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

Authors: Xian-Feng Zhou; Song Liu; Wei Jiang
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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(1)cD0,tαx(t)=f(t,x(t)),t∈J′:=J∖{t1,t2,…,tm},J:=[0,T],(2)Δx(tk)=x(tk+)-x(tk-)=Ik(x(tk)),k=1,2,…,m,(3)x(0)=x0
or
(4)Dc0,tαx(t)=Ax(t)+Bu(t),t∈J′:=J∖{t1,t2,…,tm},J:=[0,T],(5)Δx(tk)=x(tk+)-x(tk-)=Ik(x(tk)),k=1,2,…,m,(6)y(t)=Cx(t)+Du(t),(7)x(0)=x0
and so forth, where cD0,tα is the Caputo fractional derivative of order α∈(0,1) with lower limit zero, x0∈ℝ, f is jointly continuous, Ik:ℝn→ℝn, tk satisfies 0=t0<t1<t2<⋯<tm<tm+1=T, x(tk+)=limϵ→0+x(tk+ϵ), and x(tk-)=limϵ→0-x(tk+ϵ) represent, respectively, the right and the left limits of x(t) at t=tk, A, B, C, D are the known constant matrices, x(t), y(t), u(t) are vectors with appropriate dimensions.However, the functionx(t) defined on [0,T] is continuous everywhere except for finite number of points tk, k=1,2,…,m, at which the limits x(tk+) and x(tk-) exist with x(tk)=x(tk-). If there exists some k∈{1,2,…,m} such that tk∈(0,t),  0<α<1, and x(tk+)-x(tk-)≠0, then cD0,tαx(t) does not exist since x˙(t) is meaningless at the impulsive moment tk. That is to say x˙(tk) 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(8)cDti,tαx(t)=Ax(t)+Bx(t-τ)+Gu(t),t∈(ti,ti+1),i=0,1,…,k,(9)Δx(ti)=x(ti+)-x(ti-)=Ii(x(ti-)),i=1,2,…,k,(10)x(0+)=ω,(11)y(t)=Ex(t)+Fu(t),(12)x(t)=ϕ(t),-τ≤t≤0
in n-dimensional Euclidean space, where 0=t0<t1<t2<t3<⋯<tk<tk+1=T<∞, J=[0,T], cDti,tα denotes the Caputo’s derivative of order α with the lower limit ti, i=0,1,…,k, 0<α<1, ω∈ℝn, A, B, G, E, F are known constant matrices with appropriate dimensions, the state variable x(t)∈ℝn, the initial function ϕ(t)∈𝔻={ψ:[-r,0]→ℝn∣ψiscontinuouson[-τ,0]}, the delay 0<τ<∞, ∥ϕ∥𝔻=sup{∥ϕ(t)∥ℝn,t∈[-τ,0]}, the control input u(t)∈ℝp, the output y(t)∈ℝm, Ii:ℝn→ℝn, 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,cDa,tαf(t) or cDaαf(t) denotes the Caputo’s derivative of order α with the lower limit a for the function f, Iaαf(t) or Ia,tαf(t) denotes integral of order α with lower limit a for the function f, f-(s)=L[f(t);s]=∫0∞‍e-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,ℝn) be the Banach space of all continuous functions from J into ℝn with the norm ∥u∥C(J,ℝn)=sup{∥u(t)∥,t∈J}. Let the Banach space PC(J,ℝn) be
(13)PC(J,ℝn)={u:J⟶ℝn∣u∈C((ti,ti+1],ℝn),i=0,1,…,k,u(ti-),u(ti+)exist,u(ti-)=u(ti),i=1,2,…,k}
and the norm ∥u∥|PC(J,ℝn)=sup{∥u(t)∥:t∈J}.Definition 1 (see [27]).
The fractional integral of orderα with the lower limit a∈ℝ for a function f is defined as
(14)Ia,tαf(t)=1Γ(α)∫at‍(t-s)α-1f(s)ds,t>a,α>0
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 orderα with the lower limit a∈ℝ for a function f:[a,∞)→ℝ can be written as
(15)cDa,tαf(t)=1Γ(n-α)∫at‍(t-s)n-α-1f(n)(s)ds=Ia,tn-α(f(n)(t)),t>a,0<n-1<α≤n.
