On the Interval Controllability of Fractional Systems with Control Delay

Transcription

1 Journal of Mathematics Research; Vol. 9, No. 5; October 217 ISSN E-ISSN Published by Canadian Center of Science and Education On the Interval Controllability of Fractional Systems with Control Delay Jiang Wei 1 1 School of Mathematical Science, University of Anhui, P. R. China Correspondence: Jiang Wei, School of Mathematical Science, University of Anhui, Hefei, Anhui, 2339, P. R. China. jiangwei8918@126.com Received: July 19, 217 Accepted: August 8, 217 Online Published: September 17, 217 doi:1.5539/jmr.v9n5p87 URL: The research is financed by National Natural Science Foundation of China (No , No ), Anhui Provincial Natural Science Foundation (No., No MA12, 15885MA1), Program of Natural Science of Colleges of Anhui Province(No. KJ213A32). Abstract In this paper we discuss the interval controllability of fractional systems with control delay. We give the concept and some results of interval controllability. As a main result we give the necessary conditions and sufficient conditions for the interval controllability of fractional systems with control delay. Keywords: interval controllability, fractional control systems, control delay 1. Introduction The investigation of fractional differential systems have wide application in sciences and engineering such as physics, chemistry, mechanics. So in recent years, the research of fractional differential systems is extensive around the world, and there has been a significant development (Anatoly A. K., Hari M. S. & Juan, J. T., 26; Podlubny, I., 1999; Lakshmikantham, V., 28; Yong, Z., 28; Shantanu, D., 28; Jiang, W., 21; Jiang, W., 211). We also find that the phenomena of time delay exist in many systems, such as economics, biological, physiological and spaceflight systems, specially, time delay frequently exists in control. Up to now a lot of fabulous accomplishments have been given by many scholars (Hale, J., 1992; Zuxiu Z., 199; Sebakhy, O. & Bayoumi, M. M., 1971; Wansheng T. & Guangquan, L., 1995; Jiang, W. & WenZhong, S., 21; Jiang, W., 26; Jiang, W., 212; Hai, Z., Jinde, C. & Wei, J., 213; Xian-Feng, Z., Jiang, W. & Liang-Gen, H., 213; Xian-Feng, Z., Jiang, W. & Liang-Gen, H., 213; Song Liu & et al., 216; Xian-Feng, Z., Fuli, Y. & Wei, J., 215). The fractional systems with control delay have two factors (fractional order differential, and delay) synchronously. So the results of this paper must be useful. Definition 1 Riemann-Liouville s fractional integral of order α > for a function f : R + R is defined as Here Γ( ) is a Gamma-Function. D α f (t) = 1 Γ(α) t (t θ) α 1 f (θ)dθ. Definition 2 Caputo s fractional derivative of order α ( m α < m + 1) for a function f : R + R is defined as Here Γ( ) is a Gamma-Function. c D α f (t) = 1 Γ(m α + 1) t f (m+1) (θ) dθ. (t θ) α m In this paper we study the fractional control systems with control delay c D α x(t) = Ax(t) + Bu(t) + Cu(t h), t, x() = x, u(t) = ψ(t), h t, where x(t) R n is state vector; u(t) R m is control vector, A R n n, B, C R n m are any matrices;h > is time control delay; and ψ(t) the initial control function. < α 1, c D α x(t) denotes α order-caputo fractional derivative of x(t). In the investigation for controllability of fractional systems with control delay (1), we find that controllability of such systems is closely related to time interval. So we must to pay much attention to the study of the controllability for time interval. (1) 87

