# Reachability and Controllability of Fractional Singular Dynamical Systems with Control Delay

Authors: Hai Zhang; Jinde Cao; Wei Jiang
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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:(1)EcDαx(t)=Ax(t)+Bu(t)+Cu(t-τ),t≥0,u(t)=ψ(t),-τ≤t≤0,x(0)=x0,
where cDαx(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∈ℝn×n, B∈ℝn×m, C∈ℝn×m, and rank(E)<n; x∈ℝn is the state variable; u∈ℝm 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 Section3, 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 orderα>0 with the lower limit zero for a function f:R+→Rn is defined as
(2)D-αf(t)=1Γ(α)∫0t‍(t-s)α-1f(s)ds,t>0,
provided that the right side is pointwise defined on [0,+∞), where Γ(·) is the Gamma function.(b) The Caputo’s fractional derivative of orderα for a function f:R+→Rn is defined as
(3)cDαf(t)=1Γ(m-α+1)∫0t(t-s)m-αf(m+1)(s)ds,0≤m≤α<m+1.
For 0<α≤1,
(4)D-α{cDαf(t)}=f(t)-f(0).(c) The Mittag-Leffler function in two parameters is defined as(5)Eα,β(z)=∑k=0+∞‍zkΓ(kα+β),
where α>0, β>0, and z∈ℂ, ℂ denotes the complex plane. In particular, for β=1,
(6)Eα,1(z)=Eα(z)=∑k=0+∞‍zkΓ(kα+1)
has the interesting property
(7)cDαEα(λzα)=λEα(λzα),λ,z∈ℂ.(d) The Laplace transform of a functionf(t) is defined as
(8)F(s)=£[f(t)]=∫0+∞‍e-stf(t)dt,s∈ℂ,
where f(t) is n-dimensional vector-valued function. For α∈(0,1], it follows from [10] that
(9)£[(cDαf)(t)]=sα£[f(t)]-sα-1f(0).(e) Forn∈ℕ, the sequential fractional derivative for suitable function y(x) is defined by
(10)y(k):=(Dkαy)(x)=(DαD(k-1)αy)(x),
where k=1,…,n, and Dα is any fractional differential operator; here we still mention it as cDα.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∈ℝn×n, such that system (1) is equivalent to canonical system as follows:
(11)cDαx1(t)=A1x1(t)+B1u(t)+C1u(t-τ),t≥0,NcDαx2(t)=x2(t)+B2u(t)+C2u(t-τ),t≥0,u(t)=ψ(t),-τ≤t≤0,x1(0)=x10,x2(0)=x20,
where x1∈ℝn1, x2∈ℝn2, n1+n2=n, and N is nilpotent whose nilpotent index is denoted by ν; that is, Nν=0, Nν-1≠0, and
(12)PEQ=[In100N],PAQ=[A100In2],PB=[B1B2],PC=[C1C2],Q-1x(t)=[x1(t)x2(t)].Leth=[αν]+1, where [αν] denotes the integral part of αν. Let Cph([0,+∞),ℝm) be the set of h times piecewise continuously differentiable functions mapping the interval [0,+∞) into ℝm. Moreover, Ch([-τ,0],ℝm) denotes the set of h times continuously differentiable functions mapping the interval [-τ,0] into ℝm.Applying the method in [3], we can obtain the precise form of the admissible initial state set S(ψ) for system (11). For ψ∈Ch([-τ,0],ℝm), we have
(13)S(ψ):={-∑k=0ν-1‍x∈ℝn∣x=[x1x2],x1∈ℝn1,x2=-∑k=0ν-1‍[NkB2cDkαψ(0)+NkC2cDkαψ(-τ)][x1x2]}.
Thus, the set of admissible initial data is
(14)𝒜:={(x0,ψ)∣ψ∈Ch([-τ,0],ℝm),x0∈S(ψ)}.


