# Complete Controllability of Linear Fractional Differential Systems with Singularity

Authors: Qun Huang; Qiuyue Wu; Xian-Feng Zhou; Yanhua Wen; Jiang Wei
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper is concerned with the controllability of a class of linear fractional differential systems with singularity. The method which is used to deal with the fast subsystemN·cD0,tαx2(t)=x2(t)+B2u(t) and y2(t)=C2x2(t) is an improvement of the known ones. Based on the movement orbit of the state equation, we obtain several controllability criteria which are sufficient and necessary.


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## Body

## 1. Introduction and Preliminaries

Singular systems are also commonly called descriptor systems, generalized state-space systems, differential-algebraic systems, or semistate systems whose behaviors are described by differential equations (or difference equations) and algebraic equations. In the past few decades, singular systems have attracted much attention for their extensive applications in robotics [1], power systems [2], networks, economic systems [3, 4], highly interconnected large-scale systems [5], and so on. Many fundamental notions and conclusions based on regular systems have been extended to singular systems. For detail, see the monographs [6, 7].The concept of controllability plays an important role in the analysis and design of control systems. Recently, the controllability of fractional differential systems has been gaining much attention. For example, by applying Schauder’s fixed point theorems, the authors of the paper [8] obtained a set of sufficient conditions for the controllability of nonlinear fractional differential systems. Using Sadovskii’s fixed point theorem and Krasnoselskii’s fixed point theorem, respectively, and properties of characteristic solution operators, the authors of the paper [9] established the complete controllability criteria for fractional evolution systems. Without involving the compactness of characteristic solution operators, the authors of the paper [10] considered the controllability of nonlinear dynamical systems with time varying multiple delays and distributed delays, respectively. In the paper [11], the controllability of fractional impulsive neutral integrodifferential systems was investigated. By using Krasnoselskii’s fixed point theorem and the properties of resolvent operators, sufficient conditions for the controllability were established. By the representation of the state solution and construction of suitable control inputs, the authors established the controllability criteria for a class of linear neutral fractional time-invariant differential systems. These criteria are sufficient and necessary [12]. In 2013, the paper [13] was concerned with the controllability of fractional functional evolution equations of Sobolev type in Banach space. With the help of two characteristic solution operators and their properties, the authors obtained the controllability criteria corresponding to two admissible control sets via the well-known Schauder’s fixed point theorem. In 2014, the paper [14] dealt with the controllability of Sobolev type fractional evolutions and some controllability criteria were derived.However, as far as we know, little attention has been paid to singular fractional differential systems. Motivated by this fact, this paper is devoted to the controllability for the singular fractional differential system(1)E·D0,tαcxt=Axt+But,t≥0,yt=Cxt,where D0,tαc is Caputo’s derivative of order α with the lower limit 0, 0<α<1, E,A∈Rn×n are n×n constant matrices, the matrix couple (E,A) is regular which will be defined later, B∈Rn×m and C∈Rr×n are the known constant matrices, rankE≜q<n, x∈Rn is the state variable, u∈Rm is the control input, and y∈Rr is the output.Under the assumption that the matrix coupleEA is regular, there exist two nonsingular matrices Q and P satisfying (2)QEP=diag⁡In1N,QAP=diag⁡A1In2,CP=C1C2,QB=B1B2,where N∈Rn2×n2 is nilpotent whose index is h (namely, h is the smallest integer such that Nh=0), A1∈Rn1×n1, B1∈Rn1×m, B2∈Rn2×m, C1∈Rr×n1, and C2∈Rr×n2. Then system (1) can be written as (3)D0,tαcx1t=A1x1t+B1ut,(4)y1t=C1x1t,(5)N·D0,tαcx2t=x2t+B2ut,(6)y2t=C2x2t,(7)y=y1+y2,where P-1x=x1x2, x1∈Rn1, x2∈Rn2, and n1+n2=n. Subsystems (3)-(4) and (5)-(6) are called the slow subsystem and the fast subsystem, respectively.Note that subsystem (3)-(4) is a normal fractional differential linear system, whose controllability has been discussed in [15].Hence, in order to investigate the controllability of system (1), it is sufficient to investigate the controllability of subsystem (5)-(6), which is our main task in this paper.Before giving our main results, we first recall some definitions and lemmas. For more details, please refer to [6, 16, 17].Definition 1 (see [6]).
For any given two matricesE and A∈Rn×n, the matrix couple EA is called regular if there exists a constant scalar a∈C such that |aE+A|≠0 or the polynomial |sE-A|≢0.Lemma 2 (see [15]).
The slow subsystem (3)-(4) of system (1) is controllable if rank⁡B1A1B1⋯A1n1-1B1=n1.Definition 3 (see [16]).
Given an interval[a,b] of R, the fractional order integral of a function f∈L1[a,b] of order α∈R is defined by (8)Ia,tαft=1Γα∫att-sα-1fsds,t∈a,b,α>0,where Γ is the Gamma function.Definition 4 (see [16]).
Suppose that a functionf is defined on the interval [a,b] and f(n)(t)∈L1[a,b]. Caputo’s fractional derivative of order α with lower limit a for f is defined as (9)Da,tαcft=1Γn-α∫att-sn-α-1fnsds=Ia,tn-αfnt,t∈a,b,where 0<n-1<α≤n.
Particularly, when0<α<1, it holds that (10)Da,tαcft=1Γ1-α∫att-s-αf′sds=Ia,t1-αf′t,t∈a,b.The Laplace transform of Caputo’s fractional derivativeD0,tαcx(t) is (11)LD0,tαcxt;s=∫0+∞e-stD0,tαcxtdt=sαx-s-∑k=0n-1sα-k-1xk0,n-1<α≤n,where x-(s) is the Laplace transform of x(t).Particularly, for0<α<1, it holds that (12)∫0+∞e-stD0,tαcxtdt=sαx-s-sα-1x0.In addition, the Laplace transforms of thekth derivative of u(t) and the kth derivative of the Dirac function δ(t) are (13)Lukt;s=sku-s-∑i=0k-1siuk-i-10,Lδkt;s=sk,k∈N+=1,2,…and if α∈R∖N+, then the Laplace transform of t-α-1 is (14)Lt-α-1;s=Γ-αsα,α∈R∖N+,Res>0.Lemma 5 (see [17]).
Let0<Re(α)≤1. If x(t)∈AC[a,b]={x∣x(t):[a,b]→Rn is absolutely continuous or x(t)∈C[a,b]={x∣x(t):[a,b]→Rn is continuous, then (15)Ia,tαDa,tαcxt=xt-xa.Throughout this paper, “|M|” denotes the norm of the matrix “M”, “MT” denotes the transpose of the matrix “M”, C denotes the complex plane, and “⇔” denotes equivalence.