2 Journal of Mathematics Research Vol. 9, No. 5; 217 For the time interval I = [a, b], we give the concept of interval controllability. There a R and b R, b > a or b = +. Definition 3 The system (1) is said to be controllable in interval I if one can at any time t I reach any state from any admissible initial state and initial control. For systems (1), the time delay h play a very important role in the controllability. In this paper we discuss the interval controllability of fractional systems with control delay (1). In section 2, we give some preliminaries. In section 3, we give some results about controllability the interval [, h] for fractional systems with control delay (1). In section, the necessary and sufficient conditions for controllability the interval (h, ) for fractional systems with control delay (1). 2. Preliminaries In this section, we give some preliminaries. Lemma 1 The general solution of system { c D α x(t) = Ax(t) + f (t), t, x() = x, (2) can be written as: Here x(t) = Φ α,1 (A, t)x() + Φ α,β (A, t) = t k= Φ α,α (A, t τ) f (τ)dτ. (3) A k t αk+β 1 Γ(αk + β). Proof From Anatoly A.Kilbas, Hari M.Srivastava, Juan J.Trujillo(26) and I.Podlubny (1999) (page 16), we know that the Laplace Transform of Caputo fractional derivative of function f (t) is L( c D α f (t))(s) = s α L( f (t))(s) s α 1 f (). We consider the Mittag-Leffler function in two parameters (Anatoly A.Kilbas, Hari M.Srivastava, Juan J.Trujillo(26); I.Podlubny (1999) ) z k E α,β (z) =, (α >, β > ). Γ(αk + β) We have k= L(Φ α,β (A, t))(s) = e st t β 1 E α,β (At α )dt = 1 Γ(αk+β) e st t β 1 (At α ) k dt k= = A k Γ(αk+β) e st t αk+β 1 dt = k= k= = A k s (αk+β) Γ(αk+β) e h h αk+β 1 dh = k= k= Take Laplace Transform for systems (2), we have There = (As α ) k s β = (I As α ) 1 s β k= = (s α I A) 1 s α β. A k Γ(αk+β) h hαk+β 1 1 e s αk+β 1 s dh A k s (αk+β) Γ(αk+β) Γ(αk + β) s α L(x(t))(s) s α 1 x() = AL(x(t))(s) + L( f (t))(s). L(x(t))(s) = (s α I A) 1 s α 1 x() + (s α I A) 1 L( f (t))(s) = L(Φ α,1 (A, t))(s)x() + L(Φ α,α (A, t))(s)l( f (t))(s) = L(Φ α,1 (A, t))(s)x() + L(Φ α,α (A, t) f (t))(s). Φ α,α (A, t) f (t) = t Φ α,α (A, t τ) f (τ)dτ is the convolution of Φ α,α (A, t) and f (t) (Anatoly A.Kilbas, Hari M.Srivastava, Juan J.Trujillo(26); I.Podlubny (1999)). Then we have L(x(t))(s) = L(Φ α,1 (A, t))(s)x() + L( t Φ α,α(a, t τ) f (τ)dτ)(s). 88

3 Journal of Mathematics Research Vol. 9, No. 5; 217 From Lemma 1, we have Theorem 1 x(t) = Φ α,1 (A, t))x Φ α,α(a, t τ) f (τ)dτ. The general solution of system (1) can be written as that when t h when t h Let x(t) = Φ α,1 (A, t)x h (Φ α,α (A, t τ)b + Φ α,α (A, t τ h)c)u(τ)dτ Φ t h α,α(a, t τ)bu(τ)dτ + Φ h α,α(a, t τ h)cψ(τ)dτ. x(t) = Φ α,1 (A, t)x (Φ α,α(a, t τ)bu(τ)dτ Φ α,α(a, t τ h)cψ(τ)dτ. h h (a) (b) A B, C = α + Aα + A 2 α + + A n 1 α + + A+ A A n 1, where n is the order of A and α = ImageB, = ImageC. Then the space A B, C is spanned by the columns of matrix [B, AB, A 2 B,, A n 1 B, C, AC, A 2 C,, A n 1 C]. Lemma 2 For Beta function we have B(z, w) = 1 s z 1 (1 s) w 1 ds, (Re(z) >, Re(w) > ), B(z, w) = Γ(z)Γ(w) Γ(z + w). The proof of this Lemma can be seen in I.Podlubny (1999)(page 7). Lemma 3 c D α Φ α,1 (A, t) = AΦ α,1 (A, t). (5) Proof From Lemma 2, we have c D α Φ α,1 (A, t) = 1 = 1 Γ(1 α) = 1 Γ(1 α) 1 = k= = k= = k= = A k=1 t Φ α,1 (A,θ) Γ(1 α) (t θ) α t A k θ αk 1 (αk) (t θ) α Γ(αk+1) dθ t dθ A k+1 θ α(k+1) 1 (α(k+1)) (t θ) k= α Γ(α(k+1)+1) dθ A k+1 s α(k+1) 1 t α(k+1) 1 Γ(1 α)t α (1 s) α Γ(α(k+1)) tds A k+1 t αk Γ(1 α)γ(α(k+1)) k= A k+1 t αk B(α(k+1),1 α) Γ(1 α)γ(α(k+1)) A k t αk Γ(αk+1) = AΦ α,1 (A, t). For any z R n and t > h, define W(t) : R n R n by 1 sα(k+1) 1 (1 s) ( α+1) 1 ds W(t)z = t h [(Φ α,α (A, t τ)b + Φ α,α (A, t τ h)c) (Φ α,α (A, t τ)b + Φ α,α (A, t τ h)c) T ]zdτ [Φ t h α,α(a, t τ)bb T (Φ α,α (A, t τ)) T ]zdτ. 89

