The constant variation formulae for singular fractional differential systems with delay

Computers and Mathematics with Applications 59 (2010) 1184–1190

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The constant variation formulae for singular fractional differential systems with delayI Jiang Wei School of Mathematical Science, Anhui University, Hefei, Anhui, 230039, PR China

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Keywords: Singular fractional differential systems with delay α − δ function Fundamental solution Constant variation formula

abstract This paper considers the Caputo singular fractional differential systems with delay, and the Riemann–Liouville singular fractional differential systems with delay. A new function α − δ is defined. By the D−inverse matrix and α − δ function, two fundamental solutions are given. The constant variation formulae for singular fractional differential systems with delay are obtained. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Recently, fractional differential systems have gained scholar’s attention [1–9]. In some practical systems, such as economic systems, biological systems, space-light industry systems and so on, due to the transmission of the signal or the mechanical transmission, we must study time delay [4,5,10–14]. In [13–17], we see that singular differential systems have obtained considerable importance due to their applications in various sciences. The systems we consider will be singular Caputo fractional differential systems with delay:



E (c Dα x(t )) = Ax(t ) + Bx(t − 1) + f (t ), x(t ) = ϕ(t ),

t ≥ 0, −1 ≤ t ≤ 0

(1)

and singular Riemann–Liouville fractional differential systems with delay



E (Dα x(t )) = Ax(t ) + Bx(t − 1) + f (t ), x(t ) = ϕ(t ),

t ≥ 0, −1 ≤ t ≤ 0

(2)

where x(t ) ∈ Rn is a state vector; A, B, E ∈ Rn×n are constant matrices; E ∈ Rn×n is a singular matrix; ϕ(t ) is the initial control function; 0 ≤ α < 1, f (t ), ϕ(t ) ∈ Rn ; c Dα x(t ) denotes α order-Caputo fractional derivative; Dα x(t ) denotes α order-Riemann–Liouville fractional derivative. This paper considers the Caputo singular fractional differential systems with delay and the Riemann–Liouville singular fractional differential systems with delay. A new function α − δ function is defined. By the D−inverse matrix and α − δ function, two fundamental solutions are given. The constant variation formulae for singular fractional differential systems with delay are obtained.

I This research has been supported by the National Nature Science Foundation of China (No. 10771001), the Key Program of Ministry of Education of China (No. 205068) and the Innovation Group Program of Anhui University. E-mail address: [email protected].

0898-1221/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2009.07.010

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2. Preliminaries Let us start with some definitions and preliminaries. Definition 1. Riemann–Liouville’s fractional integral of order α > 0 for a function f : R+ → R is defined as D−α f (t ) =

t

Z

1

Γ (α)

(t − θ )α−1 f (θ )dθ .

0

Definition 2. Caputo’s fractional derivative of order α (0 ≤ m ≤ α < m + 1) for a function f : R+ → R is defined as c

α

D f (t ) =

Γ (m − α + 1)

f (m+1) (θ )

t

Z

1

(t − θ )α−m

0

dθ .

Definition 3. Riemann–Liouville’s fractional derivative of order α (0 ≤ m ≤ α < m + 1) for a function f : R+ → R is defined as Dα f (t ) =



1

Γ (m − α + 1)

d

m+1 Z

dt

t 0

f (θ )

(t − θ )α−m

dθ .

Remark. From [6] we have that for 0 ≤ m ≤ α < m + 1 Dα f (t ) =c Dα f (t ) +

m X

f (k) (0)

k =0

Γ (1 + k − α)

t k−α .

Specially, when 0 ≤ α < 1, Dα f (t ) = c Dα f (t ) +

f (0)

Γ (1 − α)

t −α .

From [1] we have that for 0 ≤ m ≤ α < m + 1 the Laplace transformation of D−α f (t ), c Dα f (t ) and Dα f (t ). L(D−α f (t )) = λ−α L[f (t )]. L(c Dα f (t )) = λα L[f (t )] −

m X

λα−k−1 f (k) (0).

k=0

L(Dα f (t )) = λα L[f (t )] −

m X

λk [Dα−k−1 f (0)].

k=0

Specially, when 0 ≤ α < 1, we have L(D−α f (t )) = λ−α L[f (t )].

L(c Dα f (t )) = λα L[f (t )] − λα−1 f (0). L(Dα f (t )) = λα L[f (t )] − Dα−1 f (0).