## 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)∈Cph([0,+∞),ℝm), the state response for system (11) can be represented in the following form:
(15)x1(t)=Eα,1(A1tα)x1(0)+∫t-τt(t-s)α-1Eα,α[A1(t-s)α]B1u(s)ds+∫0t-τ{(t-s)α-1Eα,α[A1(t-s)α]B1+(t-τ-s)α-1(t-s)α-1Eα,α[A1(t-s)α]×Eα,α[A1(t-τ-s)α]C1}u(s)ds+∫-τ0‍(t-τ-s)α-1Eα,α[A1(t-τ-s)α]C1ψ(s)ds,t≥0,(16)x2(t)=-∑k=0ν-1[NkB2cDkαu(t)+NkC2cDkαu(t-τ)],t≥0
where Eα,β(·) is the Mittag-Leffler function and cDkαu(·) is the sequential fractional derivative.Proof.
From [10], the Laplace transform of the α order Caputo fractional derivative of function x(t) is
(17)£[Dαx(t)]=sα£[x(t)]-sα-1x(0).
Taking the Laplace transform with respect to t in both sides of the first equation of system (11), we obtain
(18)sα£[x1(t)]-sα-1x1(0)=A1£[x1(t)]+B1£[u(t)]+C1£[u(t-τ)].
Hence,
(19)£[x1(t)]=(sαI-A1)-1sα-1x1(0)+(sαI-A1)-1£[B1u(t)+C1u(t-τ)].
From [10], we have
(20)£[tβ-1Eα,β(A1tα)]=sα-β(sαI-A1)-1.
Then (19) is equivalent to
(21)£[x1(t)]=£[Eα,1(A1tα)]x1(0)+£[tα-1Eα,α(A1tα)]£[B1u(t)+C1u(t-τ)].
The convolution theorem of the Laplace transform applied to (21) yields the form
(22)£[x1(t)]=£[Eα,1(A1tα)]x1(0)+£{∫0t‍(t-s)α-1Eα,α[A1(t-s)α]×[B1u(s)+C1u(s-τ)]ds∫0t‍}.
Applying the inverse Laplace transform, we obtain
(23)x1(t)=Eα,1(A1tα)x1(0)+∫0t‍(t-s)α-1Eα,α[A1(t-s)α]×[B1u(s)+C1u(s-τ)]ds=Eα,1(A1tα)x1(0)+∫t-τt‍(t-s)α-1Eα,α[A1(t-s)α]B1u(s)ds+∫0t-τ‍{(t-s)α-1Eα,α[A1(t-s)α]B1+(t-τ-s)α-1(t-s)α-1Eα,α[A1(t-s)α]×Eα,α[A1(t-τ-s)α]C1}u(s)ds+∫-τ0‍(t-τ-s)α-1Eα,α[A1(t-τ-s)α]C1ψ(s)ds.
On the other hand, inserting (16) into both sides of the second equation in system (11) yields
(24)NcDαx2(t)-x2(t)-B2u(t)-C2u(t-τ)=-∑k=0ν-1[Nk+1B2cD(k+1)αu(t)+Nk+1C2cD(k+1)αu(t-τ)]-B2u(t)-C2u(t-τ)+∑k=0ν-1[NkB2cDkαu(t)+NkC2cDkαu(t-τ)]=0.
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,+∞),ℝn2)→C([0,+∞),ℝn2) by Tx2(t)=cDαx2(t). Then the second equation in system (11) becomes
(25)(I-NT)x2(t)=-B2u(t)-C2u(t-τ).
Since N is nilpotent, N and T can exchange, and (I-NT)(∑k=0ν-1NkTk)=I, then we have
(26)x2(t)=-(I-NT)-1[B2u(t)+C2u(t-τ)]=-(∑k=0+∞‍NkTk)[B2u(t)+C2u(t-τ)]=-(∑k=0ν-1‍NkTk)[B2u(t)+C2u(t-τ)]=-∑k=0ν-1‍[NkB2cDkαu(t)+NkC2cDkαu(t-τ)].
Thus, (16) is indeed a solution of the second equation in system (11).
Therefore, the proof of Theorem1 is completed.For convenience sake, some notations are introduced as follows.Given matricesA∈ℝn×n and B,C∈ℝn×m, denote Im(B) as the range of B; that is,
(27)Im(B)={y∣y=Bx,∀x∈ℝm}.
