \quad t\in L_{0}, \\ y(t)=y_{0},\qquad y^{\prime }(t)=y_{1}, \quad -\rho \leq t \leq 0. \end{cases}\displaystyle \end{aligned}$$

(25)

The given system (25) is FTS with respect to $\{ \delta,\epsilon,L_{0},\alpha _{1u},\rho \}$, $\delta <\epsilon$, if it satisfies

$$\begin{aligned} \bigl\{ 1+t+\sigma _{\max }(\mathcal{A})t^{1} \bigr\} e^{\nu (t) (t+t^{2} )}+\frac{\eta _{u}}{2}t^{2}\leq \frac{\epsilon }{\delta },\quad \forall t\in L_{0}=[0,T], \end{aligned}$$

(26)

where $\eta _{u}=\frac{c\alpha _{1u}}{\delta }$, $\nu (t)=\nu _{1}(t)+\nu _{2}(t)$, $\nu _{1}(t)=\sigma _{\max }(\mathcal{A}), \nu _{2}(t)=\sigma (n+1)+K$.

Proof

The solution of (25) is given by

$$\begin{aligned} y(t)={}&y_{0}+ty_{1}-\mathcal{A}ty_{0}+ \mathcal{A} \int _{0}^{t}y(\mu ) \,d\mu + \int _{0}^{t}(t-\mu ) \Biggl[\mathcal{B}_{0}y( \mu )+\sum_{i=1}^{n}\mathcal{B}_{i}y( \mu -\rho _{i}) \\ &{} +f\bigl(\mu,y(\mu )\bigr)+\mathcal{C}u(\mu ) \Biggr]\,d\mu. \end{aligned}$$

Now computing the norm of both sides of the above equation, we get

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert \leq{} & \Vert y_{0} \Vert +t \Vert y_{1} \Vert + \Vert \mathcal{A} \Vert t \Vert y_{0} \Vert + \Vert \mathcal{A} \Vert \int _{0}^{t} \bigl\Vert y(\mu ) \bigr\Vert \,d \mu \\ &{}+ \int _{0}^{t}(t-\mu ) \Biggl[ \Biggl\Vert \mathcal{B}_{0}y(\mu ) +\sum_{i=1}^{n}\mathcal{B}_{i}y( \mu -\rho _{i}) +f\bigl(\mu,y( \mu )\bigr)+\mathcal{C}u(\mu ) \Biggr\Vert \Biggr]\,d\mu. \end{aligned}$$

Now following the steps from the proof of Theorem 3.3, we obtain

$$\begin{aligned} x(t) \leq {}& \Vert y_{0} \Vert +t \Vert y_{1} \Vert +\sigma _{ \max }(\mathcal{A})t \Vert y_{0} \Vert +\sigma _{\max }(\mathcal{A}) \int _{0}^{t}x(\mu )\,d\mu \\ &{} + \bigl(\sigma (n+1)+K \bigr) \int _{0}^{t}(t-\mu )x(\mu ) \,d\mu +c\alpha _{1u}\frac{t^{2}}{2}, \end{aligned}$$

(27)

where $\sigma _{\max }(\mathcal{A})$ denotes the largest singular value of matrix $\mathcal{A}$. Now consider the nondecreasing function $\omega (t)$ defined by

$$\begin{aligned} \omega (t)= \Vert y_{0} \Vert +t \Vert y_{1} \Vert +\sigma _{ \max }(\mathcal{A})t \Vert y_{0} \Vert \end{aligned}$$

and also let $\nu _{1}(t)=\sigma _{\max }(\mathcal{A})$, $\nu _{2}(t)=\sigma (n+1)+K$.

