Complete Controllability of Impulsive Fractional Linear Time-Invariant Systems with Delay

Abstract

Some flaws on impulsive fractional differential equations (systems) have been found. This paper is concerned with the complete controllability of impulsive fractional linear time-invariant dynamical systems with delay. The criteria on the controllability of the system, which is sufficient and necessary, are established by constructing suitable control inputs. Two examples are provided to illustrate the obtained results.

1. Introduction

Recently, a variety of problems such as the existence, uniqueness of mild solution for the initial value problem, periodic boundary value problems, antiperiodic boundary value problems, and Ulam stability for impulsive fractional differential equations have been considered due to their important role in modeling natural phenomena such as medicine, biology, and optimal control; see the paper [1–16].

The concept of controllability plays an important role in the analysis and design of control systems. With the developments of theories of impulsive fractional differential equations, there have been a few papers devoted to the controllability of impulsive fractional differential systems; see [17–20]. In [17], the author discussed the controllability of impulsive fractional linear time-invariant systems through constructing a suitable control input in time domain. By fixed point theorem, the controllability of integrodifferential systems was investigated in [18–20]. It should be mentioned that the controllability for linear fractional dynamical systems has been investigated by several scholars [21–26] while the theory of controllability for impulsive fractional linear time-invariant systems is still in the initial stage [17].

The impulsive fractional differential equations (systems) which had been investigated earlier often have the form ^𝑐𝐷^𝛼_0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡∈𝐽^′∶=𝐽⧵𝑡₁,𝑡₂,…,𝑡_𝑚,𝐽∶=[0,𝑇],Δ𝑥𝑡_𝑘=𝑥𝑡⁺_𝑘−𝑥𝑡⁻_𝑘=𝐼_𝑘𝑥𝑡_𝑘,𝑘=1,2,…,𝑚,𝑥(0)=𝑥₀ (1)(2)(3) or ^𝑐𝐷^𝛼_0,𝑡𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),𝑡∈𝐽^′∶=𝐽⧵𝑡₁,𝑡₂,…,𝑡_𝑚,𝐽∶=[0,𝑇],Δ𝑥𝑡_𝑘=𝑥𝑡⁺_𝑘−𝑥𝑡⁻_𝑘=𝐼_𝑘𝑥𝑡_𝑘,𝑘=1,2,…,𝑚,𝑦(𝑡)=𝐶𝑥(𝑡)+𝐷𝑢(𝑡),𝑥(0)=𝑥₀ (4)(5)(6)(7) and so forth, where ^𝑐𝐷^𝛼_0,𝑡 is the Caputo fractional derivative of order 𝛼∈(0,1) with lower limit zero, 𝑥₀∈ℝ, 𝑓 is jointly continuous, 𝐼_𝑘∶ℝ^𝑛→ℝ^𝑛, 𝑡_𝑘 satisfies 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑚<𝑡_𝑚+1=𝑇, 𝑥(𝑡⁺_𝑘)=lim_𝜖→0⁺𝑥(𝑡_𝑘+𝜖), and 𝑥(𝑡⁻_𝑘)=lim_𝜖→0⁻𝑥(𝑡_𝑘+𝜖) represent, respectively, the right and the left limits of 𝑥(𝑡) at 𝑡=𝑡_𝑘, 𝐴, 𝐵, 𝐶, 𝐷 are the known constant matrices, 𝑥(𝑡), 𝑦(𝑡), 𝑢(𝑡) are vectors with appropriate dimensions.

However, the function 𝑥(𝑡) defined on [0,𝑇] is continuous everywhere except for finite number of points 𝑡_𝑘, 𝑘=1,2,…,𝑚, at which the limits 𝑥(𝑡⁺_𝑘) and 𝑥(𝑡⁻_𝑘) exist with 𝑥(𝑡_𝑘)=𝑥(𝑡⁻_𝑘). If there exists some 𝑘∈{1,2,…,𝑚} such that 𝑡_𝑘∈(0,𝑡),  0<𝛼<1, and 𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘)≠0, then ^𝑐𝐷^𝛼_0,𝑡𝑥(𝑡) does not exist since 𝑥̇(𝑡) is meaningless at the impulsive moment 𝑡_𝑘. That is to say 𝑥̇(𝑡_𝑘) is meaningless. As a result, investigating (1)–(6) is meaningless.

