## Abstract

In this paper, we generalize the time-varying descriptor systems to the case of fractional order in matrix forms. Moreover, we present the general exact solutions of the linear singular and non-singular matrix fractional time-varying descriptor systems with constant coefficient matrices in Caputo sense by using a new attractive method. Finally, two illustrated examples are also given to show our new approach.

## Keywords

Time-varying descriptor system; Kronecker product; Mittag–Leffler matrix

## 1. Introduction

Matrix differential equations have been widely used in the stability, observability and controllability theories of differential equations, control theory, communication systems and many other fields of applied mathematics [1], [2], [3], [4], [5], [6], [7], [8] and [9], and also recently in the following linear time-varying system [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] and [32]:

 ${\displaystyle A(t)y^{'}(t)=B(t)y(t)+C(t)u(t):y(t_{0})=y_{0}{\mbox{,}}\quad t\geqslant 0{\mbox{,}}}$
(1-1)

where ${\textstyle A(t)\in M_{n}}$ is a time-varying singular or non-singular matrix function, ${\textstyle B(t)\in M_{m}}$ and ${\textstyle C(t)\in M_{n{\mbox{,}}m}}$ are time-varying analytic matrix functions, ${\textstyle u(t)\in M_{m{\mbox{,}}1}}$ is the output vector function and ${\textstyle y(t)\in M_{n{\mbox{,}}1}}$ is the state function vector to be solved (where ${\textstyle M_{m{\mbox{,}}n}}$ is denoted by the set of all ${\textstyle m\times n}$ matrices over the real number R and when ${\textstyle m=n}$ we write ${\textstyle M_{m}}$ instead of ${\textstyle M_{m{\mbox{,}}n}}$). This system is usually known as a non-singular (singular) descriptor system or generalized state (semi) system or system of differential-algebraic equations and plays an important role in many applications such as in electrical networks, economics, optimization problems, analysis of control systems, engineering systems, constrained mechanics aircraft and robot dynamics, biology and large-scale systems [10], [11], [12], [13], [14] and [15]. The linear time-varying descriptor system as in (1-1) has been studied and discussed by many researchers [16], [17], [18], [19] and [20]. For example, controllability and observability of this system have been studied by Wang and Liao [17], Wang [18] and Campbell and et al. [19]; the linear of matrix differential inequalities of descriptor system was established by Inoue and et al. [20]; the Weierstrass–Kronecker decomposition theorem of the regular pencil was extended to the time-varying discrete-time descriptor system by Kaczorek [32] and finally, the stability of linear time-varying descriptor system has been discussed in [21], [22], [23], [24], [25], [26] and [27]. Some special cases of the linear time-varying system as in (1-1) have been also investigated in [28], [29], [30] and [31]. For example, the stability for the special case of system (1-1) when ${\textstyle B(t)=B{\mbox{,}}\quad C(t)=C}$ and ${\textstyle A(t)=A}$ are constant matrices has been discussed in [27], [28] and [29] and also the stability analysis for the special case of system (1-1) when ${\textstyle B(t+T)=B(t){\mbox{,}}\quad C(t+T)=C(t)}$ are periodically time-varying matrices with period T   and ${\textstyle A(t)=A}$ is a constant matrix has been studied in [24] and [29]. Finally, the optimal control of system as in (1-1) has been investigated in [30] and [31].

In addition, the topic of fractional calculus has attracted many researchers because of its several applications in various fields of applied sciences, physics and economics. For a detail survey with collections of applications in various fields, see for example [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43] and [44] and numerous real-life problems are also modeled mathematically by systems of fractional differential equations [37], [39], [40], [41], [43], [45], [46], [47], [48], [49], [50], [51], [52], [53] and [54]. Since there are many definitions of fractional derivative of order ${\textstyle \alpha >0}$ most of them are used an integral or summation or limit form [33], [37], [42], [44], [48], [51], [52], [53], [54], [55], [56], [57] and [58]. One of the important and familiar definition for fractional derivative is Caputo operator which is defined by the following:

 ${\displaystyle y^{\alpha }(t)=D^{\alpha }y(t)=I^{n-\alpha }D^{n}y(t)=}$${\displaystyle {\frac {1}{\Gamma (n-\alpha )}}{\int }_{0}^{t}{\frac {y^{\left(n\right)}(s)}{{\left(t-s\right)}^{\alpha -n+1}}}ds{\mbox{,}}}$
(1-2)

where ${\textstyle \alpha >0{\mbox{,}}\quad t>0}$ and ${\textstyle n-1<\alpha \leqslant n}$${\textstyle \left(n\in N\right)}$.

Note that the fractional derivative of ${\textstyle f(x)}$ in the Caputo sense is defined for ${\textstyle 0<\alpha <1}$ as

 ${\displaystyle D^{\alpha }y(t)={\frac {1}{\Gamma (1-\alpha )}}{\int }_{0}^{t}{\frac {y^{'}(s)}{{\left(t-s\right)}^{\alpha }}}ds{\mbox{.}}}$
(1-3)

Caputo’s definition has the advantage of dealing property with initial value problems in which the initial conditions are given in terms of the field variables and their integer order which is the case most physical processes.

In the present paper, we present the general exact solutions of the singular and non-singular matrix fractional time-varying descriptor systems with constant coefficient matrices in Caputo sense based on the Kronecker product and vector-operator with two illustrated examples.

## 2. Preliminaries and basic concepts

In this section, we study some important basic results related to the Kronecker product and Mittag–Leffler function on matrices, and fractional linear system that will be useful later in our investigation of the solutions of the linear matrix fractional time-varying descriptor systems.

