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1 Introduction

Fractional calculus is a dynamic mathematical argument and suitable for analyzing various problems in evolving applied mathematical research in dealing with many real- world applications. For more than a decade, many researchers have paid attention on fractional differential equations. The study of fractional differential equation consists of key approaches to examine differential equations including fractional derivatives of unknown function. The fractional derivatives appear as attractive and powerful modeling tools in variety of areas such as bioengineering, electrical networks, signal processing, viscoelastic materials and many other physical phenomena [17, 18, 21]. In recent years, higher-order fractional systems have been widely and efficiently studied because its potential to model real time problems with more accuracy and its occurrence in control problems [16, 19]. Damping is a force within or beyond an oscillatory system that has the effect of restricting reducing or preventing its oscillations. The fractional oscillator can be modeled by establishing the fractional time derivative in standard harmonic oscillator, which explains the physical occurrence based on the fractional time evolution notion. Specifically, in the field of mechanics, fractional damping may occur towards the modeling of mechanical systems with viscoelastic components. It should be pointed out that the viscoelastic behavior of complex materials in many real time practices has been well characterized by fractional-order components; see [2, 8, 24, 29, 30].

Controllability is an essential aspect of control theory, and it acts a significant role in many control problems. The study of controllability is to verify the presence of a control function that drives the control system from its initial state to a final state in a specific time. More works for the controllability problems have been discussed in recent research; see [9, 20, 25, 27] and references therein. In recent years, the controllability of fractional damped systems has attracted much attention to researchers [11, 14, 28]. On the other hand, the stochastic analysis has gained significance and attractiveness based on its applications in wide-ranging areas of applied mathematics and engineering [4]. The research discussing the uniqueness, existence and stability of several stochastic differential equations gain more interests; see [1, 6, 12, 15] and the references therein. Recently, there has been a very important progress in the study of controllability of stochastic differential systems [13, 22, 26].

Stochastic process or noise is inevitable to model the time evolution of dynamical systems, which are related to random influences. Consequently, it is of intense importance to include the stochastic effects into the analysis of fractional-order systems. Many works have been done concerning the stochastic differential equations involving fractional derivatives in the recent years for their importance in applied sciences. Sun et al. [23] examined the controllability problem for neutral stochastic fractional integro–differential systems involving infinite delay. Guendouzi et al. [10] obtained the controllability concepts for the fractional stochastic dynamical systems involving multiple delays by means of Banach fixed point theorem. Recently, Cui and Yan [3] explored the controllability result for neutral stochastic evolution systems involving fractional Brownian motion. In [7], the authors obtained the controllability problem for fractional stochastic evolution systems involving nonlocal conditions and noncompact semigroups by means of fixed point theory. However, up to now, the controllability concept of higher-order fractional stochastic systems with damping properties and multiple delays has not been considered in the literature. Thus, this topic is an interesting one and essential to analyze it. The analysis includes the contributions, which are stated as follows.

• Most of the earlier investigations on fractional systems have been discussed with single delay. Consequently, it is essential to pay consideration to the analysis of fractional damped stochastic systems with multiple delays.

• Compared with several previous analyses, controllability of higher-order fractional stochastic system with damping effects and multiple delays is firstly presented for designing more general fractional-order model.

• The linear system of higher-order fractional damped stochastic dynamical system involving multiple delays is considered to investigate the controllability concept by utilizing Grammian matrix, it can be expressed in terms of Mittag-Leffler function.

• Further, Burkholder–Davis–Gundy’s inequality and fixed point theorem are utilized to derive the sufficient conditions for the nonlinear higher-order fractional damped system involving multiple delays.

Finally, to explain the efficiency and applicability of controllability criteria clearly, we provide two examples. A brief viewpoint on how the obtained results can be extended will be presented in the conclusion section.

2 Preliminaries

Assume the complete probability space involving filtration generated by the Wiener -dimensional process with probability measure on . Let and , the symbol represents differential operator. denotes the -dimensional Euclidean space . The state variable denoted in the Hilbert space is equipped with , where symbolizes the expectation w.r.t measure . The continuous map is defined from into satisfying Now we recall several important basic concepts.

