Reachability and Controllability of Fractional Singular Dynamical Systems with Control Delay

Abstract

This paper is concerned with the reachability and controllability of fractional singular dynamical systems with control delay. The factors of such systems including the Caputo’s fractional derivative, control delay, and singular coefficient matrix are taken into account synchronously. The state structure of fractional singular dynamical systems with control delay is characterized by analysing the state response and reachable set. A set of sufficient and necessary conditions of controllability for such systems are established based on the algebraic approach. Moreover, an example is provided to illustrate the effectiveness and applicability of the proposed criteria.

1. Introduction

Singular systems play important roles in mathematical modeling of real-life problems arising in a wide range of applications. Depending on the area of application, these models are also called descriptor systems, semistate systems, differential-algebraic systems, or generalized state-space systems. In the past few decades, singular systems with integer-order derivative have been extensively studied due to the fact that singular systems describe practical systems better than normal ones (see [1–8] and references therein).

Fractional calculus and its applications to the sciences and engineering is a recent focus of interest to many researchers. 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. Actually, fractional differential equations are considered as alternative model to integer differential equations. Many practical systems can be represented more accurately through fractional derivative formulation. For more details on fractional calculus theory, one can see the monographs of Miller and Ross [9], Podlubny [10], Diethelm [11], and Kilbas et al. [12]. Fractional differential equations involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have been paid more and more attention in [13–18] and references therein.

On the other hand, it is well-known that the issue of controllability plays an important role in engineering and control theory, which has close connections to pole assignment, structural decomposition, quadratic optimal control, and controller design, and so on. The problem of controllability for various kinds of dynamical systems with integer derivative has been extensively studied [1, 3–5, 19–22]. In particular, Dai [1] has established the theory of integer-order linear singular control systems. Yip and Sincovec [3] and Tang and Li [4] have investigated the controllability and observability of integer-order singular dynamical systems. It is worth mentioning that notable contributions have been made to fractional control systems [23–35]. The different techniques have been applied to investigate the controllability of various fractional dynamical systems, such as algebraic method [23, 24], geometrical analysis [25], fixed point theorems [26–30], and semigroup theory [31]. Recently, Sakthivel et al. [32–35] have investigated the approximate controllability problem for various kinds of fractional dynamical systems by using fixed point techniques, which have further enriched and developed the control theory of fractional dynamical systems. The earlier studies concerning the controllability of fractional dynamical systems with control delay can be found in [25, 27, 29], which especially focused on fractional normal systems. However, it should be emphasized that the control theory of fractional singular systems is not yet sufficiently elaborated, compared to that of fractional normal systems. In this regard, it is necessary and important to study the controllability problems for fractional singular dynamical systems. To the best of our knowledge, there are no relevant reports on reachability and controllability of fractional singular dynamical systems as treated in the current literature. Motivated by this consideration, in this paper, we investigate the reachability and controllability of fractional singular dynamical systems with control delay.

In this paper, we consider the reachability and controllability of the following fractional singular dynamical systems with control delay: 𝐸^𝑐𝐷^𝛼𝑥(𝑡)(𝑡)(𝑡)()𝑢(𝑡)(𝑡)𝑥(0)=𝐴𝑥+𝐵𝑢+𝐶𝑢𝑡−𝜏,𝑡≥0,=𝜓,−𝜏≤𝑡≤0,=𝑥₀, (1) where ^𝑐𝐷^𝛼𝑥(𝑡) denotes an 𝛼 order Caputo fractional derivative of 𝑥(𝑡), and 0<𝛼≤1; 𝐸, 𝐴, 𝐵, and 𝐶 are the known constant matrices, 𝐸,𝐴∈ℝ^𝑛×𝑛, 𝐵∈ℝ^𝑛×𝑚, 𝐶∈ℝ^𝑛×𝑚, and rank(𝐸)<𝑛; 𝑥∈ℝ^𝑛 is the state variable; 𝑢∈ℝ^𝑚 is the control input; 𝜏>0 is the time control delay; and 𝜓(𝑡) is the initial control function.

The main purpose of this paper is to establish reachability and controllability criteria for system (1) based on the algebraic approach. The factors of fractional singular dynamical systems including the Caputo’s fractional derivative, control delay, and singular coefficient matrices are taken into account synchronously. As discussed by Dai [1], we transform system (1) into two subsystems by the first equivalent form (FE1), which is more convenient to characterize the state reachable set. The state response, reachable set, and sufficient and necessary conditions for the controllability are obtained, respectively. The proposed criteria are applicable to a larger class of fractional dynamical systems. Therefore, we extend the known results [1, 23, 25] to a more general case.

This paper is organized as follows. In the next section, we recall some definitions and preliminary facts used in the paper. In Section 3, the state structure of system (1) is characterized by analysing the state response and reachable set. In Section 4, the necessary and sufficient conditions of controllability for two subsystems and system (1) are derived, respectively. In Section 5, an example is provided to illustrate the effectiveness and applicability of the proposed criteria. Finally, some concluding remarks are drawn in Section 6.

2. Preliminaries

In this section, we first recall some definitions of fractional calculus and preliminary facts. For more details, one can see [9–12]. Next, the first equivalent form (FE1) of system (1) is given.

(a) 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, (2) provided that the right side is pointwise defined on [0,+∞), where Γ(⋅) is the Gamma function.

