The controllability of fractional differential system with state and control delay

Abstract

In this research work, we investigate the controllability of linear fractional differential control systems with state and control delay. By using an explicit solution formula, a rank criterion for controllability is established. For the controllability criteria, we establish necessary and sufficient conditions of a fractional differential systems with state and control delay. In the end, a numerical example is constructed to support the results.

1 Introduction

The fractional differential equation is a mathematical model which is useful for the explanation of hereditary characteristics and memory of different processes and materials. A variety of research work is based on the basic study of fractional differential equations [1–6] as in further work various researchers considered control problems; for example, see [7–9].

The controllability shows a major presence in the advancement of modern mathematical control theory and engineering which has a close connection with structural decomposition, quadratic optimal and so on; see [10–17]. Controllability is a qualitative property of fractional delay dynamical system, so one needs to find its representation of a solution. He and Wei [18, 19] gave a representation of a solution and discussed the controllability and then for a fractional control delay system obtained necessary and sufficient conditions, Nirmala [11] give a representation of a solution by using Laplace transform and Mittag-Leffler function and established controllability criteria for fractional delay dynamical system. Moreover, Khusainov et al. [20] obtained the representation of a solution of a Cauchy problem for a linear differential equation with pure delay by using the delayed Mittag-Leffler function, Shukla et al. [21–24] discussed the complete and approximate controllability of semilinear stochastic systems with delays in the state and control function with non-Lipschitz coefficients, the Schauder fixed point theorem, sequence methods and by the theory of the strongly continuous z-order cosine family, and the fixed point theorem, respectively. In a most recent work [25] the authors discussed the relative controllability problem and an explicit representation of solutions is given with the use of delayed Mittag-Leffler function, Li and Wang [26] discussed the controllability criteria of a fractional differential system with state delay by using an explicit solution formula. By following this study we consider a fractional differential system with state and control delay and discussed its controllability by giving its necessary and sufficient conditions. Li and Wang [27] considered pure delay for linear fractional differential equations and gave a representation of a solution by using a delayed Mittag-Leffler type matrix:

$$\begin{aligned} \left \{ \textstyle\begin{array}{l} {}^{c}D^{\alpha}_{0^{+}}x(t)=Ax(t-h), \quad x(t)\in\mathcal{R}^{n}, t \in J:=[0,t_{1}],h > 0, \\ x(t)=\varphi(t), \quad{-}h\leq t \leq0,\varphi\in\mathcal {C}^{1}_{h}:=\mathcal{C}^{1}([-h,0],\mathcal{R}^{n}), \end{array}\displaystyle \right . \end{aligned}$$

(1)

where ${}^{c}D^{\alpha}_{0^{+}}x(t)$ stands for the αth order Caputo fractional derivative of $x(t)$ where zero is a lower limit, $t_{1}$ is the integral multiple of h, $A\in\mathcal {R}^{n\times n}$, $h >0$ is a time delay, $n \in\mathcal{N}$ stands for a constant matrix. $\mathcal{E}^{A.^{\alpha}}_{h}$ is a new notation (delayed Mittag-Leffler type matrix) being reported in Definition 2.3 [28], any solution $x \in C ([-h, t_{1}], \mathcal{R}^{n})$ of (1) can be established by Li:

$$ x(t) = \mathcal{E}^{A t^{\alpha}}_{h}\varphi(-h) + \int^{0}_{-h}\mathcal {E}^{A(t-h-\tau)^{\alpha}}_{h} \varphi'(\tau)\,d\tau. $$

(2)

Motivated by the previous study, in this research work we deal with the fractional differential systems with state and control delay by using of an explicit formula governed by

$$\begin{aligned} \left \{ \textstyle\begin{array}{l} {}^{c}D^{\alpha}_{0^{+}}x(t)=Ax(t-h)+Bu(t)+Cu(t-h),\quad x(t)\in J:=[0,t_{1}],h > 0,t_{1} \geq0, \\ x(t)=\varphi(t), \quad {-}h\leq t \leq0,\\ u(t)=\psi(t), \quad{-}h\leq t \leq0, \end{array}\displaystyle \right . \end{aligned}$$

(3)

where $x: [-h, t_{1}] \rightarrow\mathcal{R}^{n}$ is a continuous differentiable on $[0, t_{1}]$ with $t_{1} >(n-1)h$, $0<\alpha\leq1$, $A\in\mathcal{R}^{n\times n}$, $B, C\in\mathcal{R}^{n\times m}$ are any matrices, $h >0$ shows the time delay, $x(t)\in\mathcal{R}^{n}$ denotes the state vector, $u(t)\in\mathcal{R}^{m}$ shows the control vector, $\varphi(t)$ shows the initial state function and $\psi(t)$ shows the initial control function $\varphi\in\mathcal {C}^{1}_{h}:=\mathcal{C}^{1}([-h,0],\mathcal{R}^{n})$. The lay-out of this article as follows, Sect. 2 includes some useful definitions, preliminary results, and lemmas about delayed Mittag-Leffler type matrix to establish the controllability of fractional differential systems with state and control delay. In Sect. 3 we obtain necessary and sufficient conditions for controllability criteria for the above fractional differential delay system (3). Section 4 presents an example to explain the applicability of the theoretical results.

