The general solutions of singular and non-singular matrix fractional time-varying descriptor systems with constant coefficient matrices in Caputo sense

==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 [[#b0005|[1]]], [[#b0010|[2]]], [[#b0015|[3]]], [[#b0020|[4]]], [[#b0025|[5]]], [[#b0030|[6]]], [[#b0035|[7]]], [[#b0040|[8]]] and [[#b0045|[9]]], and also recently in the following linear time-varying system [[#b0050|[10]]], [[#b0055|[11]]], [[#b0060|[12]]], [[#b0065|[13]]], [[#b0070|[14]]], [[#b0075|[15]]], [[#b0080|[16]]], [[#b0085|[17]]], [[#b0090|[18]]], [[#b0095|[19]]], [[#b0100|[20]]], [[#b0105|[21]]], [[#b0110|[22]]], [[#b0115|[23]]], [[#b0120|[24]]], [[#b0125|[25]]], [[#b0130|[26]]], [[#b0135|[27]]], [[#b0140|[28]]], [[#b0145|[29]]], [[#b0150|[30]]], [[#b0155|[31]]] and [[#b0160|[32]]]:

<span id='e0005'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>A(t)y^{{'}}(t)=B(t)y(t)+C(t)u(t):y(t_0)=y_0\mbox{,}\quad t\geqslant 0\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (1-1)
|}

where <math display="inline">A(t)\in M_n</math> is a time-varying singular or non-singular matrix function, <math display="inline">B(t)\in M_m</math> and <math display="inline">C(t)\in M_{n\mbox{,}m}</math> are time-varying analytic matrix functions, <math display="inline">u(t)\in M_{m\mbox{,}1}</math> is the output vector function and <math display="inline">y(t)\in M_{n\mbox{,}1}</math> is the state function vector to be solved (where <math display="inline">M_{m\mbox{,}n}</math> is denoted by the set of all <math display="inline">m\times n</math> matrices over the real number R and when <math display="inline">m=n</math> we write <math display="inline">M_m</math> instead of <math display="inline">M_{m\mbox{,}n}</math>). 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 [[#b0050|[10]]], [[#b0055|[11]]], [[#b0060|[12]]], [[#b0065|[13]]], [[#b0070|[14]]] and [[#b0075|[15]]]. The linear time-varying descriptor system as in [[#e0005|(1-1)]] has been studied and discussed by many researchers [[#b0080|[16]]], [[#b0085|[17]]], [[#b0090|[18]]], [[#b0095|[19]]] and [[#b0100|[20]]]. For example, controllability and observability of this system have been studied by Wang and Liao [[#b0085|[17]]], Wang [[#b0090|[18]]] and Campbell and et al. [[#b0095|[19]]]; the linear of matrix differential inequalities of descriptor system was established by Inoue and et al. [[#b0100|[20]]]; the Weierstrass–Kronecker decomposition theorem of the regular pencil was extended to the time-varying discrete-time descriptor system by Kaczorek [[#b0160|[32]]] and finally, the stability of linear time-varying descriptor system has been discussed in [[#b0105|[21]]], [[#b0110|[22]]], [[#b0115|[23]]], [[#b0120|[24]]], [[#b0125|[25]]], [[#b0130|[26]]] and [[#b0135|[27]]]. Some special cases of the linear time-varying system as in [[#e0005|(1-1)]] have been also investigated in [[#b0140|[28]]], [[#b0145|[29]]], [[#b0150|[30]]] and [[#b0155|[31]]]. For example, the stability for the special case of system [[#e0005|(1-1)]] when <math display="inline">B(t)=B\mbox{,}\quad C(t)=C</math> and <math display="inline">A(t)=A</math> are constant matrices has been discussed in [[#b0135|[27]]], [[#b0140|[28]]] and [[#b0145|[29]]] and also the stability analysis for the special case of system [[#e0005|(1-1)]] when <math display="inline">B(t+T)=B(t)\mbox{,}\quad C(t+T)=C(t)</math> are periodically time-varying matrices with period ''T  '' and <math display="inline">A(t)=A</math> is a constant matrix has been studied in  [[#b0120|[24]]] and [[#b0145|[29]]]. Finally, the optimal control of system as in [[#e0005|(1-1)]] has been investigated in  [[#b0150|[30]]] and [[#b0155|[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 [[#b0165|[33]]], [[#b0170|[34]]], [[#b0175|[35]]], [[#b0180|[36]]], [[#b0185|[37]]], [[#b0190|[38]]], [[#b0195|[39]]], [[#b0200|[40]]], [[#b0205|[41]]], [[#b0210|[42]]], [[#b0215|[43]]] and [[#b0220|[44]]] and numerous real-life problems are also modeled mathematically by systems of fractional differential equations [[#b0185|[37]]], [[#b0195|[39]]], [[#b0200|[40]]], [[#b0205|[41]]], [[#b0215|[43]]], [[#b0225|[45]]], [[#b0230|[46]]], [[#b0235|[47]]], [[#b0240|[48]]], [[#b0245|[49]]], [[#b0250|[50]]], [[#b0255|[51]]], [[#b0260|[52]]], [[#b0265|[53]]] and [[#b0270|[54]]]. Since there are many definitions of fractional derivative of order <math display="inline">\alpha >0</math> most of them are used an integral or summation or limit form [[#b0165|[33]]], [[#b0185|[37]]], [[#b0210|[42]]], [[#b0220|[44]]], [[#b0240|[48]]], [[#b0255|[51]]], [[#b0260|[52]]], [[#b0265|[53]]], [[#b0270|[54]]], [[#b0275|[55]]], [[#b0280|[56]]], [[#b0285|[57]]] and [[#b0290|[58]]]. One of the important and familiar definition for fractional derivative is Caputo operator which is defined by the following:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>y^{\alpha }(t)=D^{\alpha }y(t)=I^{n-\alpha }D^ny(t)=</math><math>\frac{1}{\Gamma (n-\alpha )}{\int }_0^t\frac{y^{\left(n\right)}(s)}{{\left(t-s\right)}^{\alpha -n+1}}ds\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (1-2)
|}

where <math display="inline">\alpha >0\mbox{,}\quad t>0</math> and <math display="inline">n-1<\alpha \leqslant n</math><math display="inline">\left(n\in N\right)</math>.

