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

Abstract

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.

1. Introduction

In this paper, we consider the controllability of linear fractional differential systems with state delay and impulses as follows: 𝐷^𝛼𝑥(𝑡)(𝑡)()(𝑡),[]⧵𝑡=𝐴𝑥+𝐵𝑥𝑡−𝜏+𝐶𝑢𝑡∈0,𝑇₁,𝑡₂,…,𝑡_𝑘,Δ𝑥𝑡_𝑖=𝑥𝑡⁺_𝑖−𝑥𝑡⁻_𝑖=𝐼_𝑖𝑥𝑡_𝑖𝑥(𝑡)(𝑡)[],,𝑖=1,2,…,𝑘,=𝜑,𝑡∈−𝜏,0 (1) where 𝐷^𝛼𝑥(𝑡) denotes an 𝛼 order Caputo’s fractional derivative of 𝑥(𝑡), 0<𝛼<1, 𝐴, 𝐵, and 𝐶 are the known constant matrices and satisfy 𝐴,𝐵∈ℝ^𝑛×𝑛, 𝐶∈ℝ^𝑛×𝑚, 𝜏 is a positive constant, 𝑥∈ℝ^𝑛 is the state variable, 𝑢∈ℝ^𝑚 is the control input, 𝜑∈𝐂([−𝜏,0],ℝ^𝑛) is the initial state function, where 𝐂([−𝜏,0],ℝ^𝑛) denotes the space of all continuous functions mapping the interval [−𝜏,0] into ℝ^𝑛, 𝐼_𝑖∶ℝ^𝑛⟶ℝ^𝑛 is continuous for 𝑖=1,2,…,𝑘, and 𝑥𝑡⁺_𝑖=lim_𝜀→0+𝑥𝑡_𝑖+𝜀,𝑥𝑡⁻_𝑖=lim_𝜀→0−𝑥𝑡_𝑖+𝜀 (2) represent the right and left limits of 𝑥(𝑡) at 𝑡=𝑡_𝑖 and the discontinuous points 𝑡₁<𝑡₂<⋯<𝑡_𝑖<⋯<𝑡_𝑘, (3) where 0=𝑡₀<𝜏<𝑡₁, 𝑡_𝑘<𝑡_𝑘+1=𝑇<+∞, and 𝑥(𝑡_𝑖)=𝑥(𝑡⁻_𝑖) which implies that the solution of system (1) is left continuous at 𝑡_𝑖.

The subject of fractional differential equations is gaining much importance and attention (see [1–11] 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 [12–15].

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 Baĭnov 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 [19–36] 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 [21–24], 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 [27–29]. 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 [30–36]. 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 [31–34], 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.

2. Preliminaries

Throughout this paper, denote by 𝐂_𝑝([0,𝑇],ℝ^𝑛) the space of all piecewise left continuous functions mapping the interval [0,𝑇] into ℝ^𝑛.

Let us recall some definitions and preliminary facts. For more details, one can see [1–4].

Definition 1. The Riemann-Liouville's fractional integral of order 𝛼>0 with the lower limit zero for a function 𝑓∶𝑅⁺→𝑅^𝑛 is defined as 𝐷^−𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠,𝑡>0, (4) provided the right side is pointwise defined on [0,+∞), where Γ(⋅) is the Gamma function.

Definition 2. The Caputo's fractional derivative of order 𝛼 for a function 𝑓∶𝑅⁺→𝑅^𝑛 is defined as 𝐷^𝛼𝑓(𝑡)=1Γ(𝑚−𝛼+1)^𝑡₀(𝑡−𝑠)^𝑚−𝛼𝑓^(𝑚+1)(𝑠)𝑑𝑠,0≤𝑚≤𝛼<𝑚+1. (5)

Definition 3. The Mittag-Leffler function in two parameters is defined as 𝐸_𝛼,𝛽(𝑧)=^+∞_𝑘=0𝑧^𝑘Γ,𝑘𝛼+𝛽 (6) where 𝛼>0, 𝛽>0, and 𝑧∈ℂ, ℂ denotes the complex plane.