## Definition 2.1.

Let ${\textstyle A=\left(a_{ij}\right)\in M_{m{\mbox{,}}n}}$ and ${\textstyle B=\left(b_{kl}\right)\in M_{p{\mbox{,}}q}}$ be two rectangular matrices. Then the Kronecker product of A and B is defined by [1], [2], [3], [4], [5], [6], [7], [8], [59], [60], [61], [62], [63], [64] and [65]:

 ${\displaystyle A\otimes B={\left(a_{ij}B\right)}_{ij}\in M_{mp{\mbox{,}}nq}{\mbox{.}}}$
(2-1)

## Definition 2.2.

Let ${\textstyle A=\left(a_{ij}\right)\in M_{m{\mbox{,}}n}}$ be a rectangular matrix. Then the vector-operator of A is defined by [1], [2], [3], [4], [5], [6], [7], [8], [59], [60], [61], [62], [63], [64] and [65]:

 ${\displaystyle Vec(A)={\left(a_{11}\quad a_{21}\quad \ldots \quad a_{m1}\quad a_{12}\quad a_{22}\quad \ldots \quad a_{m2}\quad \ldots \quad a_{1n}\quad a_{2n}\quad \ldots \quad a_{mn}\right)}^{T}\in M_{mn{\mbox{,}}1}{\mbox{,}}}$
(2-2)

## Lemma 2.1.

Let  ${\textstyle A{\mbox{,}}\quad B{\mbox{,}}\quad C{\mbox{,}}\quad D}$and X be matrices with compatible orders, and  ${\textstyle I_{n}}$be the identity matrix of order  ${\textstyle n\times n}$. Then[1], [2], [3], [7], [59], [60], [61], [62], [63], [64] and [65].

 ${\displaystyle (i)\quad Vec(AXB)=(B^{T}\otimes A)VecX{\mbox{,}}}$
(2 - 3)
 ${\displaystyle (ii)\quad (A\otimes B)(C\otimes D)=AC\otimes BD{\mbox{,}}}$
(2-4)

(iv) If f is analytic function on the region containing the eigenvalues of  ${\textstyle A\in M_{m}}$such that  ${\textstyle f(A)}$exist. Then

 ${\displaystyle f(A\otimes I_{n})=f(A)\otimes I_{n}\quad {\mbox{and}}\quad f(I_{n}\otimes A)=}$${\displaystyle I_{n}\otimes f(A){\mbox{.}}}$
(2-5)

## Definition 2.3.

The one-parameter Mittag–Leffler function ${\textstyle E_{\alpha }(t)}$ and Mittag–Leffler matrix function ${\textstyle E_{\alpha }({At}^{\alpha })}$ are defined, respectively, for ${\textstyle \alpha >0}$ by [33], [42], [51], [52], [56] and [66]:

 ${\displaystyle E_{\alpha }(t)=\sum _{k=0}^{\infty }{\frac {t^{k}}{\Gamma (k\alpha +1)}}\quad {\mbox{and}}\quad E_{\alpha }({At}^{\alpha })=}$${\displaystyle \sum _{k=0}^{\infty }{\frac {A^{k}t^{\alpha k}}{\Gamma (k\alpha +1)}}{\mbox{,}}}$
(2-6)

where ${\textstyle A\in M_{n}}$ is a matrix of order ${\textstyle n\times n}$ and ${\textstyle \Gamma (\cdot )}$ is the Gamma function.

## Lemma 2.2.

Let  ${\textstyle A\in M_{m}}$be a matrix of order  ${\textstyle m\times m}$and let  ${\textstyle \left\{x_{1}{\mbox{,}}x_{2}{\mbox{,}}\ldots {\mbox{,}}x_{m}\right\}}$and  ${\textstyle \left\{y_{1}{\mbox{,}}y_{2}{\mbox{,}}\ldots {\mbox{,}}y_{m}\right\}}$be the eigenvectors corresponding to the eigenvalues  ${\textstyle \left\{{\lambda }_{1}{\mbox{,}}{\lambda }_{2}{\mbox{,}}\ldots {\mbox{,}}{\lambda }_{m}\right\}}$of A and  ${\textstyle A^{T}}$, respectively. Then the spectral decomposition of  ${\textstyle E_{\alpha }(A)}$and  ${\textstyle E_{\alpha }({At}^{\alpha })}$are given, respectively, for  ${\textstyle \alpha >0}$by[56]:

 ${\displaystyle E_{\alpha }(A)=\sum _{k=1}^{m}x_{k}y_{k}^{T}E_{\alpha }({\lambda }_{k})\quad {\mbox{and}}\quad E_{\alpha }({At}^{\alpha })=}$${\displaystyle \sum _{k=1}^{m}x_{k}y_{k}^{T}E_{\alpha }({\lambda }_{k}t^{\alpha }){\mbox{,}}}$
(2-7)

The list of nice properties for Mittag–Leffler matrix  ${\textstyle E_{\alpha }(A)}$can be found in[56], and the most important properties for Mittag–Leffler matrix  ${\textstyle E_{\alpha }(A)}$that will be used in this study are given below[56].

## Theorem 2.1.

Let  ${\textstyle A{\mbox{,}}B\in M_{m}}$and  ${\textstyle I_{n}}$be an identity matrix of order  ${\textstyle n\times n}$. Then for  ${\textstyle \alpha >0}$, we have[56]:

 ${\displaystyle (i)\quad {\mbox{If}}\quad A=diag(a_{11}{\mbox{,}}a_{22}{\mbox{,}}\cdots {\mbox{,}}a_{mm}){\mbox{,}}\quad then\quad E_{\alpha }(A)=}$${\displaystyle diag\left(E_{\alpha }(a_{11}){\mbox{,}}E_{\alpha }(a_{22}){\mbox{,}}\cdots {\mbox{,}}E_{\alpha }(a_{mm})\right){\mbox{,}}}$
(2 - 8)

 ${\displaystyle (ii)\quad E_{\alpha }(A+B)=E_{\alpha }(A)E_{\alpha }(B)\quad if\quad and\quad only\quad if\quad AB=}$${\displaystyle BA{\mbox{,}}}$
(2 - 9)
 ${\displaystyle (iii)\quad E_{\alpha }(A\otimes I_{n})=E_{\alpha }(A)\otimes I_{n}\quad and\quad E_{\alpha }(I_{n}\otimes A)=}$${\displaystyle I_{n}\otimes E_{\alpha }(A){\mbox{.}}}$
(2-10)