Now utilizing the above notations in (27), we get

$$\begin{aligned} x(t)\leq {}& \omega (t)+\nu _{1}(t) \int _{0}^{t}x(\mu )\,d\mu \\ &{} +\nu _{2}(t) \int _{0}^{t}(t-\mu )x(\mu )\,d\mu +c\alpha _{1u} \frac{t^{2}}{2}. \end{aligned}$$

(28)

From Gronwall’s inequality, we obtain

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert \leq x(t)\leq \omega (t)E_{\gamma } \bigl\{ \nu (t) \bigl(\Gamma (1)t^{1}+\Gamma (2)t^{2} \bigr) \bigr\} +c\alpha _{1u} \frac{t^{2}}{2}, \end{aligned}$$

(29)

where $\nu (t)=\nu _{1}(t)+\nu _{2}(t)$ and $\gamma =\min \{ 1,2 \} =1$. Hence we know that $E_{1}(z)=e^{z}$. Now from the condition of FTS, we get

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert \leq \delta \bigl(1+t+\sigma _{\max }( \mathcal{A})t \bigr)e^{\nu (t) (t+t^{2} )}+c\alpha _{1u} \frac{t^{2}}{2}. \end{aligned}$$

Hence

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert \leq \epsilon, \quad\forall t\in L_{0}. \end{aligned}$$

(30)

 □

3.2 Multiterm fractional-order integro-differential system

Consider the fractional integro-differential system with multistate time delay

$$\begin{aligned} \textstyle\begin{cases} {}^{C}_{0}D^{\alpha _{1}}_{t}y(t)-\mathcal{A}^{C}_{0}D^{\alpha _{2}}_{t}y(t)\\ \quad= \mathcal{B}_{0}y(t)+\sum_{i=1}^{n}\mathcal{B}_{i}y(t-\rho _{i}) +f(t,y(t),\int _{0}^{t}H(t,s,y(s))\,ds) +\mathcal{C}u(t), \\ y(t)=y_{0},\qquad y^{\prime }(t)=y_{1}, \quad -\rho \leq t \leq 0, t\in L_{0}=[0,a], \end{cases}\displaystyle \end{aligned}$$

(31)

with $0<\alpha _{2}\leq 1<\alpha _{1}\leq 2$. Here, $y(t)$, matrices $\mathcal{A},\mathcal{B}_{i}$, $\mathcal{C}$, $u(t)$, and ρ are defined the same as in (9). Also, $f\in C[L_{0} \times \mathbb{R}^{n} \times \mathbb{R}^{n}, \mathbb{R}^{n}]$ and $H\in C[L_{0} \times L_{0} \times \mathbb{R}^{n}, \mathbb{R}^{n}]$.

Theorem 3.5

Let $H(t,s,y(s))$ satisfy

$$\begin{aligned} \bigl\Vert H\bigl(t,s,y(s)\bigr) \bigr\Vert \leq N_{1} \Vert y \Vert . \end{aligned}$$

(32)

The multiterm fractional-order integro-differential system (31) is finite-time stable with respect to $\{ \delta,\epsilon,L_{0},\alpha _{1u} \}$, $\delta <\epsilon$ if

$$\begin{aligned} & \biggl\{ 1+t+ \frac{\sigma _{\max }(\mathcal{A})t^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)} \biggr\} E_{\gamma } \bigl( \nu (t) \bigl(\Gamma (\alpha _{1}-\alpha _{2})t^{ \alpha _{1}-\alpha _{2}}+ \Gamma (\alpha _{1})t^{\alpha _{1}} \bigr) \bigr) \\ &\quad{}+\frac{\eta _{u}}{\Gamma (\alpha _{1}+1)}t^{\alpha _{1}}\leq \frac{\epsilon }{\delta },\quad \forall t\in L_{0}=[0,a], \end{aligned}$$

(33)

where $\eta _{u}=\frac{c\alpha _{1u}}{\delta }$, $\nu (t)=\nu _{1}(t)+\nu _{2}(t)$; $\nu _{1}(t)= \frac{\sigma _{\max }(\mathcal{A})}{\Gamma (\alpha _{1}-\alpha _{2})}$, $\nu _{2}(t)=\frac{\sigma (n+1)+N}{\Gamma (\alpha _{1})}$; $N=D_{1}+D_{2}aN_{1}$.

