# 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.

## 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 [16] as in further work various researchers considered control problems; for example, see [79].

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 [1017]. 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. [2124] 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.

## 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)

□

## 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. □

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

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## Funding

This work was supported by National Natural Science Foundation of China (110131013, 11471015).

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Correspondence to Musarrat Nawaz.

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Nawaz, M., Wei, J. & Jiale, S. The controllability of fractional differential system with state and control delay. Adv Differ Equ 2020, 30 (2020). https://doi.org/10.1186/s13662-019-2479-4

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### Keywords

• Controllability
• Delayed Mittag-Leffler type matrix
• State delay
• Control delay