## Lemma 2.3.

Let  ${\textstyle H\in M_{n}}$be a given scalar matrix,  ${\textstyle u(t)\in M_{n{\mbox{,}}1}}$be a given vector function, and  ${\textstyle y(t)\in M_{n{\mbox{,}}1}}$be the unknown vector to be solved. Then the unique solution of the following fractional differential system[51], [52] and [56]:

 ${\displaystyle y^{\alpha }(t)=Hy(t)+u(t):\quad y(0)=y_{0}\in M_{n{\mbox{,}}1}{\mbox{,}}}$
(2-11)

is given by

 ${\displaystyle y(t)=E_{\alpha }({Ht}^{\alpha })y_{0}+{\int }_{0}^{t}{\left(t-z\right)}^{\alpha -1}E_{\alpha }\left(H{\left(t-z\right)}^{\alpha }\right)u(z)dz{\mbox{.}}}$
(2-12)

## 3. Main results

In this section, we formulate and present the general exact solutions of the singular and non-singular matrix fractional time-varying descriptor systems in Caputo sense based on the Kronecker product, vector-operator and Lemma 2.3 with two illustrated examples.

## Problem 3.1 Singular Matrix Fractional Time-Varying Descriptor System.

The linear singular matrix fractional time-varying descriptor system can be formulated by

 ${\displaystyle A(t)Y^{\alpha }(t)=B(t)Y(t)+C(t)U(t):Y(0)=Y_{0}{\mbox{,}}\quad t\geqslant 0{\mbox{,}}\quad \alpha >0{\mbox{,}}}$
(3-1)

where ${\textstyle A(t)\in M_{n}}$ is a time-varying singular matrix function, ${\textstyle B(t)\in M_{n}}$ and ${\textstyle C(t)\in M_{n}}$ are time-varying analytic matrix functions, ${\textstyle U(t)\in M_{n}}$ is the output matrix function and ${\textstyle Y(t)\in M_{n}}$ is the state function vector to be solved. Here, we will study the general solution of (3-1) when ${\textstyle A(t)=A{\mbox{,}}\quad B(t)=B}$ and ${\textstyle C(t)=C}$ are constant matrices, as a special case. For this case, suppose that the constant invertible matrices M   and ${\textstyle N\in M_{n}}$ such that:

 ${\displaystyle A=M^{-1}\left[{\begin{array}{cc}I&0\\0&0\end{array}}\right]N^{-1}{\mbox{,}}\quad B=M^{-1}\left[{\begin{array}{cc}B_{11}&B_{12}\\B_{21}&B_{22}\end{array}}\right]N^{-1}{\mbox{,}}\quad C=M^{-1}\left[{\begin{array}{c}C_{1}\\C_{2}\end{array}}\right]\quad {\mbox{and}}\quad Y(t)=N\left[{\begin{array}{c}Y_{1}(t)\\Y_{2}(t)\end{array}}\right]{\mbox{.}}}$
(3-2)

If we partition n   as ${\textstyle n=m+p}$, then ${\textstyle Y_{1}(t)\in M_{m{\mbox{,}}n}}$ and ${\textstyle Y_{2}(t)\in M_{p{\mbox{,}}n}}$. This system is restricted equivalent to:

 ${\displaystyle Y_{1}^{\alpha }(t)=B_{11}Y_{1}(t)+B_{12}Y_{2}(t)+C_{1}U(t){\mbox{,}}\quad 0=}$${\displaystyle B_{21}Y_{1}(t)+B_{22}Y_{2}(t)+C_{2}U(t){\mbox{.}}}$
(3-3)

Note that the necessary and sufficient condition for the existence of the solution of a system (3-1) is that ${\textstyle B_{22}(t)}$ is invertible.

## General Solutions of Problem 3.1.

Since ${\textstyle B_{22}(t)}$ is an invertible matrix and then from the second equation of (3-3) we have:

 ${\displaystyle Y_{2}(t)=-B_{22}^{-1}B_{21}Y_{1}(t)-B_{22}^{-1}C_{2}U(t){\mbox{.}}}$
(3-4)

By substituting this equation in the first equation of (3-3), we get:

 ${\displaystyle Y_{1}^{\alpha }(t)=S_{B_{11}}Y_{1}(t)+RU(t){\mbox{,}}}$
(3-5)

where

 ${\displaystyle R=-B_{12}B_{22}^{-1}C_{2}+C_{1}{\mbox{,}}}$
(3-6)

and

 ${\displaystyle S_{B_{11}}=B_{11}-B_{12}B_{22}^{-1}B_{21}}$
(3-7)

is called the Schur complement of ${\textstyle B_{11}}$ in a matrix ${\textstyle \left[{\begin{array}{cc}B_{11}&B_{12}\\B_{21}&B_{22}\end{array}}\right]}$.

Now, by taking ${\textstyle Vec(\cdot )}$ of both sides of (3-5), and using (2-3) in Lemma 2.1, we have:

 ${\displaystyle Vec\left(Y_{1}^{\alpha }(t)\right)=Vec\left(S_{B_{11}}Y_{1}(t)\right)+}$${\displaystyle Vec\left(RU(t)\right)=\left(I_{n}\otimes S_{B_{11}}\right)\quad Vec\left(Y_{1}(t)\right)+}$${\displaystyle \left(I_{n}\otimes R\right)Vec\left(U(t)\right){\mbox{.}}}$
(3-8)

Now by letting ${\textstyle Vec\left(Y_{1}^{\alpha }(t)\right)=y_{1}^{\alpha }(t){\mbox{,}}\quad Vec\left(Y_{1}(t)\right)=}$${\displaystyle y_{1}(t)}$ and ${\textstyle Vec\left(U(t)\right)=u(t)}$, then (3-8) can be represented as follows:

 ${\displaystyle y_{1}^{\epsilon }(t)=\left(I_{n}\otimes S_{B_{11}}\right)y_{1}(t)+}$${\displaystyle \left(I_{n}\otimes R\right)u(t){\mbox{.}}}$
(3-9)

Now by using Lemma 2.3, then the vector solution of (3-9) is given by:

 ${\displaystyle Vec\left(Y_{1}(t)\right)=y_{1}(t)=E_{\alpha }\left((I_{n}\otimes S_{B_{11}})t^{\alpha }\right)\quad y_{1}(0)+}$${\displaystyle {\int }_{0}^{t}{\left(t-z\right)}^{\alpha -1}E_{\alpha }\left((I_{n}\otimes S_{B_{11}}){\left(t-z\right)}^{\alpha }\right)\left(\left(I_{n}\otimes R\right)\quad u(z)\right)\quad dz=}$${\displaystyle E_{\alpha }\left((I_{n}\otimes S_{B_{11}})t^{\alpha }\right)\quad Vec(Y_{1}(0)+}$${\displaystyle {\int }_{0}^{t}{\left(t-z\right)}^{\alpha -1}E_{\alpha }\left((I_{n}\otimes S_{B_{11}}){\left(t-z\right)}^{\alpha }\right)\left(\left(I_{n}\otimes R\right)\quad Vec\left(U(z)\right)\right)\quad dz=}$${\displaystyle E_{\alpha }\left((I_{n}\otimes S_{B_{11}})t^{\alpha }\right)\quad Vec(Y_{1}(0)+}$${\displaystyle {\int }_{0}^{t}{\left(t-z\right)}^{\alpha -1}E_{\alpha }\left((I_{n}\otimes S_{B_{11}}){\left(t-z\right)}^{\alpha }\right)\left(Vec\left(RU(z)\right)\right)\quad dz{\mbox{,}}}$
(3-10)

where R   and ${\textstyle S_{B_{11}}}$ are constant matrices as defined in (3-6) and (3-7), respectively.

Note that the relationship between ${\textstyle Y_{1}(t)\in M_{m{\mbox{,}}n}}$ and ${\textstyle x=y_{1}(t)=Vec\left(Y_{1}(t)\right)\in M_{mn{\mbox{,}}1}}$ is given by:

 ${\displaystyle Y_{1}(t)=\left[{\begin{array}{cccc}x^{\left(1\right)}{\mbox{,}}&x^{\left(2\right)}{\mbox{,}}&\cdots &x^{\left(n\right)}\end{array}}\right]=\left[{\begin{array}{cccc}x_{1}&x_{p+1}&\cdots &x_{(n-1)p+1}\\.&.&&{\mbox{.}}\\.&.&&{\mbox{.}}\\x_{p}&x_{2p}&\cdots &x_{np}\end{array}}\right]\in M_{n{\mbox{,}}p}{\mbox{.}}}$
(3-11)

Hence, the general solution of Problem 3.1 is given by: ${\textstyle Y(t)=N\left[{\begin{array}{c}Y_{1}(t)\\Y_{2}(t)\end{array}}\right]}$, where ${\textstyle Y_{1}(t)}$ can be easily obtained from (3-10) and (3-11); and ${\textstyle Y_{2}(t)}$ is given as in (3-4).

## Problem 3.2 Non-Singular Matrix Fractional Time-Varying Descriptor System.

The linear non-singular matrix fractional time-varying descriptor system can be formulated by:

 ${\displaystyle A(t)Y^{\alpha }(t)=B(t)Y(t)+C(t)U(t):Y(0)=Y_{0}{\mbox{,}}\quad t\geqslant 0{\mbox{,}}\quad \alpha >0{\mbox{,}}}$
(3-12)

where ${\textstyle A(t)\in M_{n}}$ is a time-varying non-singular matrix function, ${\textstyle B(t)\in M_{n}}$ and ${\textstyle C(t)\in M_{n}}$ are time-varying analytic matrix functions, ${\textstyle U(t)\in M_{n}}$ is the output matrix function and ${\textstyle Y(t)\in M_{n}}$ is the state function matrix to be solved. Here, we will study the general solution of (3-12) when ${\textstyle A(t)=A{\mbox{,}}\quad B(t)=B}$ and ${\textstyle C(t)=C}$ are constant matrices, as a special case. For this case, suppose that the constant invertible matrices M   and ${\textstyle N\in M_{n}}$ such that:

 ${\displaystyle A=M^{-1}\left[{\begin{array}{cc}I&0\\0&I\end{array}}\right]N^{-1}{\mbox{,}}\quad B=M^{-1}\left[{\begin{array}{cc}B_{11}&B_{12}\\B_{21}&B_{22}\end{array}}\right]N^{-1}{\mbox{,}}\quad C=M^{-1}\left[{\begin{array}{c}C_{1}\\C_{2}\end{array}}\right]\quad {\mbox{and}}\quad Y(t)=N\left[{\begin{array}{c}Y_{1}(t)\\Y_{2}(t)\end{array}}\right]{\mbox{.}}}$
(3-13)

This system is restricted equivalent to:

 ${\displaystyle Y_{1}^{\alpha }(t)=B_{11}Y_{1}(t)+B_{12}Y_{2}+C_{1}U(t){\mbox{,}}\quad Y_{2}^{\alpha }(t)=}$${\displaystyle B_{21}Y_{1}(t)+B_{22}Y_{2}(t)+C_{2}U(t){\mbox{.}}}$
(3-14)

## General Solutions of Problem 3.2.

By taking ${\textstyle Vec(\cdot )}$ of both sides of (3-14), and using Lemma 2.1, we have:

 ${\displaystyle Vec\left(Y_{1}^{\alpha }(t)\right)=Vec\left(B_{11}Y_{1}(t)+\right.}$${\displaystyle \left.B_{12}Y_{2}(t)+C_{1}U(t)\right)=\left(I_{n}\otimes B_{11}\right)\quad Vec\left(Y_{1}(t)\right)+}$${\displaystyle \left(I_{n}\otimes B_{12}\right)\quad Vec\left(Y_{2}(t)\right)+}$${\displaystyle \left(I_{n}\otimes C_{1}\right)\quad Vec\left(U(t)\right){\mbox{,}}}$ ${\displaystyle Vec\left(Y_{2}^{\alpha }(t)\right)=Vec\left(B_{21}Y_{1}(t)+\right.}$${\displaystyle \left.B_{22}Y_{2}(t)+C_{2}U(t)\right)=\left(I_{n}\otimes B_{21}(t)\right)\quad Vec\left(Y_{1}(t)\right)+}$${\displaystyle \left(I_{n}\otimes B_{22}(t)\right)\quad Vec\left(Y_{2}(t)\right)+}$${\displaystyle \left(I_{n}\otimes C_{2}(t)\right)\quad Vec\left(U(t)\right){\mbox{.}}}$
(3-15)