Note that the fractional derivative of <math display="inline">f(x)</math> in the Caputo sense is defined for <math display="inline">0<\alpha <1</math> as

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>D^{\alpha }y(t)=\frac{1}{\Gamma (1-\alpha )}{\int }_0^t\frac{y^{{'}}(s)}{{\left(t-s\right)}^{\alpha }}ds\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (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 <math display="inline">A=\left(a_{ij}\right)\in M_{m\mbox{,}n}</math> and <math display="inline">B=\left(b_{kl}\right)\in M_{p\mbox{,}q}</math> be two rectangular matrices. Then the Kronecker product of ''A'' and ''B'' is defined by  [[#b0005|[1]]], [[#b0010|[2]]], [[#b0015|[3]]], [[#b0020|[4]]], [[#b0025|[5]]], [[#b0030|[6]]], [[#b0035|[7]]], [[#b0040|[8]]], [[#b0295|[59]]], [[#b0300|[60]]], [[#b0305|[61]]], [[#b0310|[62]]], [[#b0315|[63]]], [[#b0320|[64]]] and [[#b0325|[65]]]:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>A\otimes B={\left(a_{ij}B\right)}_{ij}\in M_{mp\mbox{,}nq}\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2-1)
|}

==Definition 2.2.                     ==

Let <math display="inline">A=\left(a_{ij}\right)\in M_{m\mbox{,}n}</math> be a rectangular matrix. Then the vector-operator of ''A'' is defined by  [[#b0005|[1]]], [[#b0010|[2]]], [[#b0015|[3]]], [[#b0020|[4]]], [[#b0025|[5]]], [[#b0030|[6]]], [[#b0035|[7]]], [[#b0040|[8]]], [[#b0295|[59]]], [[#b0300|[60]]], [[#b0305|[61]]], [[#b0310|[62]]], [[#b0315|[63]]], [[#b0320|[64]]] and [[#b0325|[65]]]:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2-2)
|}

<span id='n0020'></span>

==Lemma 2.1.                     ==

''Let  ''<math display="inline">A\mbox{,}\quad B\mbox{,}\quad C\mbox{,}\quad D</math>''and X be matrices with compatible orders, and  ''<math display="inline">I_n</math>''be the identity matrix of order  ''<math display="inline">n\times n</math>''. Then''[[#b0005|[1]]], [[#b0010|[2]]], [[#b0015|[3]]], [[#b0035|[7]]], [[#b0295|[59]]], [[#b0300|[60]]], [[#b0305|[61]]], [[#b0310|[62]]], [[#b0315|[63]]], [[#b0320|[64]]] and [[#b0325|[65]]].

<span id='e0215'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(i)\quad Vec(AXB)=(B^T\otimes A)VecX\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2 - 3)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(ii)\quad (A\otimes B)(C\otimes D)=AC\otimes BD\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2-4)
|}

''(iv) If f is analytic function on the region containing the eigenvalues of  ''<math display="inline">A\in M_m</math>''such that  ''<math display="inline">f(A)</math>''exist. Then''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>f(A\otimes I_n)=f(A)\otimes I_n\quad \mbox{and}\quad f(I_n\otimes A)=</math><math>I_n\otimes f(A)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2-5)
|}

==Definition 2.3.                     ==

The one-parameter Mittag–Leffler function <math display="inline">E_{\alpha }(t)</math> and Mittag–Leffler matrix function <math display="inline">E_{\alpha }({At}^{\alpha })</math> are defined, respectively, for <math display="inline">\alpha >0</math> by [[#b0165|[33]]], [[#b0210|[42]]], [[#b0255|[51]]], [[#b0260|[52]]], [[#b0280|[56]]] and [[#b0330|[66]]]:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>E_{\alpha }(t)=\sum_{k=0}^{\infty }\frac{t^k}{\Gamma (k\alpha +1)}\quad \mbox{and}\quad E_{\alpha }({At}^{\alpha })=</math><math>\sum_{k=0}^{\infty }\frac{A^kt^{\alpha k}}{\Gamma (k\alpha +1)}\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2-6)
|}

where <math display="inline">A\in M_n</math> is a matrix of order <math display="inline">n\times n</math> and <math display="inline">\Gamma (\cdot )</math> is the Gamma function.