2 Preliminaries and essential lemmas

This part includes some basic definitions and results used throughout this paper and some lemmas for the main results. We recall some well-known definitions. For more details, see [3, 5].

Definition 2.1

([29])

We consider a function $f:[0,\infty)\rightarrow\mathcal{R}$ where its Caputo fractional derivative of order ($0 < \alpha< 1$) is defined as

$$\bigl({}^{c}D^{\alpha}_{0^{+}} x \bigr) (t)= \frac{1}{\varGamma(1-\alpha)} \int_{0}^{t}\frac{ x'(\theta)}{(t-\theta)^{\alpha}}\,d\theta,\quad t>0. $$

Here the Gamma function is denoted by $\varGamma(\cdot)$.

Definition 2.2

([29])

We consider a function $f: [0,\infty)\rightarrow\mathcal{R}$ where its fractional integral of order $\alpha>0$ is defined as

$$\bigl(I^{\alpha}_{0^{+}}f \bigr) (t)=\frac{1}{\varGamma(\alpha)} \int_{0}^{t}(t-\theta )^{\alpha-1}f(\theta)\,d \theta. $$

Here $\varGamma(\cdot)$ denotes the Gamma function.

Definition 2.3

([26])

A matrix $\mathcal{E}^{A.^{\alpha}}_{h}:\mathcal{R}\rightarrow\mathcal {R}^{n\times n}$ known as a delayed Mittag-Leffler type matrix is defined as

$$\begin{aligned} \mathcal{E}^{A t^{\alpha}}_{h} = \left \{ \textstyle\begin{array}{l@{\quad}l} \varTheta,&- \infty< t < -h,\\ I ,&- h \leq t\leq0,\\ I+A\frac{(t)^{\alpha}}{\varGamma(\alpha+1)}+A^{2}\frac{(t-h)^{2\alpha }}{\varGamma(2\alpha+1)}+\cdots+A^{k}\frac{(t-(k-1)h)^{k\alpha}}{\varGamma (k\alpha+1)},&(k-1) h \leq t \leq k h,k \in\mathcal{N}, \end{array}\displaystyle \right . \end{aligned}$$

(4)

where zero and identity matrices are shown by Θ and I, respectively.

Definition 2.4

The system (3) is said to be controllable on $J=[0, t_{1}]$ if one can reach any state from any allowed initial state $x(t)=\varphi (t)$ and initial control $u(t)=\psi(t)$.

Lemma 2.5

([26])

Let $f: J \rightarrow\mathcal{R}^{n}$be a continuous vector value function. A solution $x \in C ([-h,t_{1}], \mathcal{R}^{n})$of the following system:

$$\begin{aligned} \left \{ \textstyle\begin{array}{l} {}^{c}D^{\alpha}_{0^{+}}x(t)=Ax(t-h)+f(t), \quad x(t)\in\mathcal{R}^{n}, t \in J:=[0,t_{1}], h > 0, \\ x(t)=\varphi(t), \quad {-}h\leq t \leq0, \varphi\in\mathcal{C}^{1}_{h}, \end{array}\displaystyle \right . \end{aligned}$$

(5)

can be written in the form of an integral equation by using the method in [26];

$$\begin{aligned} x(t) = \mathcal{E}^{A t^{\alpha}}_{h}\varphi(-h) + \int^{0}_{-h}\mathcal {E}^{A(t-h-\tau)^{\alpha}}_{h} \varphi'(\tau)\,d\tau+ \int^{t}_{0}\mathcal {E}^{A(t-h-\tau)^{\alpha}}_{h}f( \tau)\,d\tau. \end{aligned}$$

By Lemma 2.8in [26], a solution $x \in C ([-h,t_{1}], \mathcal{R}^{n})$of system (3) can be composed in the form

$$ \begin{aligned}[b]x(t) &= \mathcal{E}^{A t^{\alpha}}_{h}\varphi(-h) + \int ^{0}_{-h}\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h} \varphi'(\tau)\,d\tau \\ &\quad+ \int^{t}_{0}\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h}Bu( \tau )\,d\tau+ \int^{t}_{0}\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h}Cu( \tau-h)\,d\tau.\end{aligned} $$