Motivated by this fact, this paper is concerned with the complete controllability of the impulsive fractional linear time-invariant system with delay ^𝑐𝐷^𝛼_{𝑡𝑖,𝑡}𝑥(𝑡)(𝑡)()(𝑡),𝑡=𝐴𝑥+𝐵𝑥𝑡−𝜏+𝐺𝑢𝑡∈_𝑖,𝑡_𝑖+1,𝑖=0,1,…,𝑘,Δ𝑥𝑡_𝑖=𝑥𝑡⁺_𝑖−𝑥𝑡⁻_𝑖=𝐼_𝑖𝑥𝑡⁻_𝑖,𝑖=1,2,…,𝑘,𝑥0⁺=𝜔,𝑦(𝑡)=𝐸𝑥(𝑡)+𝐹𝑢(𝑡),𝑥(𝑡)=𝜙(𝑡),−𝜏≤𝑡≤0 (8)(9)(10)(11)(12) in 𝑛-dimensional Euclidean space, where 0=𝑡₀<𝑡₁<𝑡₂<𝑡₃<⋯<𝑡_𝑘<𝑡_𝑘+1=𝑇<∞, 𝐽=[0,𝑇], ^𝑐𝐷^𝛼_{𝑡𝑖,𝑡} denotes the Caputo’s derivative of order 𝛼 with the lower limit 𝑡_𝑖, 𝑖=0,1,…,𝑘, 0<𝛼<1, 𝜔∈ℝ^𝑛, 𝐴, 𝐵, 𝐺, 𝐸, 𝐹 are known constant matrices with appropriate dimensions, the state variable 𝑥(𝑡)∈ℝ^𝑛, the initial function 𝜙(𝑡)∈𝔻={𝜓∶[−𝑟,0]→ℝ^𝑛∣𝜓iscontinuouson[−𝜏,0]}, the delay 0<𝜏<∞, ‖𝜙‖_𝔻=sup{‖𝜙(𝑡)‖_ℝ𝑛,𝑡∈[−𝜏,0]}, the control input 𝑢(𝑡)∈ℝ^𝑝, the output 𝑦(𝑡)∈ℝ^𝑚, 𝐼_𝑖∶ℝ^𝑛→ℝ^𝑛, 𝑖=1,2,…,𝑘.

In this paper, the methods used is to construct a suitable control input function in time domain. The results obtained is sufficient and necessary, which are convenient for computation.

2. Preliminaries

In this section, we begin with some notations, definitions, and lemmas. Throughout this paper, ^𝑐𝐷^𝛼_𝑎,𝑡𝑓(𝑡) or ^𝑐𝐷^𝛼_𝑎𝑓(𝑡) denotes the Caputo’s derivative of order 𝛼 with the lower limit 𝑎 for the function 𝑓, 𝐼^𝛼_𝑎𝑓(𝑡) or 𝐼^𝛼_𝑎,𝑡𝑓(𝑡) denotes integral of order 𝛼 with lower limit 𝑎 for the function 𝑓, 𝑓(𝑠)=𝐿[𝑓(𝑡);𝑠]=∫^∞₀𝑒^−𝑠𝑡𝑓(𝑡)𝑑𝑡 denotes the Laplace transform of the function 𝑓(𝑡), and “|𝑀|” denotes the norm of the matrix “𝑀,” “𝑀^∗” denotes the transpose of the matrix “𝑀”. Let 𝐶(𝐽,ℝ^𝑛) be the Banach space of all continuous functions from 𝐽 into ℝ^𝑛 with the norm ‖𝑢‖_{𝐶(𝐽,ℝ}^𝑛₎=sup{‖𝑢(𝑡)‖,𝑡∈𝐽}. Let the Banach space PC(𝐽,ℝ^𝑛) be PC𝐽,ℝ^𝑛=𝑢∶𝐽⟶ℝ^𝑛∣𝑢∈𝐶𝑡_𝑖,𝑡_𝑖+1,ℝ^𝑛𝑢𝑡,𝑖=0,1,…,𝑘,_−𝑖,𝑢𝑡⁺_𝑖exist,𝑢𝑡⁻_𝑖=𝑢𝑡_𝑖,𝑖=1,2,…,𝑘 (13) and the norm ‖𝑢‖|_PC(𝐽,ℝ^𝑛₎=sup{‖𝑢(𝑡)‖∶𝑡∈𝐽}.