4 Journal of Mathematics Research Vol. 9, No. 5; The Controllability of (1) in Interval [, h] In this section, we discuss the controllability of (1) in interval [, h]. Let I = [, ), I 1 = [, h], I 2 = (h, ). From Definition 3, we know that system (1) is controllable in interval I 1 if and only if for any state x R n, any time t I 1, and any admissible initial state x and initial control u(t) = ψ(t)( h t ), there is a control ū(t) such that the solution x(t) of system (1) from x can reach x at time t, that is x( t) = x. If t h, systems (1) become c D α x(t) = Ax(t) + Bu(t) + Cψ(t h), t t, x() = x, u(t) = ψ(t), h t. From Theorem 1, the general solution of system (1) can be written as that when t h, (6) x(t) = Φ α,1 (A, t)x() Φ α,α(a, t τ)bu(τ)dτ Φ α,α(a, t τ h)cψ(τ)dτ. h h The control term Cu(t h) can t play a part in the controllability of (1). Consider fractional control systems without control delay c D α x(t) = Ax(t) + Bu(t), t, x() = x, u() = ψ(). (7) (8) Theorem 2 System (1) is controllable in interval I 1 if and only if system (8) is controllable in interval I 1. Proof From Section 2, we have that the solution of (8) can be written as x(t) = Φ α,1 (A, t)x() Φ α,α(a, t τ)bu(τ)dτ. We firstly prove sufficiency. We say system (8) is controllable in interval I 1. For any state x R n, any time t I 1, and any admissible initial state x and initial control u(t) = ψ(t)( h t ), let ˆx = x t h Φ h α,α(a, t τ h)cψ(τ)dτ. For system (8) is controllable in interval I 1, from (9), there is control u (t) with u () = ψ() such that ˆx = Φ α,1 (A, t)x() + t Φ α,α(a, t τ)bu (τ)dτ. x = Φ α,1 (A, t)x() + t Φ α,α(a, t τ)bu (τ)dτ + t h Φ h α,α(a, t τ h)cψ(τ)dτ. Let { u ū(t) = (t) t t, ψ(t) h t. By the definition, we know that system (1) is controllable in interval I 1. The proof of the necessity of Theorem 2 can be easily gotten from the proof of the sufficiency in inverse way. From Shantanu Das(28), we have that system (8) is controllable in interval I 1 if and only if the rank of matrix [B, AB, A 2 B,, A n 1 B] is n. So from Theorem 2, we have Theorem 3 System (1) is controllable in interval I 1 if and only if rank[b, AB, A 2 B,, A n 1 B] = n. From Shantanu Das(28), we know that system (8) is controllable in interval I 1 = [, h] if and only if it is controllable in interval I = [, ). But for system (1), this conclusion is not true. We will discuss the controllability of (1) in interval (h, ) and give an example to illustrate that in next section. (9) 9

