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]:
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:
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
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]:
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]:
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].
(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
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]:
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]:
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]:
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]:
is given by
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
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:
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:
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:
By substituting this equation in the first equation of (3-3), we get:
where
and
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:
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:
Now by using Lemma 2.3, then the vector solution of (3-9) is given by:
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:
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:
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:
This system is restricted equivalent to:
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:
This system can be represented as:
Suppose that
Now the system as in (3-16) can be rewritten as follows:
Now by using Lemma 2.3, then the solution of (3-17) is given by:
This leads to the following general vector solution of Problem 3.2:
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:
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:
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:
where
Since
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:
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:
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:
Now from (3-11), we get:
Example 3.2.
Consider the following linear non-singular matrix fractional time-varying descriptor system:
where
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:
Now,
That is by using (2-9), we have
Similarly, we have
Hence, the general solutions of system (3-29) are given by:
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
-
[1] Z. Al-Zhour; Efficient solutions of coupled matrix and matrix differential equations; Intell. Control Automat., 3 (2) (2012), pp. 176–187
-
[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
-
[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
-
[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 Article ID 736243
-
[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 Article ID 106192
-
[6] C.R. Rao, M.B. Rao; Matrix Algebra and its Applications to Statistics and Econometrics; World Scientific Publishing Co. Pte. Ltd., Singapore (1998)
-
[7] A. Graham; Kronecker Products and Matrix Calculus with Applications; (first ed.)Ellis Horwood Ltd., UK (1981)
-
[8] L. Jódar, H. Abou-Kandil; Kronecker products and coupled matrix Riccati differential systems; Linear Algebra Appl., 121 (1989), pp. 39–51
-
[9] S. Campbell; Singular Systems of Differential Equations II; Pitman, London (1982)
-
[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.
-
[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
-
[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
-
[13] E. Lewis; A survey of linear singular systems; Circ. Syst Signal Process., 5 (1) (1986), pp. 3–36
-
[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
-
[15] T. Takaba, N. Morihira, K. Katayama; A generalized Lyapunov theorem for descriptor system; Syst. Control Lett., 24 (1) (1995), pp. 49–51
-
[16] B. Brayton; Small signal stability criterion for electrical networks containing lossless transmission lines; IBMJ. Res. Develop., 12 (1968), pp. 431–440
-
[17] Chi-Jo Wang, Ho-En Liao; Impulse observability and impulse controllability of linear time-varying singular systems; Automatica (37) (2001), pp. 1867–1876
-
[18] Chi-Jo Wang; Controllability and observability of linear time-varying singular systems; IEEE Trans. Auto. Control, 44 (10) (1999), pp. 1901–1905
-
[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
-
[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.
-
[21] E. Fridman; Stability of linear descriptor systems with delay: a Lyapunov-based approach; J. Math. Anal. Appl., 273 (2002), pp. 24–44
-
[22] D. Wu, J. Zhu; On stability for a class of time-varying singular system; J. Tongji Univ., 38 (6) (2010), pp. 925–934
-
[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.
-
[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.
-
[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.
-
[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.
-
[27] P. Muller, Modified Lyapunov matrix equations with positive definite solutions for descriptor systems, in: 5th Asian Control Conference, 2004, pp. 997–1000.
-
[28] J. Ishihare, M. Terre; On the Lyapunov theorem for singular systems; IEEE Trans. Automat. Control, 47 (11) (2002), pp. 1926–1930
-
[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
-
[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.
-
[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.
-
[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
-
[33] K. Miller, B. Ross; An Introduction to the Fractional Calculus and Differential Equations; Wiley, New York (1993)
-
[34] L. Podlubny; The Laplace Transform Method for Linear Differential Equations of Fractional Order; Slovac Academy of Science, Slovac Republic (1994)
-
[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
-
[36] A. Kilbas, H. Srivastava, J. Trujillo; Theory and Applications of Fractional Differential Equations; Elsevier, Amsterdam (2006)
-
[37] S. Das; Functional Fractional Calculus for System Identification and Controls; Springer, Berlin (2008)
-
[38] R. Caponetto, G. Dongola, L. Fortuna; Fractional Order System: Modeling and Control Applications; World Scientific (2010)
-
[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
-
[40] R. Magin; Fractional calculus in bioengineering; Crit. Rev. Biomed. Eng., 32 (2004), pp. 1–104
-
[41] M. Dalir, M. Bashour; Applications of fractional calculus; Appl. Math. Sci., 4 (21) (2010), pp. 1021–1032
-
[42] B. Ross; Fractional Calculus and its Applications; Springer-Verlag, Berlin (1975)
-
[43] A. Kilicman, Z. Al-Zhour; Kronecker operational matrices for fractional calculus and some applications; Appl. Comput. Math., 187 (2007), pp. 250–265
-
[44] L. Podlubny; Fractional Differential Equations; Academic Press, New York (1999)
-
[45] G.H. He; Variational iteration method for delay differential equations; Commun. Linear Sci. Numer. Simul., 2 (4) (1997), pp. 235–236
-
[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
-
[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
-
[48] R. Hilfert; Applications of Fractional Calculus in Physics; World Scientific Publishing Company, Singapore, London (2000)
-
[49] I. Grigorenko, E. Grigorenko; Chaotic dynamics of the fractional lorenz system; Phys. Rev. Lett., 91 (3) (2003), p. 034101
-
[50] J. Wang, Y. Zhang; Network synchronization in a population of star-coupled fractional nonlinear oscillators; Phys. Lett. A, 374 (2010), pp. 1464–1468
-
[51] K. Balachandran, J. Kokila; On the controllability of fractional dynamic systems; Int. J. Appl. Math. Comput. Sci., 22 (3) (2012), pp. 523–531
-
[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
-
[53] K. Oldham, J. Spanier; The Fractional Calculus; Mathematics in Science and Engineering, vol. 198, Academic Press (1974)
-
[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.
-
[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
-
[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
-
[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 Article ID 814132
-
[58] M. Caputo; Linear models of dissipation whose Q is almost frequency independent, Part II; J. Roy. Astr. Soc., 13 (1967), pp. 529–539
-
[59] F. Zhang; Matrix Theory: Basic Results and Techniques; Springer-Verlag, New York (1999)
-
[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
-
[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
-
[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
-
[63] S. Liu; Several inequalities involving Khatri–Rao products of positive semi-definite matrices; Linear Algebra Appl., 354 (2002), pp. 175–186
-
[64] G.F. Van Loan; The ubiquitous Kronecker product; J. Comput. Appl. Math., 123 (2000), pp. 85–100
-
[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
-
[66] Z. Odibat; Analytic study on linear systems of fractional differential equations; Computer Math. Appl., 59 (2010), pp. 1171–1183