(b) The Caputo’s fractional derivative of order 𝛼 for a function 𝑓∶𝑅⁺→𝑅^𝑛 is defined as ^𝑐𝐷^𝛼𝑓(𝑡)=1Γ(𝑚−𝛼+1)^𝑡₀(𝑡−𝑠)^𝑚−𝛼𝑓^(𝑚+1)(𝑠)𝑑𝑠,0≤𝑚≤𝛼<𝑚+1. (3) For 0<𝛼≤1, 𝐷^−𝛼𝑐𝐷^𝛼𝑓(𝑡)=𝑓(𝑡)−𝑓(0). (4)

(c) The Mittag-Leffler function in two parameters is defined as 𝐸_𝛼,𝛽(𝑧)=^+∞_𝑘=0𝑧^𝑘Γ,𝑘𝛼+𝛽 (5) where 𝛼>0, 𝛽>0, and 𝑧∈ℂ, ℂ denotes the complex plane. In particular, for 𝛽=1, 𝐸_𝛼,1(𝑧)=𝐸_𝛼(𝑧)=^+∞_𝑘=0𝑧^𝑘Γ(𝑘𝛼+1) (6) has the interesting property ^𝑐𝐷^𝛼𝐸_𝛼𝜆𝑧^𝛼=𝜆𝐸_𝛼𝜆𝑧^𝛼,𝜆,𝑧∈ℂ. (7)

(d) The Laplace transform of a function 𝑓(𝑡) is defined as 𝐹(𝑠)=£𝑓(𝑡)=₀^+∞𝑒^−𝑠𝑡𝑓(𝑡)𝑑𝑡,𝑠∈ℂ, (8) where 𝑓(𝑡) is 𝑛-dimensional vector-valued function. For 𝛼∈(0,1], it follows from [10] that £^𝑐𝐷^𝛼𝑓(𝑡)=𝑠^𝛼£𝑓(𝑡)−𝑠^𝛼−1𝑓(0). (9)

(e) For 𝑛∈ℕ, the sequential fractional derivative for suitable function 𝑦(𝑥) is defined by 𝑦^(𝑘)∶=𝐃^𝑘𝛼𝑦(𝑥)=𝐃^𝛼𝐃^{(𝑘−1)𝛼}𝑦(𝑥), (10) where 𝑘=1,…,𝑛, and 𝐃^𝛼 is any fractional differential operator; here we still mention it as ^𝑐𝐷^𝛼.

Now, we introduce the first equivalent form (FE1) of system (1) by the nonsingular transform, which is also called the standard decomposition of a singular system.

Assume that (𝐸,𝐴) is regular throughout this paper. From [1, 3], there exist two nonsingular matrices 𝑃,𝑄∈ℝ^𝑛×𝑛, such that system (1) is equivalent to canonical system as follows: ^𝑐𝐷^𝛼𝑥₁(𝑡)=𝐴₁𝑥₁(𝑡)+𝐵₁𝑢(𝑡)+𝐶₁𝑢()𝑁𝑡−𝜏,𝑡≥0,_𝑐𝐷^𝛼𝑥₂(𝑡)=𝑥₂(𝑡)+𝐵₂𝑢(𝑡)+𝐶₂𝑢()𝑢(𝑡)(𝑡)𝑥𝑡−𝜏,𝑡≥0,=𝜓,−𝜏≤𝑡≤0,₁(0)=𝑥₁₀,𝑥₂(0)=𝑥₂₀, (11) where 𝑥₁∈ℝ^𝑛1, 𝑥₂∈ℝ^𝑛2, 𝑛₁+𝑛₂=𝑛, and 𝑁 is nilpotent whose nilpotent index is denoted by 𝜈; that is, 𝑁^𝜈=0, 𝑁^𝜈−1≠0, and 𝐼𝑃𝐸𝑄=^𝑛10𝐴0𝑁,𝑃𝐴𝑄=¹00𝐼_𝑛2,𝐵𝑃𝐵=¹𝐵₂𝐶,𝑃𝐶=¹𝐶₂,𝑄⁻¹𝑥(𝑡)=𝑥₁(𝑡)𝑥₂(𝑡). (12)

Let ℎ=[𝛼𝜈]+1, where [𝛼𝜈] denotes the integral part of 𝛼𝜈. Let 𝐂^ℎ_𝑝([0,+∞),ℝ^𝑚) be the set of ℎ times piecewise continuously differentiable functions mapping the interval [0,+∞) into ℝ^𝑚. Moreover, 𝐂^ℎ([−𝜏,0],ℝ^𝑚) denotes the set of ℎ times continuously differentiable functions mapping the interval [−𝜏,0] into ℝ^𝑚.

Applying the method in [3], we can obtain the precise form of the admissible initial state set 𝑆(𝜓) for system (11). For 𝜓∈𝐂^ℎ([−𝜏,0],ℝ^𝑚), we have 𝑆𝜓∶=𝑥∈ℝ^𝑛𝑥∣𝑥=¹𝑥₂,𝑥₁∈ℝ^𝑛1,𝑥₂=−^𝜈−1_𝑘=0𝑁^𝑘𝐵₂^𝑐𝐷^𝑘𝛼𝜓(0)+𝑁^𝑘𝐶₂^𝑐𝐷^𝑘𝛼𝜓(−𝜏). (13) Thus, the set of admissible initial data is 𝒜∶=𝑥₀,𝜓∣𝜓∈𝐂^ℎ[−𝜏,0],ℝ^𝑚,𝑥₀∈𝑆𝜓. (14)

3. State Response and State Reachability

In this section, we characterize the state structure of system (11) by analysing the state response and reachable set.