Definition 4. If det(λE − A) 6≡ 0, we call matrix couple (A, E ) regular. If (A, E ) is regular, we call system (1) regular. From [1], it is evident that if (E , A) is regular, the system (1) is solvable. From [11], we have Lemma 1. For any square matrix E, the Drazin inverse matrix E d exists and is unique, and if the Jordan normalized form is

 E=T

J1 0

0 J0



T −1 .

Here J0 is a nilpotent matrix, J1 and T is an invertible matrix. Then Ed = T



J1−1 0



0 T −1 . 0

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J. Wei / Computers and Mathematics with Applications 59 (2010) 1184–1190

3. Constant variation formula of system (1) In this section we give the constant variation formula of the singular Caputo fractional differential systems with delay. We separate system (1) into three systems

 

E (c Dα x(t )) = Ax(t ) + Bx(t − 1) + EE d f (t ), x(t ) ≡ 0,

t ≥0 −1 ≤ t ≤ 0,

E (c Dα x(t )) = Ax(t ) + Bx(t − 1) + (I − EE d )f (t ), x(t ) ≡ 0,

t ≥0 −1 ≤ t ≤ 0,

(3)

(4)

and



E (c Dα x(t )) = Ax(t ) + Bx(t − 1), x(t ) = ϕ(t ),

t ≥0 −1 ≤ t ≤ 0 .

(5)

It is easy to see that Lemma 2. Assume that x1 (t ), x2 (t ), x3 (t ) is the solution of system (3)–(5) respectively, we have that x(t ) = x1 (t )+ x2 (t )+ x3 (t ) is the solution of system (1). Definition 5. Let X (t ) ∈ Rn×n , and satisfy

 c α E ( D X (t )) = AX (t ) + BX (t − 1), EE d , t = 0, X (t ) = 0, −1 ≤ t < 0

(6)

then X (t ) is called the corresponding fundamental solution of system (3). We also call X (t ) the first fundamental solution of system (1), the first fundamental solution for short. To give the corresponding fundamental solution of system (4), we define a new function: Definition 6. Let α (0 ≤ α < 1), the function

δ α (t ) =

Γ (1 − α)

δ(θ ) dθ (t − θ )α

t

Z

1

0

is called α − δ function, here δ(t ) is the δ -function. Lemma 3. The Laplace transformation of α − δ function is L(δ α (t )) =

1

λ(1−α)

.

Proof. α

L(δ (t )) = L



δ(θ ) dθ (t − θ )α

−α




L (δ(t ))

L t −α




Γ (1 − α) 1

=

Γ (1 − α)

=

Γ (1 − α)

=

t

Z

1

1

L t

+∞

Z

1

0

Γ (1 − α)



t −α e−λt dt . 0

Let ξ = λt, we have

 −α ξ 1 L(δ (t )) = e−ξ dξ Γ (1 − α) 0 λ λ Z +∞ 1 1 = ξ (1−α)−1 e−ξ dξ Γ (1 − α) λ(1−α) 0 1

α

=

1

λ(1−α)

Z

. 

+∞

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Definition 7. Let Y (t ) ∈ Rn×n , and satisfy

 c α E ( D Y (t )) = AY (t ) + BY (t − 1) + (I − EE d )δ α (t ), I − EE d , t = 0, Y ( t ) = 0, −1 ≤ t < 0

(7)

then Y (t ) is called the corresponding fundamental solution of system (4). We also call Y (t ) as the second fundamental solution of system (1), the second fundamental solution for short. There δ α (t ) is the α − δ function. Let H (λ) = λα E − A − e−λ B and L−1 denote inverse transformation of Laplace-transformation, from (6) we have X (t ) = L−1 [λα−1 H −1 (λ)EEE d ].

(8)

From (7), we have

(λα E − A − e−λ B)L[Y (t )] = λα−1 (E + I )(I − EE d ), H (λ)L[Y (t )] = λα−1 (E + I )(I − EE d ), that is Y (t ) = L−1 [λα−1 H −1 (λ)(E + I )(I − EE d )].

(9)

Theorem 1. Assume that (E , A) is regular, X (t ) is the first fundamental solution, we have the solution of system (3) x(t ) =

t

Z

(c D1−α X )(t − θ)E d f (θ )dθ + E d D−α [f (t )]. 0

Proof. Take the Laplace transformation for system (3), we have L[x(t )] = H −1 (λ)EE d L[f (t )]

λ1−α λα−1 H −1 (λ)EEE d E d L[f (t )] λ1−α L[X (t )]E d L[f (t )] (L[(c D1−α X )(t )] + λ−α EE d )E d L[f (t )] L[(c D1−α X )(t )]E d L[f (t )] + E d L[D−α f (t )].