(6)

Lemma 2.6

([18])

From Lemma 2.5for system (3), a general solution can be composed as

$$ \begin{aligned}[b] x(t) &= \mathcal{E}^{A t^{\alpha}}_{h}\varphi(-h) + \int^{0}_{-h}\mathcal {E}^{A(t-h-\tau)^{\alpha}}_{h} \varphi'(\tau)\,d\tau+ \int^{t-h}_{0}\mathcal {E}^{A(t-h-\tau)^{\alpha}}_{h}Bu( \tau)\,d\tau \\ &\quad + \int^{t}_{t-h}\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h}Bu( \tau )\,d\tau+ \int^{t-h}_{0}\mathcal{E}^{A(t-2h-\tau)^{\alpha}}_{h}Cu( \tau )\,d\tau \\ &\quad+ \int^{0}_{-h}\mathcal{E}^{A(t-2h-\tau)^{\alpha}}_{h}C \psi(\tau)\,d\tau.\end{aligned} $$

(7)

Definition 2.7

We call the set in [18] $R(\varphi,\psi)$ = $\{\nu\mid$ there exists $t_{1} > 0, u(t)\in C^{l-1}$, such that the solution of the system (3) $x(t,\varphi ,\psi)$ satisfies $x(t_{1},\varphi,\psi)=\nu\}$ the reachable set of (3) with $x(t)=\varphi(t)$ and $u(t)=\psi (t)$ at $-h \leq t \leq0$.

Lemma 2.8

([18])

For the beta function

$$\mathcal{B}(p,q) = \int^{1}_{0}s^{p-1}(1-s)^{q-1}\,ds\quad \bigl(Re(p)>0, Re(q)>0 \bigr), $$

we have

$$\mathcal{B}(p,q) =\frac{\varGamma(p)\varGamma(q)}{\varGamma(p+q)}. $$

Lemma 2.9

([28])

Let $(k-1) h \leq t \leq k h$, $k \in\mathcal{N}$, we have

$$\begin{aligned} \int^{t}_{(k-1)h}(t-s)^{-\alpha} \bigl(s- (k-1)h \bigr)^{k\alpha-1} \,ds =& \bigl(t-(k-1)h \bigr)^{(k-1)\alpha}\mathcal{B}[1- \alpha, k\alpha], \end{aligned}$$

where $\mathcal{B}$is the beta function; see Lemma 2.8.

Lemma 2.10

For a delayed Mittag-Leffler type matrix $\mathcal{E}^{A . ^{\alpha}}_{h} : \mathcal{R} \rightarrow\mathcal{R}^{n\times n}$, one has

$$ {}^{c}D^{\alpha}_{0^{+}} \bigl( \mathcal{E}^{A t^{\alpha}}_{h} \bigr)= A \mathcal {E}^{A(t-h)^{\alpha}}_{h}, $$

(8)

i.e., $\mathcal{E}^{A t^{\alpha}}_{h}$is a solution of $({}^{c}D^{\alpha }_{0^{+}}x)(t) = A x(t-h)$that satisfies the initial conditions $\mathcal{E}^{ A t^{\alpha}}_{h}= I$, $- h \leq t\leq0$.

Proof

For arbitrary $t \in(-\infty,-h]$, $\mathcal{E}^{ At^{\alpha}}_{h} = \mathcal{E}^{A(t-h)^{\alpha}}_{h} = \varTheta$. Obviously, (8) holds. Next for $t \in(-h, 0]$, $\mathcal{E}^{ At^{\alpha}}_{h}= I$ and $\mathcal{E}^{A(t-h)^{\alpha}}_{h} = \varTheta$. which shows ${}^{c}D^{\alpha }_{0^{+}}I = \varTheta= A \varTheta$. Thus, (8) holds.

For arbitrary $t \in((k-1) h, Kh]$, $k \in\mathcal{N}$, we follow mathematical induction to establish our result.