Theorem 1. For any admissible initial data (𝑥₀,𝜓)∈𝒜 and the control function 𝑢(𝑡)∈𝐂^ℎ_𝑝([0,+∞),ℝ^𝑚), the state response for system (11) can be represented in the following form: 𝑥₁(𝑡)=𝐸_𝛼,1𝐴₁𝑡^𝛼𝑥₁(0)+^𝑡_𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁𝑢(𝑠)+𝑑𝑠_0𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁+(𝑡−𝜏−𝑠)^𝛼−1×𝐸_𝛼,𝛼𝐴₁(𝑡−𝜏−𝑠)^𝛼𝐶₁𝑢(𝑠)𝑑𝑠+⁰_−𝜏(𝑡−𝜏−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝜏−𝑠)^𝛼𝐶₁𝜓(𝑠)𝑥𝑑𝑠,𝑡≥0,₂(𝑡)=−^𝜈−1_𝑘=0𝑁^𝑘𝐵₂^𝑐𝐷^𝑘𝛼𝑢(𝑡)+𝑁^𝑘𝐶₂^𝑐𝐷^𝑘𝛼𝑢(𝑡−𝜏),𝑡≥0 (15)(16) where 𝐸_𝛼,𝛽(⋅) is the Mittag-Leffler function and ^𝑐𝐷^𝑘𝛼𝑢(⋅) is the sequential fractional derivative.

Proof. From [10], the Laplace transform of the 𝛼 order Caputo fractional derivative of function 𝑥(𝑡) is £𝐷^𝛼𝑥(𝑡)=𝑠^𝛼£[𝑥(𝑡)]−𝑠^𝛼−1𝑥(0). (17) Taking the Laplace transform with respect to 𝑡 in both sides of the first equation of system (11), we obtain 𝑠^𝛼£𝑥₁(𝑡)−𝑠^𝛼−1𝑥₁(0)=𝐴₁£𝑥₁(𝑡)+𝐵₁£[𝑢(𝑡)]+𝐶₁£[𝑢(𝑡−𝜏)]. (18) Hence, £𝑥₁(𝑡)=𝑠^𝛼𝐼−𝐴₁⁻¹𝑠^𝛼−1𝑥₁(0)+𝑠^𝛼𝐼−𝐴₁⁻¹£𝐵₁𝑢(𝑡)+𝐶₁𝑢(𝑡−𝜏). (19) From [10], we have £𝑡^𝛽−1𝐸_𝛼,𝛽𝐴₁𝑡^𝛼=𝑠^𝛼−𝛽𝑠^𝛼𝐼−𝐴₁⁻¹. (20) Then (19) is equivalent to £𝑥₁(𝑡)=£𝐸_𝛼,1𝐴₁𝑡^𝛼𝑥₁(0)+£𝑡^𝛼−1𝐸_𝛼,𝛼𝐴₁𝑡^𝛼£𝐵₁𝑢(𝑡)+𝐶₁𝑢(𝑡−𝜏). (21) The convolution theorem of the Laplace transform applied to (21) yields the form £𝑥₁(𝑡)=£𝐸_𝛼,1𝐴₁𝑡^𝛼𝑥₁(0)+£^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼×𝐵₁𝑢(𝑠)+𝐶₁𝑢(𝑠−𝜏)𝑑𝑠. (22) Applying the inverse Laplace transform, we obtain 𝑥₁(𝑡)=𝐸_𝛼,1𝐴₁𝑡^𝛼𝑥₁(0)+^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼×𝐵₁𝑢(𝑠)+𝐶₁𝑢(𝑠−𝜏)𝑑𝑠=𝐸_𝛼,1𝐴₁𝑡^𝛼𝑥₁(0)+^𝑡_𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁𝑢(𝑠)+𝑑𝑠_0𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁+(𝑡−𝜏−𝑠)^𝛼−1×𝐸_𝛼,𝛼𝐴₁(𝑡−𝜏−𝑠)^𝛼𝐶₁𝑢(𝑠)𝑑𝑠+⁰_−𝜏(𝑡−𝜏−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝜏−𝑠)^𝛼𝐶₁𝜓(𝑠)𝑑𝑠. (23) On the other hand, inserting (16) into both sides of the second equation in system (11) yields 𝑁^𝑐𝐷^𝛼𝑥₂(𝑡)−𝑥₂(𝑡)−𝐵₂𝑢(𝑡)−𝐶₂𝑢(𝑡−𝜏)=−^𝜈−1_𝑘=0𝑁^𝑘+1𝐵₂^𝑐𝐷^(𝑘+1)𝛼𝑢(𝑡)+𝑁^𝑘+1𝐶₂^𝑐𝐷^(𝑘+1)𝛼𝑢(𝑡−𝜏)−𝐵₂𝑢(𝑡)−𝐶₂𝑢()+𝑡−𝜏_{𝜈−1𝑘=0}𝑁^𝑘𝐵₂^𝑐𝐷^𝑘𝛼𝑢(𝑡)+𝑁^𝑘𝐶₂^𝑐𝐷^𝑘𝛼𝑢(𝑡−𝜏)=0. (24) Then (16) is a solution of the second equation in system (11).
Next, we will show that any solution of the second equation in system (11) has the form of (16). Define linear operator 𝑇∶𝐂¹([0,+∞),ℝ^𝑛2)→𝐂([0,+∞),ℝ^𝑛2) by 𝑇𝑥₂(𝑡)=^𝑐𝐷^𝛼𝑥₂(𝑡). Then the second equation in system (11) becomes (𝐼−𝑁𝑇)𝑥₂(𝑡)=−𝐵₂𝑢(𝑡)−𝐶₂𝑢(𝑡−𝜏). (25) Since 𝑁 is nilpotent, 𝑁 and 𝑇 can exchange, and (𝐼−𝑁𝑇)(∑^𝜈−1_𝑘=0𝑁^𝑘𝑇^𝑘)=𝐼, then we have 𝑥₂(𝑡)=−(𝐼−𝑁𝑇)⁻¹𝐵₂𝑢(𝑡)+𝐶₂𝑢(𝑡−𝜏)=−^+∞_𝑘=0𝑁^𝑘𝑇^𝑘𝐵₂𝑢(𝑡)+𝐶₂𝑢(𝑡−𝜏)=−^𝜈−1_𝑘=0𝑁^𝑘𝑇^𝑘𝐵₂𝑢(𝑡)+𝐶₂𝑢(𝑡−𝜏)=−^𝜈−1_𝑘=0𝑁^𝑘𝐵₂^𝑐𝐷^𝑘𝛼𝑢(𝑡)+𝑁^𝑘𝐶₂^𝑐𝐷^𝑘𝛼𝑢(𝑡−𝜏). (26) Thus, (16) is indeed a solution of the second equation in system (11).
Therefore, the proof of Theorem 1 is completed.

