Computers and Mathematics with Applications. The controllability of fractional control systems with control delay

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1 Computers and Mathematics with Applications 64 (212) Contents lists available at SciVerse ScienceDirect Computers and Mathematics with Applications journal homepage: The controllability of fractional control systems with control delay Jiang Wei School of Mathematical Science, University of Anhui, Hefei, Anhui, 2339, PR China a r t i c l e i n f o a b s t r a c t Keywords: Controllability Fractional control systems Control delay This paper discusses the controllability of fractional control systems with control delay. We firstly give the solution expression for fractional control systems with control delay, then give the necessary and sufficient conditions for the controllability of fractional control systems with control delay. 212 Elsevier Ltd. All rights reserved. 1. Introduction Fractional differential systems have been ardently investigated in recent years [1 13], and many interesting results have been given. We notice that a lot of practical systems, such as economic, biological, physiological and spaceflight systems, have the phenomenon of time delay, and many scholars have paid much attention to time delay systems [12 22] and have also achieved many great accomplishments. In particular, time delay frequently exists in control [16,17,2]. For the systems we study in this paper, we consider two factors (fractional order differential, and delay) synchronously. The results of this paper should be useful. Definition 1. Riemann Liouville s fractional integral of order α > for a function f : R R is defined as D α f (t) 1 Γ (α) (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 c D α 1 f (t) Γ (m α 1) f (m1) (θ) dθ. (t θ) α m The fractional control systems with control delay we study in this paper are ( c D α x(t)) Ax(t) Bu(t) Cu(t h), t, x() x, u(t) ψ(t), t, where x(t) R n is a state vector; u(t) R m is a control vector, a sufficiently order differentiable function; A R n n, B, C R n m are any matrices; det(λe A) ; h > is the time control delay; and ψ(t) is the initial control function. < α 1, c D α x(t) denotes an α order-caputo fractional derivative of x(t). Definition 3. The system (1) is said to be controllable if one can reach any state from any admissible initial state and initial control. (1) This research had been supported by National Natural Science Foundation of China (No ) Doctoral Fund of Ministry of Education of China (No ) and Major Program of Educational Commission of Anhui Province of China (No. KJ21ZD2). address: jiangwei@ahu.edu.cn /$ see front matter 212 Elsevier Ltd. All rights reserved. doi:1.116/j.camwa

2 3154 J. Wei / Computers and Mathematics with Applications 64 (212) This paper discusses the controllability of fractional control systems with control delay. We firstly give the solution expression for fractional control systems with control delay, then give the necessary and sufficient conditions for the controllability of fractional control systems with control delay. 2. The solution expression for fractional control systems with control delay In this section, we firstly give the solution expression of canonical fractional control systems with control delay (1). Now we give two lemmas. Theorem 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) k A k t αkβ 1 is the state transfer matrix and Γ ( ) is a Gamma-function. Φ α,α (A, t τ)f (τ)dτ. (3) Proof. From [1,2] (p. 16) we know that the Laplace Transform of the 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 [1,2] z k E α,β (z), (α >, β > ). We have k L(Φ α,β (A, t))(s) e st t β 1 E α,β (At α )dt 1 e st t β 1 (At α ) k dt k A k e st t αkβ 1 A k dt k k A k s (αkβ) e h αkβ 1 A k s (αkβ) dh k k (As α ) k s β (I As α ) 1 s β k (s α I A) 1 s α β. Taking the Laplace Transform for systems (2), we have That is Here 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) Φ α,α (A, t τ)f (τ)dτ hαkβ 1 1 e s αkβ 1 s dh