(1) For $k = 1$, $0 \leq t\leq h$, we have

$$\begin{aligned} x(t)= \mathcal{E}^{ At^{\alpha}}_{h} = I + \frac{A(t)^{\alpha}}{\varGamma(\alpha +1)},\qquad x'(t)=\frac{\alpha A(t)^{\alpha-1}}{\varGamma(\alpha+1)}. \end{aligned}$$

(9)

Next by using the Caputo fractional differentiation expression of $\mathcal{E}^{ A.^{\alpha}}_{h}$ via (9) and Lemma 2.9, we obtain

$$ {}^{c}D^{\alpha}_{0^{+}} \bigl( \mathcal{E}^{A s^{\alpha}}_{h} \bigr) (t)= \frac{\alpha A}{\varGamma(\alpha+1)\varGamma(1-\alpha)} \int^{t}_{0}(t-s)^{-\alpha }(s)^{\alpha-1}\,ds =A. $$

(10)

(2) For $k = 2$, $h \leq t\leq2h$, we have

$$\begin{aligned} \begin{gathered}x(t)= \mathcal{E}^{ At^{\alpha}}_{h} = I + \frac{A(t)^{\alpha}}{\varGamma(\alpha+1)}+\frac{A^{2}(t-h)^{2\alpha}}{\varGamma(2\alpha+1)}, \\ x'(t)=\frac{\alpha A(t)^{\alpha-1}}{\varGamma(\alpha+1)} + \frac{2\alpha A^{2}(t-h)^{2\alpha-1}}{\varGamma(2\alpha+1)}.\end{gathered} \end{aligned}$$

(11)

Next by using the Caputo fractional differentiation expression of $\mathcal{E}^{ A.^{\alpha}}_{h}$ via (11), (10) and Lemma 2.9, we obtain

$$\begin{aligned} {}^{c}D^{\alpha}_{0^{+}} \bigl(\mathcal{E}^{A s^{\alpha}}_{h} \bigr) (t) =& A +\frac {2\alpha A^{2}}{\varGamma(2\alpha+1)\varGamma(1-\alpha)} \int ^{t}_{h}(t-s)^{-\alpha}(s-h)^{2\alpha-1}\,ds \\ =& A +\frac {A^{2}(t-h)^{\alpha}}{\varGamma(\alpha+1)}. \end{aligned}$$

(3) Let $k = M$, $(M-1)h \leq t\leq M h$ and $M \in\mathcal{N;}$ the following relation holds:

$$\begin{aligned} {}^{c}D^{\alpha}_{0^{+}} \bigl(\mathcal{E}^{A s^{\alpha}}_{h} \bigr) (t) =& A +\frac {A^{2}(t-h)^{\alpha}}{\varGamma(\alpha+1)}+\frac{A^{3}(t-2h)^{2\alpha }}{\varGamma(2\alpha+1)}+\cdots \\ &{}+\frac{A^{M}(t-(M-1)h)^{(M-1)\alpha }}{\varGamma((M-1)\alpha+1)}. \end{aligned}$$

Next let $k = M+1, Mh \leq t\leq(M+1)h$; by elementary computation, we get

$$ \begin{aligned}[b]{x'}(t)&=\frac{\alpha A(t)^{\alpha-1}}{\varGamma(\alpha+1)} + \frac {2\alpha A^{2}(t-h)^{2\alpha-1}}{\varGamma(2\alpha+1)}+ \cdots \\ &\quad+\frac{(M+1)\alpha A^{(M+1)}(t-M h)^{(M+1)\alpha-1}}{\varGamma ((M+1)\alpha+1)}.\end{aligned} $$

(12)

Now taking the Caputo fractional differentiation expression of $\mathcal {E}^{ A.^{\alpha}}_{h}$ via (12) and Lemma 2.9, we obtain

$$\begin{aligned} &{}^{c}D^{\alpha}_{0^{+}} \bigl(\mathcal{E}^{A s^{\alpha}}_{h} \bigr) (t) \\ &\quad= \frac{\alpha A}{\varGamma(\alpha+1)\varGamma(1-\alpha)} \int ^{t}_{0}(t-s)^{-\alpha}s^{\alpha-1}\,ds \\ &\qquad{}+ \frac{2\alpha A^{2}}{\varGamma (2\alpha+1)\varGamma(1-\alpha)} \int^{t}_{h}(t-s)^{-\alpha}(s-h)^{2\alpha -1}\,ds+\cdots \\ &\qquad{}+\frac{(M+1)\alpha A^{(M+1)}}{\varGamma(1-\alpha)\varGamma ((M+1)\alpha+1)} \int^{t}_{Mh}(t-s)^{-\alpha}(s-Mh)^{(M+1)\alpha-1}\,ds \\ &\quad= A+\frac{A^{2}(t-h)^{\alpha}}{\varGamma(\alpha+1)}+\frac {A^{3}(t-2h)^{2\alpha}}{\varGamma(2\alpha+1)}+\cdots+\frac{A^{(M+1)}(t-M h)^{M\alpha}}{\varGamma(M\alpha+1)}. \end{aligned}$$

This shows that Eq. (8) is satisfied for any $(k-1)h \leq t\leq kh$ and $k \in\mathcal{N}$. The proof is completed. From Lemma 2.10, we have

$$ {}^{c}D^{\alpha}_{0^{+}} \bigl( \mathcal{E}^{A (t-h-\tau)^{\alpha}}_{h} \bigr)= A \mathcal {E}^{A(t-2h-\tau)^{\alpha}}_{h}. $$

(13)

 □

3 Main results

In this part for the controllability of system (3) necessary and sufficient conditions are given. Firstly we prove a lemma, then by using this lemma the main results are constructed.