Particularly, when 0<α<1, it holds
(16)cDa,tαf(t)=1Γ(1-α)∫at‍(t-s)-αf′(s)ds=Ia,t1-αf′(t),t>a.The Laplace transform ofcD0,tαf(t) is
(17)L[cD0,tαf(t);s]=∫0+∞‍e-st(cD0,tαf(t))dt=sαf-(s)-∑k=0n-1‍sα-k-1f(k)(0),n-1<α≤n,
where f-(s) is the Laplace transform of f(t).In particular, for0<α<1, it holds
(18)∫0+∞‍e-st(cD0,tαf(t))dt=sαf-(s)-sα-1f(0).Definition 3 (see [27]).
The two-parameter Mittag-Leffler function is defined as(19)Eα,β(z)=∑k=0∞‍zkΓ(αk+β),α>0,β>0,z∈ℂ.
The Laplace transform of Mittag-Leffler function is(20)L[tαk+β-1Eα,β(k)(±atα);s]=∫0∞‍e-sttαk+β-1Eα,β(k)(±atα)dt=k!sα-β(sα∓a)k+1,Re(s)>|a|1/α,
where Re(s) denotes the real parts of s.In addition, the Laplace transform oftα-1 is
(21)L[tα-1;s]=Γ(α)s-α,Re(s)>0.Lemma 4 (see [28]).
Let0<Re(α)≤1. If x(t)∈C[a,b], then
(22)Ia,tα(cDa,tαx(t))=x(t)-x(a),
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∈ℝn, 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 variablex(t) of the system (8)−(12) can be written as
(23)x(t)={ωEα,1(Atα)+∫0t‍(t-s)α-1Eα,α(A(t-s)α)×(Bx(s-τ)+Gu(s))ds,t∈(0,t1];(x(t1)+I1(x(t1)))Eα,1(Atα)+∫t1t‍(t-s)α-1Eα,α(A(t-s)α)×(Bx(s-τ)+Gu(s))ds,t∈(t1,t2];⋮(x(ti)+Ii(x(ti)))Eα,1(Atα)+∫tit‍(t-s)α-1Eα,α(A(t-s)α)×(Bx(s-τ)+Gu(s))ds,t∈(ti,ti+1],i=2,3,…,k.Theorem 7.
The system (8)–(12) is completely controllable on [0,T] if and only if the controllability matrices
(24)Wc[ti,ti+1]=∫titi+1‍(ti+1-s)α-1Eα,α(A(ti+1-s)α)×GG*Eα,α(A*(ti+1-s)α)ds
are nonsingular, i=0,1,2,…,k.Proof.
Sufficiency. Suppose that Wc[ti,ti+1] is nonsingular; then Wc-1[ti,ti+1] is well defined, i=0,1,2,…,k.
Fort∈(0,t1], it follows from the formula (23) that
(25)x(t)=ωEα,1(atα)+∫0t‍(t-s)α-1Eα,α(A(t-s)α)×(Bx(s-τ)+Gu(s))ds,t∈(0,t1].
For all Z0∈ℝn, choosing
(26)u(t)=G*Eα,α(A*(t1-t)α)Wc-1[0,t1]·[∫0t1‍Z0-ωEα,1(At1α)(At1α)-∫0t1‍(t1-s)α-1Eα,α(A(t1-s)α)Bx(s-τ)ds]
and inserting (26) into (25) yields x(t1)=Z0. Thus, the system (8)–(12) is completely controllable on [0,t1].
Similarly, fort∈(t1,t2], it follows from the formula (23) that
(27)x(t)=(x(t1)+I1(x(t1)))Eα,1(Atα)+∫t1t‍(t-s)α-1Eα,α(A(t-s)α)×(Bx(s-τ)+Gu(s))ds,t∈(t1,t2].
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
(28)x(t)=I1(0)Eα,1(Atα)+∫t1t‍(t-s)α-1Eα,α(A(t-s)α)×(Bx(s-τ)+Gu(s))ds,t∈(t1,t2].
For allZ1∈ℝn, choosing
(29)u(t)=G*Eα,α(A*(t2-t)α)Wc-1[t1,t2]·[(A(t2-t)α)∫t1t2‍Z1-I1(0)Eα,1(At2α)-∫t1t2‍(t2-s)α-1Eα,α(A(t2-t)α)Bx(s-τ)ds],
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 [ti,ti+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].