3 J. Wei / Computers and Mathematics with Applications 64 (212) is the convolution of Φ α,α (A, t) and f (t) [1,2]. Then we have L(x(t))(s) L(Φ α,1 (A, t))(s)x() L Φ α,α (A, t τ)f (τ)dτ (s). That is x(t) Φ α,1 (A, t)x() The proof of Theorem 1 is completed. Φ α,α (A, t τ)f (τ)dτ. From Theorem 1, we have the following theorem. Theorem 2. The general solution of system (1) can be written as x(t) Φ α,1 (A, t)x() t Φ α,α (A, t τ)bu(τ)dτ (Φ α,α (A, t τ)b Φ α,α (A, t τ h)c)u(τ)dτ Φ α,α (A, t τ h)cψ(τ)dτ. 3. The necessary and sufficient conditions for the controllability of systems (1) Now we have the main result of this paper. Theorem 3. The fractional control systems with control delay (1) are controllable if and only if rank[b, AB, A 2 B,..., A n 1 B, C, AC, A 2 C,..., A n 1 C] n. Remark 1. Let A B, C α Aα A 2 α A n 1 α β Aβ A 2 β A n 1 β, where n is the order of A and α Image B, β Image C. Then the space A B, C is spanned by the columns of the matrix [B, AB, A 2 B,..., A n 1 B, C, AC, A 2 C,..., A n 1 C]. That is, the conditions in Theorem 3 are equivalent to A B, C R n. To prove Theorem 3, we give three lemmas. Lemma 1. For the Beta function we have B(z, w) B(z, w) 1 s z 1 (1 s) w 1 ds, (Re(z) >, Re(w) > ), Γ (z)γ (w) Γ (z w). The proof of this Lemma can be seen in [2] (p. 7). Lemma 2. c D α Φ α,β (A, t) AΦ α,β (A, t). (4) Proof. From Lemma 1, we have c D α Φ α,β (A, t) 1 Γ (1 α) 1 Γ (1 α) 1 Γ (1 α) Φ α,β (A, θ) (t θ) dθ α k1 k A k θ αkβ 1 1 (αk β 1) dθ (t θ) α A k1 θ α(k1)β 2 (α(k 1) β 1) dθ (t θ) α Γ (α(k 1) β)

4 3156 J. Wei / Computers and Mathematics with Applications 64 (212) A k k k k 1 A k1 s α(k1)β 2 t α(k1)β 2 Γ (1 α)t α (1 s) α Γ (α(k 1) β 1) tds A k1 t αkβ 1 Γ (1 α)γ (α(k 1) β 1) 1 A k1 t αkβ 1 B(α(k 1) β 1, 1 α) Γ (1 α)γ (α(k 1) β 1) A k t αkβ 1 AΦ α,β (A, t). s α(k1)β 2 (1 s) ( α1) 1 ds Lemma 3. For any z R n, define W(t) : R n R n by W(t)z [(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 t [Φ α,α (A, t τ)bb T (Φ α,α (A, t τ)) T ]zdτ, then ImW(t) A B, C. Proof. Showing ImW(t) A B, C is equivalent to showing that n 1 n 1 Ker W(t) Ker B T (A T ) i Ker C T (A T ) j. (5) i j If x Ker W(t) and x, then x T W(t)x (Φ α,α (A, t τ)b Φ α,α (A, t τ h)c) T x 2 dτ t B T Φ α,α (A, t τ) T x 2 dτ, that is (Φα,α(A, t τ)b Φα,α(A, t τ h)c) T x, τ t h, B T Φ α,α (A, t τ) T x, t h < τ t. Taking Caputo s fractional derivative for the second equation of (6), from Lemma 2 we have B T ( c D α Φ α,α (A, t τ) T )x, B T Φ α,α (A, t τ) T A T x. Letting τ t, we have B T A T x. Repeatedly taking Caputo s fractional derivative for the second equation of (6), and letting τ t, we have (6) (7) B T (A T k )k x, for k, 1, 2,..., n 1. (8) From the Cayley Hamilton Theorem [18], we have A k t αkα 1 n 1 Φ α,α (A, t) Γ (αk α) γ k (t)a k. (9) k k Then when τ t h, n 1 B T Φ α,α (A, t τ) T A T x γ k (t τ)b T (A T ) k x. k Taking it into the first equation of (6), we have C T Φ α,α (A, t τ h) T x, τ t h.