Definition 4. The Laplace transform of a function 𝑓(𝑡) is defined as 𝐹(𝑠)=£𝑓(𝑡)=₀^+∞𝑒^−𝑠𝑡𝑓(𝑡)𝑑𝑡,𝑠∈ℂ, (7) where 𝑓(𝑡) is 𝑛-dimensional vector-valued function.

Remark 5. If 𝛼∈(0,1), then £𝐷^𝛼𝑓(𝑡)=𝑠^𝛼£𝑓(𝑡)−𝑠^𝛼−1𝑓(0). (8)

Lemma 6 (see [2]). Let ℂ be complex plane, for any 𝛼>0, 𝛽>0, and 𝐴∈ℂ^𝑛×𝑛; then £𝑡^𝛽−1𝐸_𝛼,𝛽𝐴𝑡^𝛼=𝑠^𝛼−𝛽𝑠^𝛼𝐼−𝐴⁻¹,ℜ(𝑠)>‖𝐴‖^1/𝛼 (9) holds, where ℜ(𝑠) represents the real part of the complex number 𝑠 and 𝐼 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: 𝐷^𝛼𝑥(𝑡)(𝑡)()(𝑡)[],𝑥(𝑡)(𝑡)[].=𝐴𝑥+𝐵𝑥𝑡−𝜏+𝑓,𝑡∈0,𝑇=𝜑,𝑡∈−𝜏,0 (10)

Lemma 7. Let 0<𝛼<1; if 𝑓∶[0,𝑇]→ℝ^𝑛 is continuous and exponentially bounded, then the solution of system (10) can be represented as 𝑥(𝑡)=𝜑(0)+^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×(0)()(𝑠[]𝐴𝜑+𝐵𝑥𝑠−𝜏+𝑓)𝑑𝑠,𝑡∈0,𝑇 (11) and 𝑥(𝑡)=𝜑(𝑡), 𝑡∈[−𝜏,0].

Proof. Applying the method of steps which has been presented in [12], then there exists a unique solution to system (10).
For 𝑡∈[0,𝑇], taking the Laplace transform with respect to 𝑡 in both sides of system (10), we obtain 𝑠^𝛼£[𝑥(𝑡)]−𝑠^𝛼−1𝜑(0)=𝐴£[𝑥(𝑡)]+£𝐵𝑥(𝑡−𝜏)+𝑓(𝑡). (12) Then (12) can be written as £[𝑥(𝑡)]=𝑠^𝛼𝐼−𝐴⁻¹𝑠^𝛼−1𝜑(0)+𝑠^𝛼𝐼−𝐴⁻¹£𝐵𝑥(𝑡−𝜏)+𝑓(𝑡). (13) From Definition 4 and Lemma 6, then (13) is equivalent to £[𝑥(𝑡)]=𝑠^𝛼𝐼−𝐴⁻¹𝑠^𝛼−1𝜑(0)+𝑠^𝛼𝐼−𝐴⁻¹£()(𝑡=𝑠𝐵𝑥𝑡−𝜏+𝑓)_𝛼𝐼−𝐴⁻¹𝑠^𝛼£𝜑(0+𝑠_𝛼)𝐼−𝐴⁻¹£()(𝑡𝜑(0+𝑠𝐵𝑥𝑡−𝜏+𝑓)=£)^𝛼𝐼−𝐴⁻¹(0)()(𝑡𝜑(0𝑡×£𝐴𝜑+𝐵𝑥𝑡−𝜏+𝑓)=£)+£^𝛼−1𝐸_𝛼,𝛼𝐴𝑡^𝛼×£𝐴𝜑(0)+𝐵𝑥(𝑡−𝜏)+𝑓(𝑡). (14) The convolution theorem of the Laplace transform applied to (14) yields the form £[𝑥(𝑡)]=£𝜑(0)+£^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)+𝑓(𝑠)𝑑𝑠. (15) Applying the inverse Laplace transform, we obtain 𝑥(𝑡)=𝜑(0)+^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×(0)()(𝑠[].𝐴𝜑+𝐵𝑥𝑠−𝜏+𝑓)𝑑𝑠,𝑡∈0,𝑇 (16) Therefore, we have the stated result.