Definition 1 (see [27]). The fractional integral of order 𝛼 with the lower limit 𝑎∈ℝ for a function 𝑓 is defined as 𝐼^𝛼_𝑎,𝑡𝑓(𝑡)=1Γ(𝛼)^𝑡_𝑎(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠,𝑡>𝑎,𝛼>0 (14) Provided that the right-hand side is pointwise defined on [𝑎,+∞), where Γ is the Gamma function.

Definition 2 (see [27]). The Caputo’s derivative of order 𝛼 with the lower limit 𝑎∈ℝ for a function 𝑓∶[𝑎,∞)→ℝ can be written as ^𝑐𝐷^𝛼_𝑎,𝑡𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡_𝑎(𝑡−𝑠)^{𝑛−𝛼−1}𝑓^(𝑛)(𝑠)𝑑𝑠=𝐼^𝑛−𝛼_𝑎,𝑡𝑓^(𝑛)(𝑡),𝑡>𝑎,0<𝑛−1<𝛼≤𝑛. (15) Particularly, when 0<𝛼<1, it holds ^𝑐𝐷^𝛼_𝑎,𝑡𝑓(𝑡)=1Γ(1−𝛼)^𝑡_𝑎(𝑡−𝑠)^−𝛼𝑓^′(𝑠)𝑑𝑠=𝐼^1−𝛼_𝑎,𝑡𝑓^′(𝑡),𝑡>𝑎. (16)

The Laplace transform of ^𝑐𝐷^𝛼_0,𝑡𝑓(𝑡) is 𝐿^𝑐𝐷^𝛼_0,𝑡𝑓(𝑡)=;𝑠_0+∞𝑒^{−𝑠𝑡𝑐}𝐷^𝛼_0,𝑡𝑓(𝑡)𝑑𝑡=𝑠^𝛼𝑓(𝑠)−^𝑛−1_𝑘=0𝑠^{𝛼−𝑘−1}𝑓^(𝑘)(0),𝑛−1<𝛼≤𝑛, (17) where 𝑓(𝑠) is the Laplace transform of 𝑓(𝑡).

In particular, for 0<𝛼<1, it holds ₀^+∞𝑒^{−𝑠𝑡𝑐}𝐷^𝛼_0,𝑡𝑓(𝑡)𝑑𝑡=𝑠^𝛼𝑓(𝑠)−𝑠^𝛼−1𝑓(0). (18)

Definition 3 (see [27]). The two-parameter Mittag-Leffler function is defined as 𝐸_𝛼,𝛽(𝑧)=^∞_𝑘=0𝑧^𝑘Γ𝛼𝑘+𝛽,𝛼>0,𝛽>0,𝑧∈ℂ. (19)
The Laplace transform of Mittag-Leffler function is 𝐿𝑡^{𝛼𝑘+𝛽−1}𝐸^(𝑘)_𝛼,𝛽±𝑎𝑡^𝛼=;𝑠_∞0𝑒^−𝑠𝑡𝑡^{𝛼𝑘+𝛽−1}𝐸^(𝑘)_𝛼,𝛽±𝑎𝑡^𝛼=𝑑𝑡𝑘!𝑠^𝛼−𝛽(𝑠^𝛼∓𝑎)^𝑘+1,Re(𝑠)>|𝑎|^1/𝛼, (20) where Re(𝑠) denotes the real parts of 𝑠.

In addition, the Laplace transform of 𝑡^𝛼−1 is 𝐿𝑡^𝛼−1;𝑠=Γ(𝛼)𝑠^−𝛼,Re(𝑠)>0. (21)

Lemma 4 (see [28]). Let 0<Re(𝛼)≤1. If 𝑥(𝑡)∈𝐶[𝑎,𝑏], then 𝐼^𝛼_𝑎,𝑡^𝑐𝐷^𝛼_𝑎,𝑡𝑥(𝑡)=𝑥(𝑡)−𝑥(𝑎), (22) where 𝐶[𝑎,𝑏] denotes the set of continuous functions on [𝑎,𝑏].