Definition 1. Fractional derivative with Caputo sense of order for a function is stated as

The Laplace Transform (LT) of fractional derivative with Caputo sense is

Definition 2. The Mittag-Leffler function involving is stated as

The Mittag-Leffler function involving is stated as

The LT of is

For , we have

Lemma 1[Burkholder–Davis–Gundy’s inequality]. (See [4, 5].) For any and for arbitrary -valued predictable process , one has

where

Let the Cauchy fractional problem

with and

Here is a continuous function, and is a matrix. Applying LT to (1), we get

Applying inverse LT to the above equation, then utilizing LT of Mittag-Leffler function and convolution operator, we obtain

3 Controllability result for linear system

Consider the linear damped fractional stochastic system involving multiple delays of the form

where and . represents a state variable, are constant matrices, denotes a control input, are constant delays, and represents the initial control function represents -dimensional Wiener process involving generated by , and is a continuous function.

The solution of fractional system (2)–(4) takes the form

For

For

Controllability Grammian matrix is as follows:

Definition 3. System (2)–(4) is known as controllable on if there exists a control for every . Then the solution of system (2)–(4) satisfies

Theorem 1.The linear fractional system (2)-(4) is controllable on if and only if the Grammian matrix

is nonsingular.

Proof. Assume that is nonsingular. For every and , we can take the following input function:

where

for

for

for

At , the solution of system (2)–(4) can be written in the following form:

Formula

Therefore, system (2)–(4) is controllable on .

On the other hand, assume that system (2)–(4) is controllable, but the matrix is singular. Then there exists a vector such that

Hence

and

for .

Since system (2)–(4) is controllable, it can be driven from the initial points to the final point So there exists a control that drives the initial state to

Thus

Then, taking into account that

and

tend to zero, it follows that This implies the contradiction to Hence the matrix is nonsingular.

4Controllability result for nonlinear system

In this section, we analyze the controllability criteria of nonlinear fractional damped stochastic dynamical system (5)–(7) based on contraction mapping principle. Consider the nonlinear damped fractional stochastic system involving multiple delays of the form

where and and are defined as in previous section,are constant delays, and Then the solution of system (5)–(7) is defined as

and

for

for

for

We impose the following assumptions.

(H1) The linear damped fractional stochastic system involving multiple delays (2)–(4) is controllable on .

(H2) There exist the constants such that the continuous functions and satisfy the following:

(H3) For every and , there exist constants such that the functions and satisfy the following Lipschitz form:

For transience, we present the following representations:

Theorem 2. Assume that (H1)-(H3) hold, then the nonlinear fractional system (5)-(7) is controllable on .

Proof. Define an operator as follows:

where the control function is defined as in (9).

By Theorem 1, the control (9) transfers (8) from the initial state to the final state , provided that the operators has a fixed point in . So, if the operator has a fixed point, then system (5)–(7) is controllable. As mentioned before, to prove the controllability of system (5)–(7), it is enough to show that has a fixed point in . To do this, we can employ the contraction mapping principle. In the following, we will divide the proof into two steps.

Based on contraction mapping principle, we shall prove that maps into itself. By Eq. (11), we have

Using Hölder inequality, Burkholder–Davis–Gundy’s inequality (here ) and (10), we have the following estimates:

Then

Here is a constant, which gives that maps into itself.

Next, for any , we shall prove that is a contraction mapping on ,

Hence, if

then is a contraction mapping on . Now the Banach contraction fixed point theorem guarantees that has a unique fixed point. Therefore, the solution of system (5)–(7) is that given by (8), and we can see that Moreover, the control drives the state of system (5)–(7) from to final state on . Consequently, system (5)–(7) is controllable on .

5 Examples

Example 1. Consider the following linear damped fractional stochastic system involving multiple delays:

where and

By Theorem 1, the Grammian matrix defined as

The Mittag-Leffler function is given by

where

Similarly,

where

where

Thus, the matrix is obtained by

From the above, we have shown that is a nonsingular matrix. Thus, we can ensure that system (12) is controllable on .

Example 2. Consider the following nonlinear damped fractional stochastic system involving multiple delays:

where and and are as above,

Since the corresponding linear fractional system is controllable by Example and satisfy the assumptions of Theorem 2. Based on that, we conclude that system (13) is controllable on .

6 Conclusion

In this paper, we have analyzed the higher-order fractional damped stochastic system involving multiple delays in both linear and nonlinear cases. Based on controllability Grammian matrix, controllability results for the considered linear damped fractional stochastic system have been attained under suitable assumptions. Some sufficient conditions, which ensure the controllability of nonlinear damped fractional stochastic system containing multiple delays in control have been derived by applying the fixed point technique. Examples were provided to verify the established criteria. The proposed technique could be implemented to other type of fractional-order dynamical systems. An interesting extension would be to study the controllability concept for the fractional damped nonlinear equation involving time-varying delay or fractional damped stochastic system with various delay effects. This area will be the future focus of our research.

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