Lemma 8. Let 0<𝛼<1 and 𝑢∈𝐂_𝑝([0,𝑇],ℝ^𝑚); then state response of system (1) can be represented as follows.
For 𝑡∈[−𝜏,0], 𝑥(𝑡)=𝜑(𝑡). (17) For 𝑡∈[0,𝑡₁], 𝑥(𝑡)=𝜑(0)+^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)+𝐶𝑢(𝑠)𝑑𝑠. (18) For 𝑡∈(𝑡₁,𝑡₂], 𝑥(𝑡)=𝜑(0)+𝐼₁𝑥𝑡⁻₁+^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)+𝐶𝑢(𝑠)𝑑𝑠. (19) For 𝑡∈(𝑡_𝑖,𝑡_𝑖+1], 𝑖=1,2,…,𝑘, 𝑥(𝑡)=𝜑(0)+^𝑖_𝑗=1𝐼_𝑗𝑥𝑡⁻_𝑗+^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)+𝐶𝑢(𝑠)𝑑𝑠. (20)

Proof. If 𝑡∈[−𝜏,0], then the conclusion obviously holds. If 𝑡∈[0,𝑡₁], then, from Lemma 7, 𝑥(𝑡)=𝜑(0)+^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×(0)()(𝑠𝑥𝑡𝐴𝜑+𝐵𝑥𝑠−𝜏+𝐶𝑢)𝑑𝑠,₁=𝜑(0)+^𝑡1₀𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼×𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)+𝐶𝑢(𝑠)𝑑𝑠. (21) If 𝑡∈(𝑡₁,𝑡₂], applying the idea used in [18], we have 𝑥(𝑡)=𝑥𝑡⁺₁−^𝑡1₀𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼×(0)()(𝑠+𝐴𝜑+𝐵𝑥𝑠−𝜏+𝐶𝑢)𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×(0)()(𝑠𝑡𝐴𝜑+𝐵𝑥𝑠−𝜏+𝐶𝑢)𝑑𝑠=𝑥⁻₁+𝐼₁𝑥𝑡⁻₁−^𝑡1₀𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼×(0)()(𝑠+𝐴𝜑+𝐵𝑥𝑠−𝜏+𝐶𝑢)𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×(0)()(𝑠(0)𝐴𝜑+𝐵𝑥𝑠−𝜏+𝐶𝑢)𝑑𝑠=𝜑+𝐼₁𝑥𝑡⁻₁+^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)+𝐶𝑢(𝑠)𝑑𝑠. (22) If 𝑡∈(𝑡₂,𝑡₃], then 𝑥(𝑡)=𝑥𝑡⁺₂−^𝑡2₀𝑡₂−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₂−𝑠^𝛼×(0)()(𝑠+𝐴𝜑+𝐵𝑥𝑠−𝜏+𝐶𝑢)𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×(0)()(𝑠𝑡𝐴𝜑+𝐵𝑥𝑠−𝜏+𝐶𝑢)𝑑𝑠=𝑥⁻₂+𝐼₂𝑥𝑡⁻₂−^𝑡2₀𝑡₂−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₂−𝑠^𝛼×(0)()(𝑠+𝐴𝜑+𝐵𝑥𝑠−𝜏+𝐶𝑢)𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×(0)()(𝑠(0)+𝐴𝜑+𝐵𝑥𝑠−𝜏+𝐶𝑢)𝑑𝑠=𝜑²_𝑗=1𝐼_𝑗𝑥𝑡⁻_𝑗+^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)+𝐶𝑢(𝑠)𝑑𝑠. (23) If 𝑡∈(𝑡_𝑖,𝑡_𝑖+1](𝑖=1,2,…,𝑘), then the same argument implies the following expression: 𝑥(𝑡)=𝜑(0)+^𝑖_𝑗=1𝐼_𝑗𝑥𝑡⁻_𝑗+^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)+𝐶𝑢(𝑠)𝑑𝑠. (24) Thus, the proof is completed.