Proof

The solution of (31) can be obtained in the following form:

$$\begin{aligned} y(t)={}&y_{0}+ty_{1}- \frac{\mathcal{A}t^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)}y_{0} +\frac{\mathcal{A}}{\Gamma (\alpha _{1}-\alpha _{2})} \int _{0}^{t}(t- \mu )^{\alpha _{1}-\alpha _{2}-1}y(\mu )\,d\mu \\ &{} +\frac{1}{\Gamma (\alpha _{1})} \int _{0}^{t}(t-\mu )^{ \alpha _{1}-1} \Biggl[ \mathcal{B}_{0}y(\mu )+\sum_{i=1}^{n} \mathcal{B}_{i}y(\mu -\rho _{i}) \\ &{} +f\biggl(\mu,y(\mu ), \int _{0}^{t}H\bigl(\mu,s,y(s)\bigr)\,ds\biggr) + \mathcal{C}u(\mu ) \Biggr]\,d\mu. \end{aligned}$$

(34)

Then the above equation (34) implies

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert \leq {}& \Vert y_{0} \Vert +t \Vert y_{1} \Vert + \frac{ \Vert \mathcal{A} \Vert (t)^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)} \Vert y_{0} \Vert + \frac{ \Vert \mathcal{A} \Vert }{\Gamma (\alpha _{1}-\alpha _{2})} \\ &{} \times \int _{0}^{t}(t-\mu )^{\alpha _{1}-\alpha _{2}-1} \bigl\Vert y( \mu ) \bigr\Vert \,d\mu +\frac{1}{\Gamma (\alpha _{1})} \int _{0}^{t}(t- \mu )^{\alpha _{1}-1} \Biggl\Vert \mathcal{B}_{0}y(\mu ) \\ &{}+\sum_{i=1}^{n} \mathcal{B}_{i}y(\mu -\rho _{i})+f\biggl(\mu,y( \mu ), \int _{0}^{t}H\bigl(\mu,s,y(s)\bigr)\,ds\biggr) + \mathcal{C}u(\mu ) \Biggr\Vert \,d\mu. \end{aligned}$$

(35)

Now,

$$\begin{aligned} & \Biggl\Vert \mathcal{B}_{0}y(\mu )+\sum _{i=1}^{n}\mathcal{B}_{i}y( \mu -\rho _{i})+f\biggl(\mu,y(\mu ), \int _{0}^{t}H\bigl(\mu,s,y(s)\bigr)\,ds\biggr) + \mathcal{C}u(\mu ) \Biggr\Vert \\ &\quad \leq \Vert \mathcal{B}_{0} \Vert \bigl\Vert y(t) \bigr\Vert +\sum_{i=1}^{n} \Vert \mathcal{B}_{i} \Vert \bigl\Vert y(t- \rho _{i}) \bigr\Vert \\ &\qquad{}+ \biggl\Vert f\biggl(t,y(t), \int _{0}^{t}H\bigl(t,s,y(s)\bigr) \,ds\biggr) \biggr\Vert + \Vert \mathcal{C} \Vert \bigl\Vert u(t) \bigr\Vert . \end{aligned}$$

(36)

From Hypothesis ${{(\mathrm{H}_{2})}}$, we have

$$\begin{aligned} \biggl\Vert f\biggl(t,y(t), \int _{0}^{t}H\bigl(t,s,y(s)\bigr)\,ds\biggr) \biggr\Vert \leq D_{1} \bigl\Vert y(t) \bigr\Vert +D_{2} \int _{0}^{t} \bigl\Vert H\bigl(t,s,y(s)\bigr) \bigr\Vert \,ds. \end{aligned}$$

(37)

Using the condition (32), for $t\leq a$, we get

$$\begin{aligned} \biggl\Vert f\biggl(t,y(t), \int _{0}^{t}H\bigl(t,s,y(s)\bigr)\,ds\biggr) \biggr\Vert \leq D_{1} \bigl\Vert y(t) \bigr\Vert +aD_{2}N_{1} \bigl\Vert y(t) \bigr\Vert \leq N \bigl\Vert y(t) \bigr\Vert , \end{aligned}$$

(38)

where $N=D_{1}+aD_{2}N_{1}$.

