(Created page with "==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 solution...")
 
m (Clara moved page Draft García 207472726 to Al-Zhour 2016a)
 
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Latest revision as of 11:43, 11 April 2017

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]:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A(t)\in M_n}

is a time-varying singular or non-singular matrix function, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B(t)\in M_m}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C(t)\in M_{n\mbox{,}m}}
are time-varying analytic matrix functions, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u(t)\in M_{m\mbox{,}1}}
is the output vector function and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y(t)\in M_{n\mbox{,}1}}
is the state function vector to be solved (where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{m\mbox{,}n}}
is denoted by the set of all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m\times n}
matrices over the real number R and when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m=n}
we write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_m}
instead of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B(t)=B\mbox{,}\quad C(t)=C}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B(t+T)=B(t)\mbox{,}\quad C(t+T)=C(t)}
are periodically time-varying matrices with period T   and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): y^{\alpha }(t)=D^{\alpha }y(t)=I^{n-\alpha }D^ny(t)=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha >0\mbox{,}\quad t>0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n-1<\alpha \leqslant n}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left(n\in N\right)} .

Note that the fractional derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(x)}

in the Caputo sense is defined for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0<\alpha <1}
as
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A=\left(a_{ij}\right)\in M_{m\mbox{,}n}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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]:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): A\otimes B={\left(a_{ij}B\right)}_{ij}\in M_{mp\mbox{,}nq}\mbox{.}
(2-1)

Definition 2.2.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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]:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A\mbox{,}\quad B\mbox{,}\quad C\mbox{,}\quad D} and X be matrices with compatible orders, and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_n} be the identity matrix of order  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n\times n} . Then[1], [2], [3], [7], [59], [60], [61], [62], [63], [64] and [65].

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (i)\quad Vec(AXB)=(B^T\otimes A)VecX\mbox{,}
(2 - 3)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (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  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A\in M_m} such that  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(A)} exist. Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f(A\otimes I_n)=f(A)\otimes I_n\quad \mbox{and}\quad f(I_n\otimes A)=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): I_n\otimes f(A)\mbox{.}

(2-5)

Definition 2.3.

The one-parameter Mittag–Leffler function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle E_{\alpha }(t)}

and Mittag–Leffler matrix function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle E_{\alpha }({At}^{\alpha })}
are defined, respectively, for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha >0}
by [33], [42], [51], [52], [56] and [66]:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): E_{\alpha }(t)=\sum_{k=0}^{\infty }\frac{t^k}{\Gamma (k\alpha +1)}\quad \mbox{and}\quad E_{\alpha }({At}^{\alpha })=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \sum_{k=0}^{\infty }\frac{A^kt^{\alpha k}}{\Gamma (k\alpha +1)}\mbox{,}

(2-6)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A\in M_n}

is a matrix of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n\times n}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Gamma (\cdot )}
is the Gamma function.

Lemma 2.2.

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A\in M_m} be a matrix of order  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m\times m} and let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left\{x_1\mbox{,}x_2\mbox{,}\ldots \mbox{,}x_m\right\}} and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left\{y_1\mbox{,}y_2\mbox{,}\ldots \mbox{,}y_m\right\}} be the eigenvectors corresponding to the eigenvalues  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left\{{\lambda }_1\mbox{,}{\lambda }_2\mbox{,}\ldots \mbox{,}{\lambda }_m\right\}} of A and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A^T} , respectively. Then the spectral decomposition of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle E_{\alpha }(A)} and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle E_{\alpha }({At}^{\alpha })} are given, respectively, for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha >0} by[56]:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): E_{\alpha }(A)=\sum_{k=1}^mx_ky_k^TE_{\alpha }({\lambda }_k)\quad \mbox{and}\quad E_{\alpha }({At}^{\alpha })=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \sum_{k=1}^mx_ky_k^TE_{\alpha }({\lambda }_kt^{\alpha })\mbox{,}

(2-7)

The list of nice properties for Mittag–Leffler matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle E_{\alpha }(A)} can be found in[56], and the most important properties for Mittag–Leffler matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle E_{\alpha }(A)} that will be used in this study are given below[56].

Theorem 2.1.