5 J. Wei / Computers and Mathematics with Applications 64 (212) Repeatedly taking Caputo s fractional derivative and letting τ t h, we have C T (A T ) k x, for k, 1, 2,..., n 1. (1) From (8) and (1) we have x n 1 i Ker BT (A T ) i n 1 j Ker C T (A T ) j. That is n 1 n 1 Ker W(t) Ker B T (A T ) i Ker C T (A T ) j. (11) i j Conversely, letting x n 1 i Ker BT (A T ) i n 1 j Ker C T (A T ) j, (8) and (1) are true. Then from (6) and (9), for t h < τ t, n 1 B T Φ α,α (A, t τ) T A T x γ k (t τ)b T (A T ) k x and for τ t h, k n 1 n 1 (Φ α,α (A, t τ)b Φ α,α (A, t τ h)c) T x γ k (t τ)b T (A T 1 )k x γ k (t τ h)c T (A T ) k x therefore x Ker W(t). That is k n 1 n 1 Ker W(t) Ker B T (A T ) i Ker C T (A T ) j. (12) i j From (11) and (12), we have that (5) is true and the proof of Lemma 3 is completed. Definition 4. We call the set R(x, ψ) {v There exists t 1 >, u(t) C l 1, such that the solution of (1) x(t, x, ψ) satisfies that x(t 1, x, ψ) v} the reachable set of (1) from initial values x() x and u(t) ψ(t), t. Now we prove Theorem 3. Proof of Theorem 3. First we prove that R(, ) A B, C. Let x R(, ), then from Theorem 2 and (9), we have x 1 1 n 1 i (Φ α,α (A, t 1 τ)b Φ α,α (A, t 1 τ h)c)u(τ)dτ Φ α,α (A, t 1 τ)bu(τ)dτ 1 That is x A B, C. So, R(, ) A B, C. γ i (t 1 s)a i Bu(s)ds 1 n 1 1 j 1 t 1 Φ α,α (A, t 1 τ h)cu(τ)dτ γ j (t 1 s)a j Cu(s)ds. k Φ α,α (A, t 1 τ)bu(τ)dτ Contrarily, we prove R(, ) A B, C. Suppose ˆx A B, C. Letting x(t) be a solution of (1) at t >, from Theorem 2 we have x(t) (Φ α,α (A, t τ)b Φ α,α (A, t τ h)c)u(τ)dτ For ˆx A B, C, from Lemma 3, there exists z R n, such that W(t)z ˆx Let (B T Φ α,α (A, t s) T C T Φ α,α (A, t s h) T )z, s t h, u(s) B T Φ α,α (A, t s) T z, t h < s t,, s. t Φ α,α (A, t τ)bu(τ)dτ. (13)

6 3158 J. Wei / Computers and Mathematics with Applications 64 (212) Then t That is Φ α,α (A, t s)bu(s)ds Φ α,α (A, t s)cu(s h)ds [(Φ α,α (A, t s)b Φ α,α (A, t s h)c) (Φ α,α (A, t s)b Φ α,α (A, t s h)c) T ]zds W(t)z ˆx. R(, ) A B, C. From (13) and (14) we have R(, ) A B, C. [Φ α,α (A, t s)bb T Φ α,α (A, t s) T ]zds, t Now we prove the necessity of Theorem 3. If system (1) is controllable, for any x R n, by Definition 3, to initial state x and initial control ψ, there exists a control u(s) such that x (Φ α,α (A, t s)b Φ α,α (A, t s h)c)u(s)ds Φ α,α (A, t s)bu(s)ds. t From (9) we have x A B, C. That is R n A B, C. So that R n A B, C, and the conditions of Theorem 3 are true. Finally, we prove the sufficiency. If the conditions of Theorem 3 are true, then R n A B, C. For any x R n and any initial state x and initial control ψ, let k x Φ α,1 (A, t)x t Φ α,α (A, t s)bdsψ() (Φ α,α (A, t s)b Φ α,α (A, t s h)c)dsψ() Φ α,α (A, t s h)cψ(s)ds. For k R n A B, C, that is k R(, ). Then there exists a control u (s) such that k (Φ α,α (A, t s)b Φ α,α (A, t s h)c)u (s)ds t Φ α,α (A, t s)bu(s)ds Φ α,α (A, t s h)cψ(s)ds. Let u(s) u (s) ψ(), then we have x Φ α,1 (A, t)x t Φ α,α (A, t s)bu(s)ds (Φ α,α (A, t s)b Φ α,α (A, t s h)c)u(s)ds Φ α,α (A, t s h)cψ(s)ds. The fractional control systems with control delay (1) are controllable. The sufficiency is shown. And the proof of Theorem 3 is completed. References [1] Anatoly A. Kilbas, Hari M. Srivastava, Juan J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B.V., 26. [2] I. Podlubny, Fractional Differential Equations, San Diego Academic Press, [3] K.S. Miller, B. Boss, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, [4] V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Analysis 69 (1) (28) [5] Yong Zhou, Existence and uniqueness of fractional functional differential equations with unbounde delay, International Journal of Dynamiacal Systems and Differential Equations 4 (1) (28) [6] B. Bonilla, M. Rivero, J.J. Trujillo, On systems of linear fractional differential equations with constant coefficients, Applied Mathematics and Computation 187 (1) (27) [7] Aytac Arikoglu, Ibrahim Ozkol, Solution of fractional differential equations by using differential transform method, Chaos, Solitons, Fractals 34 (5) (27) [8] Rabha W. Ibrahim, Shaher Momani, On the existence and uniqueness of solutions of a class of fractional differential equations, Journal of Mathematical Analysis and Applications 334 (1) (27) 1 1. (14)

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