Remark 3.1

Let

$$\begin{aligned} \langle A|B,C \rangle= \alpha+ A \alpha+ A^{2} \alpha+ \cdots+ A^{n-1}\alpha+ \beta+ B\beta+ B^{2}\beta+ \cdots+ B^{n-1}\beta, \end{aligned}$$

where $\alpha= \operatorname{Image} B$, $\beta= \operatorname{Image} C$ and n stands for order of A. Then the space $\langle A|B,C \rangle$ is spanned by the columns of the matrix

$$\bigl[B, AB, A^{2}B,\ldots, A^{n-1}B, C, AC, A^{2}C, A^{3}C,\ldots,A^{n-1}C \bigr]. $$

Lemma 3.2

For any $z\in\mathcal{R}^{n}$, define $W(t) : \mathcal{R}^{n} \rightarrow \mathcal{R}^{n}$by

$$ \begin{aligned}[b]W(t) &= \int^{t-h}_{0} \bigl[ \bigl(\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h}B + \mathcal {E}^{A(t-2h-\tau)^{\alpha}}_{h}C \bigr) \bigl(\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h}B + \mathcal{E}^{A(t-2h-\tau)^{\alpha}}_{h}C \bigr)^{T} \bigr]z\,d\tau \\ &\quad+ \int^{t}_{t-h} \bigl[ \bigl(\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h} \bigr)BB^{T} \bigl(\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h} \bigr)^{T} \bigr]z\,d\tau.\end{aligned} $$

(14)

Then

$$ \operatorname{Im} W(t) = \langle A|B,C \rangle . $$

(15)

Proof

Showing $\operatorname{Im} W(t) = \langle A|B,C \rangle$ is equivalent to

$$ \operatorname{Ker} W(t) = \bigcap^{n-1}_{i=0} \operatorname{Ker} B^{T} \bigl(A^{T} \bigr)^{i} \bigcap^{n-1}_{j=0} \operatorname{Ker} C^{T} \bigl(A^{T} \bigr)^{j} . $$

(16)

If $x \in\operatorname{ker} W(t)$ and $x\neq0$ then

$$\begin{aligned} 0 =& x^{T} W(t) x \\ =& \int^{t-h}_{0} \bigl\Vert \bigl( \mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h}B + \mathcal{E}^{A(t-2h-\tau)^{\alpha}}_{h}C \bigr)^{T} x \bigr\Vert ^{2}\,d\tau \\ &{}+ \int^{t}_{t-h} \bigl\Vert B^{T} \bigl( \mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h} \bigr)^{T} x \bigr\Vert ^{2}\,d\tau, \end{aligned}$$

that is

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l} 0 = (\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h}B + \mathcal{E}^{A(t-2h-\tau )^{\alpha}}_{h}C)^{T}x, & 0 \leq\tau\leq t-h, \\ 0 = B^{T}(\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h})^{T}x ,& t-h \leq\tau< t. \end{array}\displaystyle \right . \end{aligned}$$

(17)

For the second equation of (17) by taking its Caputo derivative from Lemma 2.10 we have

$$ \begin{aligned}[b]0 &= B^{T} \bigl({}^{c}D^{\alpha}_{0^{+}} \mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h} \bigr)^{T}x \\ &= B^{T} \bigl(\mathcal{E}^{A(t-2h-\tau)^{\alpha}}_{h} \bigr)^{T} A^{T} x.\end{aligned} $$

(18)

Let $\tau= t-h$; we have

$$\begin{aligned} 0 = B^{T} A^{T} x. \end{aligned}$$

For the second equation of (17) by performing repeatedly Caputo’s differentiation, we get

$$\begin{aligned} 0 = B^{T} A^{T} x, \quad \text{for }k = 0, 1, 2, 3, \ldots, n-1. \end{aligned}$$

(19)

Using the Cayley–Hamiltonian theorem [18]

$$ \mathcal{E}^{A u ^{\alpha}}_{h} = \sum ^{n-1}_{k=0}\frac {A^{k}(u-(k-1)h)^{(k+1)\alpha-1}}{\varGamma(k\alpha+ \beta)}, $$

