Controllability Criteria for Linear Fractional Differential Systems with State Delay and Impulses

This paper is concerned with the controllability of linear fractional differential systems with delay in state and impulses.The factors of such systems including fractional derivative, impulses, and delay are taken into account synchronously. The expression of state response for such systems is derived, and the sufficient and necessary conditions of controllability criteria are established. Both the proposed criteria and illustrative examples show that the controllability property of the linear systems is dependent neither on the order of fractional derivative, on delay nor on impulses.

The subject of fractional differential equations is gaining much importance and attention (see [1][2][3][4][5][6][7][8][9][10][11] and references therein).Fractional differential equations have been proved to be an excellent tool in the modelling of many phenomena in various fields of engineering, physics, and economics.In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations.At the same time, time delay is one of the inevitable problems in practical engineering applications, which has an important effect on the stability and performance of system.In the last few years, the results with regard to the fractional delay differential systems have been presented in [12][13][14][15].
Although most dynamical systems are analyzed in either the continuous or discrete-time domain, many real systems in physics, chemistry, biology, engineering, and information science may experience abrupt changes as certain instants during the continuous dynamical processes.This kind of impulsive behaviors can be modeled by impulsive systems.The basic theory of impulsive differential equations can be found in the monographs of Baȋnov and Simeonov [16], Benchohra et al. [17], and the paper of Fečkan et al. [18].
On the other hand, controllability is the most fundamental concept in modern control theory, which has close connections to pole assignment, structural decomposition, quadratic optimal control, and so forth.Some important results concerning the control theory for various kinds of systems have been obtained in [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] and references therein.Kalman et al. [19] have investigated the controllability of linear dynamical systems based on the algebraic approach.Wonham and Morse [20] have discussed the pole assignment problems of linear systems based on the geometric approach.In [21][22][23][24], the authors have discussed the controllability of integer derivative delay systems.In [25,26], the controllability of the descriptor (singular) systems has been considered.Impulsive control systems with integer derivative have been investigated in [27][28][29].For integer derivative control systems with state delay and impulses, Zhang et al. [27] have derived the sufficient conditions for the controllability based on the fixed point theorem.It is worth pointing out that notable contributions have been made to fractional control systems in [30][31][32][33][34][35][36].The different techniques have been developed to investigate the control problems of fractional differential systems, such as fractional sliding manifold approach [30], fixed point theorems [31][32][33][34], functional analysis method [33,34], and algebraic method [35,36].To the best of our knowledge, there are no relevant reports on the controllability of fractional differential systems with state delay and impulses as treated in the current literature.In this paper, the factors of control systems including the Caputo's fractional derivative, impulses, and delay are taken into account synchronously.The purpose of this paper is to establish the sufficient and necessary conditions of controllability for system (1) based on the algebraic approach.The recent research surge in developing the theory of fractional control systems has motivated and inspired our present work.
This paper is organized as follows.In Section 2, we recall some definitions and preliminary facts, and the expression of state response for system (1) is derived.In Section 3, the sufficient and necessary conditions of controllability criteria are established.In Section 4, some examples are given to illustrate the effectiveness and applicability of controllability criteria.Finally, some concluding remarks are drawn in Section 5.

Preliminaries
Throughout this paper, denote by C  ([0, ], R  ) the space of all piecewise left continuous functions mapping the interval [0, ] into R  .
Definition 1.The Riemann-Liouville's fractional integral of order  > 0 with the lower limit zero for a function  :  + →   is defined as provided the right side is pointwise defined on [0, +∞), where Γ(⋅) is the Gamma function.
Definition 2. The Caputo's fractional derivative of order  for a function  :  + →   is defined as ( Definition 3. The Mittag-Leffler function in two parameters is defined as where  > 0,  > 0, and  ∈ C, C denotes the complex plane. Definition 4. The Laplace transform of a function () is defined as where () is -dimensional vector-valued function.
Lemma 6 (see [2]).Let C be complex plane, for any  > 0,  > 0, and  ∈ C × ; then holds, where R() represents the real part of the complex number  and  denotes the identity matrix.
In order to obtain the state response of system (1), we firstly consider the representation of solution for linear fractional delay differential systems without impulses as follows: and exponentially bounded, then the solution of system (10) can be represented as and Proof.Applying the method of steps which has been presented in [12], then there exists a unique solution to system (10).
For  ∈ [0, ], taking the Laplace transform with respect to  in both sides of system (10), we obtain Then ( 12) can be written as From Definition 4 and Lemma 6, then ( 13) is equivalent to The convolution theorem of the Laplace transform applied to (14) yields the form Applying the inverse Laplace transform, we obtain Therefore, we have the stated result.

Controllability Criteria for System (1)
In this section, we establish the sufficient and necessary conditions of controllability criteria for system (1) based on the algebraic approach.For  ∈ ( 1 ,  2 ], we take the control function as Substituting  =  in (19) and inserting (28) yield Thus system ( 1) is controllable on [0, ],  ∈ ( 1 ,  2 ].
Theorem 10 presents a geometric type criterion.By the algebraic transform and computation, we can obtain an algebraic criterion which is similar to the famous Kalman's rank condition [19].
Remark 12. System (1) is controllable if and only if the resolvent condition ( +   ) −1 → 0 as  → 0 holds (here   is the respective Gramian matrix in the nonfractional, nondelay, and nonimpulsive case) since this is equivalent to the rank condition in the finite dimensional case [19,35,36].

Illustrative Examples
In this section, we give two examples to illustrate the presented criteria.
Example 13.Consider the controllability of linear fractional differential systems with state delay and impulses as follows: Now, we apply Theorem 10 to prove that system (47) is controllable on [0, 4].Let us take By computation, we have

Conclusions
In this paper, the controllability criteria for linear fractional differential systems with delay in the state and impulses have been investigated.The sufficient and necessary conditions for the controllability of such systems have been established.Furthermore, both the proposed criteria and illustrative examples have shown that the controllability property of the linear systems is dependent neither on the order of fractional derivative, on delay nor on impulses.