This system can be represented as:

 ${\displaystyle \left[{\begin{array}{c}Vec\left(Y_{1}^{\alpha }(t)\right)\\Vec\left(Y_{2}^{\alpha }(t)\right)\end{array}}\right]=\left[{\begin{array}{cc}I_{n}\otimes B_{11}&I_{n}\otimes B_{12}\\I_{n}\otimes B_{21}&I_{n}\otimes B_{22}\end{array}}\right]\left[{\begin{array}{c}Vec\left(Y_{1}(t)\right)\\Vec\left(Y_{2}(t)\right)\end{array}}\right]+\left[{\begin{array}{c}\left(I_{n}\otimes C_{1}\right)Vec\left(U(t)\right)\\\left(I_{n}\otimes C_{2}\right)Vec\left(U(t)\right)\end{array}}\right]{\mbox{.}}}$
(3-16)

Suppose that

 ${\displaystyle T^{\alpha }(t)=\left[{\begin{array}{c}Vec\left(Y_{1}^{\alpha }(t)\right)\\Vec\left(Y_{2}^{\alpha }(t)\right)\end{array}}\right]{\mbox{,}}\quad H=\left[{\begin{array}{cc}I_{n}\otimes B_{11}&I_{n}\otimes B_{12}\\I_{n}\otimes B_{21}&I_{n}\otimes B_{22}\end{array}}\right]{\mbox{,}}}$
 ${\displaystyle T(t)=\left[{\begin{array}{c}Vec\left(Y_{1}(t)\right)\\Vec\left(Y_{2}(t)\right)\end{array}}\right]{\mbox{,}}\quad D(t)=\left[{\begin{array}{c}\left(I_{n}\otimes C_{1}\right)Vec\left(U(t)\right)\\\left(I_{n}\otimes C_{2}\right)Vec\left(U(t)\right)\end{array}}\right]{\mbox{.}}}$

Now the system as in (3-16) can be rewritten as follows:

 ${\displaystyle T^{\alpha }(t)=HT(t)+D(t){\mbox{.}}}$
(3-17)

Now by using Lemma 2.3, then the solution of (3-17) is given by:

 ${\displaystyle T(t)=E_{\alpha }(H\quad t^{\alpha })\quad T(0)+{\int }_{0}^{t}{\left(t-z\right)}^{\alpha -1}E_{\alpha }\left(H{\left(t-z\right)}^{\alpha }\right)\quad D(z)\quad dz{\mbox{.}}}$
(3-18)

This leads to the following general vector solution of Problem 3.2:

 ${\displaystyle \left[{\begin{array}{c}Vec\left(Y_{1}(t)\right)\\Vec\left(Y_{2}(t)\right)\end{array}}\right]\quad =E_{\alpha }\left(\left[{\begin{array}{cc}I_{n}\otimes B_{11}&I_{n}\otimes B_{12}\\I_{n}\otimes B_{21}&I_{n}\otimes B_{22}\end{array}}\right]\quad t^{\alpha }\right)\quad N^{-1}\left[{\begin{array}{c}Vec\left(Y_{1}(0)\right)\\Vec\left(Y_{2}(0)\right)\end{array}}\right]}$ ${\displaystyle \quad +{\int }_{0}^{t}{\left(t-z\right)}^{\alpha -1}E_{\alpha }\left(\left[{\begin{array}{cc}I_{n}\otimes B_{11}&I_{n}\otimes B_{12}\\I_{n}\otimes B_{21}&I_{n}\otimes B_{22}\end{array}}\right]\quad {\left(t-z\right)}^{\alpha }\right)\quad \left[{\begin{array}{c}\left(I_{n}\otimes C_{1}\right)Vec\left(U(z)\right)\\\left(I_{n}\otimes C_{2}\right)Vec\left(U(z)\right)\end{array}}\right]\quad dz{\mbox{.}}}$ ${\displaystyle \quad =E_{\alpha }\left(\left[{\begin{array}{cc}I_{n}\otimes B_{11}&I_{n}\otimes B_{12}\\I_{n}\otimes B_{21}&I_{n}\otimes B_{22}\end{array}}\right]\quad t^{\alpha }\right)\quad N^{-1}\left[{\begin{array}{c}Vec\left(Y_{1}(0)\right)\\Vec\left(Y_{2}(0)\right)\end{array}}\right]}$ ${\displaystyle \quad +{\int }_{0}^{t}{\left(t-z\right)}^{\alpha -1}E_{\alpha }\left(\left[{\begin{array}{cc}I_{n}\otimes B_{11}&I_{n}\otimes B_{12}\\I_{n}\otimes B_{21}&I_{n}\otimes B_{22}\end{array}}\right]\quad {\left(t-z\right)}^{\alpha }\right)\quad \left[{\begin{array}{c}Vec\left(C_{1}U(z)\right)\\Vec\left(C_{2}U(z)\right)\end{array}}\right]\quad dz{\mbox{.}}}$
(3-19)

Another special case of (3-12) is when ${\textstyle A(t)=A}$ and ${\textstyle B(t)=B}$ are constant matrices and ${\textstyle U(t)=0}$. Then the general solution of this case is given by ${\textstyle Y(t)=E_{\alpha }((A^{-1}B)t^{\alpha })Y_{0}}$.