For convenience sake, some notations are introduced as follows.

Given matrices 𝐴∈ℝ^𝑛×𝑛 and 𝐵,𝐶∈ℝ^𝑛×𝑚, denote Im(𝐵) as the range of 𝐵; that is, Im(𝐵)=𝑦∣𝑦=𝐵𝑥,∀𝑥∈ℝ^𝑚. (27) Let ⟨𝐴∣𝐵,𝐶⟩=Im(𝐵)+Im(𝐴𝐵)+⋯+Im𝐴^𝑛−1𝐵+Im(𝐶)+Im(𝐴𝐶)+⋯+Im𝐴^𝑛−1𝐶. (28) Then the space ⟨𝐴₁∣𝐵₁,𝐶₁⟩ is spanned by the columns of 𝐵₁,𝐴₁𝐵₁,…,𝐴^𝑛1₁⁻¹𝐵₁,𝐶₁,𝐴₁𝐶₁,…,𝐴^𝑛1₁⁻¹𝐶₁, (29) and the space ⟨𝑁|𝐵₂,𝐶₂⟩ is spanned by the columns of 𝐵₂,𝑁𝐵₂,…,𝑁^𝑛2−1𝐵₂,𝐶₂,𝑁𝐶₂,…,𝑁^𝑛2−1𝐶₂. (30) For any nonzero polynomial 𝑓(𝑠), we define 𝑊(𝑓,𝑡)∶ℝ^𝑛1→ℝ^𝑛1 by 𝑊𝑧=𝑓,𝑡_{𝑡𝑡−𝜏}𝑓(𝑠)(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼×𝐵₁𝐵^∗₁(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)^𝛼𝑓(𝑠)𝑧𝑑𝑠+₀^𝑡−𝜏𝑓(𝑠)(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁+(𝑡−𝜏−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝜏−𝑠)^𝛼𝐶₁×𝐶^∗₁(𝑡−𝜏−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝜏−𝑠)^𝛼+𝐵^∗₁(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)^𝛼𝑓(𝑠)𝑧𝑑𝑠, (31) where ∗ denotes the matrix transpose.

The following lemma is required for the main result, which is a natural extension of Lemma 2-1.1 in [1].

Lemma 2. Given matrices 𝐴₁∈ℝ^𝑛1×𝑛1 and 𝐵₁,𝐶₁∈ℝ^𝑛1×𝑚1, then Im𝑊𝑓,𝑡=𝐴₁∣𝐵₁,𝐶₁. (32)

