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

and Applied Analysis 3 In addition, the Laplace transform of tα−1 is L [t α−1 ; s] = Γ (α) s −α , Re (s) > 0. (21) Lemma 4 (see [28]). Let 0 < Re(α) ≤ 1. If x(t) ∈ C[a, b], then I α

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.
Definition 1 (see [27]).The fractional integral of order  with the lower limit  ∈ R for a function  is defined as Provided that the right-hand side is pointwise defined on [, +∞), where Γ is the Gamma function.
Definition 2 (see [27]).The Caputo's derivative of order  with the lower limit  ∈ R for a function  : [, ∞) → R can be written as Particularly, when 0 <  < 1, it holds The Laplace transform of    0, () is where () is the Laplace transform of ().
In particular, for 0 <  < 1, it holds Definition 3 (see [27]).The two-parameter Mittag-Leffler function is defined as The Laplace transform of Mittag-Leffler function is where Re() denotes the real parts of .
where [, ] denotes the set of continuous functions on [, ].
Using the Laplace transform method, we can easily obtain the following lemma.Lemma 6.The movement orbit of the state variable () of the system (8)−( 12) can be written as . . .
Remark 9. From Theorem 8, we can conclude that the complete controllability of the system (8)-( 12) is unrelated to the matrix  and initial function ().The matrices ,  determine if the the system (8)-( 12) possesses complete controllability.