3. 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.

Definition 9. System (1) is called controllable on [0,𝜔](𝜔∈(0,𝑇]); for any initial function 𝜑∈𝐂([−𝜏,0],ℝ^𝑛) and any state 𝑥_𝜔∈ℝ^𝑛, there exists a control input 𝑢(𝑡)∈𝐂_𝑝([0,𝜔],ℝ^𝑚), such that the corresponding solution of (1) satisfies 𝑥(𝜔)=𝑥_𝜔.

Theorem 10. System (1) is controllable on [0,𝜔] if and only if the Gramian matrix 𝑊_𝑐[0,𝜔]=^𝜔₀(𝜔−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝑠)^𝛼×𝐶𝐶^∗𝐸_𝛼,𝛼𝐴^∗(𝜔−𝑠)^𝛼𝑑𝑠 (25) is nonsingular for some 𝜔∈[0,𝑇], where 𝐸_𝛼,𝛼(⋅) is the Mittag-Leffler function and ∗ denotes the matrix transpose.

Proof. We firstly prove sufficiency of Theorem 10. If 𝑊_𝑐[0,𝜔] is nonsingular, then 𝑊_𝑐⁻¹[0,𝜔] is well defined. For any initial state 𝜑∈𝐂([−𝜏,0],ℝ^𝑛), when 𝜔∈[0,𝑡₁], we take the control function as 𝑢(𝑡)=𝐶^∗𝐸_𝛼,𝛼𝐴^∗(𝜔−𝑡)^𝛼𝑊_𝑐⁻¹[]×𝑥0,𝜔_𝜔−𝜑(0)−^𝜔₀(𝜔−𝜃)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝜃)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝜃−𝜏)𝑑𝜃. (26) Substituting 𝑡=𝜔 in (18) and inserting (26) yield 𝑥(𝜔)=𝜑(0)+^𝜔₀(𝜔−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝑠)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)+𝐶𝐶^∗𝐸_𝛼,𝛼𝐴^∗(𝜔−𝑠)^𝛼×𝑊_𝑐⁻¹[]×𝑥0,𝜔_𝜔−𝜑(0)−^𝜔₀(𝜔−𝜃)^𝛼−1×𝐸_𝛼,𝛼𝐴(𝜔−𝜃)^𝛼×(0)((0)+𝐴𝜑+𝐵𝑥𝜃−𝜏))𝑑𝜃𝑑𝑠=𝜑^𝜔₀(𝜔−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝑠)^𝛼×(0)(+𝑥𝐴𝜑+𝐵𝑥𝑠−𝜏)𝑑𝑠_𝜔−𝜑(0)−^𝜔₀(𝜔−𝜃)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝜃)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝜃−𝜏)𝑑𝜃=𝑥_𝜔. (27) Thus system (1) is controllable on [0,𝜔], 𝜔∈[0,𝑡₁].
For 𝜔∈(𝑡₁,𝑡₂], we take the control function as 𝑢(𝑡)=𝐶^∗𝐸_𝛼,𝛼𝐴^∗(𝜔−𝑡)^𝛼𝑊_𝑐⁻¹[]×𝑥0,𝜔_𝜔−𝜑(0)−𝐼₁𝑥𝑡⁻₁−^𝜔₀(𝜔−𝜃)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝜃)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝜃−𝜏)𝑑𝜃. (28) Substituting 𝑡=𝜔 in (19) and inserting (28) yield 