Proof. The formula (32) holds if and only if Ker𝑊𝑓,𝑡=^𝑛1−1_𝑖=0Ker𝐵^∗₁𝐴^∗₁^{𝑖𝑛1−1}_𝑖=0Ker𝐶^∗₁𝐴^∗₁^𝑖. (33) If 𝑧₀∈Ker𝑊(𝑓,𝑡) and 𝑧₀≠0, 𝑧^∗₀𝐺(𝑡)𝑧₀=0, that is, ^𝑡_𝑡−𝜏𝐵^∗₁(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼[𝐴^∗₁(𝑡−𝑠)𝑓(𝑠)𝑧₀²+𝑑𝑠_0𝑡−𝜏𝐵^∗₁(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)+𝐶^∗₁(𝑡−𝜏−𝑠)^𝛼−1×𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝜏−𝑠)𝑓(𝑠)𝑧₀²𝑑𝑠=0; (34) then we have 𝐵^∗₁(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)𝑓(𝑠)𝑧₀[],𝐵=0,𝑠∈𝑡−𝜏,𝑡_∗1(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)+𝐶^∗₁(𝑡−𝜏−𝑠)^𝛼−1×𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝜏−𝑠)𝑓(𝑠)𝑧₀=0,𝑠∈[0,𝑡−𝜏]. (35)(36) Since polynomial 𝑓(𝑠) only has a finite number of zero on 𝑡−𝜏≤𝑠≤𝑡 and 0≤𝑠≤𝑡−𝜏, we immediately have 𝐵^∗₁(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)𝑧₀[],𝐵=0,𝑠∈𝑡−𝜏,𝑡_∗1(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)+𝐶^∗₁(𝑡−𝜏−𝑠)^𝛼−1×𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝜏−𝑠)𝑧₀=0,𝑠∈[0,𝑡−𝜏]. (37)(38) Repeatedly taking the Caputo’s derivative on both sides of (37), we have 𝐵^∗₁(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)𝐴^∗₁^𝑖𝑧₀=0,𝑖=1,2,…,𝑛₁−1,𝑠∈[𝑡−𝜏,𝑡]. (39) Substituting 𝑠=𝑡 in (39) yields 𝐵^∗𝐴^∗𝑖𝑧₀=0,𝑖=1,2,…,𝑛₁−1, (40) which implies that 𝑧₀∈Ker𝐵^∗₁(𝐴^∗₁)^𝑖, 𝑖=1,2,…,𝑛₁−1. According to Cayley-Hamilton theorem [4], (𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼[𝐴₁(𝑡−𝑠)^𝛼] can be represented as (𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼=^+∞_𝑘=0(𝑡−𝑠)^{𝑘𝛼+𝛼−1}Γ()𝐴𝑘𝛼+𝛼^𝑘1=^𝑛1−1_𝑘=0𝛾_𝑘(𝑡−𝑠)𝐴^𝑘₁. (41) For 𝑠∈[0,𝑡−𝜏], it follows from (39) and (41) that 𝐵^∗₁(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)𝑧₀=^𝑛1−1_𝑘=0𝛾_𝑘(𝑡−𝑠)𝐵^∗₁𝐴^∗₁^𝑘𝑧₀=0. (42) Inserting (42) into (38) yields 𝐶^∗₁(𝑡−𝜏−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝜏−𝑠)𝑧₀=0. (43) Repeatedly taking the Caputo’s fractional derivative on both sides of (43) and letting 𝑠=𝑡−𝜏, we have 𝐶^∗₁𝐴^∗₁^𝑘𝑧₀=0,𝑘=1,2,…,𝑛₁−1, (44) which implies that 𝑧₀∈Ker𝐶^∗₁(𝐴^∗₁)^𝑖, 𝑖=1,2,…,𝑛₁−1. Therefore, we have Ker𝑊𝑓,𝑡⊆^𝑛1−1_𝑖=0Ker𝐵^∗₁𝐴^∗₁^{𝑖𝑛1−1}_𝑖=0Ker𝐶^∗₁𝐴^∗₁^𝑖. (45)
Conversely, if 𝑧₀∈⋂^𝑛1−1_𝑖=0Ker𝐵^∗₁(𝐴^∗₁)^𝑖⋂^𝑛1−1_𝑖=0Ker𝐶^∗₁(𝐴^∗₁)^𝑖 and 𝑧₀≠0, then 𝑧₀∈Ker𝐵^∗₁(𝐴^∗₁)^𝑖, 𝑧₀∈Ker𝐶^∗₁(𝐴^∗₁)^𝑖, 𝑖=1,2,…,𝑛₁−1, and 𝐵^∗₁𝐴^∗₁^𝑖𝑧₀=𝐶^∗₁𝐴^∗₁^𝑖𝑧₀=0,𝑖=1,2,…,𝑛₁−1. (46) For 𝑠∈[𝑡−𝜏,𝑡], it follows from (41) that 𝐵^∗₁(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)𝑧₀=^𝑛1−1_𝑘=0𝛾_𝑘(𝑡−𝑠)𝐵^∗₁𝐴^∗₁^𝑘𝑧₀=0. (47) For 𝑠∈[0,𝑡−𝜏], the same argument yields 𝐵^∗₁(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)+𝐶^∗₁(𝑡−𝜏−𝑠)^𝛼−1×𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝜏−𝑠)𝑧₀=^𝑛1−1_𝑘=0𝛾_𝑘(𝑡−𝑠)𝐵^∗₁𝐴^∗₁^𝑘𝑧₀+^𝑛1−1_𝑘=0𝜆_𝑘(𝑡−𝜏−𝑠)𝐶^∗₁𝐴^∗₁^𝑘𝑧₀=0. (48) Hence, 𝑧₀∈Ker𝑊(𝑓,𝑡), which implies Ker𝑊𝑓,𝑡⊇^𝑛1−1_𝑖=0Ker𝐵^∗₁𝐴^∗₁^{𝑖𝑛1−1}_𝑖=0Ker𝐶^∗₁𝐴^∗₁^𝑖. (49) From (45) and (49), we know that (32) is true. The proof is therefore completed.

The reachable set for system (1) (or (11)) may be defined as follows.

Definition 3. Any vector 𝜔∈ℝ^𝑛 in 𝑛-dimensional vector space is said to reachable if there exists an admissible initial data (𝑥₀,𝜓)∈𝒜, admissible control input 𝑢(𝑡)∈𝐂^ℎ_𝑝([0,+∞),ℝ^𝑚), and 𝑡_𝑓>0 such that the solution of system (1) (or (11)) satisfies 𝑥(𝑡_𝑓,𝑥₀,𝜓)=𝜔.
Let ℜ(𝑥₀,𝜓) be the reachable set from any admissible initial data (𝑥₀,𝜓)∈𝒜. Then ℜ𝑥₀=,𝜓𝜔∈ℝ^𝑛∣𝑥₀,𝜓∈𝒜,𝑢(𝑡)∈𝐂^ℎ_𝑝[0,+∞),ℝ^𝑚,𝑡_𝑓>0,𝑥𝑡_𝑓,𝑥₀,𝜓=𝜔. (50)

Considering the reachable set from the initial conditions 𝑥₀=0 and 𝜓≡0, we have the following theorem.

Theorem 4. For system (11), the reachable set ℜ(0,0) can be represented as ℜ(0,0)=𝐴₁∣𝐵₁,𝐶₁⊕𝑁∣𝐵₂,𝐶₂, (51) where “⊕” is the direct sum in vector space.