= = = = That is x(t ) =

t

Z

(c D1−α X )(t − θ )E d f (θ )dθ + E d D−α [f (t )].  0

Theorem 2. Assume that (E , A) is regular, Y (t ) is the second fundamental solution, we have the solution of system (4) x(t ) =

t

Z

(c D1−α Y )(t − θ )(I + E (I − EE d ))−1 f (θ )dθ + (I − EE d )(I + E (I − EE d ))−1 D−α [f (t )]. 0

Proof. From Lemma 1, we have that if E = T −1 I − EE d = T −1 E+I =T

−1





0 0

J1 + I 0



0 0

J1 0

0 J0



T,



0 T, I



0 J0 + I

(E + I )(I − EE d ) = T −1 I − EE d = T −1





0 0

0 J0 + I

T,



0 J0 + I



TT −1



T.

I 0

0

 −1

J0 + I

= (E + I )(I − EE d )(I + E (I − EE d ))−1 .

T

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J. Wei / Computers and Mathematics with Applications 59 (2010) 1184–1190

Take the Laplace transformation for system (4), we have L[x(t )] = H −1 (λ)(I − EE d )L[f (t )] H −1 (λ)(E + I )(I − EE d )(I + E (I − EE d ))−1 L[f (t )]

= = = = = =

λ1−α λα−1 H −1 (λ)(E + I )(I − EE d )(I + E (I − EE d ))−1 L[f (t )] λ1−α L[Y (t )](I + E (I − EE d ))−1 L[f (t )] (L[(c D1−α Y )(t )] + λ−α (I − EE d ))(I + E (I − EE d ))−1 L[f (t )] L[(c D1−α Y )(t )](I + E (I − EE d ))−1 L[f (t )] + λ−α (I − EE d )(I + E (I − EE d ))−1 L[f (t )] L[(c D1−α Y )(t )](I + E (I − EE d ))−1 L[f (t )] + (I − EE d )(I + E (I − EE d ))−1 L[D−α f (t )].

That is x(t ) =

t

Z

(c D1−α Y )(t − θ )(I + E (I − EE d ))−1 f (θ )dθ + (I − EE d )(I + E (I − EE d ))−1 D−α [f (t )].  0

From Lemma 2, Theorems 1 and 2, we have the constant variation formula for singular fractional differential systems with delay (1). Theorem 3. Assume that x(t , ϕ(t ), 0) is the solution of system (5), we have that the solution of system (1) x(t , ϕ(t ), f (t )) can be written as x(t ) =

t

Z

(c D1−α )[X (t − θ )E d + Y (t − θ )(I + E (I − EE d ))−1 ]f (θ )dθ 0

+ (E + (I − EE d ))−1 D−α [f (t )] + x(t , ϕ(t ), 0),

(10)

there X (t ) is the first fundamental solution, Y (t ) is the second fundamental solution. Proof. x(t ) =

t

Z

(c D1−α X )(t − θ )E d f (θ )dθ + E d D−α [f (t )] +

t

Z

0

(c D1−α Y )(t − θ )(I + E (I − EE d ))−1 f (θ )dθ 0

+ (I − EE d )(I + E (I − EE d ))−1 D−α [f (t )] + x(t , ϕ(t ), 0) Z t = (c D1−α )[X (t − θ )E d + Y (t − θ )(I + E (I − EE d ))−1 ]f (θ )dθ 0

+ (E d + (I − EE d )(I + E (I − EE d ))−1 )D−α [f (t )] + x(t , ϕ(t ), 0)  −1     −1 I 0 0 d d d −1 −1 J 1 −1 0 0 E + (I − EE )(I + E (I − EE )) = T T +T T 0 I 0 J0 + I 0 0  −1    0 0 J 0 = T −1 1 T + T −1 T 0 (J0 + I )−1 0 0  −1  J 0 = T −1 1 T 0 (J0 + I )−1  −1 J 0 = T −1 1 T 0 J0 + I    −1 J1 0 0 0 −1 = T + T 0

J0

0

I

= (E + (I − EE d ))−1 . That is x(t ) =

t

Z

(c D1−α )[X (t − θ )E d + Y (t − θ )(I + E (I − EE d ))−1 ]f (θ )dθ 0

+ (E + (I − EE d ))−1 D−α [f (t )] + x(t , ϕ(t ), 0).  4. Constant variation formula of system (2) In this section, we give the constant variation formula of the singular Riemann–Liouville fractional differential systems with delay (2).