## 2. State Response

In this section, the state response of system (5)-(6) is given by the following theorem.Theorem 6.
Assume thath is the nilpotent index of the matrix N. Suppose further that the function u(t) is h times continuously differentiable and u(i)(0)=0, i=1,2,…. Then the distribution solution of the state equation (5) has the following form: (16)x2t=-∑k=1h-1Nkgktx20-∑k=0h-1NkB2fkt,where(17)f0t=ut,gkt=L-1skα-1;t,fkt=L-1skαu-s;t,k=1,2,…,h-1.Here x2(t)|t=0=x2(0) and u-(s) is the Laplace transform of u(t). Namely, u-(s)=L[u(t);s]. L-1[u-(s);t] denotes the inverse Laplace transform of  u-(s).Proof.
An application of the Laplace transform on both sides of (5) yields (18)N·sαx-2s-sα-1x20=x-2s+B2u-s,where x-2(s) and u-(s) are the Laplace transform of x2(t) and u(t), respectively. That is (19)x-2s=Nsα-I-1Nsα-1x20+Nsα-I-1B2u-s.Since N∈Rn2×n2 is nilpotent whose index is h and (20)
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.Now, we consider the following two cases.
Case (i). ν∈N+={1,2,…} and ui(0)=0, i=1,2,…,ν. By formula (13), we have (22)L-1sν;t=δνt,L-1sνu-s;t=uνt.Case (ii). ν∈R∖N+. By formula (14), we have (23)L-1sν;t=1Γ-νt-ν-1.Applying the inverse Laplace transform and Convolution Theorem yields (24)L-1sνu-s;t=1Γ-ν∫0tt-τ-ν-1uτdτ.Finally, in order to obtain the desired results, it is sufficient to apply the inverse Laplace transform on both sides of (21) together with formulas (22)–(24). This completes the proof.Remark 7.
Obviously, the output responsey2(t) is given by (25)y2t=C2x2t=-∑k=1h-1C2Nkgktx20-∑k=0h-1C2NkB2fkt.