Now substituting (38) into (36), we have

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert \leq {}& \Vert y_{0} \Vert +t \Vert y_{1} \Vert + \frac{ \Vert \mathcal{A} \Vert (t)^{\alpha _{1}-\alpha _{2}}}{\Gamma (\alpha _{1}-\alpha _{2}+1)} \Vert y_{0} \Vert \\ &{}+ \frac{ \Vert \mathcal{A} \Vert }{\Gamma (\alpha _{1}-\alpha _{2})} \int _{0}^{t}(t-\mu )^{\alpha _{1}-\alpha _{2}-1} \bigl\Vert y( \mu ) \bigr\Vert \,d\mu \\ &{}+\frac{1}{\Gamma (\alpha _{1})} \int _{0}^{t}(t-\mu )^{\alpha _{1}-1} \Biggl[ \Vert \mathcal{B}_{0} \Vert \bigl\Vert y(\mu ) \bigr\Vert +\sum _{i=1}^{n} \Vert \mathcal{B}_{i} \Vert \bigl\Vert y(\mu - \rho _{i}) \bigr\Vert \\ &{}+N \bigl\Vert y(\mu ) \bigr\Vert + \Vert \mathcal{C} \Vert \bigl\Vert u( \mu ) \bigr\Vert \Biggr]\,d\mu. \end{aligned}$$

By following the procedure of Theorem 3.3 and from (33), we have $\Vert y(t) \Vert < \epsilon, \forall t\in L_{0}$. Hence the proof is complete. □

Corollary 3.6

When $\alpha _{1}=2$, $\alpha _{2}=1$ and in the absence of delay, the system (31) becomes the second-order integro-differential system without time delay which is given by

$$\begin{aligned} \textstyle\begin{cases} \frac{d^{2}y}{dt^{2}}-\mathcal{A}\frac{dy}{dt}=\mathcal{B}_{0}y(t)+f(t,y(t), \int _{0}^{t}H(t,s,y(s))\,ds) +\mathcal{C}u(t),\quad t\in L_{0}=[0,a], \\ y(t)=y_{0},\qquad y^{\prime }(t)=y_{1},\quad -\rho \leq t \leq 0, \end{cases}\displaystyle \end{aligned}$$

(39)

which is FTS with respect to $\{ \delta,\epsilon,L_{0},\alpha _{1u} \}$, $\delta <\epsilon$ if

$$\begin{aligned} \bigl\{ 1+t+\sigma _{\mathrm{max}}(\mathcal{A})t \bigr\} e^{\nu (t) (t+t^{2} )}+ \frac{\eta _{u}}{2}t^{2}\leq \frac{\epsilon }{\delta },\quad \forall t\in L_{0}=[0,a], \end{aligned}$$

(40)

where $\eta _{u}=\frac{c\alpha _{1u}}{\delta }$, $\nu (t)=\nu _{1}(t)+\nu _{2}(t)$; $\nu _{1}(t)= \frac{\sigma _{\max }(\mathcal{A})}{\Gamma (\alpha _{1}-\alpha _{2}}$, $\nu _{2}(t)= \frac{N+\sigma _{\max }(\mathcal{B}_{0})}{\Gamma (\alpha _{1})}$; $N=D_{1}+D_{2}aN_{1}$.

Proof

The solution of (39) is given by

$$\begin{aligned} y(t)={}&y_{0}+ty_{1}- \mathcal{A}ty_{0}+\mathcal{A} \int _{0}^{t}y(\mu ) \,d\mu + \int _{0}^{t}(t-\mu ) \biggl[\mathcal{B}_{0}y( \mu ) \\ &{} +f\biggl(\mu,y(\mu ), \int _{0}^{t}H\bigl(\mu,s,y(s)\bigr)\,ds\biggr) + \mathcal{C}u(\mu ) \biggr]\,d\mu. \end{aligned}$$

Now taking the norm of both sides, we get

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert ={}& \Vert y_{0} \Vert +t \Vert y_{1} \Vert + \Vert \mathcal{A} \Vert t \Vert y_{0} \Vert + \Vert \mathcal{A} \Vert \int _{0}^{t} \bigl\Vert y(\mu ) \bigr\Vert \,d \mu \\ &{}+ \int _{0}^{t}(t-\mu ) \biggl\Vert \mathcal{B}_{0}y(\mu ) +f\biggl(\mu,y(\mu ), \int _{0}^{t}H\bigl(\mu,s,y(s)\bigr)\,ds\biggr) + \mathcal{C}u( \mu ) \biggr\Vert \,d\mu. \end{aligned}$$