𝑥(𝜔)=𝜑(0)+𝐼₁𝑥𝑡⁻₁+^𝜔₀(𝜔−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝑠)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)+𝐶𝐶^∗𝐸_𝛼,𝛼𝐴^∗(𝜔−𝑠)^𝛼𝑊_𝑐⁻¹[]×𝑥0,𝜔_𝜔−𝜑(0)−𝐼₁𝑥𝑡⁻₁−^𝜔₀(𝜔−𝜃)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝜃)^𝛼×(0)()(0)𝐴𝜑+𝐵𝑥𝜃−𝜏𝑑𝜃𝑑𝑠=𝜑+𝐼₁𝑥𝑡⁻₁+^𝜔₀(𝜔−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝑠)^𝛼×(0)(+𝑥𝐴𝜑+𝐵𝑥𝑠−𝜏)𝑑𝑠_𝜔−𝜑(0)−𝐼₁𝑥𝑡⁻₁−^𝜔₀(𝜔−𝜃)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝜃)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝜃−𝜏)𝑑𝜃=𝑥_𝜔. (29) Thus system (1) is controllable on [0,𝜔], 𝜔∈(𝑡₁,𝑡₂].
For 𝜔∈(𝑡_𝑖,𝑡_𝑖+1], 𝑖=1,2,…,𝑘, we take the control function as 𝑢(𝑡)=𝐶^∗𝐸_𝛼,𝛼𝐴^∗(𝜔−𝑡)^𝛼𝑊_𝑐⁻¹[]×𝑥0,𝜔_𝜔−𝜑(0)−^𝑖_𝑗=1𝐼_𝑗𝑥𝑡⁻_𝑗−^𝜔₀(𝜔−𝜃)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝜃)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝜃−𝜏)𝑑𝜃. (30) Substituting 𝑡=𝜔 in (20) and inserting (30), then the same argument implies 𝑥(𝜔)=𝑥_𝜔. Therefore system (1) is controllable on [0,𝜔].
Next, we prove necessity of Theorem 10. Suppose 𝑊_𝑐[0,𝜔] is singular, without loss of generality; for 𝜔∈(𝑡_𝑖,𝑡_𝑖+1], 𝑖=1,2,…,𝑘, there exists a nonzero vector 𝑧₀ such that 𝑧^∗₀𝑊_𝑐[0,𝜔]𝑧₀=0. (31) That is, ^𝜔₀𝑧^∗₀(𝜔−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝑠)^𝛼×𝐶𝐶^∗𝐸_𝛼,𝛼𝐴^∗(𝜔−𝑠)^𝛼𝑧₀𝑑𝑠=0. (32) Then it follows 𝑧^∗₀𝐸_𝛼,𝛼𝐴(𝜔−𝑠)^𝛼𝐶=0 (33) on 𝑠∈[0,𝜔]. Since system (1) is controllable, there exist control inputs 𝑢₁(𝑡) and 𝑢₂(𝑡) such that 𝑥(𝜔)=𝜑(0)+^𝑖_𝑗=1𝐼_𝑗𝑥𝑡⁻_𝑗+^𝜔₀(𝜔−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝑠)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)+𝐶𝑢₁(𝑠𝑧)𝑑𝑠=0,₀=𝜑(0)+^𝑖_𝑗=1𝐼_𝑗𝑥𝑡⁻_𝑗+^𝜔₀(𝜔−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝑠)^𝛼×𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)+𝐶𝑢₂(𝑠)𝑑𝑠. (34)(35) Combining (34) and (35) yields 𝑧₀−^𝜔₀(𝜔−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝑠)^𝛼𝐶𝑢₂(𝑠)−𝑢₁(𝑠)𝑑𝑠=0. (36) Multiplying 𝑧^∗₀ on both sides of (36), we get 𝑧^∗₀𝑧₀−^𝜔₀(𝜔−𝑠)^𝛼−1𝑧^∗₀𝐸_𝛼,𝛼𝐴(𝜔−𝑠)^𝛼×𝐶𝑢₂(𝑠)−𝑢₁(𝑠)𝑑𝑠=0. (37) According to 𝑧^∗₀𝐸_𝛼,𝛼[𝐴(𝜔−𝑠)^𝛼]𝐶=0, we have 𝑧^∗₀𝑧₀=0. Thus 𝑧₀=0. This contradiction therefore completes the proof.