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From Section 1, we have

ϕ(t )(0) −α Dα x(t ) =c Dα x(t ) + t . Γ (1 − α) Take it to system (2), system (2) will become  

E (c Dα x(t )) = Ax(t ) + Bx(t − 1) + f (t ) −

x(t ) = ϕ(t ),

ϕ(0) t −α , Γ (1 − α)

t ≥ 0,

−1 ≤ t ≤ 0.

(11)

Theorem 4. Assume that x(t , ϕ(t ), 0) is the solution of system (5), we have that the solution x(t , ϕ(t ), f (t )) of the singular Riemann–Liouville fractional differential systems with delay (2) can be written as x(t ) =

t

Z

(c D1−α )[X (t − θ )E d + Y (t − θ )(I + E (I − EE d ))−1 ][f (θ )]dθ

0

+ (E + (I − EE d ))−1 D−α [f (t )] + x(t , ϕ(t ), 0) Z t ϕ(0) (c D1−α )[X (t − θ )E d + Y (t − θ )(I + E (I − EE d ))−1 ][θ −α ]dθ − ϕ(0)(E + (I − EE d ))−1 , − Γ (1 − α) 0 there X (t ) is the first fundamental solution, Y (t ) is the second fundamental solution. Proof. From Theorem 3, (11), we have x(t ) =

t

Z

(c D1−α )[X (t − θ )E d + Y (t − θ )(I + E (I − EE d ))−1 ][f (θ ) −

0

ϕ(0) θ −α ]dθ Γ (1 − α)

ϕ(0) + (E + (I − EE d ))−1 D−α [f (t ) − t −α ] + x(t , ϕ(t ), 0) Γ (1 − α) Z t = (c D1−α )[X (t − θ )E d + Y (t − θ )(I + E (I − EE d ))−1 ][f (θ )]dθ + (E + (I − EE d ))−1 D−α [f (t )] + x(t , ϕ(t ), 0) 0 Z t ϕ(0) − (c D1−α )[X (t − θ )E d + Y (t − θ )(I + E (I − EE d ))−1 ][θ −α ]dθ Γ (1 − α) 0 ϕ(0) − (E + (I − EE d ))−1 D−α [t −α ]. Γ (1 − α) For D−α [t −α ] =

t

Z

1

Γ (α)

(t − θ )α−1 θ −α dθ ,

0

let ξ = θt , we have D−α [t −α ] =

=

1

Z

1

Γ (α)

(1 − ξ )α−1 ξ −α dξ =

0

1

Γ (α)

B(α, 1 − α)

1 Γ (α)Γ (1 − α) = Γ (1 − α). Γ (α) Γ (α + (1 − α))

There B(a, b) is the β − function [1]. So, we have x(t ) =

Z

t

(c D1−α )[X (t − θ )E d + Y (t − θ )(I + E (I − EE d ))−1 ][f (θ )]dθ + (E + (I − EE d ))−1 D−α [f (t )] + x(t , ϕ(t ), 0) Z t ϕ(0) − (c D1−α )[X (t − θ )E d + Y (t − θ )(I + E (I − EE d ))−1 ][θ −α ]dθ − ϕ(0)(E + (I − EE d ))−1 .  Γ (1 − α) 0 0

From Theorems 3 and 4, it is easy to prove the following theorem. Theorem 5. Assume that x(t , ϕ(t ), f (t )) is the solution of the singular Caputo fractional differential systems with delay (1), x¯ (t , ϕ(t ), f (t )) is the solution of the singular Riemann–Liouville fractional differential systems with delay (2), we have

ϕ(0) Γ (1 − α) − ϕ(0)(E + (I − EE d ))−1 ,

x¯ (t , ϕ(t ), f (t )) = x(t , ϕ(t ), f (t )) −

t

Z

(c D1−α )[X (t − θ )E d + Y (t − θ )(I + E (I − EE d ))−1 ][θ −α ]dθ 0

there X (t ) is the first fundamental solution, Y (t ) is the second fundamental solution.

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J. Wei / Computers and Mathematics with Applications 59 (2010) 1184–1190

Acknowledgement The author is very grateful to Professor Zheng Zuxiu for his helpful and valuable comments and suggestions for improving this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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