5 Journal of Mathematics Research Vol. 9, No. 5; 217. The Controllability of (1) in Interval (h, ) In this section we discuss the controllability of (1) in interval (h, ). Firstly we give a conclusion about the relationship of ImW(t 1 ) and A B, C. Theorem Proof If Φ α,1 (A, h) is nonsingular, and system (1) is controllable at any time t 1 > h, then ImW(t 1 ) = A B, C. To show ImW(t 1 ) = A B, C is equivalence to showing that If x KerW(t 1 ) and x,then KerW(t 1 ) = KerB T (A T ) i KerC T (A T ) j. (1) i= = x T W(t 1 ) x = t 1 h (Φ α,α (A, t 1 τ)b + Φ α,α (A, t 1 τ h)c) T x 2 dτ 1 t 1 h BT Φ α,α (A, t 1 τ) T x 2 dτ, j= that is { = (Φα,α (A, t 1 τ)b + Φ α,α (A, t 1 τ h)c) T x, τ t 1 h, = B T Φ α,α (A, t 1 τ) T x, t 1 h < τ t 1, (11) For (1) is controllable, from theorem 1, for any t 1 and any ˆx, for x(t 1 ) = h Φ α,α(a, t 1 τ h)c)ψ(τ)dτ there is ū such that = Φ α,1 (A, t 1 ) ˆx 1 h (Φ α,α (A, t 1 τ)b + Φ α,α (A, t 1 τ h)c)ū(τ)dτ 1 Φ t 1 h α,α(a, t 1 τ)bū(τ)dτ. Left time x T, from (11), we have Let t 1 h and ˆx = Φ 1 α,1 (A, h)bbt x, we have = x T Φ α,1 (A, t 1 ) ˆx. (12) = x T BB T x. B T x =. Take Caputo s fractional derivative (12) for t 1, from Lemma 3 we have Let t 1 h and ˆx = Φ 1 α,1 (A, h)bbt A T x, we have = x T AΦ α,1 (A, t 1 ) ˆx. = x T ABB T A T x. B T A T x =. Repeatedly take k times Caputo s fractional derivative for the (12), and let t 1 h, ˆx = Φ 1 α,1 (A, h)bbt (A T ) k x we have = x T A k BB T (A T ) k x, B T (A T ) k x =, f or k =, 1, 2,, n 1. (13) For (12), let t 1 h and ˆx = Φ 1 α,1 (A, h)cct x, we have = x T CC T x. C T x =. 91

6 Journal of Mathematics Research Vol. 9, No. 5; 217 Take Caputo s fractional derivative (12) for t 1, from Lemma 3 we have Let t 1 h and ˆx = Φ 1 α,1 (A, h)cct A T x, we have = x T Φ α,1 (A, t 1 )A ˆx. = x T ACC T A T x. C T A T x =. Repeatedly take k times Caputo s fractional derivative for the (12), and let t 1 h, ˆx = Φ 1 α,1 (A, h)cct (A T ) k x we have = x T A k CC T (A T ) k x, C T (A T ) k x =, f or k =, 1, 2,, n 1. (1) From (13) and (1) we have x n 1 Conversely, let x n 1 i= i= KerB T (A T ) i n 1 KerB T (A T ) i n 1 j= j= KerC T (A T ) j. KerW(t 1 ) KerB T (A T ) i KerC T (A T ) j. (15) i= j= KerC T (A T ) j, (13) and (1) is true. From Cayley Hamilton Theorem ( Zheng Zuxiu (199) ), we have Φ α,α (A, t 1 ) = k= A k t 1 αk+α 1 Γ(αk+α) = n 1 γ k (t 1 )A k (16) k= Then from (13),(1) and (16), for t 1 h < τ t 1, n 1 B T Φ α,α (A, t 1 τ) T A T x = γ k (t 1 τ)b T (A T ) k x =. and for τ t 1 h, k= (Φ α,α (A, t 1 τ)b + Φ α,α (A, t 1 τ h)c) T x = n 1 γ k (t 1 τ)b T (A T ) k x + n 1 γ k (t 1 τ h)c T (A T ) k x = k= therefore x KerW(t 1 ). KerW(t 1 ) KerB T (A T ) i KerC T (A T ) j. (17) k= i= j= From (15) and (17), we have (1) is true and the proof of Theorem is over. Theorem 5 If Φ α,1 (A, h) is nonsingular, the system (1) is controllable in (h, ), that is for any time t 1 > h system (1) is controllable at t 1 if and only if W(t 1 ) is nonsingular. Proof If for any time t 1 h, W(t 1 ) is nonsingular, for any x R n, let z = x Φ α,1 (A, t 1 )x() h Φ α,α (A, t 1 τ h)cψ(τ)dτ. (18) From (a), let ū(τ) = (B T Φ α,α (A, t 1 τ) T +C T Φ α,α (A, t 1 τ h) T )W(t 1 ) 1 z, τ t 1 h, B T Φ α,α (A, t 1 τ) T W(t 1 ) 1 z, t 1 h < τ t 1, 92