3. Main Results

Definition 5 (complete controllability). The system (8)–(12) is said to be completely controllable on the interval 𝐽=[0,𝑇] if, for any 𝑡_∗>0 (𝑡_∗∈(0,𝑇]), 𝜙(𝑡)∈𝔻, and 𝑍∈ℝ^𝑛, there exists an admissible control input 𝑢(𝑡) such that the state variable 𝑥(𝑡) of the system (8)–(12) satisfies 𝑥(𝑡_∗)=𝑍.
Using the Laplace transform method, we can easily obtain the following lemma.

Lemma 6. The movement orbit of the state variable 𝑥(𝑡) of the system (8)−(12) can be written as 𝑥(𝑡)=𝜔𝐸_𝛼,1𝐴𝑡^𝛼+^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×(𝐵𝑥(𝑠−𝜏)+𝐺𝑢(𝑠))𝑑𝑠,𝑡∈0,𝑡₁;𝑥𝑡₁+𝐼₁𝑥𝑡₁𝐸_𝛼,1𝐴𝑡^𝛼+^𝑡_𝑡1(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×(𝐵𝑥(𝑠−𝜏)+𝐺𝑢(𝑠))𝑑𝑠,𝑡∈𝑡₁,𝑡₂;⋮𝑥𝑡_𝑖+𝐼_𝑖𝑥𝑡_𝑖𝐸_𝛼,1𝐴𝑡^𝛼+^𝑡_𝑡𝑖(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×(𝐵𝑥(𝑠−𝜏)+𝐺𝑢(𝑠))𝑑𝑠,𝑡∈𝑡_𝑖,𝑡_𝑖+1,𝑖=2,3,…,𝑘. (23)

Theorem 7. The system (8)–(12) is completely controllable on [0,𝑇] if and only if the controllability matrices 𝑊_𝑐𝑡_𝑖,𝑡_𝑖+1=^𝑡𝑖+1_𝑡𝑖𝑡_𝑖+1−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡_𝑖+1−𝑠^𝛼×𝐺𝐺^∗𝐸_𝛼,𝛼𝐴^∗𝑡_𝑖+1−𝑠^𝛼𝑑𝑠 (24) are nonsingular, 𝑖=0,1,2,…,𝑘.