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].

Theorem 11. System (1) is controllable on [0,𝜔] if and only if rank𝐶∣𝐴𝐶∣⋯∣𝐴^𝑛−1𝐶=𝑛. (38)

Proof. According to Cayley-Hamilton theorem, 𝑡^𝛼−1𝐸_𝛼,𝛼(𝐴𝑡^𝛼) can be represented as 𝑡^𝛼−1𝐸_𝛼,𝛼𝐴𝑡^𝛼=^+∞_𝑘=0𝑡^{𝑘𝛼+𝛼−1}Γ()𝐴𝑘𝛼+𝛼^𝑘=^𝑛−1_𝑘=0𝐺_𝑘(𝑡)𝐴^𝑘. (39) For 𝜔∈[0,𝑡₁], 𝑥(𝜔)=𝜑(0)+^𝜔₀(𝜔−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝑠)^𝛼×(0)()(𝑠(0)+𝐴𝜑+𝐵𝑥𝑠−𝜏+𝐶𝑢)𝑑𝑠=𝜑^𝑛−1_𝑘=0^𝜔₀𝐺_𝑘(𝜔−𝑠)𝐴^𝑘×𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)+𝐶𝑢(𝑠)𝑑𝑠. (40) Let Ψ=𝜑(0)+^𝑛−1_𝑘=0^𝜔₀𝐺_𝑘(𝜔−𝑠)𝐴^𝑘𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)𝑑𝑠. (41) Then combining (40) with (41) yields 𝑥(𝜔)−Ψ=^𝑛−1_𝑘=0𝐴^𝑘𝐶^𝜔₀𝐺_𝑘()𝑢(𝑠)=𝜔−𝑠𝑑𝑠𝐶∣𝐴𝐶∣⋯∣𝐴^𝑛−1𝐶𝑑₀𝑑₁⋮𝑑_𝑛−1, (42) where 𝑑_𝑘=∫^𝜔₀𝐺_𝑘(𝜔−𝑠)𝑢(𝑠)𝑑𝑠, 𝑘=0,1,…,𝑛−1. Note that, for arbitrary 𝜑∈𝐂([−𝜏,0],ℝ^𝑛) and 𝑥(𝜔)∈ℝ^𝑛, the sufficient and necessary condition to have a control input 𝑢(𝑡) satisfying (42) is that rank𝐶∣𝐴𝐶∣⋯∣𝐴^𝑛−1𝐶=𝑛. (43)
For 𝜔∈(𝑡_𝑖,𝑡_𝑖+1],𝑖=1,2,…,𝑘, 𝑥(𝜔)=𝜑(0)+^𝑖_𝑗=1𝐼_𝑗𝑥𝑡⁻_𝑗+^𝜔₀(𝜔−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝜔−𝑠)^𝛼×(0)()(𝑠(0)+𝐴𝜑+𝐵𝑥𝑠−𝜏+𝐶𝑢)𝑑𝑠=𝜑^𝑖_𝑗=1𝐼_𝑗𝑥𝑡⁻_𝑗+^𝑛−1_𝑘=0^𝜔₀𝐺_𝑘(𝜔−𝑠)𝐴^𝑘×𝐴𝜑(0)+𝐵𝑥(𝑠−𝜏)+𝐶𝑢(𝑠)𝑑𝑠. (44) Combining (41) with (44) yields 𝑥(𝜔)−Ψ−^𝑖_𝑗=1𝐼_𝑗𝑥𝑡⁻_𝑗=^𝑛−1_𝑘=0𝐴^𝑘𝐶^𝜔₀𝐺_𝑘()𝑢(𝑠)=𝜔−𝑠𝑑𝑠𝐶∣𝐴𝐶∣⋯∣𝐴^𝑛−1𝐶𝑑₀𝑑₁⋮𝑑_𝑛−1. (45) Note that, for arbitrary 𝜑∈𝐂([−𝜏,0],ℝ^𝑛) and 𝑥(𝜔)∈ℝ^𝑛, the sufficient and necessary condition to have a control input 𝑢(𝑡) satisfying (45) is that rank𝐶∣𝐴𝐶∣⋯∣𝐴^𝑛−1𝐶=𝑛. (46) Thus, the proof is completed.