The main problem in the solution of Problem 3.2 as in (3-19) is how to compute the following Mittag–Leffler matrix:

 ${\displaystyle E_{\alpha }\left(\left[{\begin{array}{cc}I_{n}\otimes B_{11}&I_{n}\otimes B_{12}\\I_{n}\otimes B_{21}&I_{n}\otimes B_{22}\end{array}}\right]\right){\mbox{.}}}$
(3-20)

As a special case, if ${\textstyle B_{11}B_{12}=B_{12}B_{22}}$ and ${\textstyle B_{21}B_{11}=B_{22}B_{21}}$, then by using the same procedure in the proof of Theorem 2 in [56] and Theorem 2.1, we have:

 ${\displaystyle E_{\alpha }\left(\left[{\begin{array}{cc}I_{n}\otimes B_{11}&I_{n}\otimes B_{12}\\I_{n}\otimes B_{21}&I_{n}\otimes B_{22}\end{array}}\right]\right)=\left[{\begin{array}{cc}I_{n}\otimes E_{\alpha }\left(B_{11}\right)\left\{{\frac {I_{n}\otimes E_{\alpha }\left(B_{12}\right)-I_{n}\otimes E_{\alpha }\left(B_{21}\right)}{2}}\right\}&I_{n}\otimes E_{\alpha }\left(B_{11}\right)\left\{{\frac {I_{n}\otimes E_{\alpha }\left(B_{12}\right)+I_{n}\otimes E_{\alpha }\left(B_{21}\right)}{2}}\right\}\\I_{n}\otimes E_{\alpha }\left(B_{22}\right)\left\{{\frac {I_{n}\otimes E_{\alpha }\left(B_{12}\right)+I_{n}\otimes E_{\alpha }\left(B_{21}\right)}{2}}\right\}&I_{n}\otimes E_{\alpha }\left(B_{22}\right)\left\{{\frac {I_{n}\otimes E_{\alpha }\left(B_{12}\right)-I_{n}\otimes E_{\alpha }\left(B_{21}\right)}{2}}\right\}\end{array}}\right]}$
(3-21)

Now, it is easy to get ${\textstyle Y_{1}(t)}$ and ${\textstyle Y_{2}(t)}$ of this case by substituting (3-21) in (3-19) and then the general solution of this problem is given by ${\textstyle Y(t)=N\left[{\begin{array}{c}Y_{1}(t)\\Y_{2}(t)\end{array}}\right]}$.

## Example 3.1.

Consider the following linear singular matrix fractional time-varying descriptor system:

 ${\displaystyle {AY}^{\alpha }(t)=BY(t)+CU(t):Y(0)=Y_{0}{\mbox{,}}\quad t\geqslant 0{\mbox{,}}\quad \alpha >0{\mbox{,}}}$
(3-22)

where

 ${\displaystyle A=\left[{\begin{array}{cc}I_{2}&0\\0&0\end{array}}\right]=\left[{\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&0&0\\0&0&0&0\end{array}}\right]{\mbox{,}}\quad B=\left[{\begin{array}{cc}B_{11}&B_{12}\\B_{21}&B_{22}\end{array}}\right]=\left[{\begin{array}{cc}I_{2}&I_{2}\\I_{2}&I_{2}\end{array}}\right]=\left[{\begin{array}{cccc}1&0&1&0\\0&1&0&1\\1&0&1&0\\0&1&0&1\end{array}}\right]{\mbox{,}}}$
 ${\displaystyle C=\left[{\begin{array}{c}C_{1}\\C_{2}\end{array}}\right]=\left[{\begin{array}{cccc}1&0&1&0\\0&1&0&1\\-&-&-&-\\0&1&0&1\\1&0&1&0\end{array}}\right]{\mbox{,}}\quad U(t)=\left[{\begin{array}{cccc}t&0&0&0\\0&t&0&0\\0&0&t&0\\0&0&0&t\end{array}}\right]{\mbox{,}}\quad Y(0)=\left[{\begin{array}{c}Y_{1}(0)\\Y_{2}(0)\end{array}}\right]=\left[{\begin{array}{cccc}1&0&1&0\\0&1&0&1\\-&-&-&-\\1&0&1&0\\0&1&0&1\end{array}}\right]\quad {\mbox{and}}}$
 ${\displaystyle M=N=I_{4}=\left[{\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}}\right]{\mbox{.}}}$

Since

 ${\displaystyle S_{B_{11}}=I_{2}-I_{2}I_{2}^{-1}I_{2}=I_{2}=\left[{\begin{array}{cc}1&0\\0&1\end{array}}\right]{\mbox{,}}}$
 ${\displaystyle R=-I_{2}I_{2}^{-1}\left[{\begin{array}{cccc}0&1&0&1\\1&0&1&0\end{array}}\right]+\left[{\begin{array}{cccc}1&0&1&0\\0&1&0&1\end{array}}\right]=\left[{\begin{array}{cccc}1&1&1&1\\1&1&1&1\end{array}}\right]{\mbox{.}}}$

Then ${\textstyle Y_{1}(t)\in M_{2{\mbox{,}}4}}$ and ${\textstyle Y_{2}(t)\in M_{2{\mbox{,}}4}}$ by using (3-4) and (3-10), respectively, are given by:

 ${\displaystyle Vec\left(Y_{1}(t)\right)=E_{\alpha }\left((I_{4}\otimes I_{2})t^{\alpha }\right)\quad Vec(Y_{1}(0)+}$${\displaystyle {\int }_{0}^{t}{\left(t-z\right)}^{\alpha -1}E_{\alpha }\left((I_{4}\otimes I_{2}){\left(t-z\right)}^{\alpha }\right)\left(Vec\left(RU(z)\right)\right)\quad dz=}$${\displaystyle E_{\alpha }\left((I_{8}t^{\alpha }\right)\quad Vec(Y_{1}(0)+}$${\displaystyle {\int }_{0}^{t}{\left(t-z\right)}^{\alpha -1}E_{\alpha }\left(I_{8}{\left(t-z\right)}^{\alpha }\right)\left(Vec\left(RU(z)\right)\right)\quad dz=}$${\displaystyle \left[{\begin{array}{c}E_{\alpha }(t^{\alpha })\\0\\E_{\alpha }(t^{\alpha })\\0\\0\\E_{\alpha }(t^{\alpha })\\0\\E_{\alpha }(t^{\alpha })\end{array}}\right]+{\int }_{0}^{t}{\left(t-z\right)}^{\alpha -1}\left[{\begin{array}{l}{zE}_{\alpha }{\left(t-z\right)}^{\alpha }\\{zE}_{\alpha }{\left(t-z\right)}^{\alpha }\\{zE}_{\alpha }{\left(t-z\right)}^{\alpha }\\{zE}_{\alpha }{\left(t-z\right)}^{\alpha }\\{zE}_{\alpha }{\left(t-z\right)}^{\alpha }\\{zE}_{\alpha }{\left(t-z\right)}^{\alpha }\\{zE}_{\alpha }{\left(t-z\right)}^{\alpha }\\{zE}_{\alpha }{\left(t-z\right)}^{\alpha }\end{array}}\right]dz{\mbox{.}}}$
(3-23)
 ${\displaystyle Y_{2}(t)=-I_{2}^{-1}I_{2}Y_{1}(t)-I_{2}C_{2}U(t)=-Y_{1}(t)-C_{2}U(t)=}$${\displaystyle -Y_{1}(t)-\left[{\begin{array}{cccc}0&t&0&t\\t&0&t&0\end{array}}\right]{\mbox{,}}}$
(3-24)