7 Journal of Mathematics Research Vol. 9, No. 5; 217 we have From the definition of W(t 1 ) and (18), we have x(t 1 ) = Φ α,1 (A, t 1 )x() + Φ h α,α(a, t 1 τ h)cψ(τ)dτ 1 h (Φ α,α (A, t 1 τ)b + Φ α,α (A, t 1 τ h)c)ū(τ)dτ 1 Φ t 1 h α,α(a, t 1 τ)bū(τ)dτ = Φ α,1 (A, t 1 )x() + Φ h α,α(a, t 1 τ h)cψ(τ)dτ 1 h (Φ α,α (A, t 1 τ)b + Φ α,α (A, t 1 τ h)c) (B T Φ α,α (A, t 1 τ) T + C T Φ α,α (A, t 1 τ h) T )W(t 1 ) 1 zdτ 1 Φ t 1 h α,α(a, t 1 τ)bb T Φ α,α (A, t 1 τ) T W(t 1 ) 1 zdτ. x(t 1 ) = Φ α,1 (A, t 1 )x() + Φ h α,α(a, t 1 τ h)cψ(τ)dτ +W(t 1 )W(t 1 ) 1 z = Φ α,1 (A, t 1 )x() + Φ h α,α(a, t 1 τ h)cψ(τ)dτ + z = x. From Definition 3, we have that the system (1) is controllable in (h, ). Now we prove that if system (1) is controllable in (h, ) then for any time t 1 > h such that W(t 1 ) is nonsingular. If this is not true, then there is x such that = x T W(t 1 ) x = t 1 h (Φ α,α (A, t 1 τ)b + Φ α,α (A, t 1 τ h)c) T x 2 dτ 1 t 1 h BT Φ α,α (A, t 1 τ) T x 2 dτ, that is { = (Φα,α (A, t 1 τ)b + Φ α,α (A, t 1 τ h)c) T x, τ t 1 h, = B T Φ α,α (A, t 1 τ) T x, t 1 h < τ t 1, (19) For (1) is controllable, from theorem 1, for x(t 1 ) = h Φ α,α(a, t 1 τ h)c)ψ(τ)dτ there is ū such that Time x T, from (19) we have = Φ α,1 (A, t 1 ) x 1 h (Φ α,α (A, t 1 τ)b + Φ α,α (A, t 1 τ h)c)ū(τ)dτ 1 Φ t 1 h α,α(a, t 1 τ)bū(τ)dτ. = x T Φ α,1 (A, t 1 ) x. We have that Φ α,1 (A, t 1 ) is singular for any time t 1 h. a contradiction with Φ α,1 (A, h) is nonsingular. The proof of Theorem 5 is over. From the proof of Theorem 5 we can see that Φ α,1 (A, h) is nonsingular can be changed as that if there is t [h, ), Φ α,1 (A, t) is nonsingular. Theorem 5 can be generalized as Theorem 6 For system (1) we have 1. If for any time t 1 (h, ), W(t 1 ) is nonsingular, then system (1) is controllable in (h, ). 2. If there is t (h, ), Φ α,1 (A, t) is nonsingular and for any time t 1 (h, ) system (1) is controllable at t 1, then W(t 1 ) is nonsingular. Now we discuss the algebraic property for the controllability of system (1). Theorem 7 Proof If Φ α,1 (A, h) is nonsingular and the system (1) is controllable in (h, ), then rank[b, AB, A 2 B,, A n 1 B, C, AC, A 2 C,, A n 1 C] = n. From Theorem, for any time t 1 h, ImW(t 1 ) = A B, C. From Theorem 5, ImW(t 1 ) = R n. So, we have 93