Let
(28)〈A∣B,C〉=Im(B)+Im(AB)+⋯+Im(An-1B)+Im(C)+Im(AC)+⋯+Im(An-1C).
Then the space 〈A1∣B1,C1〉 is spanned by the columns of
(29)[B1,A1B1,…,A1n1-1B1,C1,A1C1,…,A1n1-1C1],
and the space 〈N|B2,C2〉 is spanned by the columns of
(30)[B2,NB2,…,Nn2-1B2,C2,NC2,…,Nn2-1C2].
For any nonzero polynomial f(s), we define W(f,t):ℝn1→ℝn1 by
(31)W(f,t)z=∫t-τtf(s)(t-s)α-1Eα,α[A1(t-s)α]×B1B1*(t-s)α-1Eα,α[A1*(t-s)α]f(s)zds+∫0t-τf(s){(t-s)α-1Eα,α[A1(t-s)α]B1+(t-τ-s)α-1Eα,α[A1(t-τ-s)α]C1}×{C1*(t-τ-s)α-1Eα,α[A1*(t-τ-s)α]+B1*(t-s)α-1Eα,α[A1*(t-s)α]f(s)}zds,
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 matricesA1∈ℝn1×n1 and B1,C1∈ℝn1×m1, then
(32)ImW(f,t)=〈A1∣B1,C1〉.Proof.
The formula (32) holds if and only if
(33)KerW(f,t)=⋂i=0n1-1‍KerB1*(A1*)i⋂i=0n1-1‍KerC1*(A1*)i.
If z0∈KerW(f,t) and z0≠0, z0*G(t)z0=0, that is,
(34)∫t-τt∥B1*(t-s)α-1Eα,α[(A1*(t-s)]f(s)z0∥2ds+∫0t-τ∥[B1*(t-s)α-1Eα,α(A1*(t-s))+C1*(t-τ-s)α-1×Eα,α(A1*(t-τ-s))B1*(t-s)α-1]f(s)z0∥2ds=0;
then we have
(35)B1*(t-s)α-1Eα,α[A1*(t-s)]f(s)z0=0,s∈[t-τ,t],(36)[B1*(t-s)α-1Eα,α(A1*(t-s))+C1*(t-τ-s)α-1B1*(t-s)α-1Eα,α(A1*(t-s))+C1*(t-τ-s)α-1×Eα,α(A1*(t-τ-s))]f(s)z0=0,s∈[0,t-τ].
Since polynomial f(s) only has a finite number of zero on t-τ≤s≤t and 0≤s≤t-τ, we immediately have
(37)B1*(t-s)α-1Eα,α[A1*(t-s)]z0=0,s∈[t-τ,t],(38)[B1*(t-s)α-1Eα,α(A1*(t-s))+C1*(t-τ-s)α-1B1*(t-s)α-1Eα,α(A1*(t-s))+C1*(t-τ-s)α-1×Eα,α(A1*(t-τ-s))]z0=0,s∈[0,t-τ].
Repeatedly taking the Caputo’s derivative on both sides of (37), we have
(39)B1*(t-s)α-1Eα,α[A1*(t-s)](A1*)iz0=0,i=1,2,…,n1-1,s∈[t-τ,t].
Substituting s=t in (39) yields
(40)B*(A*)iz0=0,i=1,2,…,n1-1,
which implies that z0∈KerB1*(A1*)i, i=1,2,…,n1-1. According to Cayley-Hamilton theorem [4], (t-s)α-1Eα,α[A1(t-s)α] can be represented as
(41)(t-s)α-1Eα,α[A1(t-s)α]=∑k=0+∞(t-s)kα+α-1Γ(kα+α)A1k=∑k=0n1-1γk(t-s)A1k.
For s∈[0,t-τ], it follows from (39) and (41) that
(42)B1*(t-s)α-1Eα,α(A1*(t-s))z0=∑k=0n1-1‍γk(t-s)B1*A1*kz0=0.
Inserting (42) into (38) yields
(43)C1*(t-τ-s)α-1Eα,α(A1*(t-τ-s))z0=0.