==Lemma 2.2.                     ==

''Let  ''<math display="inline">A\in M_m</math>''be a matrix of order  ''<math display="inline">m\times m</math>''and let  ''<math display="inline">\left\{x_1\mbox{,}x_2\mbox{,}\ldots \mbox{,}x_m\right\}</math>''and  ''<math display="inline">\left\{y_1\mbox{,}y_2\mbox{,}\ldots \mbox{,}y_m\right\}</math>''be the eigenvectors corresponding to the eigenvalues  ''<math display="inline">\left\{{\lambda }_1\mbox{,}{\lambda }_2\mbox{,}\ldots \mbox{,}{\lambda }_m\right\}</math>''of A and  ''<math display="inline">A^T</math>'', respectively. Then the spectral decomposition of  ''<math display="inline">E_{\alpha }(A)</math>''and  ''<math display="inline">E_{\alpha }({At}^{\alpha })</math>''are given, respectively, for  ''<math display="inline">\alpha >0</math>''by''[[#b0280|''[''56'']'']]'':''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>E_{\alpha }(A)=\sum_{k=1}^mx_ky_k^TE_{\alpha }({\lambda }_k)\quad \mbox{and}\quad E_{\alpha }({At}^{\alpha })=</math><math>\sum_{k=1}^mx_ky_k^TE_{\alpha }({\lambda }_kt^{\alpha })\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2-7)
|}

''The list of nice properties for Mittag–Leffler matrix  ''<math display="inline">E_{\alpha }(A)</math>''can be found in''[[#b0280|''[''56'']'']]'', and the most important properties for Mittag–Leffler matrix  ''<math display="inline">E_{\alpha }(A)</math>''that will be used in this study are given below''[[#b0280|''[''56'']'']]''.''

<span id='n0015'></span>

==Theorem 2.1.                     ==

''Let  ''<math display="inline">A\mbox{,}B\in M_m</math>''and  ''<math display="inline">I_n</math>''be an identity matrix of order  ''<math display="inline">n\times n</math>''. Then for  ''<math display="inline">\alpha >0</math>'', we have''[[#b0280|''[''56'']'']]'':''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(i)\quad \mbox{If}\quad A=diag(a_{11}\mbox{,}a_{22}\mbox{,}\cdots \mbox{,}a_{mm})\mbox{,}\quad then\quad E_{\alpha }(A)=</math><math>diag\left(E_{\alpha }(a_{11})\mbox{,}E_{\alpha }(a_{22})\mbox{,}\cdots \mbox{,}E_{\alpha }(a_{mm})\right)\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2 - 8)
|}

<span id='e0230'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(ii)\quad E_{\alpha }(A+B)=E_{\alpha }(A)E_{\alpha }(B)\quad if\quad and\quad only\quad if\quad AB=</math><math>BA\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2 - 9)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(iii)\quad E_{\alpha }(A\otimes I_n)=E_{\alpha }(A)\otimes I_n\quad and\quad E_{\alpha }(I_n\otimes A)=</math><math>I_n\otimes E_{\alpha }(A)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2-10)
|}

<span id='n0030'></span>

==Lemma 2.3.                     ==

''Let  ''<math display="inline">H\in M_n</math>''be a given scalar matrix,  ''<math display="inline">u(t)\in M_{n\mbox{,}1}</math>''be a given vector function, and  ''<math display="inline">y(t)\in M_{n\mbox{,}1}</math>''be the unknown vector to be solved. Then the unique solution of the following fractional differential system''[[#b0255|[51]]], [[#b0260|[52]]] and [[#b0280|[56]]]'':''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>y^{\alpha }(t)=Hy(t)+u(t):\quad y(0)=y_0\in M_{n\mbox{,}1}\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2-11)
|}

''is given by''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (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 [[#n0030|Lemma 2.3]] with two illustrated examples.

<span id='n0050'></span>

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

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

<span id='e0055'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-1)
|}

where <math display="inline">A(t)\in M_n</math> is a time-varying singular matrix function, <math display="inline">B(t)\in M_n</math> and <math display="inline">C(t)\in M_n</math> are time-varying analytic matrix functions, <math display="inline">U(t)\in M_n</math> is the output matrix function and <math display="inline">Y(t)\in M_n</math> is the state function vector to be solved. Here, we will study the general solution of [[#e0055|(3-1)]] when <math display="inline">A(t)=A\mbox{,}\quad B(t)=B</math> and <math display="inline">C(t)=C</math> are constant matrices, as a special case. For this case, suppose that the constant invertible matrices ''M  '' and <math display="inline">N\in M_n</math> such that:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-2)
|}

If we partition ''n  '' as <math display="inline">n=m+p</math>, then <math display="inline">Y_1(t)\in M_{m\mbox{,}n}</math> and <math display="inline">Y_2(t)\in M_{p\mbox{,}n}</math>. This system is restricted equivalent to:

<span id='e0065'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>Y_1^{\alpha }(t)=B_{11}Y_1(t)+B_{12}Y_2(t)+C_1U(t)\mbox{,}\quad 0=</math><math>B_{21}Y_1(t)+B_{22}Y_2(t)+C_2U(t)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-3)
|}

Note that the necessary and sufficient condition for the existence of the solution of a system [[#e0055|(3-1)]] is that <math display="inline">B_{22}(t)</math> is invertible.

==General Solutions of Problem 3.1.                     ==

Since <math display="inline">B_{22}(t)</math> is an invertible matrix and then from the second equation of [[#e0065|(3-3)]] we have:

<span id='e0070'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>Y_2(t)=-B_{22}^{-1}B_{21}Y_1(t)-B_{22}^{-1}C_2U(t)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-4)
|}

By substituting this equation in the first equation of [[#e0065|(3-3)]], we get:

<span id='e0075'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>Y_1^{\alpha }(t)=S_{B_{11}}Y_1(t)+RU(t)\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-5)
|}

where

<span id='e0080'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>R=-B_{12}B_{22}^{-1}C_2+C_1\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-6)
|}

and

<span id='e0085'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>S_{B_{11}}=B_{11}-B_{12}B_{22}^{-1}B_{21}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-7)
|}

is called the Schur complement of <math display="inline">B_{11}</math> in a matrix <math display="inline">\left[\begin{array}{cc}
B_{11} & B_{12}\\
B_{21} & B_{22}
\end{array}\right]</math>.