Remark 12. System (1) is controllable if and only if the resolvent condition 𝜆(𝜆𝐼+𝑄_𝜔)⁻¹→0 as 𝜆→0 holds (here 𝑄_𝜔 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].

4. 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: 𝐷^1/2𝑥(𝑡)=𝑥(𝑡)+𝑥01100−1231𝑡−3+21𝑢(𝑡)[]⧵{},𝑡,𝑡∈0,41,2,3Δ𝑥_𝑖=12𝑥𝑡⁻_𝑖,𝑡_𝑖𝑥(𝑡)=𝑖,𝑖=1,2,3,=𝑒^𝑡−1,𝑡∈3,0. (47) Now, we apply Theorem 10 to prove that system (47) is controllable on [0,4]. Let us take 1𝛼=221,𝐴=0100,𝐵=−1231,𝐶=. (48) By computation, we have 𝐶𝐶^∗=21=𝐸214221_1/2,1/2,𝐴(4−𝑠)^1/2=¹_𝑘=0𝐴^𝑘(4−𝑠)^𝑘/2Γ()=1𝑘/2+1/2√𝜋()14−𝑠201√𝜋,𝐸_1/2,1/2𝐴^∗(4−𝑠)^1/2=¹_𝑘=0𝐴^𝑘(4−𝑠)^𝑘/2Γ()=1𝑘/2+1/2√𝜋0(4−𝑠)^1/21√𝜋. (49)(50)(51) Substituting 𝜔=4 in (25) and combining (25) with (49)–(51) yield 𝑊_𝑐[0,4]=⁴₀(4−𝑠)^−1/2𝐸_1/2,1/2𝐴(4−𝑠)^1/2×𝐶𝐶^∗𝐸_1/2,1/2𝐴^∗(4−𝑠)^1/2=𝑑𝑠16𝜋+16√𝜋+1638𝜋+4√𝜋8𝜋+4√𝜋4𝜋. (52) Obviously, 𝑊_𝑐[0,4] is nonsingular. Thus by Theorem 10, system (47) is controllable on [0,4].

Example 14. Consider the controllability of linear fractional differential systems with state delay and impulses as follows: 𝐷^1/3𝑥(𝑡)=𝑥(𝑡)+𝑥𝜋010001−2−4−31012112−10𝑡−3+𝑢(𝑡),51001−11𝑡∈0,2𝜋⧵𝜋23,𝜋,2,𝑡𝜋,2𝜋Δ𝑥_𝑖=13𝑥𝑡⁻_𝑖,𝑡_𝑖=𝑖𝜋2𝑥(𝑡)−𝜋,𝑖=1,2,3,4,=sin𝑡,𝑡∈3,0. (53) Now, we apply Theorem 11 to prove that system (53) is controllable on [0,(5/2)𝜋]. Let us take 1𝛼=3,.,𝐴=010001−2−4−3𝐵=1012112−10,𝐶=1001−11 (54) Then one can obtain rank𝐶∣𝐴𝐶∣⋯∣𝐴^𝑛−1𝐶=rank100⋆⋆⋆01−1⋆⋆⋆002⋆⋆⋆=3. (55) Thus by Theorem 11, system (53) is controllable on [0,(5/2)𝜋].

5. 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.

Acknowledgments

The authors are very grateful to the Associate Editor, Professor Francisco J. Marcellán, and the two anonymous reviewers for their helpful and valuable comments and suggestions, which significantly contributed to improving the quality of this paper. This work is jointly supported by the National Natural Science Foundation of China under Grant nos. 61272530 and 11072059, the Natural Science Foundation of Jiangsu Province of China under Grant no. BK2012741, and the Key Program of Educational Commission of Anhui Province of China under Grant no. KJ2011A197.