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A\mbox{,}B\in M_m} and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_n} be an identity matrix of order  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n\times n} . Then for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha >0} , we have[56]:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (i)\quad \mbox{If}\quad A=diag(a_{11}\mbox{,}a_{22}\mbox{,}\cdots \mbox{,}a_{mm})\mbox{,}\quad then\quad E_{\alpha }(A)=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): diag\left(E_{\alpha }(a_{11})\mbox{,}E_{\alpha }(a_{22})\mbox{,}\cdots \mbox{,}E_{\alpha }(a_{mm})\right)\mbox{,}

(2 - 8)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (ii)\quad E_{\alpha }(A+B)=E_{\alpha }(A)E_{\alpha }(B)\quad if\quad and\quad only\quad if\quad AB=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): BA\mbox{,}

(2 - 9)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (iii)\quad E_{\alpha }(A\otimes I_n)=E_{\alpha }(A)\otimes I_n\quad and\quad E_{\alpha }(I_n\otimes A)=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): I_n\otimes E_{\alpha }(A)\mbox{.}

(2-10)

Lemma 2.3.

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H\in M_n} be a given scalar matrix,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u(t)\in M_{n\mbox{,}1}} be a given vector function, and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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]:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): y^{\alpha }(t)=Hy(t)+u(t):\quad y(0)=y_0\in M_{n\mbox{,}1}\mbox{,}
(2-11)

is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A(t)\in M_n}

is a time-varying singular matrix function, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B(t)\in M_n}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C(t)\in M_n}
are time-varying analytic matrix functions, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U(t)\in M_n}
is the output matrix function and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A(t)=A\mbox{,}\quad B(t)=B}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C(t)=C}
are constant matrices, as a special case. For this case, suppose that the constant invertible matrices M   and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N\in M_n}
such that:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n=m+p} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Y_1(t)\in M_{m\mbox{,}n}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Y_2(t)\in M_{p\mbox{,}n}}

. This system is restricted equivalent to:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Y_1^{\alpha }(t)=B_{11}Y_1(t)+B_{12}Y_2(t)+C_1U(t)\mbox{,}\quad 0=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): B_{21}Y_1(t)+B_{22}Y_2(t)+C_2U(t)\mbox{.}

(3-3)

Note that the necessary and sufficient condition for the existence of the solution of a system (3-1) is that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_{22}(t)}

is invertible.

General Solutions of Problem 3.1.

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_{22}(t)}

is an invertible matrix and then from the second equation of (3-3) we have:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Y_2(t)=-B_{22}^{-1}B_{21}Y_1(t)-B_{22}^{-1}C_2U(t)\mbox{.}
(3-4)

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Y_1^{\alpha }(t)=S_{B_{11}}Y_1(t)+RU(t)\mbox{,}
(3-5)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): R=-B_{12}B_{22}^{-1}C_2+C_1\mbox{,}
(3-6)

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): S_{B_{11}}=B_{11}-B_{12}B_{22}^{-1}B_{21}
(3-7)

is called the Schur complement of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_{11}}

in a matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left[\begin{array}{cc} B_{11} & B_{12}\\ B_{21} & B_{22} \end{array}\right]}

.

Now, by taking Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Vec(\cdot )}

of both sides of (3-5), and using (2-3) in Lemma 2.1, we have:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Vec\left(Y_1^{\alpha }(t)\right)=Vec\left(S_{B_{11}}Y_1(t)\right)+

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Vec\left(RU(t)\right)=\left(I_n\otimes S_{B_{11}}\right)\quad Vec\left(Y_1(t)\right)+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(I_n\otimes R\right)Vec\left(U(t)\right)\mbox{.}

(3-8)

Now by letting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Vec\left(Y_1^{\alpha }(t)\right)=y_1^{\alpha }(t)\mbox{,}\quad Vec\left(Y_1(t)\right)=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): y_1(t)

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Vec\left(U(t)\right)=u(t)}

, then (3-8) can be represented as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): y_1^{\epsilon }(t)=\left(I_n\otimes S_{B_{11}}\right)y_1(t)+

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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)+

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): E_{\alpha }\left((I_n\otimes S_{B_{11}})t^{\alpha }\right)\quad Vec(Y_1(0)+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): E_{\alpha }\left((I_n\otimes S_{B_{11}})t^{\alpha }\right)\quad Vec(Y_1(0)+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle S_{B_{11}}}

are constant matrices as defined in  (3-6) and (3-7), respectively.