Now, by taking <math display="inline">Vec(\cdot )</math> of both sides of [[#e0075|(3-5)]], and using [[#e0215|(2-3)]] in [[#n0020|Lemma 2.1]], we have:

<span id='e0090'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>Vec\left(Y_1^{\alpha }(t)\right)=Vec\left(S_{B_{11}}Y_1(t)\right)+</math><math>Vec\left(RU(t)\right)=\left(I_n\otimes S_{B_{11}}\right)\quad Vec\left(Y_1(t)\right)+</math><math>\left(I_n\otimes R\right)Vec\left(U(t)\right)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-8)
|}

Now by letting <math display="inline">Vec\left(Y_1^{\alpha }(t)\right)=y_1^{\alpha }(t)\mbox{,}\quad Vec\left(Y_1(t)\right)=</math><math>y_1(t)</math> and <math display="inline">Vec\left(U(t)\right)=u(t)</math>, then [[#e0090|(3-8)]] can be represented as follows:

<span id='e0095'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>y_1^{\epsilon }(t)=\left(I_n\otimes S_{B_{11}}\right)y_1(t)+</math><math>\left(I_n\otimes R\right)u(t)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-9)
|}

Now by using [[#n0030|Lemma 2.3]], then the vector solution of [[#e0095|(3-9)]] is given by:

<span id='e0100'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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)+</math><math>{\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=</math><math>E_{\alpha }\left((I_n\otimes S_{B_{11}})t^{\alpha }\right)\quad Vec(Y_1(0)+</math><math>{\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=</math><math>E_{\alpha }\left((I_n\otimes S_{B_{11}})t^{\alpha }\right)\quad Vec(Y_1(0)+</math><math>{\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{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-10)
|}

where ''R  '' and <math display="inline">S_{B_{11}}</math> are constant matrices as defined in  [[#e0080|(3-6)]] and [[#e0085|(3-7)]], respectively.

Note that the relationship between <math display="inline">Y_1(t)\in M_{m\mbox{,}n}</math> and <math display="inline">x=y_1(t)=Vec\left(Y_1(t)\right)\in M_{mn\mbox{,}1}</math> is given by:

<span id='e0105'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-11)
|}

Hence, the general solution of [[#n0050|Problem 3.1]] is given by: <math display="inline">Y(t)=N\left[\begin{array}{c}
Y_1(t)\\
Y_2(t)
\end{array}\right]</math>, where <math display="inline">Y_1(t)</math> can be easily obtained from [[#e0100|(3-10)]] and [[#e0105|(3-11)]]; and <math display="inline">Y_2(t)</math> is given as in [[#e0070|(3-4)]].

<span id='n0055'></span>

==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:

<span id='e0110'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-12)
|}

where <math display="inline">A(t)\in M_n</math> is a time-varying non-singular matrix function, <math display="inline">B(t)\in M_n</math> and <math display="inline">C(t)\in M_n</math> are time-varying analytic matrix functions, <math display="inline">U(t)\in M_n</math> is the output matrix function and <math display="inline">Y(t)\in M_n</math> is the state function matrix to be solved. Here, we will study the general solution of [[#e0110|(3-12)]] when <math display="inline">A(t)=A\mbox{,}\quad B(t)=B</math> and <math display="inline">C(t)=C</math> are constant matrices, as a special case. For this case, suppose that the constant invertible matrices ''M  '' and <math display="inline">N\in M_n</math> such that:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-13)
|}

This system is restricted equivalent to:

<span id='e0120'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>Y_1^{\alpha }(t)=B_{11}Y_1(t)+B_{12}Y_2+C_1U(t)\mbox{,}\quad Y_2^{\alpha }(t)=</math><math>B_{21}Y_1(t)+B_{22}Y_2(t)+C_2U(t)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-14)
|}

==General Solutions of Problem 3.2.                     ==

By taking <math display="inline">Vec(\cdot )</math> of both sides of [[#e0120|(3-14)]], and using [[#n0020|Lemma 2.1]], we have:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>Vec\left(Y_1^{\alpha }(t)\right)=Vec\left(B_{11}Y_1(t)+\right. </math><math>\left. B_{12}Y_2(t)+C_1U(t)\right)=\left(I_n\otimes B_{11}\right)\quad Vec\left(Y_1(t)\right)+</math><math>\left(I_n\otimes B_{12}\right)\quad Vec\left(Y_2(t)\right)+</math><math>\left(I_n\otimes C_1\right)\quad Vec\left(U(t)\right)\mbox{,}</math>
|-
|<math>Vec\left(Y_2^{\alpha }(t)\right)=Vec\left(B_{21}Y_1(t)+\right. </math><math>\left. B_{22}Y_2(t)+C_2U(t)\right)=\left(I_n\otimes B_{21}(t)\right)\quad Vec\left(Y_1(t)\right)+</math><math>\left(I_n\otimes B_{22}(t)\right)\quad Vec\left(Y_2(t)\right)+</math><math>\left(I_n\otimes C_2(t)\right)\quad Vec\left(U(t)\right)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-15)
|}

This system can be represented as:

<span id='e0130'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-16)
|}

Suppose that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

Now the system as in [[#e0130|(3-16)]] can be rewritten as follows:

<span id='e0135'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>T^{\alpha }(t)=HT(t)+D(t)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-17)
|}

Now by using [[#n0030|Lemma 2.3]], then the solution of [[#e0135|(3-17)]] is given by:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-18)
|}

This leads to the following general vector solution of [[#n0055|Problem 3.2]]:

<span id='e0145'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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]</math>
|-
|<math>\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{.}</math>
|-
|<math>\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]</math>
|-
|<math>\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_1U(z)\right)\\
Vec\left(C_2U(z)\right)
\end{array}\right]\quad dz\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-19)
|}

Another special case of [[#e0110|(3-12)]] is when <math display="inline">A(t)=A</math> and <math display="inline">B(t)=B</math> are constant matrices and <math display="inline">U(t)=0</math>. Then the general solution of this case is given by <math display="inline">Y(t)=E_{\alpha }((A^{-1}B)t^{\alpha })Y_0</math>.

The main problem in the solution of [[#n0055|Problem 3.2]] as in [[#e0145|(3-19)]] is how to compute the following Mittag–Leffler matrix:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-20)
|}

As a special case, if <math display="inline">B_{11}B_{12}=B_{12}B_{22}</math> and <math display="inline">B_{21}B_{11}=B_{22}B_{21}</math>, then by using the same procedure in the proof of Theorem 2 in [[#b0280|[56]]] and [[#n0015|Theorem 2.1]], we have:

<span id='e0155'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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]</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-21)
|}

Now, it is easy to get <math display="inline">Y_1(t)</math> and <math display="inline">Y_2(t)</math> of this case by substituting [[#e0155|(3-21)]] in [[#e0145|(3-19)]] and then the general solution of this problem is given by <math display="inline">Y(t)=N\left[\begin{array}{c}
Y_1(t)\\
Y_2(t)
\end{array}\right]</math>.

==Example 3.1.                     ==

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

<span id='e0160'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>{AY}^{\alpha }(t)=BY(t)+CU(t):Y(0)=Y_0\mbox{,}\quad t\geqslant 0\mbox{,}\quad \alpha >0\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-22)
|}

where

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

Since

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>S_{B_{11}}=I_2-I_2I_2^{-1}I_2=I_2=\left[\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right]\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>R=-I_2I_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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

Then <math display="inline">Y_1(t)\in M_{2\mbox{,}4}</math> and <math display="inline">Y_2(t)\in M_{2\mbox{,}4}</math> by using [[#e0070|(3-4)]] and [[#e0100|(3-10)]], respectively, are given by:

<span id='e0165'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>Vec\left(Y_1(t)\right)=E_{\alpha }\left((I_4\otimes I_2)t^{\alpha }\right)\quad Vec(Y_1(0)+</math><math>{\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=</math><math>E_{\alpha }\left((I_8t^{\alpha }\right)\quad Vec(Y_1(0)+</math><math>{\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=</math><math>\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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-23)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>Y_2(t)=-I_2^{-1}I_2Y_1(t)-I_2C_2U(t)=-Y_1(t)-C_2U(t)=</math><math>-Y_1(t)-\left[\begin{array}{cccc}
0 & t & 0 & t\\
t & 0 & t & 0
\end{array}\right]\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-24)
|}

where <math display="inline">Y_1(t)\in M_{2\mbox{,}4}</math> is given as a vector solution as in [[#e0165|(3-23)]].

Finally the general solution of system as in [[#e0160|(3-22)]] is given by:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-25)
|}

As a special case of system [[#e0160|(3-22)]], if <math display="inline">U(t)=0</math>, then <math display="inline">Y_1(t)\in M_{2\mbox{,}4}\mbox{,}\quad Y_2(t)\in M_{2\mbox{,}4}</math> and <math display="inline">Y(t)\in M_4</math> are given, respectively, as in [[#e0180|(3-26)]], [[#e0185|(3-27)]] and [[#e0190|(3-28)]] below:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

Now from [[#e0105|(3-11)]], we get:

<span id='e0180'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-26)
|}

<span id='e0185'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-27)
|}

<span id='e0190'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-28)
|}

<span id='n0010'></span>

==Example 3.2.                     ==

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

<span id='e0195'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>A(t)Y^{\alpha }(t)=B(t)Y(t):Y(0)=Y_0\mbox{,}\quad t\geqslant 0\mbox{,}\quad \alpha >0\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-29)
|}

where

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

Since <math display="inline">B_{11}B_{12}=B_{12}B_{22}</math> and <math display="inline">B_{21}B_{11}=B_{22}B_{21}</math>, then by applying [[#e0145|(3-19)]] and [[#e0155|(3-21)]] and [[#n0015|Theorem 2.1]], we get:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

Now,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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_2t^{\alpha }\right)\right)\quad Vec\left(Y_2(0)\right)=</math><math>\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)=</math><math>Vec\left\{\left(E_{\alpha }\left(B_{11}(t)\right)E_{\alpha }\left(I_2\right)t^{\alpha }\right)\quad Y_2(0)\right\}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | 
|}

That is by using [[#e0230|(2-9)]], we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>Y_1(t)=\left(E_{\alpha }\left(B_{11}(t)\right)E_{\alpha }\left(I_2\right)t^{\alpha }\right)Y_2(0)=</math><math>\left\{E_{\alpha }\left(\left(B_{11}(t)+E_{\alpha }\left(I_2\right)\right)t^{\alpha }\right)\right\}\quad Y_2(0)=</math><math>\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}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-30)
|}

Similarly, we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-31)
|}

Hence, the general solutions of system [[#e0195|(3-29)]] are given by:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3-32)
|}

Note that if <math display="inline">A(t)=A</math> and <math display="inline">B(t)=B</math> are constant matrices in [[#n0010|Example 3.2]], then the general solution given by <math display="inline">Y(t)=E_{\alpha }((A^{-1}B)t^{\alpha })Y_0</math>.