(20)

where $u = t - h - \tau$. Then when $0 \leq\tau\leq t-h$

$$\begin{aligned} 0 = B^{T} \bigl(\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h} \bigr)^{T} A^{T} x = \sum^{n-1}_{k=0} \gamma_{k}(t-h-\tau) B^{T} \bigl(A^{T} \bigr)^{k} x = 0. \end{aligned}$$

By taking it into the first equation of (17)

$$\begin{aligned} 0 = C^{T} \bigl(\mathcal{E}^{A(t-2h-\tau)^{\alpha}}_{h} \bigr)^{T} x , \quad 0\leq\tau \leq t-h. \end{aligned}$$

By taking its Caputo derivative and letting $\tau= t- 2h$, we get

$$\begin{aligned} 0 = C^{T} \bigl(\mathcal{E}^{A(t-3h-\tau)^{\alpha}}_{h} \bigr)^{T} A^{T} x. \end{aligned}$$

By performing repeatedly Caputo’s differentiation, we get

$$\begin{aligned} 0 = C^{T} A^{T} x, \quad \text{for }k = 0, 1, 2, 3, \ldots, n-1. \end{aligned}$$

(21)

Using (19) and (21) we get

$$\begin{aligned} x \in\bigcap^{n-1}_{i=0} \operatorname{ker} B^{T} \bigl(A^{T} \bigr)^{i} \bigcap ^{n-1}_{j=0} \operatorname{ker} C^{T} \bigl(A^{T} \bigr)^{j}. \end{aligned}$$

That is,

$$ \operatorname{ker} W(t) \subset\bigcap ^{n-1}_{i=0} \operatorname{ker} B^{T} \bigl(A^{T} \bigr)^{i} \bigcap^{n-1}_{j=0} \operatorname{ker} C^{T} \bigl(A^{T} \bigr)^{j}. $$

(22)

Conversely, suppose

$$\begin{aligned} x \in\bigcap^{n-1}_{i=0} \operatorname{ker} B^{T} \bigl(A^{T} \bigr)^{i} \bigcap ^{n-1}_{j=0} \operatorname{Ker} C^{T} \bigl(A^{T} \bigr)^{j}, \end{aligned}$$

then (19) and (21) hold.

For $t-h \leq\tau< t$, from (17 and 20),

$$\begin{aligned} B^{T} \bigl(\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h} \bigr)^{T} A^{T} x = \sum^{n-1}_{k=0} \gamma_{k}(t-h-\tau) B^{T} \bigl(A^{T} \bigr)^{k} x = 0, \end{aligned}$$

for $0 \leq\tau\leq t-h$,

$$\begin{aligned} \bigl(\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h}B + \mathcal{E}^{A(t-2h-\tau )^{\alpha}}_{h}C \bigr)^{T}x &= \sum^{n-1}_{k=0} \gamma_{k}(t-h-\tau) B^{T} \bigl(A^{T} \bigr)^{k} x \\ &\quad+\sum^{n-1}_{k=0}\gamma_{k}(t-2h- \tau) C^{T} \bigl(A^{T} \bigr)^{k} x\\& =0.\end{aligned} $$

Therefore, $x \in\operatorname{ker} W(t)$, that is,

$$ \operatorname{Ker} W(t) \supset\bigcap ^{n-1}_{i=0} \operatorname{Ker} B^{T} \bigl(A^{T} \bigr)^{i} \bigcap^{n-1}_{j=0} \operatorname{Ker} C^{T} \bigl(A^{T} \bigr)^{j}. $$

(23)

From (22) and (23), it is proven that (16) holds, completing the proof of the lemma. □

Theorem 3.3

([18])

For system (3) the fractional differential control system with state and control delay is controllable iff

$$\operatorname{rank} \bigl[B, AB ,A^{2}B,\ldots, A^{n-1}B,C,AC,A^{2}C,A^{3}C, \ldots,A^{n-1}C \bigr] = n. $$

That is, in Theorem 3.3 the conditions are equivalent to $\langle A|B,C \rangle= \mathcal{R}^{n}$.

By using Lemmas 2.8, 2.10, 3.2 we will prove Theorem 3.3.

Proof of Theorem 3.3

Firstly we show that $R(0,0) = \langle A|B,C \rangle$.