Repeatedly taking the Caputo’s fractional derivative on both sides of (43) and letting s=t-τ, we have
(44)C1*A1*kz0=0,k=1,2,…,n1-1,
which implies that z0∈KerC1*(A1*)i, i=1,2,…,n1-1. Therefore, we have
(45)KerW(f,t)⊆⋂i=0n1-1‍KerB1*(A1*)i⋂i=0n1-1‍KerC1*(A1*)i.
Conversely, ifz0∈⋂i=0n1-1KerB1*(A1*)i⋂i=0n1-1KerC1*(A1*)i and z0≠0, then z0∈KerB1*(A1*)i, z0∈KerC1*(A1*)i, i=1,2,…,n1-1, and
(46)B1*(A1*)iz0=C1*(A1*)iz0=0,i=1,2,…,n1-1.
For s∈[t-τ,t], it follows from (41) that
(47)B1*(t-s)α-1Eα,α(A1*(t-s))z0=∑k=0n1-1‍γk(t-s)B1*A1*kz0=0.
For s∈[0,t-τ], the same argument yields
(48)[B1*(t-s)α-1Eα,α(A1*(t-s))+C1*(t-τ-s)α-1×Eα,α(A1*(t-τ-s))(t-s)α-1]z0=∑k=0n1-1‍γk(t-s)B1*A1*kz0+∑k=0n1-1‍λk(t-τ-s)C1*A1*kz0=0.
Hence, z0∈KerW(f,t), which implies
(49)KerW(f,t)⊇⋂i=0n1-1‍KerB1*(A1*)i⋂i=0n1-1‍KerC1*(A1*)i.
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 vectorω∈ℝn in n-dimensional vector space is said to reachable if there exists an admissible initial data (x0,ψ)∈𝒜, admissible control input u(t)∈Cph([0,+∞),ℝm), and tf>0 such that the solution of system (1) (or (11)) satisfies x(tf,x0,ψ)=ω.
Letℜ(x0,ψ) be the reachable set from any admissible initial data (x0,ψ)∈𝒜. Then
(50)ℜ(x0,ψ)={ω∈ℝn∣(x0,ψ)∈𝒜,u(t)∈Cph([0,+∞),ℝm),tf>0,x(tf,x0,ψ)=ω}.Considering the reachable set from the initial conditionsx0=0 and ψ≡0, we have the following theorem.Theorem 4.
For system (11), the reachable set ℜ(0,0) can be represented as
(51)ℜ(0,0)=〈A1∣B1,C1〉⊕〈N∣B2,C2〉,
where “⊕” is the direct sum in vector space.Proof.
Letω∈ℜ(0,0), from (15), (16), and (50); then there exists tf>0 and u(t)∈Cph([0,+∞),ℝm) such that
(52)ω1=∫0tf(tf-s)α-1Eα,α[A1(tf-s)α]B1u(s)ds+∫0tf-τ(tf-τ-s)α-1Eα,α[A1(tf-τ-s)α]×C1u(s)ds.
Combining (41) and (52) yields
(53)ω1=∑k=0n1-1∫0tfφk(tf-s)A1kB1u(s)ds+∑k=0n1-1∫0tf-τϕk(tf-τ-s)A1kC1u(s)ds.
Therefore, ω1∈〈A1∣B1,C1〉. Moreover, we also have
(54)ω2=-∑k=0ν-1[NkB2cDkαu(tf)+NkC2cDkαu(tf-τ)],
which implies that ω2∈〈N∣B2,C2〉. Thus,
(55)ℜ(0,0)⊆〈A1∣B1,C1〉⊕〈N∣B2,C2〉.
On the other hand, we need to prove(56)ℜ(0,0)⊇〈A1∣B1,C1〉⊕〈N∣B2,C2〉.
Suppose that
(57)ω=[ω1ω2]∈〈A1∣B1,C1〉⊕〈N∣B2,C2〉,
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
(58)x1(t)=∫t-τt(t-s)α-1Eα,α[A1(t-s)α]B1[u1(s)+u2(s)]ds+∫0t-τ(t-s)α-1Eα,α[A1(t-s)α]×B1[u1(s)+u2(s)]ds+∫0t-τ(t-τ-s)α-1Eα,α[A1(t-τ-s)α]×C1[u1(s)+u2(s)]ds,(59)x2(t)=-∑k=0ν-1[NkB2cDkαu1(t)+NkC2cDkαu1(t-τ)]-∑k=0ν-1[NkB2cDkαu2(t)+NkC2cDkαu2(t-τ)].