==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.

==References==

<ol style='list-style-type: none;margin-left: 0px;'><li><span id='b0005'></span>
[[#b0005|[1]]] Z. Al-Zhour; Efficient solutions of coupled matrix and matrix differential equations; Intell. Control Automat., 3 (2) (2012), pp. 176–187</li>
<li><span id='b0010'></span>
[[#b0010|[2]]] A. Kilicman, Z. Al-Zhour; The general common exact solutions of coupled linear matrix and matrix differential equations; J. Anal. Comput., 1 (1) (2005), pp. 15–30</li>
<li><span id='b0015'></span>
[[#b0015|[3]]] A. Kilicman, Z. Al-Zhour; Vector least-squares solutions of coupled singular matrix equations; J. Comput. Appl. Math., 206 (2) (2007), pp. 1051–1069</li>
<li><span id='b0020'></span>
[[#b0020|[4]]] A. Kilicman, Z. Al-Zhour; On the connection between Kronecker and Hadamard convolution products of matrices and some applications; J. Ineq. Appl., 2009 (2009), p. 10 [http://dx.doi.org/10.1155/2009/736243 http://dx.doi.org/10.1155/2009/736243] Article ID 736243</li>
<li><span id='b0025'></span>
[[#b0025|[5]]] A. Kilicman, Z. Al-Zhour; Note on the numerical solutions of the general matrix convolution equations by using the iterative methods and box convolution products; Abstract Appl. Anal., 2010 (2010), p. 16 [http://dx.doi.org/10.1155/2010/106192 http://dx.doi.org/10.1155/2010/106192] Article ID 106192</li>
<li><span id='b0030'></span>
[[#b0030|[6]]] C.R. Rao, M.B. Rao; Matrix Algebra and its Applications to Statistics and Econometrics; World Scientific Publishing Co. Pte. Ltd., Singapore (1998)</li>
<li><span id='b0035'></span>
[[#b0035|[7]]] A. Graham; Kronecker Products and Matrix Calculus with Applications;  (first ed.)Ellis Horwood Ltd., UK (1981)</li>
<li><span id='b0040'></span>
[[#b0040|[8]]] L. Jódar, H. Abou-Kandil; Kronecker products and coupled matrix Riccati differential systems; Linear Algebra Appl., 121 (1989), pp. 39–51</li>
<li><span id='b0045'></span>
[[#b0045|[9]]] S. Campbell; Singular Systems of Differential Equations II; Pitman, London (1982)</li>
<li><span id='b0050'></span>
[[#b0050|[10]]] N. Kablar, D. Lj. Debeljkovic, Finite-time stability of time-varying linear singular systems, in: Proceeding of the 37th IEEE Conference on Decision and Control, Tampa, Florida, USA, 1998, pp. 3831–3836.</li>
<li><span id='b0055'></span>
[[#b0055|[11]]] X. Su, Y. Zhi; Sufficient conditions of asymptotic stability of the time-varying descriptor systems; Int. J. Eng. Math., 2013 (2013), p. 4 Article ID 920851</li>
<li><span id='b0060'></span>
[[#b0060|[12]]] L.S. Campell, L.R. Petzold; Canonical forms and solvable singular systems of differential equations; SIAM J. Algebr. Discr. Methods, 4 (2) (1983), pp. 517–521</li>
<li><span id='b0065'></span>
[[#b0065|[13]]] E. Lewis; A survey of linear singular systems; Circ. Syst Signal Process., 5 (1) (1986), pp. 3–36</li>
<li><span id='b0070'></span>
[[#b0070|[14]]] K. Wei; Stabilization of linear time-invariant interval systems via constant state feedback control; IEEE Trans. Auto. Control., 39 (1) (1994), pp. 22–32</li>
<li><span id='b0075'></span>
[[#b0075|[15]]] T. Takaba, N. Morihira, K. Katayama; A generalized Lyapunov theorem for descriptor system; Syst. Control Lett., 24 (1) (1995), pp. 49–51</li>
<li><span id='b0080'></span>
[[#b0080|[16]]] B. Brayton; Small signal stability criterion for electrical networks containing lossless transmission lines; IBMJ. Res. Develop., 12 (1968), pp. 431–440</li>
<li><span id='b0085'></span>
[[#b0085|[17]]] Chi-Jo Wang, Ho-En Liao; Impulse observability and impulse controllability of linear time-varying singular systems; Automatica (37) (2001), pp. 1867–1876</li>
<li><span id='b0090'></span>
[[#b0090|[18]]] Chi-Jo Wang; Controllability and observability of linear time-varying singular systems; IEEE Trans. Auto. Control, 44 (10) (1999), pp. 1901–1905</li>
<li><span id='b0095'></span>
[[#b0095|[19]]] S. Campbell, N. Nichols, W. Terrell; Duality observability and controllability for linear time-varying descriptor systems; Circ. Syst. Signal Process., 10 (4) (1991), pp. 455–459</li>
<li><span id='b0100'></span>
[[#b0100|[20]]] M. Inoue, T. Wada, M. Ikeda, E. Uezato, Stabilization of linear time-varying descriptor systems: a linear matrix differential inequality approach, in: Proceedings of the International Conference on Control, Automation and Systems (ICCAS’08) COEX, Seoul, Korea, 2008, pp. 1855–1860.</li>
<li><span id='b0105'></span>