Proof. Let 𝜔∈ℜ(0,0), from (15), (16), and (50); then there exists 𝑡_𝑓>0 and 𝑢(𝑡)∈𝐂^ℎ_𝑝([0,+∞),ℝ^𝑚) such that 𝜔₁=^𝑡𝑓₀𝑡_𝑓−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴₁𝑡_𝑓−𝑠^𝛼𝐵₁𝑢(𝑠)+𝑑𝑠_{𝑡𝑓0−𝜏}𝑡_𝑓−𝜏−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴₁𝑡_𝑓−𝜏−𝑠^𝛼×𝐶₁𝑢(𝑠)𝑑𝑠. (52) Combining (41) and (52) yields 𝜔₁=^𝑛1−1_𝑘=0^𝑡𝑓₀𝜑_𝑘𝑡_𝑓−𝑠𝐴^𝑘₁𝐵₁𝑢(𝑠)+𝑑𝑠_{𝑛1−1𝑘=0𝑡𝑓0−𝜏}𝜙_𝑘𝑡_𝑓−𝜏−𝑠𝐴^𝑘₁𝐶₁𝑢(𝑠)𝑑𝑠. (53) Therefore, 𝜔₁∈⟨𝐴₁∣𝐵₁,𝐶₁⟩. Moreover, we also have 𝜔₂=−^𝜈−1_𝑘=0𝑁^𝑘𝐵₂^𝑐𝐷^𝑘𝛼𝑢𝑡_𝑓+𝑁^𝑘𝐶₂^𝑐𝐷^𝑘𝛼𝑢𝑡_𝑓−𝜏, (54) which implies that 𝜔₂∈⟨𝑁∣𝐵₂,𝐶₂⟩. Thus, ℜ(0,0)⊆𝐴₁∣𝐵₁,𝐶₁⊕𝑁∣𝐵₂,𝐶₂. (55)
On the other hand, we need to prove ℜ(0,0)⊇𝐴₁∣𝐵₁,𝐶₁⊕𝑁∣𝐵₂,𝐶₂. (56) Suppose that 𝜔𝜔=¹𝜔₂∈𝐴₁∣𝐵₁,𝐶₁⊕𝑁∣𝐵₂,𝐶₂, (57) where 𝜔₁∈⟨𝐴₁∣𝐵₁,𝐶₁⟩, 𝜔₂∈⟨𝑁∣𝐵₂,𝐶₂⟩, and 𝜔₁≠0, 𝜔₂≠0. For the initial conditions 𝑥₀=0 and 𝜓(𝑡)≡0,𝑡∈[−𝜏,0], let 𝑢(𝑠)=𝑢₁(𝑠)+𝑢₂(𝑠); then we have 𝑥₁(𝑡)=^𝑡_𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁𝑢₁(𝑠)+𝑢₂(𝑠+)𝑑𝑠_0𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼×𝐵₁𝑢₁(𝑠)+𝑢₂(𝑠+)𝑑𝑠_0𝑡−𝜏(𝑡−𝜏−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝜏−𝑠)^𝛼×𝐶₁𝑢₁(𝑠)+𝑢₂(𝑠𝑥)𝑑𝑠,₂(𝑡)=−^𝜈−1_𝑘=0𝑁^𝑘𝐵₂^𝑐𝐷^𝑘𝛼𝑢₁(𝑡)+𝑁^𝑘𝐶₂^𝑐𝐷^𝑘𝛼𝑢₁(−𝑡−𝜏)_{𝜈−1𝑘=0}𝑁^𝑘𝐵₂^𝑐𝐷^𝑘𝛼𝑢₂(𝑡)+𝑁^𝑘𝐶₂^𝑐𝐷^𝑘𝛼𝑢₂(𝑡−𝜏). (58)(59) As discussed by Yip and Sincovec [3], we choose 𝑢₁(𝑠)=𝑓(𝑠)𝑦(𝑠) to satisfy ₀^𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁𝑓(𝑠)𝑔(𝑠)+𝑑𝑠_𝑡𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐶₁𝑓(𝑠−𝜏)𝑔(𝑠−𝜏)𝑑𝑠=𝜔₁−₀^𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁𝑢₂(𝑠)−𝑑𝑠_0𝑡−𝜏(𝑡−𝜏−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝜏−𝑠)^𝛼×𝐶₁𝑢₂(𝑠)𝑑𝑠≡𝜔₁,_𝑐𝐷^𝑘𝛼𝑢₁(𝑡)=^𝑐𝐷^𝑘𝛼𝑢₁(𝑡−𝜏)=0,𝑘=0,1,2,…,𝜐−1. (60) Then 𝑢₁(𝑠) contributes nothing to 𝑥₂(𝑡) at 𝑠=𝑡 and 𝑠=𝑡−𝜏. Now, we prove that (60) is true.
In fact, for 𝜔₁∈⟨𝐴₁∣𝐵₁,𝐶₁⟩, from Lemma 2, there exists 𝑧∈ℝ^𝑛 such that 𝑊(𝑓,𝑡)𝑧=𝜔₁.
Let 𝑦(𝑠)=𝑓(𝑠)𝐵^∗₁(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)+𝐶^∗₁(𝑡−𝜏−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁()𝑓(𝑠)𝐵𝑡−𝜏−𝑠𝑧,0≤𝑠≤𝑡−𝜏,_∗1(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)𝑧,𝑡−𝜏≤𝑠≤𝑡,−𝜏≤𝑠≤0. (61) Then ₀^𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁𝑢₁(𝑠)+𝑑𝑠_𝑡𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐶₁𝑢₁()=𝑠−𝜏𝑑𝑠_{𝑡𝑡−𝜏}𝑓(𝑠)(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼×𝐵₁𝐵^∗₁(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)^𝛼𝑓(𝑠)+𝑧𝑑𝑠_0𝑡−𝜏𝑓(𝑠)(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁+(𝑡−𝜏−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝜏−𝑠)^𝛼𝐶₁×𝐶^∗₁(𝑡−𝜏−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝜏−𝑠)^𝛼+𝐵^∗₁(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴^∗₁(𝑡−𝑠)^𝛼𝑓(𝑠𝜔)𝑧𝑑𝑠=𝑊𝑓,𝑡𝑧=₁. (62) Therefore, (60) is true.
For 𝜔₂∈⟨𝑁∣𝐵₂,𝐶₂⟩, there exist 𝑦_𝑘,𝑧_𝑘 such that 𝜔₂=−^𝜈−1_𝑘=0𝑁^𝑘𝐵₂𝑦_𝑘−^𝜈−1_𝑘=0𝑁^𝑘𝐶₂𝑧_𝑘. (63) There exists a function 𝑝(𝑠) such that ^𝑐𝐷^𝑘𝛼𝑝(0)=0, ^𝑐𝐷^𝑘𝛼𝑝(𝑡)=𝑦_𝑘, and ^𝑐𝐷^𝑘𝛼𝑝(𝑡−𝜏)=𝑧_𝑘. Let 𝑢₂(𝑠)=𝑝(𝑠),0≤𝑠≤𝑡,0,−𝜏≤𝑠≤0; (64) then we have 𝑥₂(𝑡)=−^𝜈−1_𝑘=0𝑁^𝑘𝐵₂^𝑐𝐷^𝑘𝛼𝑢₂(𝑡)−^𝜈−1_𝑘=0𝑁^𝑘𝐶₂^𝑐𝐷^𝑘𝛼𝑢₂(𝑡−𝜏)=𝜔₂, (65) and 𝑥₂(0)=0. It follows from (60) and (65) that 𝜔∈ℜ(0,0). Thus, (56) holds. Combining (55) and (56), we know that (51) is true. Then the proof of Theorem 4 is completed.