Now following similar steps as in the proof of Theorem 3.5, we obtain

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert \leq {}& \Vert y_{0} \Vert +t \Vert y_{1} \Vert + \Vert \mathcal{A} \Vert (t) \Vert y_{0} \Vert + \Vert \mathcal{A} \Vert \int _{0}^{t} \bigl\Vert y(\mu ) \bigr\Vert \,d \mu \\ &{}+ \int _{0}^{t}(t-\mu ) \bigl[ \Vert \mathcal{B}_{0} \Vert \bigl\Vert y(\mu ) \bigr\Vert +N \bigl\Vert y(\mu ) \bigr\Vert + \Vert \mathcal{C} \Vert \bigl\Vert u(\mu ) \bigr\Vert \bigr]\,d\mu, \end{aligned}$$

(41)

where $N=D_{1}+D_{2}aN_{1}$.

Now following the same steps which proved in Corollary 3.4, we obtain

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert \leq \bigl\{ 1+t+\sigma _{\max }( \mathcal{A})t \bigr\} e^{\nu (t) (t+t^{2} )}+\frac{c\alpha _{1u}}{2}t^{2}. \end{aligned}$$

(42)

Hence

$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert < \epsilon,\quad \forall t\in L_{0}. \end{aligned}$$

 □

Remark 3.7

It is noted that for a nonnegative function $f(t)$, the fractional integral $\int ^{t}_{0}(t-s)^{\alpha _{1}-\alpha _{2}-1}f(s)\,ds$ may be monotonically increasing or decreasing with respect to t for $0<\alpha _{1}-\alpha _{2}<1$ (see [3, 9, 11, 30]). To prove that the integral term $\int ^{t}_{0}(t-s)^{\alpha _{1}-\alpha _{2}-1}f(s)\,ds$ is monotonically increasing for $f(t)\geq 0$, there is an alternative approach found in [10] (Lemma 5). It is noted that the results of Lemma 5 in [10] can also be used for proving that the fractional integrals in (22) are monotonically increasing for $0<\alpha _{1}-\alpha _{2}<1$.

Remark 3.8

The fractional oscillation equation

$$\begin{aligned} D^{2}y(t)+a_{2n-1}D^{2-\frac{1}{n}}y(t)+ \cdots +a_{1}D^{\frac{1}{n}}y(t)+y(t)=0 \end{aligned}$$

(43)

reduces to the harmonic oscillation equation $D^{2}y(t)+y(t)=0$ when $a_{n}=0$, $n=1,2, \dots, 2n-1$. In fact, this equation states $mD^{2}y(t)+ky(t)=0$. Here m is the mass, and k is the spring constant of the oscillator. Based on this system, many researchers studied various characteristics and effects of fractional oscillator models [28, 39]. In mechanical systems, damping is generated by several friction processes, like air resistance, viscous and dry friction, etc. It is well known that the damping force is related to the velocity of the process, which means that the friction force may be consistently interchanged with a viscous damping force.

When $n=2$ and $a_{1}\neq 0$, $a_{2}=a_{3}=0$, equation (43) becomes $D^{2}y(t)+a_{1}D^{\frac{1}{2}}y(t)+y(t)=0$, $a_{1}>0$. From this we get the following system with a forcing function $f(t)$, $mD^{2}y(t)+k_{2}D^{\alpha }y(t)+k_{1}y(t)=f(t)$, $\alpha \in (0,1)$. This equation represents a mechanical system with a mass, a spring and viscoelastic damping. Here $m, k_{1}$, and $k_{2}$ denote the mechanical constants. This model has been used in various studies and many results have been established for it [2, 13]. It is important to note that the FTS analysis for this type of fractional nonlinear systems with multistate time delay has been analyzed for the first time. Also we note that the available results related to stability of fractional systems were discussed with a single-delay in state and without damping behavior. So the results which were obtained in this work are new and will be more useful in practice.