As discussed by Yip and Sincovec [3], we choose u1(s)=f(s)y(s) to satisfy
(60)∫0t-τ(t-s)α-1Eα,α[A1(t-s)α]B1f(s)g(s)ds+∫τt(t-s)α-1Eα,α[A1(t-s)α]C1f(s-τ)g(s-τ)ds=ω1-∫0t-τ(t-s)α-1Eα,α[A1(t-s)α]B1u2(s)ds-∫0t-τ(t-τ-s)α-1Eα,α[A1(t-τ-s)α]×C1u2(s)ds≡ω^1,cDkαu1(t)=cDkαu1(t-τ)=0,k=0,1,2,…,υ-1.
Then u1(s) contributes nothing to x2(t) at s=t and s=t-τ. Now, we prove that (60) is true.
In fact, forω^1∈〈A1∣B1,C1〉, from Lemma 2, there exists z∈ℝn such that W(f,t)z=ω^1.
Let(61)y(s)={f(s)[B1*(t-s)α-1Eα,α(A1*(t-s))+C1*(t-τ-s)α-1Eα,α(A1*(t-τ-s))]z,0≤s≤t-τ,f(s)B1*(t-s)α-1Eα,α(A1*(t-s))z,t-τ≤s≤t,-τ≤s≤0.
Then
(62)∫0t-τ(t-s)α-1Eα,α[A1(t-s)α]B1u1(s)ds+∫τt(t-s)α-1Eα,α[A1(t-s)α]C1u1(s-τ)ds=∫t-τtf(s)(t-s)α-1Eα,α[A1(t-s)α]×B1B1*(t-s)α-1Eα,α[A1*(t-s)α]f(s)zds+∫0t-τf(s){(t-s)α-1Eα,α[A1(t-s)α]B1+(t-τ-s)α-1Eα,α[A1(t-τ-s)α]C1}×{C1*(t-τ-s)α-1Eα,α[A1*(t-τ-s)α]+B1*(t-s)α-1Eα,α[A1*(t-s)α]f(s)}zds=W(f,t)z=ω^1.
Therefore, (60) is true.
Forω2∈〈N∣B2,C2〉, there exist yk,zk such that
(63)ω2=-∑k=0ν-1‍NkB2yk-∑k=0ν-1‍NkC2zk.
There exists a function p(s) such that cDkαp(0)=0, cDkαp(t)=yk, and cDkαp(t-τ)=zk. Let
(64)u2(s)={p(s),0≤s≤t,0,-τ≤s≤0;
then we have
(65)x2(t)=-∑k=0ν-1‍NkB2cDkαu2(t)-∑k=0ν-1‍NkC2cDkαu2(t-τ)=ω2,
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 tf>0 if one can reach any state at tf from any admissible initial data (x0,ψ)∈𝒜.Theorem 6.
Canonical system (11) is controllable if and only if
(66)rank[B1,A1B1,…,A1n1-1B1,C1,A1C1,…,A1n1-1C1]=n1,(67)rank[B2,NB2,…,Nn2-1B2,C2,NC2,…,Nn2-1C2]=n2.Proof.
We firstly prove the necessity of Theorem6. If system (11) is controllable, for any ω=[ω1ω2]∈ℝn, to the initial state x0=0 and initial control function ψ≡0, there exists tf>0 and a control function u(·)∈Cph([0,+∞) such that ω1,ω2 could be written in the form of (58) and (59). That is
(68)ω∈〈A1∣B1,C1〉⊕〈N∣B2,C2〉,
which implies that
(69)ℝn⊆〈A1∣B1,C1〉⊕〈N∣B2,C2〉.
Obviously,
(70)ℝn⊇〈A1∣B1,C1〉⊕〈N∣B2,C2〉.