where ${\textstyle Y_{1}(t)\in M_{2{\mbox{,}}4}}$ is given as a vector solution as in (3-23).

Finally the general solution of system as in (3-22) is given by:

 ${\displaystyle Y(t)=I_{4}\left[{\begin{array}{c}Y_{1}(t)\\Y_{2}(t)\end{array}}\right]=\left[{\begin{array}{c}Y_{1}(t)\\Y_{2}(t)\end{array}}\right]\in M_{4}{\mbox{.}}}$
(3-25)

As a special case of system (3-22), if ${\textstyle U(t)=0}$, then ${\textstyle Y_{1}(t)\in M_{2{\mbox{,}}4}{\mbox{,}}\quad Y_{2}(t)\in M_{2{\mbox{,}}4}}$ and ${\textstyle Y(t)\in M_{4}}$ are given, respectively, as in (3-26), (3-27) and (3-28) below:

 ${\displaystyle Vec\left(Y_{1}(t)\right)=\left[{\begin{array}{c}E_{\alpha }(t^{\alpha })\\0\\E_{\alpha }(t^{\alpha })\\0\\0\\E_{\alpha }(t^{\alpha })\\0\\E_{\alpha }(t^{\alpha })\end{array}}\right]{\mbox{.}}}$

Now from (3-11), we get:

 ${\displaystyle Y_{1}(t)=\left[{\begin{array}{cccc}E_{\alpha }(t^{\alpha })&0&E_{\alpha }(t^{\alpha })&0\\0&E_{\alpha }(t^{\alpha })&0&E_{\alpha }(t^{\alpha })\end{array}}\right]\in M_{2{\mbox{,}}4}{\mbox{.}}}$
(3-26)

 ${\displaystyle Y_{2}(t)=-Y_{1}(t)-\left[{\begin{array}{cccc}0&t&0&t\\t&0&t&0\end{array}}\right]=\left[{\begin{array}{cccc}-E_{\alpha }(t^{\alpha })&-t&-E_{\alpha }(t^{\alpha })&-t\\-t&-E_{\alpha }(t^{\alpha })&-t&-E_{\alpha }(t^{\alpha })\end{array}}\right]\in M_{2{\mbox{,}}4}{\mbox{.}}}$
(3-27)

 ${\displaystyle Y(t)=\left[{\begin{array}{c}Y_{1}(t)\\Y_{2}(t)\end{array}}\right]=\left[{\begin{array}{cccc}E_{\alpha }(t^{\alpha })&0&E_{\alpha }(t^{\alpha })&0\\0&E_{\alpha }(t^{\alpha })&0&E_{\alpha }(t^{\alpha })\\-E_{\alpha }(t^{\alpha })&-t&-E_{\alpha }(t^{\alpha })&-t\\-t&-E_{\alpha }(t^{\alpha })&-t&-E_{\alpha }(t^{\alpha })\end{array}}\right]\in M_{4}{\mbox{.}}}$
(3-28)

## Example 3.2.

Consider the following linear non-singular matrix fractional time-varying descriptor system:

 ${\displaystyle A(t)Y^{\alpha }(t)=B(t)Y(t):Y(0)=Y_{0}{\mbox{,}}\quad t\geqslant 0{\mbox{,}}\quad \alpha >0{\mbox{,}}}$
(3-29)

where

 ${\displaystyle M=N=I_{4}=\left[{\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}}\right]{\mbox{,}}\quad A(t)=\left[{\begin{array}{cc}I_{2}(t)&0\\0&0\end{array}}\right]=\left[{\begin{array}{cccc}t&0&0&0\\0&t&0&0\\0&0&t&0\\0&0&0&t\end{array}}\right]{\mbox{,}}}$
 ${\displaystyle B(t)=\left[{\begin{array}{cc}B_{11}(t)&B_{12}(t)\\B_{21}(t)&B_{22}(t)\end{array}}\right]=\left[{\begin{array}{ccccc}-t&0&\quad &1&0\\0&1&\quad &0&1\\-&-&\quad &-&-\\1&0&\quad &-t&0\\0&1&\quad &0&1\end{array}}\right]\quad {\mbox{and}}}$
 ${\displaystyle Y(0)=\left[{\begin{array}{c}Y_{1}(0)\\Y_{2}(0)\end{array}}\right]=\left[{\begin{array}{cccc}1&0&1&0\\0&1&0&1\\-&-&-&-\\1&0&1&0\\0&1&0&1\end{array}}\right]{\mbox{.}}}$

Since ${\textstyle B_{11}B_{12}=B_{12}B_{22}}$ and ${\textstyle B_{21}B_{11}=B_{22}B_{21}}$, then by applying (3-19) and (3-21) and Theorem 2.1, we get:

 ${\displaystyle \left[{\begin{array}{c}Vec\left(Y_{1}(t)\right)\\Vec\left(Y_{2}(t)\right)\end{array}}\right]=E_{\alpha }\left(\left[{\begin{array}{cc}I_{2}\otimes B_{11}(t)&I_{2}\otimes I_{2}\\I_{2}\otimes I_{2}&I_{2}\otimes B_{22}(t)\end{array}}\right]t^{\alpha }\right)\quad \left[{\begin{array}{c}Vec\left(Y_{1}(0)\right)\\Vec\left(Y_{2}(0)\right)\end{array}}\right]=\left[{\begin{array}{cc}I_{2}\otimes E_{\alpha }\left(B_{11}(t)\right)\left\{{\frac {I_{2}\otimes E_{\alpha }\left(I_{2}\right)-I_{2}\otimes E_{\alpha }\left(I_{2}\right)}{2}}\right\}&I_{2}\otimes E_{\alpha }\left(B_{11}(t)\right)\left\{{\frac {I_{2}\otimes E_{\alpha }\left(I_{2})\right)+I_{2}\otimes E_{\alpha }\left(I_{2}\right)}{2}}\right\}\\I_{2}\otimes E_{\alpha }\left(B_{22}(t)\right)\left\{{\frac {I_{2}\otimes E_{\alpha }\left(I_{2}\right)+I_{2}\otimes E_{\alpha }\left(I_{2}\right)}{2}}\right\}&I_{2}\otimes E_{\alpha }\left(B_{22}(t)\right)\left\{{\frac {I_{2}\otimes E_{\alpha }\left(I_{2}\right)-I_{2}\otimes E_{\alpha }\left(I_{2}\right)}{2}}\right\}\end{array}}\right]t^{\alpha }\times \quad \left[{\begin{array}{l}Vec\left(Y_{1}(0)\right)\\Vec\left(Y_{2}(0)\right)\end{array}}\right]{\mbox{.}}}$

Now,

 ${\displaystyle Vec\left(Y_{1}(t)\right)=\left(I_{2}\otimes E_{\alpha }\left(B_{11}(t)\right)\right)\left(I_{2}\otimes E_{\alpha }\left(I_{2}t^{\alpha }\right)\right)\quad Vec\left(Y_{2}(0)\right)=}$${\displaystyle \left(I_{2}\otimes E_{\alpha }\left(B_{11}(t)\right)E_{\alpha }\left(I_{2}\right)t^{\alpha }\right)\quad Vec\left(Y_{2}(0)\right)=}$${\displaystyle Vec\left\{\left(E_{\alpha }\left(B_{11}(t)\right)E_{\alpha }\left(I_{2}\right)t^{\alpha }\right)\quad Y_{2}(0)\right\}}$

That is by using (2-9), we have

 ${\displaystyle Y_{1}(t)=\left(E_{\alpha }\left(B_{11}(t)\right)E_{\alpha }\left(I_{2}\right)t^{\alpha }\right)Y_{2}(0)=}$${\displaystyle \left\{E_{\alpha }\left(\left(B_{11}(t)+E_{\alpha }\left(I_{2}\right)\right)t^{\alpha }\right)\right\}\quad Y_{2}(0)=}$${\displaystyle \left[{\begin{array}{cc}E_{\alpha }\left(t^{\alpha }-t^{\alpha +1}\right)&0\\0&E_{\alpha }\left(2t^{\alpha }\right)\end{array}}\right]Y_{2}(0)=\left[{\begin{array}{cc}E_{\alpha }\left(t^{\alpha }-t^{\alpha +1}\right)&0\\0&E_{\alpha }\left(2t^{\alpha }\right)\end{array}}\right]\left[{\begin{array}{cccc}1&0&1&0\\0&1&0&1\end{array}}\right]=\left[{\begin{array}{cccc}E_{\alpha }\left(t^{\alpha }-t^{\alpha +1}\right)&0&E_{\alpha }\left(t^{\alpha }-t^{\alpha +1}\right)&0\\0&E_{\alpha }\left(2t^{\alpha }\right)&0&E_{\alpha }\left(2t^{\alpha }\right)\end{array}}\right]\in M_{2{\mbox{,}}4}}$
(3-30)

Similarly, we have

 ${\displaystyle Y_{2}(t)=\left[{\begin{array}{cc}E_{\alpha }\left(t^{\alpha }-t^{\alpha +1}\right)&0\\0&E_{\alpha }\left(2t^{\alpha }\right)\end{array}}\right]Y_{1}(0)=\left[{\begin{array}{cccc}E_{\alpha }\left(t^{\alpha }-t^{\alpha +1}\right)&0&E_{\alpha }\left(t^{\alpha }-t^{\alpha +1}\right)&0\\0&E_{\alpha }\left(2t^{\alpha }\right)&0&E_{\alpha }\left(2t^{\alpha }\right)\end{array}}\right]\in M_{2{\mbox{,}}4}{\mbox{.}}}$
(3-31)

Hence, the general solutions of system (3-29) are given by:

 ${\displaystyle Y(t)=I_{4}\left[{\begin{array}{c}Y_{1}(t)\\Y_{2}(t)\end{array}}\right]=\left[{\begin{array}{c}Y_{1}(t)\\Y_{2}(t)\end{array}}\right]=\left[{\begin{array}{cccc}E_{\alpha }\left(t^{\alpha }-t^{\alpha +1}\right)&0&E_{\alpha }\left(t^{\alpha }-t^{\alpha +1}\right)&0\\0&E_{\alpha }(2t^{\alpha })&0&E_{\alpha }(2t^{\alpha })\\E_{\alpha }\left(t^{\alpha }-t^{\alpha +1}\right)&0&E_{\alpha }\left(t^{\alpha }-t^{\alpha +1}\right)&0\\0&E_{\alpha }(2t^{\alpha })&0&E_{\alpha }(2t^{\alpha })\end{array}}\right]\in M_{4}{\mbox{.}}}$
(3-32)

Note that if ${\textstyle A(t)=A}$ and ${\textstyle B(t)=B}$ are constant matrices in Example 3.2, then the general solution given by ${\textstyle Y(t)=E_{\alpha }((A^{-1}B)t^{\alpha })Y_{0}}$.

## 4. Conclusion

The general exact solutions of the singular and non-singular matrix fractional time-varying descriptor systems in Caputo sense with constant coefficient matrices are presented by a new attractive method with two illustrated examples. How to find the general solutions of these problems with non-constant coefficient matrices and also how to find the sufficient conditions, stability, controllability and observability of these problems still require further research.

## Acknowledgments

The author expresses his sincere thanks to referees for very careful reading and helpful suggestion of this paper.

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