Proof. Sufficiency. Suppose that 𝑊_𝑐[𝑡_𝑖,𝑡_𝑖+1] is nonsingular; then 𝑊_𝑐⁻¹[𝑡_𝑖,𝑡_𝑖+1] is well defined, 𝑖=0,1,2,…,𝑘.
For 𝑡∈(0,𝑡₁], it follows from the formula (23) that 𝑥(𝑡)=𝜔𝐸_𝛼,1𝑎𝑡^𝛼+^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×(𝐵𝑥(𝑠−𝜏)+𝐺𝑢(𝑠))𝑑𝑠,𝑡∈0,𝑡₁. (25) For all 𝑍₀∈ℝ^𝑛, choosing 𝑢(𝑡)=𝐺^∗𝐸_𝛼,𝛼𝐴^∗𝑡₁−𝑡^𝛼𝑊_𝑐⁻¹0,𝑡₁⋅𝑍₀−𝜔𝐸_𝛼,1𝐴𝑡^𝛼₁−^𝑡1₀𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼𝐵𝑥(𝑠−𝜏)𝑑𝑠 (26) and inserting (26) into (25) yields 𝑥(𝑡₁)=𝑍₀. Thus, the system (8)–(12) is completely controllable on [0,𝑡₁].
Similarly, for 𝑡∈(𝑡₁,𝑡₂], it follows from the formula (23) that 𝑥(𝑡)=𝑥𝑡₁+𝐼₁𝑥𝑡₁𝐸_𝛼,1𝐴𝑡^𝛼+^𝑡_𝑡1(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×(𝐵𝑥(𝑠−𝜏)+𝐺𝑢(𝑠))𝑑𝑠,𝑡∈𝑡₁,𝑡₂. (27)
Since the system (8)–(12) is completely controllable on [0,𝑡₁], there exists a control input 𝑢₁(𝑡) such that 𝑥(𝑡₁)=0. By (27), it follows that 𝑥(𝑡)=𝐼₁(0)𝐸_𝛼,1𝐴𝑡^𝛼+^𝑡_𝑡1(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×(𝐵𝑥(𝑠−𝜏)+𝐺𝑢(𝑠))𝑑𝑠,𝑡∈𝑡₁,𝑡₂. (28)
For all 𝑍₁∈ℝ^𝑛, choosing 𝑢(𝑡)=𝐺^∗𝐸_𝛼,𝛼𝐴^∗𝑡₂−𝑡^𝛼𝑊_𝑐⁻¹𝑡₁,𝑡₂⋅𝑍₁−𝐼₁(0)𝐸_𝛼,1𝐴𝑡^𝛼₂−^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₂−𝑡^𝛼𝐵𝑥(𝑠−𝜏)𝑑𝑠, (29) together with (28) yields 𝑥(𝑡₂)=𝑍₁. Thus, the system (8)–(12) is completely controllable on [𝑡₁,𝑡₂].
By similar arguments, we can prove that the system (8)–(12) is completely controllable on [𝑡_𝑖,𝑡_𝑖+1],  𝑖=2,…,𝑘.
Consequently, the system (8)–(12) is completely controllable on 𝐽=[0,𝑇].
Necessity. Suppose that the system (8)–(12) is completely controllable on 𝐽=[0,𝑇].
If 𝑊_𝑐[𝑡₀,𝑡₁] is singular, then there exists a nonzero vector 𝑍₀ such that 𝑍^∗₀𝑊_𝑐0,𝑡₁𝑍₀=0. (30) That is ^𝑡1_𝑡0𝑍^∗₀𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼𝐺𝐺^∗𝐸_𝛼,𝛼×𝐴^∗𝑡₁−𝑠^𝛼𝑍₀𝑑𝑠=0. (31) Then we have 𝑍^∗₀𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼𝐺=0 (32) on 𝑠∈[0,𝑡₁]. By the assumption that the system (8)–(12) is completely controllable on 𝐽, the system (8)–(12) is completely controllable on [𝑡_𝑖,𝑡_𝑖+1], 𝑖=0,1,2,…,𝑘. There exist control inputs 𝑢₀(𝑡) and 𝑢₀(𝑡) such that 