[[#b0105|[21]]] E. Fridman; Stability of linear descriptor systems with delay: a Lyapunov-based approach; J. Math. Anal. Appl., 273 (2002), pp. 24–44</li>
<li><span id='b0110'></span>
[[#b0110|[22]]] D. Wu, J. Zhu; On stability for a class of time-varying singular system; J. Tongji Univ., 38 (6) (2010), pp. 925–934</li>
<li><span id='b0115'></span>
[[#b0115|[23]]] W. Chen, W. Fu, R. Du, X. Lu, Exponential stability of a class of linear time-varying singularly perturbed systems, in: Proceedings of the International Conference on Information Science and Technology (ICIST’11) Jiansu, China, 2011, pp. 778–783.</li>
<li><span id='b0120'></span>
[[#b0120|[24]]] X. Su, M. Lv, H. Shi, Q. Zhang, Stability analysis for linear time-varying periodically descriptor systems: a generalized Lyapunov approach, in: Proceedings of the 6th World Congress on Intelligent Control and Automation (WCICA’06) Dalian, China, 2006, pp. 728–732.</li>
<li><span id='b0125'></span>
[[#b0125|[25]]] S. Sh. Alaviani, M. Shafiee, Exponential stability and stabilization of linear time-varying singular system, in: Proceedings of the International Multi Conference of Engineers and Computer Scientists, Hong Kong, vol. 2, 2009.</li>
<li><span id='b0130'></span>
[[#b0130|[26]]] J. Ishihare, M. Terre, Generalized Lyapunov theorems for rectangular descriptor systems, in: Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida, USA, 2001, pp. 2858–2859.</li>
<li><span id='b0135'></span>
[[#b0135|[27]]] P. Muller, Modified Lyapunov matrix equations with positive definite solutions for descriptor systems, in: 5th Asian Control Conference, 2004, pp. 997–1000.</li>
<li><span id='b0140'></span>
[[#b0140|[28]]] J. Ishihare, M. Terre; On the Lyapunov theorem for singular systems; IEEE Trans. Automat. Control, 47 (11) (2002), pp. 1926–1930</li>
<li><span id='b0145'></span>
[[#b0145|[29]]] X. Liu, X. Wang, D. Ho; Input-output block decoupling of linear time-varying singular systems; IEEE Trans. Automat. Control, 45 (2) (2000), pp. 312–318</li>
<li><span id='b0150'></span>
[[#b0150|[30]]] Y. Jiuxi, C. Zhaolin, Y. Yi, The linear quadratic optimal regulation for continuous time-varying singular systems, in: Proceedings of the 34th Conference on Decision and Control, New Orleans, LA, 1995, pp. 3920–3921.</li>
<li><span id='b0155'></span>
[[#b0155|[31]]] G. Kurina, R. Mirz, On linear quadratic optimal control problems for time-varying descriptor systems, in: 43th IEEE Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, 2004, pp. 1830–1835.</li>
<li><span id='b0160'></span>
[[#b0160|[32]]] T. Kaczorek; Positive descriptor time-varying discrete-time linear systems and their asymptotic stability; Int. J. Mar. Navig. Safety Sea Transp., 9 (1) (2015), pp. 85–91</li>
<li><span id='b0165'></span>
[[#b0165|[33]]] K. Miller, B. Ross; An Introduction to the Fractional Calculus and Differential Equations; Wiley, New York (1993)</li>
<li><span id='b0170'></span>
[[#b0170|[34]]] L. Podlubny; The Laplace Transform Method for Linear Differential Equations of Fractional Order; Slovac Academy of Science, Slovac Republic (1994)</li>
<li><span id='b0175'></span>
[[#b0175|[35]]] A. Kilbas, M. Saigo; Fractional integrals and derivatives of functions of Mittag Leffler type; Doki. Akad. Nauk Belar., 39 (4) (1995), pp. 22–26</li>
<li><span id='b0180'></span>
[[#b0180|[36]]] A. Kilbas, H. Srivastava, J. Trujillo; Theory and Applications of Fractional Differential Equations; Elsevier, Amsterdam (2006)</li>
<li><span id='b0185'></span>
[[#b0185|[37]]] S. Das; Functional Fractional Calculus for System Identification and Controls; Springer, Berlin (2008)</li>
<li><span id='b0190'></span>
[[#b0190|[38]]] R. Caponetto, G. Dongola, L. Fortuna; Fractional Order System: Modeling and Control Applications; World Scientific (2010)</li>
<li><span id='b0195'></span>
[[#b0195|[39]]] Y. Rossikhin, M. Shitikova; Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids; Appl. Mech. Rev., 50 (1997), pp. 15–67</li>
<li><span id='b0200'></span>
[[#b0200|[40]]] R. Magin; Fractional calculus in bioengineering; Crit. Rev. Biomed. Eng., 32 (2004), pp. 1–104</li>
<li><span id='b0205'></span>
[[#b0205|[41]]] M. Dalir, M. Bashour; Applications of fractional calculus; Appl. Math. Sci., 4 (21) (2010), pp. 1021–1032</li>
<li><span id='b0210'></span>
[[#b0210|[42]]] B. Ross; Fractional Calculus and its Applications; Springer-Verlag, Berlin (1975)</li>
<li><span id='b0215'></span>
[[#b0215|[43]]] A. Kilicman, Z. Al-Zhour; Kronecker operational matrices for fractional calculus and some applications; Appl. Comput. Math., 187 (2007), pp. 250–265</li>