IfWc[t0,t1] is singular, then there exists a nonzero vector Z0 such that
(30)Z0*Wc[0,t1]Z0=0.
That is
(31)∫t0t1‍Z0*(t1-s)α-1Eα,α(A(t1-s)α)GG*Eα,α×(A*(t1-s)α)Z0ds=0.
Then we have
(32)Z0*Eα,α(A(t1-s)α)G=0
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 [ti,ti+1], i=0,1,2,…,k. There exist control inputs u0(t) and u^0(t) such that
(33)x(t1)=ωEα,1(At1α)+∫0t1‍(t1-s)α-1Eα,α(A(t1-s)α)Bx(s-τ)ds+∫0t1‍(t1-s)α-1Eα,α(A(t1-s)α)Gu0(s)ds=0,(34)x(t1)=ωEα,1(At1α)+∫0t1‍(t1-s)α-1Eα,α(A(t1-s)α)Bx(s-τ)ds+∫0t1‍(t1-s)α-1Eα,α(A(t1-s)α)Gu^0(s)ds=Z0.
By (34), we have
(35)ωEα,1(At1α)+∫0t1‍(t1-s)α-1Eα,α(A(t1-s)α)Bx(s-τ)ds=Z0-∫0t1‍(t1-s)α-1Eα,α(A(t1-s)α)Gu^0(s)ds.
Inserting (35) into (33) yields
(36)Z0+∫0t1‍(t1-s)α-1Eα,α(A(t1-s)α)×G(u0(s)-u^0(s))ds=0.
Multiplying Z0* on both side of (36) yields
(37)Z0*Z0+∫0t1‍(t1-s)α-1Z0*Eα,α(A(t1-s)α)×G(u0(s)-u^0(s))ds=0.
By (32) and (37), we have Z0*Z0=0. Thus, Z0=0. This is a contradiction.
IfWc[ti,ti+1] is singular for some i∈{1,…,k}, then there exists a nonzero vector Zi such that
(38)Zi*Wc[ti,ti+1]Zi=0.
That is
(39)∫titi+1‍Zi*(ti+1-s)α-1Eα,α(A(ti+1-s)α)×GG*Eα,α(A*(ti+1-s)α)Zids=0.
Then, it follows that
(40)Zi*Eα,α(A(ti+1-s)α)G=0
on s∈[ti,ti+1]. By formula (23) and the assumption that the system (8)–(12) is completely controllable, there exist control inputs ui-1(t) and ui(t) such that x(ti)=0 and
(41)x(ti+1)=Ii(0)Eα,1(Ati+1α)+∫titi+1‍(ti+1-s)α-1Eα,α(A(ti+1-s)α)·(Bx(s-τ)+Gui(s))ds=0.
Similarly, there exists a control inputu^i(t) such that
(42)x(ti+1)=Ii(0)Eα,1(Ati+1α)+∫titi+1‍(ti+1-s)α-1Eα,α(A(ti+1-s)α)·(Bx(s-τ)+Gu^i(s))ds=Zi.
By (42), we have
(43)Ii(0)Eα,1(Ati+1α)+∫titi+1‍(ti+1-s)α-1Eα,α(A(ti+1-s)α)Bx(s-τ)ds=Zi-∫titi+1‍(ti+1-s)α-1Eα,α(A(ti+1-s)α)Gu^i(s)ds.
Inserting (43) into (41) yields
(44)Zi+∫titi+1(ti+1-s)‍α-1×Eα,α(A(ti+1-s)α)G(ui(s)-u^i(s))ds=0.
Multiplying Zi* on both side of (44) yields
(45)Zi*Zi+∫titi+1‍(ti+1-s)α-1Zi*×Eα,α(A(ti+1-s)α)G(ui(s)-u^i(s))ds=0.
Combining (45) with (40) yields Zi*Zi=0. Thus, Zi=0. This is a contradiction.
Thus,Wc[ti,ti+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
(46)rank(G∣AG∣⋯∣An-1G)=n.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∈ℝn, there exists a control input u0(t) such that x(t1)=Z0. By the formula (23), it follows that
(47)Z0=x(t1)=ωEα,1(At1α)+∫0t1‍(t1-s)α-1Eα,α(A(t1-s)α)×(Bx(s-τ)+Gu(s))ds.