𝑥𝑡₁=𝜔𝐸_𝛼,1𝐴𝑡^𝛼₁+^𝑡1₀𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼()+𝐵𝑥𝑠−𝜏𝑑𝑠_𝑡10𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼𝐺𝑢₀(𝑠)𝑑𝑠=0,𝑥𝑡₁=𝜔𝐸_𝛼,1𝐴𝑡^𝛼₁+^𝑡1₀𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼()+𝐵𝑥𝑠−𝜏𝑑𝑠_𝑡10𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼𝐺𝑢₀(𝑠)𝑑𝑠=𝑍₀. (33)(34) By (34), we have 𝜔𝐸_𝛼,1𝐴𝑡^𝛼₁+^𝑡1₀𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼𝐵𝑥(𝑠−𝜏)𝑑𝑠=𝑍₀−^𝑡1₀𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼𝐺𝑢₀(𝑠)𝑑𝑠. (35) Inserting (35) into (33) yields 𝑍₀+^𝑡1₀𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼×𝐺𝑢₀(𝑠)−𝑢₀(𝑠)𝑑𝑠=0. (36) Multiplying 𝑍^∗₀ on both side of (36) yields 𝑍^∗₀𝑍₀+^𝑡1₀𝑡₁−𝑠^𝛼−1𝑍^∗₀𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼×𝐺𝑢₀(𝑠)−𝑢₀(𝑠)𝑑𝑠=0. (37) By (32) and (37), we have 𝑍^∗₀𝑍₀=0. Thus, 𝑍₀=0. This is a contradiction.
If 𝑊_𝑐[𝑡_𝑖,𝑡_𝑖+1] is singular for some 𝑖∈{1,…,𝑘}, then there exists a nonzero vector 𝑍_𝑖 such that 𝑍^∗_𝑖𝑊_𝑐𝑡_𝑖,𝑡_𝑖+1𝑍_𝑖=0. (38) That is ^𝑡𝑖+1_𝑡𝑖𝑍^∗_𝑖𝑡_𝑖+1−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡_𝑖+1−𝑠^𝛼×𝐺𝐺^∗𝐸_𝛼,𝛼𝐴^∗𝑡_𝑖+1−𝑠^𝛼𝑍_𝑖𝑑𝑠=0. (39) Then, it follows that 𝑍^∗_𝑖𝐸_𝛼,𝛼𝐴𝑡_𝑖+1−𝑠^𝛼𝐺=0 (40) on 𝑠∈[𝑡_𝑖,𝑡_𝑖+1]. By formula (23) and the assumption that the system (8)–(12) is completely controllable, there exist control inputs 𝑢_𝑖−1(𝑡) and 𝑢_𝑖(𝑡) such that 𝑥(𝑡_𝑖)=0 and 𝑥𝑡_𝑖+1=𝐼_𝑖(0)𝐸_𝛼,1𝐴𝑡^𝛼_𝑖+1+^𝑡𝑖+1_𝑡𝑖𝑡_𝑖+1−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡_𝑖+1−𝑠^𝛼⋅𝐵𝑥(𝑠−𝜏)+𝐺𝑢_𝑖(𝑠)𝑑𝑠=0. (41)
Similarly, there exists a control input 𝑢_𝑖(𝑡) such that 𝑥𝑡_𝑖+1=𝐼_𝑖(0)𝐸_𝛼,1𝐴𝑡^𝛼_𝑖+1+^𝑡𝑖+1_𝑡𝑖𝑡_𝑖+1−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡_𝑖+1−𝑠^𝛼⋅𝐵𝑥(𝑠−𝜏)+𝐺𝑢_𝑖(𝑠)𝑑𝑠=𝑍_𝑖. (42) By (42), we have 𝐼_𝑖(0)𝐸_𝛼,1𝐴𝑡^𝛼_𝑖+1+^𝑡𝑖+1_𝑡𝑖𝑡_𝑖+1−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡_𝑖+1−𝑠^𝛼𝐵𝑥(𝑠−𝜏)𝑑𝑠=𝑍_𝑖−^𝑡𝑖+1_𝑡𝑖𝑡_𝑖+1−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡_𝑖+1−𝑠^𝛼𝐺𝑢_𝑖(𝑠)𝑑𝑠. (43) Inserting (43) into (41) yields 𝑍_𝑖+^𝑡𝑖+1_𝑡𝑖𝑡_𝑖+1−𝑠^𝛼−1×𝐸_𝛼,𝛼𝐴𝑡_𝑖+1−𝑠^𝛼𝐺𝑢_𝑖(𝑠)−𝑢_𝑖(𝑠)𝑑𝑠=0. (44) Multiplying 𝑍^∗_𝑖 on both side of (44) yields 𝑍^∗_𝑖𝑍_𝑖+^𝑡𝑖+1_𝑡𝑖𝑡_𝑖+1−𝑠^𝛼−1𝑍^∗_𝑖×𝐸_𝛼,𝛼𝐴𝑡_𝑖+1−𝑠^𝛼𝐺𝑢_𝑖(𝑠)−𝑢_𝑖(𝑠)𝑑𝑠=0. (45) Combining (45) with (40) yields 𝑍^∗_𝑖𝑍_𝑖=0. Thus, 𝑍_𝑖=0. This is a contradiction.
Thus, 𝑊_𝑐[𝑡_𝑖,𝑡_𝑖+1] is nonsingular for 𝑖=0,1,…,𝑘. This completes the proof.