Actually, let $x \in R(0,0)$, from Lemma 2.6 and Eq. (20), we get

$$\begin{aligned}& x= \int^{t_{1} -h}_{0} \bigl(\mathcal{E}^{A(t_{1}-h-\tau)^{\alpha}}_{h}B +\mathcal {E}^{A(t_{1}-2h-\tau)^{\alpha}}_{h}C \bigr)u(\tau)\,d\tau \\& \phantom{x=}{}+ \int^{t_{1}}_{t_{1}-h}\mathcal{E}^{A(t_{1}-h-\tau)^{\alpha}}_{h}Bu( \tau )\,d\tau, \\& x = \int^{t_{1}}_{0}\mathcal{E}^{A(t_{1}-h-\tau)^{\alpha}}_{h}Bu( \tau)\,d\tau + \int^{t_{1} -h}_{0}\mathcal{E}^{A(t_{1}-2h-\tau)^{\alpha}}_{h}Cu( \tau)\,d\tau \\& \phantom{x}= \sum^{n-1}_{i=0} \int^{t_{1}}_{0}\gamma_{i}(t_{1}-h-s)A^{i}Bu(s)\,ds + \sum^{n-1}_{j=0} \int^{t_{1} -h}_{0}\gamma_{j}(t_{1}-2h-s)A^{j}Cu(s)\,ds, \end{aligned}$$

which implies $x \in \langle A|B,C \rangle$.

Thus,

$$\begin{aligned} \langle A|B,C \rangle\supset R(0,0). \end{aligned}$$

(24)

On the other hand, we show $\langle A|B,C \rangle\subset R(0,0)$. Let $\hat{x} \in\langle A|B,C \rangle$, let $x(t)$ be a solution of system (3) at $t > 0$ from Lemma 2.6 we get

$$\begin{aligned} x(t) &= \int^{t-h}_{0} \bigl(\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h}B + \mathcal {E}^{A(t-2h-\tau)^{\alpha}}_{h}C \bigr)u(\tau)\,d\tau \\ &\quad+ \int^{t}_{t-h}\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h}Bu( \tau)\,d\tau.\end{aligned} $$

For $x \in \langle A|B, C \rangle$ from Lemma 3.2 there exists $z \in\mathcal{R}^{n}$, s.t.

$$\hat{x} = W(t)z. $$

Let

$$\begin{aligned} u(s)= \left \{ \textstyle\begin{array}{l@{\quad}l} (\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h}B + \mathcal{E}^{A(t-2h-\tau )^{\alpha}}_{h}C)^{T}z, & 0 \leq s \leq t-h, \\ B^{T}\mathcal{E}^{A(t-h-\tau)^{\alpha}}_{h} Z, & t-h \leq s < t,\\ 0, & -h \leq s \leq0. \end{array}\displaystyle \right . \end{aligned}$$

Then

$$\begin{aligned} & \int^{t}_{0}\mathcal{E}^{A(t-h-s)^{\alpha}}_{h}Bu(s)\,ds + \int ^{t}_{0}\mathcal{E}^{A(t-h-s)^{\alpha}}_{h}Cu(s-h)\,ds \\ &\quad= \int^{t-h}_{0} \bigl[ \bigl(\mathcal{E}^{A(t-h-s)^{\alpha}}_{h}B + \mathcal{E}^{A(t-2h-s)^{\alpha}}_{h}C \bigr)+ \bigl(\mathcal{E}^{A(t-h-s)^{\alpha}}_{h}B + \mathcal{E}^{A(t-2h-s)^{\alpha}}_{h}C \bigr) \bigr]^{T} z \,ds \\ &\qquad{}+ \int^{t}_{t-h} \bigl(\mathcal{E}^{A(t-h-s)^{\alpha}}_{h} \bigr)BB^{T} \bigl(\mathcal {E}^{A(t-h-s)^{\alpha}}_{h} \bigr) z \,ds \\ &\quad= W(t)z = \hat{x}. \end{aligned}$$

That is

$$\begin{aligned} R(0, 0) \supset\langle A|B, C \rangle. \end{aligned}$$

(25)

Using (24) and (25) we get

$$\begin{aligned} R(0,0) =& \langle A|B, C \rangle. \end{aligned}$$

Immediately we show the necessity of Theorem 3.3. Assuming that, for any $x \in\mathcal{R}^{n}$, system (3) is controllable, by Definition 2.4, via the initial state $\varphi=0$ and the initial control $\psi= 0$, there occurs a control $u(s)$ such that

$$ = \int^{t-h}_{0} \bigl(\mathcal{E}^{A(t-h-s)^{\alpha}}_{h}B + \mathcal {E}^{A(t-2h-s)^{\alpha}}_{h}C \bigr)u(s)\,ds + \int^{t}_{t-h} \bigl(\mathcal{E}^{A(t-h-s)^{\alpha}}_{h} \bigr)B u(s) \,ds. $$