By Cayley-Hamilton theorem, we have
(48)tα-1Eα,α(Atα)=∑j=0n-1‍cj(t)Aj,
where cj(t) are functions in t, j=1,2,…,n-1. Combining the formula (48) and the equality (47), we have
(49)x(t1)-ωEα,1(At1α)-∫0t1‍(t1-s)α-1Eα,α(A(t1-s)α)Bx(s-τ)ds=∫0t1‍(t1-s)α-1Eα,α(A(t1-s)α)Gu(s)ds=∑j=0n-1‍AjG∫0t1‍cj(t1-s)u(s)ds=(G∣AG∣⋯∣An-1G).(d0d1⋮dn-1),
where dj=∫0t1‍cj(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
(50)rank(G∣AG∣⋯∣An-1G)=n.Sufficiency. Suppose that rank(G∣AG∣⋯∣An-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 [ti,ti+1],  i=0,1,…,k, respectively.
The formula (23) together with (48) yields (49). By the assumption that rank(G∣AG∣⋯∣An-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],
(51)x(t)=I1(0)Eα,1(Atα)+∫t1t‍(t-s)α-1Eα,α(A(t-s)α)×(Bx(s-τ)+Gu(s))ds,t∈(t1,t2].
Thus, it follows
(52)x(t2)=I1(0)Eα,1(At2α)+∫t1t2‍(t2-s)α-1Eα,α(A(t2-s)α)×(Bx(s-τ)+Gu(s))ds.
By (48) it follows taht
(53)x(t2)-I1(0)Eα,1(At2α)-∫t1t2‍(t2-s)α-1Eα,α(A(t2-s)α)Bx(s-τ)ds=∫t1t2‍(t2-s)α-1Eα,α(A(t2-s)α)Gu(s)ds=∑j=0n-1‍AjG∫t1t2‍cj(t2-s)u(s)ds=(G∣AG∣⋯∣An-1G).(d0′d1′⋮dn-1′),
where dj′=∫t1t2‍cj(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(ti,ti+1], respectively, we can prove that the system (8)–(12) is completely controllable on (ti,ti+1], i=2,…,k. In conclusion, the system (8)–(12) is completely controllable on J=[0,T]. This completes the proof.Remark 9.
From Theorem8, 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(t1+)-x(t1-)=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(54)E1/2,1/2(A(1-s)1/2)=∑k=0∞‍Ak(1-s)k/2Γ(k/2+1/2)=1Γ(1/2)I+1Γ(1)(0100)(1-s)1/2,(55)GG*=(21)(21)=(4221),(56)E1/2,1/2(A*(1-s)1/2)=∑k=0∞‍(A*)k(1-s)k(1/2)Γ(k(1/2)+1/2)=1Γ(1/2)I+1Γ(1)(0010)(1-s)1/2,(57)E1/2,1/2(A(2-s)1/2)=∑k=0∞‍Ak(2-s)k/2Γ(k/2+1/2)=1Γ(1/2)I+1Γ(1)(0100)(2-s)1/2,(58)E1/2,1/2(A*(2-s)1/2)=∑k=0∞‍(A*)k(2-s)k/2Γ(k/2+1/2)=1Γ(1/2)I+1Γ(1)(0010)(2-s)1/2.
By the formula (24)
(59)Wc[ti,ti+1]=∫titi+1‍(ti+1-s)-1/2[E1/2,1/2(A(ti+1-s)1/2)G)]×[G*E1/2,1/2(A*(ti+1-s)1/2)]ds,
we have
(60)Wc[0,1]=(8π+4π0.5+234π+1π0.54π+1π0.52π),Wc[1,2]=(4π+6π0.5+232π+1π0.52π+1π0.51π).
It is obvious that Wc[0,1] and Wc[1,2] are nonsingular. By Theorem 7, the system is completely controllable.On the other hand,(61)rank(G∣AG)=rank(2110)=2.
By Theorem 8, the system is completely controllable.Example 2.
Consider the time-invariant system (8)–(12). Choose
(62)A=(-1-4-206-1171),α=13,G=(201).
By computation, we have(63)rank(G∣AG∣A2G)=rank(2-460-1711-12)=3.
By Theorem 8, the system is completely controllable.



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