Theorem 8. The system (8)–(12) is completely controllable on [0,𝑇] if and only if rank𝐺∣𝐴𝐺∣⋯∣𝐴^𝑛−1𝐺=𝑛. (46)

Proof. Necessity. Suppose that system (8)–(12) is completely controllable on [0,𝑇]. Then, the system (8)–(12) is completely controllable on [0,𝑡₁]. Then, for any 𝑍₀∈ℝ^𝑛, there exists a control input 𝑢₀(𝑡) such that 𝑥(𝑡₁)=𝑍₀. By the formula (23), it follows that 𝑍₀=𝑥𝑡₁=𝜔𝐸_𝛼,1𝐴𝑡^𝛼₁+^𝑡1₀𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼×(𝐵𝑥(𝑠−𝜏)+𝐺𝑢(𝑠))𝑑𝑠. (47) By Cayley-Hamilton theorem, we have 𝑡^𝛼−1𝐸_𝛼,𝛼𝐴𝑡^𝛼=^𝑛−1_𝑗=0𝑐_𝑗(𝑡)𝐴^𝑗, (48) where 𝑐_𝑗(𝑡) are functions in 𝑡, 𝑗=1,2,…,𝑛−1. Combining the formula (48) and the equality (47), we have 𝑥𝑡₁−𝜔𝐸_𝛼,1𝐴𝑡^𝛼₁−^𝑡1₀𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼()=𝐵𝑥𝑠−𝜏𝑑𝑠_𝑡10𝑡₁−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₁−𝑠^𝛼(𝑠)=𝐺𝑢𝑑𝑠_{𝑛−1𝑗=0}𝐴^𝑗𝐺^𝑡1₀𝑐_𝑗𝑡₁𝑢(𝑠)=−𝑠𝑑𝑠𝐺∣𝐴𝐺∣⋯∣𝐴^𝑛−1𝐺.𝑑₀𝑑₁⋮𝑑_𝑛−1, (49) where 𝑑_𝑗=∫^𝑡1₀𝑐_𝑗(𝑡₁−𝑠)𝑢(𝑠)𝑑𝑠, 𝑗=0,1,2,…,𝑛−1. For arbitrary state 𝑍₀ and initial function 𝜙(𝑡), the system (8)–(12) is completely controllable on [0,𝑡₁] if and only if there exists a control input 𝑢(𝑡) such that (47) or (49) holds. Obviously, for arbitrary initial function 𝜙(𝑡) and 𝑍₀, the sufficient and necessary condition to have a control input 𝑢(𝑡) satisfying (49) is that rank𝐺∣𝐴𝐺∣⋯∣𝐴^𝑛−1𝐺=𝑛. (50)Sufficiency. Suppose that rank(𝐺∣𝐴𝐺∣⋯∣𝐴^𝑛−1𝐺)=𝑛. In order to prove that the system (8)–(12) is completely controllable on [0,𝑇], it is sufficient to prove that the system (8)–(12) is completely controllable on [𝑡_𝑖,𝑡_𝑖+1],  𝑖=0,1,…,𝑘, respectively.
The formula (23) together with (48) yields (49). By the assumption that rank(𝐺∣𝐴𝐺∣⋯∣𝐴^𝑛−1𝐺)=𝑛, the system (8)–(12) is completely controllable on [0,𝑡₁].
Now we prove that the system (8)–(12) is completely controllable on [𝑡₁,𝑡₂]. The complete controllability of the system (8)–(12) on [0,𝑡₁] implies that there exists a control input 𝑢₀(𝑡) such that 𝑥(𝑡₁)=0. Inserting 𝑥(𝑡₁)=0 into the formula (23), we have, for 𝑡∈(𝑡₁,𝑡₂], 𝑥(𝑡)=𝐼₁(0)𝐸_𝛼,1𝐴𝑡^𝛼+^𝑡_𝑡1(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝑠)^𝛼×(𝐵𝑥(𝑠−𝜏)+𝐺𝑢(𝑠))𝑑𝑠,𝑡∈𝑡₁,𝑡₂. (51) Thus, it follows 𝑥𝑡₂=𝐼₁(0)𝐸_𝛼,1𝐴𝑡^𝛼₂+^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₂−𝑠^𝛼×(𝐵𝑥(𝑠−𝜏)+𝐺𝑢(𝑠))𝑑𝑠. (52) By (48) it follows taht 𝑥𝑡₂−𝐼₁(0)𝐸_𝛼,1𝐴𝑡^𝛼₂−^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₂−𝑠^𝛼()=𝐵𝑥𝑠−𝜏𝑑𝑠_𝑡2𝑡1𝑡₂−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡₂−𝑠^𝛼(𝑠)=𝐺𝑢𝑑𝑠_{𝑛−1𝑗=0}𝐴^𝑗𝐺^𝑡2_𝑡1𝑐_𝑗𝑡₂𝑢(𝑠)=−𝑠𝑑𝑠𝐺∣𝐴𝐺∣⋯∣𝐴^𝑛−1𝐺.𝑑^′₀𝑑^′₁⋮𝑑^′_𝑛−1, (53) where 𝑑^′_𝑗=∫^𝑡2_𝑡1𝑐_𝑗(𝑡₂−𝑠)𝑢(𝑠)𝑑𝑠, 𝑗=0,1,2,…,𝑛−1. Similar to the previous arguments, we can conclude that system (8)–(12) is completely controllable on (𝑡₁,𝑡₂].
Repeating the process on (𝑡_𝑖,𝑡_𝑖+1], respectively, we can prove that the system (8)–(12) is completely controllable on (𝑡_𝑖,𝑡_𝑖+1], 𝑖=2,…,𝑘. In conclusion, the system (8)–(12) is completely controllable on 𝐽=[0,𝑇]. This completes the proof.