Using Eq. (20) we get $x \in\langle A|B, C \rangle$. That is, $\mathcal{R}^{n} \subset \langle A|B, C \rangle$. Thus $\mathcal{R}^{n} = \langle A|B, C \rangle$, and the conditions of Theorem 3.3 are satisfied. At last, we show the sufficiency. Suppose the conditions of Theorem 3.3 are satisfied, then $\mathcal{R}^{n} = \langle A|B, C \rangle$. For any $\overline{x} \in\mathcal{R}^{n}$ and any initial state φ and initial control ψ, let

$$\begin{aligned} k =& \overline{x} - \mathcal{E}^{A t^{\alpha}}_{h}\varphi(-h) - \int ^{0}_{-h}\mathcal{E}^{A(t-h-s)^{\alpha}}_{h} \varphi'(s)\,ds \\ &{}- \int ^{t-h}_{0} \bigl(\mathcal{E}^{A(t-h-s)^{\alpha}}_{h}B +\mathcal {E}^{A(t-2h-s)^{\alpha}}_{h}C \bigr)\psi(0)\,ds \\ &{}- \int^{t}_{t-h}\mathcal{E}^{A(t-h-s)^{\alpha}}_{h}B \psi(0)\,ds - \int ^{0}_{-h}\mathcal{E}^{A(t-2h-s)^{\alpha}}_{h}C \psi(s)\,ds. \end{aligned}$$

For $k \in\mathcal{R}^{n} = \langle A|B, C \rangle$, that is, $k \in R(0,0)$, there exists a control $u^{*}(s)$ such that

$$\begin{aligned} k =& \int^{t-h}_{0} \bigl(\mathcal{E}^{A(t-h-s)^{\alpha}}_{h}B +\mathcal {E}^{A(t-2h-s)^{\alpha}}_{h}C \bigr)u^{*}(s)\,ds \\ &{}+ \int^{t}_{t-h}\mathcal{E}^{A(t-h-s)^{\alpha}}_{h}B u^{*}(s)\,ds + \int ^{0}_{-h}\mathcal{E}^{A(t-2h-s)^{\alpha}}_{h}C \psi(s)\,ds. \end{aligned}$$

Let $u(s) = u^{*}(s) + \psi(0)$ then we have

$$\begin{aligned} \overline{x} =&\mathcal{E}^{A t^{\alpha}}_{h}\varphi(-h) + \int ^{0}_{-h}\mathcal{E}^{A(t-h-s)^{\alpha}}_{h} \varphi'(s)\,ds \\ &{}+ \int^{t-h}_{0} \bigl(\mathcal{E}^{A(t-h-s)^{\alpha}}_{h}B+ \mathcal {E}^{A(t-2h-s)^{\alpha}}_{h}C \bigr)u(s)\,ds \\ &{}+ \int^{t}_{t-h}\mathcal{E}^{A(t-h-s)^{\alpha}}_{h}B u(s)\,ds+ \int ^{0}_{-h}\mathcal{E}^{A(t-2h-s)^{\alpha}}_{h}C \psi(s)\,ds. \end{aligned}$$

So the fractional control system (3) with state and control delay is controllable. Sufficiency is proved. This completes the result of Theorem 3.3. □

4 Example

Now, we will apply the conditions which we obtained in the previous section for a fractional differential system with state and control delay;

$${}^{c}D^{\alpha}_{0^{+}}x(t)=Ax(t-h)+Bu(t)+Cu(t-h), $$

$\alpha= 0.5$, $h=1$, where

$$\begin{gathered}A=\left ( \textstyle\begin{array}{c@{\quad}c} 3 & 0 \\ 0 & 4 \end{array}\displaystyle \right ),\quad\quad B=\left ( \textstyle\begin{array}{c} 5 \\ 0 \end{array}\displaystyle \right ),\qquad C=\left ( \textstyle\begin{array}{c} 0 \\ 3 \end{array}\displaystyle \right ),\qquad \\ {}^{c}D^{0.5}_{0^{+}}x(t)=\left ( \textstyle\begin{array}{c@{\quad}c} 3 & 0 \\ 0 & 4 \end{array}\displaystyle \right )x(t-1)+\left ( \textstyle\begin{array}{c} 5 \\ 0 \end{array}\displaystyle \right )u(t)+\left ( \textstyle\begin{array}{c} 0 \\ 3 \end{array}\displaystyle \right )u(t-1),\end{gathered} $$

where $x \in\mathcal{R}^{n}$ by simple calculations shows that

$$(B AB C AC) =\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 & 15 & 0 & 0 \\ 0 & 0 & 3 & 12 \end{array}\displaystyle \right ) $$

and $\operatorname{rank}(B AB C AC) = 2$.