4 Numerical examples

Example 4.1

Consider the multiterm fractional-order multistate time delay system (9) with $\alpha _{1}=1.25$, $\alpha _{2}=0.75$,

$$\begin{aligned} &\mathcal{A}= \begin{bmatrix} 3 &1 &8 \\ 0 &5 &2 \\ 0 &0 &4 \end{bmatrix},\qquad \mathcal{B}_{0}= \begin{bmatrix} 1 &3 &2 \\ 5 &0 &1 \\ 4 &7 &6 \end{bmatrix},\qquad \mathcal{B}_{1}= \begin{bmatrix} 1 &0 &2 \\ 0 &0 &5 \\ 5 &1 &3 \end{bmatrix},\\ & \mathcal{B}_{2}= \begin{bmatrix} 1 &0 &5 \\ 2 &-1 &3 \\ -1 &5 &6 \end{bmatrix},\qquad \mathcal{C}= \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}, \end{aligned}$$

and also take the nonlinear term

$$f(t,x(t))= \begin{bmatrix} \sin x_{1}(t) \\ \sin x_{2}(t) \\ \sin x_{3}(t) \end{bmatrix}. $$

Then we can calculate that $\sigma _{\max }(\mathcal{A})=9.7843$, $\sigma _{\max }(\mathcal{B}_{0})=11.0497$, $\sigma _{\max }(\mathcal{B}_{1})=7.1136$, and $\sigma _{\max }(\mathcal{B}_{0})=9.1027$. Hence $\sigma =11.0497$, $K=1$, and $c=2$. Let $\delta =0.1$, $\epsilon =100$, $\alpha _{1u}=1$. The aim is to validate the FTS condition (10) with respect to

$$ \{ t_{0}=0, \delta =0.1, \epsilon =100, \alpha _{1u}=1,\rho _{1}=0.1, \rho _{2}=0.01 \}. $$

Then by the FTS condition of Theorem 3.3, we obtain $T_{e}=0.1$.

Example 4.2

Consider the multiterm fractional-order integro-differential system (31) with $\alpha _{1}=1.25$, $\alpha _{2}=0.75$,

$$\begin{aligned} &\mathcal{A}= \begin{bmatrix} 1 &2 \\ 0 &3 \end{bmatrix},\qquad \mathcal{B}_{0}= \begin{bmatrix} -1 &0 \\ 1 &2 \end{bmatrix},\qquad \mathcal{B}_{1}= \begin{bmatrix} 0 &4 \\ 1 &2 \end{bmatrix},\\ & \mathcal{B}_{2}= \begin{bmatrix} 1 &0 \\ 0 &-1 \end{bmatrix},\qquad \mathcal{C}= \begin{bmatrix} -1 \\ 1 \end{bmatrix}, \end{aligned}$$

and also take $f(t,y(t),\int _{0}^{t}H(t,s,y(s))\,ds)=y(t)+\int _{0}^{t} \sin y(s)\,ds$. Then we get $\sigma _{\max }(\mathcal{A})=3.6503$, $\sigma _{\max }(\mathcal{B}_{0})=2.2883$, $\sigma _{\max }(\mathcal{B}_{1})=4.4954$, and $\sigma _{\max }(\mathcal{B}_{2})=1$. Hence $\sigma =4.4954$, $N_{1}=1$, $D_{1}=1$, and $D_{2}=1$. Hence $N=3$. Let $\delta =0.1$, $\epsilon =100$, $\alpha _{1u}=1$. The aim is to validate the FTS condition (33) with respect to $\{t_{0}=0, \delta =0.1, \epsilon =100, \alpha _{1u}=1, \rho _{1}=0.1, \rho _{2}=0.01 \}$. Then by the FTS condition of Theorem 3.5, we obtain $T_{e}=0.35$.

5 Conclusion

The problem of FTS of multiterm fractional nonlinear and integro-differential system between $0<\alpha _{2}\leq 1<\alpha _{1}\leq 2$ with multistate time delay is emphasized in this work. For this, we obtained new conditions that guarantee the FTS of both given systems by means of generalized Gronwall inequality. The importance and efficacy of our results are demonstrated by numerical examples. Furthermore, this work can be also extended to stochastic systems with various effects, like impulses, various delay situations, and so on, which makes the results more significant, and they will be considered in our future work.