<li><span id='b0220'></span>
[[#b0220|[44]]] L. Podlubny; Fractional Differential Equations; Academic Press, New York (1999)</li>
<li><span id='b0225'></span>
[[#b0225|[45]]] G.H. He; Variational iteration method for delay differential equations; Commun. Linear Sci. Numer. Simul., 2 (4) (1997), pp. 235–236</li>
<li><span id='b0230'></span>
[[#b0230|[46]]] F. Mainardi, Y. Luchko, G. Pagnini; The fundamental solution of the space-time fractional diffusion equation; Fract. Calc. Appl. Anal., 4 (2001), pp. 153–192</li>
<li><span id='b0235'></span>
[[#b0235|[47]]] K. Diethein, N. Ford, A. Feed; A predictor-corrector approach for the numerical solution of fractional differential equations; Nonlinear Dynam., 29 (2002), pp. 3–22</li>
<li><span id='b0240'></span>
[[#b0240|[48]]] R. Hilfert; Applications of Fractional Calculus in Physics; World Scientific Publishing Company, Singapore, London (2000)</li>
<li><span id='b0245'></span>
[[#b0245|[49]]] I. Grigorenko, E. Grigorenko; Chaotic dynamics of the fractional lorenz system; Phys. Rev. Lett., 91 (3) (2003), p. 034101</li>
<li><span id='b0250'></span>
[[#b0250|[50]]] J. Wang, Y. Zhang; Network synchronization in a population of star-coupled fractional nonlinear oscillators; Phys. Lett. A, 374 (2010), pp. 1464–1468</li>
<li><span id='b0255'></span>
[[#b0255|[51]]] K. Balachandran, J. Kokila; On the controllability of fractional dynamic systems; Int. J. Appl. Math. Comput. Sci., 22 (3) (2012), pp. 523–531</li>
<li><span id='b0260'></span>
[[#b0260|[52]]] K. Balachandran, J. Kokila, J. Trujillo; Relative controllability of fractional dynamic systems with multiple delays in control; Comput. Math. Appl., 64 (2012), pp. 3037–3045</li>
<li><span id='b0265'></span>
[[#b0265|[53]]] K. Oldham, J. Spanier; The Fractional Calculus; Mathematics in Science and Engineering, vol. 198, Academic Press (1974)</li>
<li><span id='b0270'></span>
[[#b0270|[54]]] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in: Carpinteri, Mainardi (Eds.) Fractals and Fractional Calculus, New York, 1997.</li>
<li><span id='b0275'></span>
[[#b0275|[55]]] R. Khalil, M. Al-Horani, A. Yousef, M. Sababheh; A new definition of fractional derivative; J. Comput. Appl. Math., 264 (2014), pp. 65–70</li>
<li><span id='b0280'></span>
[[#b0280|[56]]] Z. Al-Zhour; The general (vector) solutions of such linear (coupled) matrix fractional differential equations by using Kronecker structures; Appl. Math. Comput., 232 (2014), pp. 498–510</li>
<li><span id='b0285'></span>
[[#b0285|[57]]] S. Rida, A. Arafa; New method for solving linear fractional differential equations; Int. J. Diff. Eq., 2011 (2011), p. 8 [http://dx.doi.org/10.1155/2011/814132 http://dx.doi.org/10.1155/2011/814132] Article ID 814132</li>
<li><span id='b0290'></span>
[[#b0290|[58]]] M. Caputo; Linear models of dissipation whose Q is almost frequency independent, Part II; J. Roy. Astr. Soc., 13 (1967), pp. 529–539</li>
<li><span id='b0295'></span>
[[#b0295|[59]]] F. Zhang; Matrix Theory: Basic Results and Techniques; Springer-Verlag, New York (1999)</li>
<li><span id='b0300'></span>
[[#b0300|[60]]] Z. Al-Zhour, A. Kilicman; Matrix equalities and inequalities involving Khatri-Rao and Tracy-Singh sums; J. Ineq. Pure Appl. Math., 7 (1) (2006), pp. 496–513 Article 34</li>
<li><span id='b0305'></span>
[[#b0305|[61]]] Z. Al-Zhour, A. Kilicman; Extensions and Generalization inequalities involving the Khatri–Rao product of several positive matrices; J. Ineq. Appl., 80878 (2006), pp. 1–21</li>
<li><span id='b0310'></span>
[[#b0310|[62]]] Z. Al-Zhour, A. Kilicman; Some new connections between matrix products for partitioned and non-partitioned matrices; Comput. Math. Appl., 54 (2007), pp. 763–784</li>
<li><span id='b0315'></span>
[[#b0315|[63]]] S. Liu; Several inequalities involving Khatri–Rao products of positive semi-definite matrices; Linear Algebra Appl., 354 (2002), pp. 175–186</li>
<li><span id='b0320'></span>
[[#b0320|[64]]] G.F. Van Loan; The ubiquitous Kronecker product; J. Comput. Appl. Math., 123 (2000), pp. 85–100</li>
<li><span id='b0325'></span>
[[#b0325|[65]]] G. Visick; A quantitative version of the observation that the Hadamard product is a principle submatrix of the Kronecker product; Linear Algebra Appl., 304 (2000), pp. 45–68</li>
<li><span id='b0330'></span>
[[#b0330|[66]]] Z. Odibat; Analytic study on linear systems of fractional differential equations; Computer Math. Appl., 59 (2010), pp. 1171–1183</li>
</ol>