On the Interval Controllability of Fractional Systems with Control Delay

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., 2006; Podlubny, I., 1999; Lakshmikantham, V., 2008; Yong, Z., 2008; Shantanu, D., 2008; Jiang, W., 2010; Jiang, W., 2011). 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., 1994; Sebakhy, O. & Bayoumi, M. M., 1971; Wansheng T. & Guangquan, L., 1995; Jiang, W. & WenZhong, S., 2001; Jiang, W., 2006; Jiang, W., 2012; Hai, Z., Jinde, C. & Wei, J., 2013; Xian-Feng, Z., Jiang, W. & Liang-Gen, H., 2013; Xian-Feng, Z., Jiang, W. & Liang-Gen, H., 2013; Song Liu & et al., 2016; Xian-Feng, Z., Fuli, Y. & Wei, J., 2015). 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 > 0 for a function f : R → R is defined as

Definition 1 Riemann-Liouville's fractional integral of order α > 0 for a function f : R + → R is defined as Here Γ(•) is a Gamma-Function.
In this paper we study the fractional control systems with control delay 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 > 0 is time control delay; and ψ(t) the initial control function.0 < 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.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 [0, h] for fractional systems with control delay (1).In section 4, the necessary and sufficient conditions for controllability the interval (h, ∞) for fractional systems with control delay (1).

Preliminaries
In this section, we give some preliminaries.
Then we have From Lemma 1, we have Theorem 1 The general solution of system (1) can be written as that when t ≥ h where n is the order of A and α = ImageB, = ¯ImageC.Then the space ⟨A|B, C⟩ is spanned by the columns of matrix Lemma 2 For Beta function The proof of this Lemma can be seen in I.Podlubny (1999)(page 7).
Proof From Lemma 2, we have For any z ∈ R n and t > h, define W(t) : R n → R n by

The Controllability of (1) in Interval [0, h]
In this section, we discuss the controllability of (1) in interval [0, 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 0 and initial control u(t) = ψ(t)(−h ≤ t ≤ 0), there is a control ū(t) such that the solution x(t) of system (1) from x 0 can reach x at time t, that is From Theorem 1, the general solution of system (1) can be written as that when 0 ≤ t ≤ h, The control term Cu(t − h) can't play a part in the controllability of (1).
Consider fractional control systems without control delay 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 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 0 and initial control 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(2008), we have that system (8) is controllable in interval I 1 if and only if the rank of matrix From Shantanu Das(2008), we know that system (8) is controllable in interval I 1 = [0, h] if and only if it is controllable in interval I = [0, ∞).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.

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⟩.
Proof To show ImW(t 1 ) = ⟨A|B, C⟩ is equivalence to showing that If x ∈ KerW(t 1 ) and x 0,then For ( 1) is controllable, from theorem 1, for any t 1 and any x, for Left time xT , from (11), we have 0 Let t 1 → h and x = Φ −1 α,1 (A, h)BB T x , we have 0 = xT BB T x.
Let t 1 → h and x = Φ −1 α,1 (A, h)BB T A T x , we have That is Repeatedly take k times Caputo's fractional derivative for the (12), and let For ( 12), let t 1 → h and x = Φ −1 α,1 (A, h)CC T x , we have That is we have From the definition of W(t 1 ) and ( 18), we have 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 0 such that 0 For ( 1) is controllable, from theorem 1, for x(t 1 ) = ∫ 0 Time xT , from (19) we have 0 = xT Φ α,1 (A, t 1 ) x.
We have that Φ α,1 (A, t 1 ) is singular for any time t 1 ≥ h.That is 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 0 .If for any time t 1 ∈ (h, ∞), W(t 1 ) is nonsingular, then system (1) is controllable in (h, ∞).

For
the time interval I = [a, b], we give the concept of interval controllability.There a ∈ R and b ∈ R, b > a or b = +∞.