4. Controllability Criteria

In this section, we proceed to investigate the controllability criteria of system (1) and system (11) based on the reachable set. A set of sufficient conditions and necessary conditions for the controllability are derived based on the algebraic approach.

Definition 5. System (1) (or (11)) is said to be controllable at 𝑡_𝑓>0 if one can reach any state at 𝑡_𝑓 from any admissible initial data (𝑥₀,𝜓)∈𝒜.

Theorem 6. Canonical system (11) is controllable if and only if rank𝐵₁,𝐴₁𝐵₁,…,𝐴^𝑛1₁⁻¹𝐵₁,𝐶₁,𝐴₁𝐶₁,…,𝐴^𝑛1₁⁻¹𝐶₁=𝑛₁,rank𝐵₂,𝑁𝐵₂,…,𝑁^𝑛2−1𝐵₂,𝐶₂,𝑁𝐶₂,…,𝑁^𝑛2−1𝐶₂=𝑛₂. (66)(67)

Proof. We firstly prove the necessity of Theorem 6. If system (11) is controllable, for any 𝜔=^𝜔1_𝜔2∈ℝ^𝑛, to the initial state 𝑥₀=0 and initial control function 𝜓≡0, there exists 𝑡_𝑓>0 and a control function 𝑢(⋅)∈𝐂^ℎ_𝑝([0,+∞) such that 𝜔₁,𝜔₂ could be written in the form of (58) and (59). That is 𝜔∈𝐴₁∣𝐵₁,𝐶₁⊕𝑁∣𝐵₂,𝐶₂, (68) which implies that ℝ^𝑛⊆𝐴₁∣𝐵₁,𝐶₁⊕𝑁∣𝐵₂,𝐶₂. (69) Obviously, ℝ^𝑛⊇𝐴₁∣𝐵₁,𝐶₁⊕𝑁∣𝐵₂,𝐶₂. (70) Consequently, we have ℝ^𝑛=𝐴₁∣𝐵₁,𝐶₁⊕𝑁∣𝐵₂,𝐶₂,𝐴₁∣𝐵₁,𝐶₁=ℝ^𝑛1,𝑁∣𝐵₂,𝐶₂=ℝ^𝑛2. (71) Thus, (66) and (67) are true.
Next, we prove the sufficiency of Theorem 6. If (66) and (67) are true, then we know that (71) holds. For any 𝜔∈ℝ^𝑛 and any initial state 𝑥₀ and initial control function 𝜓(⋅), let 𝑘₁=𝜔₁−𝐸_𝛼,1𝐴₁𝑡^𝛼𝑥₁(0)−^𝑡_𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁𝜓(0)−𝑑𝑠_0𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁+(𝑡−𝜏−𝑠)^𝛼−1×𝐸_𝛼,𝛼𝐴₁(𝑡−𝜏−𝑠)^𝛼𝐶₁𝜓(0)𝑑𝑠−⁰_−𝜏(𝑡−𝜏−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝜏−𝑠)^𝛼𝐶₁𝜓(𝑠)𝑘𝑑𝑠,₂=𝜔₂+𝐵₂𝜓(0)+𝐶₂𝜓(0). (72) For 𝑘=^𝑘1_𝑘2∈ℝ^𝑛=⟨𝐴₁∣𝐵₁,𝐶₁⟩⊕⟨𝑁∣𝐵₂,𝐶₂⟩, we have 𝑘∈ℜ(0,0). Then there exists a control 𝑢(𝑠) such that 𝑘₁=^𝑡_𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁𝑢(𝑠)+𝑑𝑠_0𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁+(𝑡−𝜏−𝑠)^𝛼−1×𝐸_𝛼,𝛼𝐴₁(𝑡−𝜏−𝑠)^𝛼𝐶₁𝑢(𝑠)𝑘𝑑𝑠,₂=−^𝜈−1_𝑘=0𝑁^𝑘𝐵₂^𝑐𝐷^𝑘𝛼𝑢(𝑡)+𝑁^𝑘𝐶₂^𝑐𝐷^𝑘𝛼𝑢(𝑡−𝜏). (73) Let 𝑢(𝑠)=𝑢(𝑠)+𝜓(0), then we have 𝜔₁=𝐸_𝛼,1𝐴₁𝑡^𝛼𝑥₁(0)+^𝑡_𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁𝑢(𝑠)+𝑑𝑠_0𝑡−𝜏(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝑠)^𝛼𝐵₁+(𝑡−𝜏−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝜏−𝑠)^𝛼𝐶₁𝑢(𝑠)𝑑𝑠+⁰_−𝜏(𝑡−𝜏−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴₁(𝑡−𝜏−𝑠)^𝛼𝐶₁𝜓(𝑠)𝜔𝑑𝑠,₂=−^𝜈−1_𝑘=0𝑁^𝑘𝐵₂^𝑐𝐷^𝑘𝛼𝑢(𝑡)+𝑁^𝑘𝐶₂^𝑐𝐷^𝑘𝛼𝑢(𝑡−𝜏). (74)(75) Thus, by Definition 5, system (11) is controllable. Therefore, the proof is completed.