Remark 9. From Theorem 8, we can conclude that the complete controllability of the system (8)–(12) is unrelated to the matrix 𝐵 and initial function 𝜙(𝑡). The matrices 𝐴,𝐺 determine if the the system (8)–(12) possesses complete controllability.

4. Examples

Example 1. Consider the system (8)–(12). Choose 𝛼=1/2, 𝐽=[0,2],  𝑡₀=0, 𝑡₁=1, 𝑡₂=2, Δ𝑥(𝑡₁)=𝑥(𝑡⁺₁)−𝑥(𝑡⁻₁)=3, 𝐴=⁰¹₀₀, 𝐺=²₁. Now, we employ Theorems 7 and 8 to prove if that the system (8)–(10) is completely controllable, respectively.
By computation, we have 𝐸_1/2,1/2𝐴(1−𝑠)^1/2=^∞_𝑘=0𝐴^𝑘(1−𝑠)^𝑘/2Γ()=1𝑘/2+1/2Γ()11/2𝐼+Γ(1)()01001−𝑠^1/2,𝐺𝐺^∗=2121=,4221𝐸_1/2,1/2𝐴^∗(1−𝑠)^1/2=^∞_𝑘=0𝐴^∗𝑘(1−𝑠)^𝑘(1/2)Γ(𝑘())=11/2+1/2Γ()11/2𝐼+Γ(1)()00101−𝑠^1/2,𝐸_1/2,1/2𝐴(2−𝑠)^1/2=^∞_𝑘=0𝐴^𝑘(2−𝑠)^𝑘/2Γ()=1𝑘/2+1/2Γ()11/2𝐼+Γ(1)()01002−𝑠^1/2,𝐸_1/2,1/2𝐴^∗(2−𝑠)^1/2=^∞_𝑘=0𝐴^∗𝑘(2−𝑠)^𝑘/2Γ()=1𝑘/2+1/2Γ()11/2𝐼+Γ(1)()00102−𝑠^1/2. (54)(55)(56)(57)(58) By the formula (24) 𝑊_𝑐𝑡_𝑖,𝑡_𝑖+1=^𝑡𝑖+1_𝑡𝑖𝑡_𝑖+1−𝑠^−1/2𝐸_1/2,1/2𝐴𝑡_𝑖+1−𝑠^1/2𝐺×𝐺^∗𝐸_1/2,1/2𝐴^∗𝑡_𝑖+1−𝑠^1/2𝑑𝑠, (59) we have 𝑊_𝑐[]=80,1𝜋+4𝜋^0.5+234𝜋+1𝜋^0.54𝜋+1𝜋^0.52𝜋,𝑊_𝑐[]=41,2𝜋+6𝜋^0.5+232𝜋+1𝜋^0.52𝜋+1𝜋^0.51𝜋. (60) It is obvious that 𝑊_𝑐[0,1] and 𝑊_𝑐[1,2] are nonsingular. By Theorem 7, the system is completely controllable.

On the other hand, rank(𝐺∣𝐴𝐺)=rank2110=2. (61) By Theorem 8, the system is completely controllable.

Example 2. Consider the time-invariant system (8)–(12). Choose 1𝐴=−1−4−206−1171,𝛼=3201,𝐺=. (62)
By computation, we have rank𝐺∣𝐴𝐺∣𝐴²𝐺=rank2−460−1711−12=3. (63) By Theorem 8, the system is completely controllable.

Acknowledgments

The authors would like to thank the referee for his or her valuable comments, which help us to improve the quality of the paper. This paper is supported by National Natural Science Foundation of China (11071001), Research Fund for Doctoral Program of Educational Ministry of China (20103401120002 and 20123401120001), Program of Natural Science Research in Anhui Universities (KJ2011A020 and KJ2013A032), Scientific Research Starting Fund for Dr. of Anhui University (023033190001, 023033190181), and the 211 Project of Anhui University (KJQN1001, 023033050055).