## 3. Controllability

In this section, we will establish some controllability criteria for subsystem (5)-(6) and system (1). We begin with the concept of the controllability of subsystem (5)-(6).Definition 8.
System (5)-(6) is called controllable if, for any t1>0, x2(0)∈Rn2, and w∈Rn2, there exists a control input u(t)∈Cph-1, x2(t1)=w. Here Cph-1 denotes the set of the h-1 times piecewise continuously differentiable functions.The following lemma is well-known.Lemma 9.
Consider two given constant matrices:A∈Rn×n and B∈Rn×m. The following statements are equivalent: (1)
rank⁡sI-AB=n, ∀s∈C, where C represents the complex plane;(2)
rank⁡λiI-AB=n, where λi are the eigenvalues of matrix A;(3)
rank⁡BAB⋯An-1B=n.The following two theorems and a corollary are our main results of this paper.Theorem 10.
The slow subsystem (3)-(4) of system (1) is controllable if and only if one of the following holds: (a)
rank⁡B1A1B1⋯A1n1-1B1=n1;(b)
rank⁡sI-A1B1=n1;(c)
rank⁡sE-AB=n, ∀s∈C, where C represents the complex plane.Proof.
By Lemma2, the slow subsystem (3)-(4) is controllable if and only if (a) holds.(a)⇔(b). By Lemma 9, (a) and (b) are equivalent.(b)⇔(c). In fact, (26)rank⁡sE-A,B=rank⁡sQEP-QAPQB=rank⁡sI-A1OB1OsN-IB2=rank⁡sI-A1B1OOOsN-I=n2+rank⁡sI-A1B1,where the matrices Q and P satisfy (2). Obviously, (b) and (c) are equivalent. This completes the proof.Theorem 11.
The fast subsystem (5)-(6) of system (1) is controllable if and only if one of the following holds: (a)
rank⁡B2NB2⋯Nh-1B2=n2;(b)
rank⁡NB2=n2;(c)
rank⁡EB=n.Proof.
We first prove that subsystem (5)-(6) is controllable if and only if (a) holds. According to the state response of the fast subsystem (5)-(6), we have (27)x2t+∑k=1h-1Nkgktx20=-∑k=0h-1NkB2fkt=B2NB2⋯Nh-1B2f0tf1t⋮fh-1t,where fk(t) and gk(t) are satisfying (17). For any t1>0, x2(0)∈Rn2 and w2∈Rn2, the sufficient and necessary condition to have a control input u(t)∈Cph-1 such that x2(t1)=w2 is rankB2NB2⋯Nh-1B2=n2. Thus subsystem (5)-(6) is controllable if and only if (a) holds.
Now we prove the equivalence between (a) and (b). SinceN is nilpotent, Δ(N)=λi∣λi∈C,|λiI-N|=0={0}. By the equivalence of (2) and (3) in Lemma 9, rankB2NB2⋯Nh-1B2=n2 holds if and only if rank-NB2=rankNB2=n2.
Finally, we prove the equivalence between (b) and (c). Note the fact that(28)rank⁡EB=rank⁡QEPQB=rank⁡In1OB1ONB2=n1+rank⁡NB2,where the matrices Q and P satisfy (2). Obviously, (b) and (c) are equivalent. This completes the proof.By Theorems10 and 11, the following corollary yields.Corollary 12.
System (1) is controllable if and only if ranksE-AB=n and rankEB=n, ∀s∈C.


## 4. Illustrative Examples

Example 1.
Consider system (1). Assume that α=1/2, and (29)E=3000050000-200000,A=00010010-11001003,B=000-1.By computation, we have(30)rank⁡sE-AB=rank⁡3s00-1005s-1001-1-2s00-100-3-1=rank⁡000-1000-100100000000-1=4,rank⁡EB=rank⁡300000500000-2000000-1=4.By Corollary 12, system (1) is controllable.Example 2.
Consider system (1). Assume that α=1/2, and (31)E=010101000,A=100010001,B=010,C=010.By computation, we have (32)rank⁡sE-AB=rank⁡-1s00s-1s100-10=rank⁡100000010010=3,rank⁡EB=rank⁡010010110000=2<3.By Corollary 12, system (1) is not controllable.



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