Applying the results of Yip and Sincovec [3], Tang and Li [4], and Theorem 6 to the pair of matrices, we can obtain the following results.

Theorem 7. Canonical system (11) is controllable if and only if rank𝑠𝐼−𝐴₁,𝐵₁,𝐶₁=𝑛₁,∀𝑠∈ℂ,𝑠𝑓𝑖𝑛𝑖𝑡𝑒,rank𝑁,𝐵₂,𝐶₂=𝑛₂. (76)

Theorem 8. System (1) is controllable if and only if [][]rank𝑠𝐸−𝐴,𝐵,𝐶=𝑛,∀𝑠∈ℂ,𝑠𝑓𝑖𝑛𝑖𝑡𝑒,rank𝐸,𝐵,𝐶=𝑛. (77)

Remark 9. If we choose 𝛼=1 and 𝐶=0, then system (1) reduces to an integer-order singular system without control delay. From Theorem 8, system (1) is controllable if and only if [][]rank𝑠𝐸−𝐴,𝐵=𝑛,∀𝑠∈ℂ,𝑠finite,rank𝐸,𝐵=𝑛, (78) which is just the result in [1].

If we choose 𝐸=𝐼, 𝐶=0, then system (1) reduces to a fractional normal system without control delay. From Theorem 8, system (1) is controllable if and only if rank𝐵,𝐴𝐵,…,𝐴^𝑛−1𝐵=𝑛, (79) which is just Corollary  3.4 in [23].

If we choose 𝐸=𝐼, then system (1) reduces to a fractional normal system with control delay. From Theorem 8, system (1) is controllable if and only if rank𝐵,𝐴𝐵,…,𝐴^𝑛−1𝐵,𝐶,𝐴𝐶,…,𝐴^𝑛−1𝐶=𝑛, (80) which is just Theorem  3 in [25]. Therefore, our results extend the existing results [1, 23, 25].

Remark 10. Theorem 8 actually offers an algebraic criterion of the exact controllability for linear fractional singular dynamical systems, which is concise and convenient to check the controllability of such systems. Motivated and inspired by the works of Sakthivel et al. [32–35], the approximate controllability of linear (nonlinear) fractional singular dynamical systems with the control delay will become our future investigative work.

5. An Illustrative Example

In this section, we give an example to illustrate the applications of Theorem 8.

Example 11. Consider the controllability of fractional singular dynamical systems as follows: 𝐷100001000^1/2𝑥(𝑡)=𝑥(𝑡)+𝑢(𝑡)+𝑢𝜋100011101101101100111𝑡−4[),𝑢(𝑡)(𝑡)−𝜋,𝑡∈0,+∞=𝜓,𝑡∈4,𝑥(0),0=𝑥₀, (81) where the admissible initial data (𝑥₀,𝜓)∈𝒜. Now, we apply Theorem 8 to prove that system (81) is controllable. Let us take 1𝛼=2,.,𝐸=100001000,𝐴=100011101𝐵=101101,𝐶=100111 (82) If we choose 𝜆=1 such that det(𝜆𝐸+𝐴)=2≠0, then (𝐸,𝐴) is regular. By using the elementary transformation of matrix, one can obtain [][]rank𝑠𝐸−𝐴,𝐵,𝐶=rank101⋆⋆⋆⋆01−1⋆⋆⋆⋆002⋆⋆⋆⋆=3,rank𝐸,𝐵,𝐶=rank101⋆⋆⋆⋆010⋆⋆⋆⋆001⋆⋆⋆⋆=3. (83) Thus, by Theorem 8, system (81) is controllable.

6. Conclusions

In this paper, the reachability and controllability of fractional singular dynamical systems with control delay have been investigated. The state structure of fractional singular dynamical systems with control delay has been characterized by analysing the state response and reachable set. A set of sufficient and necessary conditions of controllability criteria for such systems has been established based on the algebraic approach. An example is also presented to illustrate the effectiveness and applicability of the results obtained. Both the proposed criteria and the example show that the controllability property of fractional linear singular dynamical systems is dependent neither on the order of fractional derivative nor on control delay. Comparing with some existing results [1, 23, 25], we find that the algebraic approach has been extended to consider the controllability of more general fractional singular dynamical systems. Motivated and inspired by the works of Sakthivel et al. [32–35], the approximate controllability of linear (nonlinear) fractional singular dynamical systems will become our future investigative work.

Conflicts of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the Associate Editor, Professor Han H. Choi, and the anonymous reviewers for their helpful and valuable comments and suggestions which improved the quality of the paper. This work is jointly supported by the National Natural Science Foundation of China under Grant nos. 11071001, 11072059, and 61272530, the Natural Science Foundation of Jiangsu Province of China under Grant no. BK2012741, and the Programs of Educational Commission of Anhui Province of China under Grant nos. KJ2011A197 and KJ2013Z186. This work is also funded by the Deanship of Scientific Research, King Abdulaziz University (KAU), under Grant no. 3-130/1434/HiCi. Therefore, the authors acknowledge technical and financial support of KAU.