Note that the relationship between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Y_1(t)\in M_{m\mbox{,}n}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x=y_1(t)=Vec\left(Y_1(t)\right)\in M_{mn\mbox{,}1}}
is given by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Y(t)=N\left[\begin{array}{c} Y_1(t)\\ Y_2(t) \end{array}\right]} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Y_1(t)}

can be easily obtained from (3-10) and (3-11); and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A(t)\in M_n}

is a time-varying non-singular matrix function, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B(t)\in M_n}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C(t)\in M_n}
are time-varying analytic matrix functions, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U(t)\in M_n}
is the output matrix function and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A(t)=A\mbox{,}\quad B(t)=B}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C(t)=C}
are constant matrices, as a special case. For this case, suppose that the constant invertible matrices M   and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N\in M_n}
such that:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Y_1^{\alpha }(t)=B_{11}Y_1(t)+B_{12}Y_2+C_1U(t)\mbox{,}\quad Y_2^{\alpha }(t)=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): B_{21}Y_1(t)+B_{22}Y_2(t)+C_2U(t)\mbox{.}

(3-14)

General Solutions of Problem 3.2.

By taking Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Vec(\cdot )}

of both sides of (3-14), and using Lemma 2.1, we have:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Vec\left(Y_1^{\alpha }(t)\right)=Vec\left(B_{11}Y_1(t)+\right.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. B_{12}Y_2(t)+C_1U(t)\right)=\left(I_n\otimes B_{11}\right)\quad Vec\left(Y_1(t)\right)+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(I_n\otimes B_{12}\right)\quad Vec\left(Y_2(t)\right)+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(I_n\otimes C_1\right)\quad Vec\left(U(t)\right)\mbox{,}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Vec\left(Y_2^{\alpha }(t)\right)=Vec\left(B_{21}Y_1(t)+\right.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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)+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(I_n\otimes B_{22}(t)\right)\quad Vec\left(Y_2(t)\right)+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left(I_n\otimes C_2(t)\right)\quad Vec\left(U(t)\right)\mbox{.}

(3-15)

This system can be represented as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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{,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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{.}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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{.}
(3-19)

Another special case of (3-12) is when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A(t)=A}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B(t)=B}
are constant matrices and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U(t)=0}

. Then the general solution of this case is given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_{11}B_{12}=B_{12}B_{22}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Y_1(t)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {AY}^{\alpha }(t)=BY(t)+CU(t):Y(0)=Y_0\mbox{,}\quad t\geqslant 0\mbox{,}\quad \alpha >0\mbox{,}
(3-22)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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{,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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{,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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{.}

Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Y_1(t)\in M_{2\mbox{,}4}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Y_2(t)\in M_{2\mbox{,}4}}
by using (3-4) and (3-10), respectively, are given by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Vec\left(Y_1(t)\right)=E_{\alpha }\left((I_4\otimes I_2)t^{\alpha }\right)\quad Vec(Y_1(0)+

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): E_{\alpha }\left((I_8t^{\alpha }\right)\quad Vec(Y_1(0)+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Y_2(t)=-I_2^{-1}I_2Y_1(t)-I_2C_2U(t)=-Y_1(t)-C_2U(t)=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -Y_1(t)-\left[\begin{array}{cccc} 0 & t & 0 & t\\ t & 0 & t & 0 \end{array}\right]\mbox{,}

(3-24)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U(t)=0} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Y_1(t)\in M_{2\mbox{,}4}\mbox{,}\quad Y_2(t)\in M_{2\mbox{,}4}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Y(t)\in M_4}
are given, respectively, as in (3-26), (3-27) and (3-28) below:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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{,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_{11}B_{12}=B_{12}B_{22}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_{21}B_{11}=B_{22}B_{21}}

, then by applying (3-19) and (3-21) and Theorem 2.1, we get:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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)=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Y_1(t)=\left(E_{\alpha }\left(B_{11}(t)\right)E_{\alpha }\left(I_2\right)t^{\alpha }\right)Y_2(0)=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left\{E_{\alpha }\left(\left(B_{11}(t)+E_{\alpha }\left(I_2\right)\right)t^{\alpha }\right)\right\}\quad Y_2(0)= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A(t)=A}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B(t)=B}
are constant matrices in Example 3.2, then the general solution given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\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.

References

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