Consequently, we have
(71)ℝn=〈A1∣B1,C1〉⊕〈N∣B2,C2〉,〈A1∣B1,C1〉=ℝn1,〈N∣B2,C2〉=ℝn2.
Thus, (66) and (67) are true.
Next, we prove the sufficiency of Theorem6. If (66) and (67) are true, then we know that (71) holds. For any ω∈ℝn and any initial state x0 and initial control function ψ(·), let
(72)k1=ω1-Eα,1(A1tα)x1(0)-∫t-τt(t-s)α-1Eα,α[A1(t-s)α]B1ψ(0)ds-∫0t-τ{(t-s)α-1Eα,α[A1(t-s)α]B1+(t-τ-s)α-1(t-s)α-1Eα,α[A1(t-s)α]×Eα,α[A1(t-τ-s)α]C1}ψ(0)ds-∫-τ0(t-τ-s)α-1Eα,α[A1(t-τ-s)α]C1ψ(s)ds,k2=ω2+B2ψ(0)+C2ψ(0).
For k=[k1k2]∈ℝn=〈A1∣B1,C1〉⊕〈N∣B2,C2〉, we have k∈ℜ(0,0). Then there exists a control u~(s) such that
(73)k1=∫t-τt(t-s)α-1Eα,α[A1(t-s)α]B1u~(s)ds+∫0t-τ[(t-s)α-1Eα,α[A1(t-s)α]B1+(t-τ-s)α-1(t-s)α-1Eα,α[A1(t-s)α]×Eα,α[A1(t-τ-s)α]C1]u~(s)ds,k2=-∑k=0ν-1‍[NkB2cDkαu~(t)+NkC2cDkαu~(t-τ)].
Let u(s)=u~(s)+ψ(0), then we have
(74)ω1=Eα,1(A1tα)x1(0)+∫t-τt(t-s)α-1Eα,α[A1(t-s)α]B1u(s)ds+∫0t-τ{(t-s)α-1Eα,α[A1(t-s)α]B1+(t-τ-s)α-1Eα,α[A1(t-τ-s)α]C1}u(s)ds+∫-τ0(t-τ-s)α-1Eα,α[A1(t-τ-s)α]C1ψ(s)ds,(75)ω2=-∑k=0ν-1[NkB2cDkαu(t)+NkC2cDkαu(t-τ)].
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
(76)rank[sI-A1,B1,C1]=n1,∀s∈ℂ,sfinite,rank[N,B2,C2]=n2.Theorem 8.
System (1) is controllable if and only if
(77)rank[sE-A,B,C]=n,∀s∈ℂ,sfinite,rank[E,B,C]=n.Remark 9.
If we chooseα=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
(78)rank[sE-A,B]=n,∀s∈ℂ,sfinite,rank[E,B]=n,
which is just the result in [1].If we chooseE=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
(79)rank[B,AB,…,An-1B]=n,
which is just Corollary  3.4 in [23].If we chooseE=I, then system (1) reduces to a fractional normal system with control delay. From Theorem 8, system (1) is controllable if and only if
(80)rank[B,AB,…,An-1B,C,AC,…,An-1C]=n,
which is just Theorem  3 in [25]. Therefore, our results extend the existing results [1, 23, 25].Remark 10.
Theorem8 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 Theorem8.Example 11.
Consider the controllability of fractional singular dynamical systems as follows:(81)[100001000]D1/2x(t)=[100011101]x(t)+[101101]u(t)+[100111]u(t-π4),t∈[0,+∞),u(t)=ψ(t),t∈[-π4,0],x(0)=x0,
where the admissible initial data (x0,ψ)∈𝒜. Now, we apply Theorem 8 to prove that system (81) is controllable. Let us take
(82)α=12,E=[100001000],A=[100011101],B=[101101],C=[100111].
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
(83)rank[sE-A,B,C]=rank[101⋆⋆⋆⋆01-1⋆⋆⋆⋆002⋆⋆⋆⋆]=3,rank[E,B,C]=rank[101⋆⋆⋆⋆010⋆⋆⋆⋆001⋆⋆⋆⋆]=3.
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.



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