8 Journal of Mathematics Research Vol. 9, No. 5; 217 rank[b, AB, A 2 B,, A n 1 B, C, AC, A 2 C,, A n 1 C] = n. Remark If Φ α,1 (A, h) is nonsingular, and rank[b, AB, A 2 B,, A n 1 B, C, AC, A 2 C,, A n 1 C] n, from Theorem 7 we can judge that the system (1) is uncontrollable in (h, ). Now we give two examples to illustrate the use of Theorem 6 and Theorem 7. Example 1 Consider the fractional control systems with control delay c D 3 x 1 (t) = x 2 (t) + u 1 (t), t, c D 3 x 2 (t) = u 2 (t h), t, x() = x, u(t) = ψ(t), h t. (2) Compare (2) with (1), we have A = ( 1 ) ( 1, B = ) (, C = 1 ), α = 3. Then Φ α,β (A, t) = A k t αk+β 1 k= Γ(αk+β) = t β 1 Γ(β) t α+β 1 Γ(α+β) t β 1 Γ(β) = tβ 1 ( 1 Γ(β) I +. ) t α+β 1 Γ(α+β) So Φ 3,1(A, t) = Φ 3, 3 (A, t) = 3 t Γ( 3 +1) 1 1 t 1 Γ( 3 ) 2t 1 2 Γ( 1 2 ) t 1 Γ( 3 ) 3 t 3Γ( 3 ) = 1 1 = t 1 Γ( 3 ) 2t 1 2 π t 1 Γ( 3 ),. Φ 3, 3 (A, t τ)b + Φ 3, 3 (A, t τ h)c = (t τ) 1 1 2(t τ h) 2 Γ( 3 ) π (t τ h) 1 Γ( 3 ), [Φ 3, 3 (A, t τ)b + Φ 3, 3 (A, t τ h)c][φ 3, 3 (A, t τ)b + Φ 3, 3 (A, t τ h)c]t = (t τ) 1 2 (Γ( 3 ))2 + (t τ h) 2(t τ h) 1 π πγ( 3 ) 2(t τ h) 1 (t τ h) 1 2 πγ( 3 ) (Γ( 3 ))2, [Φ 3, 3 (A, t τ)b][φ 3, 3 (A, t τ)b]t = (t τ) 1 2 = (Γ( 3 ))2. (t τ) 1 Γ( 3 ) (t τ) 1 Γ( 3 ) T 9

9 Journal of Mathematics Research Vol. 9, No. 5; 217 For when t > h, W(t) = t h [(Φ 3, 3 (A, t τ)b + Φ 3, 3 (A, t τ h)c), 3 (A, t τ)b + Φ 3, 3 (A, t τ h)c)t ]dτ (Φ 3 [Φ t h 3, 3 (A, t τ)bbt (Φ 3, 3 (A, t τ))t ]dτ 2(t 2 1 h 1 2 ) + 2 (Γ( = 3 ))2 π (t 5 h)2 8(t h) 5 πγ( 3 ) 8(t h) (t h) 2 + πγ( 3 ) (Γ( 3 ))2 2t (Γ( = 3 ))2 π (t 5 h)2 8(t h) 5 πγ( 3 ) 8(t h) W(t) = ( 2t 1 2 (Γ( 3 ))2 5 2(t h) πγ( 3 ) (Γ( 3 ))2 = t 1 2 (t h) 1 2 (Γ( 3 )) + 2 π (t h)2 ) 2(t h) 1 2 (Γ( 3 ))2 + 36(t h) 5 2 >, 25π(Γ( 3 ))2 2h 1 2 ) (Γ( 3 )2 ( 8(t h) 5 5 πγ( 3 ))2 we know that W(t) is nonsingular. From Theorem 6, we know that system (2) is controllable in (h, ). Example 2 For the fractional control systems with control delay c D 1 2 x 1 (t) = x 2 (t) + u 1 (t) + u 2 (t h), t, c D 1 2 x 2 (t) =, t, x() = x, u(t) = ψ(t), h t. (21) Compare (2) with (1), we have Then Φ α,1 (A, h) = 1, and A = ( 1 [B, AB, C, AC] = ) ( 1, B = Φ α,1 (A, h) = 1 1 ) ( 1, C = 1 h 2 Γ( 3 2 ), ( 1 1 rank[b, AB, C, AC] = 1 2, from Theorem 7 we have that the system (21) is uncontrollable in (h, ). References Anatoly, A. K., Hari, M. S., & Juan, J. T. (26). Theory and Applications of Fractional differential Equations Elsevier B. V. Hai Z., Jinde C., & Wei, J. (213). Controllability Criteria for Linear Fractional Differential Systems with State Delay and Impulses. Journal of Applied Mathematics, Volume 213, Article ID 161, 9. Hale, J. (1992). Introduction to Functional Differential Equations, Springer Verlay. Jiang, W., & WenZhong, S. (21). Controllability of sinular systems with control delay. Automatica, 37, Jiang, W. (26). Function-controllability of nonlinear singular delay differential control systems. Acta Mathematica Sinica, 9(5), Jiang, W. (21). The constant variation formulae for singular fractional differential systems with delay. Computers and Mathematics with Applications, 59(3), Jiang, W. (211). Variation formulae for time varying singular fractional delay differential systems. Fractional Differential Equations, 1(1), ). ). 95

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