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]:

(1-1)

where is a time-varying singular or non-singular matrix function, and are time-varying analytic matrix functions, is the output vector function and is the state function vector to be solved (where is denoted by the set of all matrices over the real number R and when we write instead of ). 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 and 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 are periodically time-varying matrices with period T   and 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 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:

(1-2)

where and .

Note that the fractional derivative of in the Caputo sense is defined for as

(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 and 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]:

(2-1)

Definition 2.2.

Let 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]:

(2-2)

Lemma 2.1.

Let  and X be matrices with compatible orders, and  be the identity matrix of order  . Then[1], [2], [3], [7], [59], [60], [61], [62], [63], [64] and [65].

(2 - 3)
(2-4)

(iv) If f is analytic function on the region containing the eigenvalues of  such that  exist. Then

(2-5)

Definition 2.3.

The one-parameter Mittag–Leffler function and Mittag–Leffler matrix function are defined, respectively, for by [33], [42], [51], [52], [56] and [66]:

(2-6)

where is a matrix of order and is the Gamma function.

Lemma 2.2.

Let  be a matrix of order  and let  and  be the eigenvectors corresponding to the eigenvalues  of A and  , respectively. Then the spectral decomposition of  and  are given, respectively, for  by[56]:

(2-7)

The list of nice properties for Mittag–Leffler matrix  can be found in[56], and the most important properties for Mittag–Leffler matrix  that will be used in this study are given below[56].

Theorem 2.1.

Let  and  be an identity matrix of order  . Then for  , we have[56]:

(2 - 8)

(2 - 9)
(2-10)

Lemma 2.3.

Let  be a given scalar matrix,  be a given vector function, and  be the unknown vector to be solved. Then the unique solution of the following fractional differential system[51], [52] and [56]:

(2-11)

is given by

(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

(3-1)

where is a time-varying singular matrix function, and are time-varying analytic matrix functions, is the output matrix function and is the state function vector to be solved. Here, we will study the general solution of (3-1) when and are constant matrices, as a special case. For this case, suppose that the constant invertible matrices M   and such that:

(3-2)

If we partition n   as , then and . This system is restricted equivalent to:

(3-3)

Note that the necessary and sufficient condition for the existence of the solution of a system (3-1) is that is invertible.

General Solutions of Problem 3.1.

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

(3-4)

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

(3-5)

where

(3-6)

and

(3-7)

is called the Schur complement of in a matrix .

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

(3-8)

Now by letting and , then (3-8) can be represented as follows:

(3-9)

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

(3-10)

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

Note that the relationship between and is given by:

(3-11)

Hence, the general solution of Problem 3.1 is given by: , where can be easily obtained from (3-10) and (3-11); and 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:

(3-12)

where is a time-varying non-singular matrix function, and are time-varying analytic matrix functions, is the output matrix function and is the state function matrix to be solved. Here, we will study the general solution of (3-12) when and are constant matrices, as a special case. For this case, suppose that the constant invertible matrices M   and such that:

(3-13)

This system is restricted equivalent to:

(3-14)

General Solutions of Problem 3.2.

By taking of both sides of (3-14), and using Lemma 2.1, we have:

(3-15)

This system can be represented as:

(3-16)

Suppose that

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

(3-17)

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

(3-18)

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

(3-19)

Another special case of (3-12) is when and are constant matrices and . Then the general solution of this case is given by .

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

(3-20)

As a special case, if and , then by using the same procedure in the proof of Theorem 2 in [56] and Theorem 2.1, we have:

(3-21)

Now, it is easy to get and of this case by substituting (3-21) in (3-19) and then the general solution of this problem is given by .

Example 3.1.

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

(3-22)

where

Since

Then and by using (3-4) and (3-10), respectively, are given by:

(3-23)
(3-24)

where is given as a vector solution as in (3-23).

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

(3-25)

As a special case of system (3-22), if , then and are given, respectively, as in (3-26), (3-27) and (3-28) below:

Now from (3-11), we get:

(3-26)

(3-27)

(3-28)

Example 3.2.

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

(3-29)

where

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

Now,

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

(3-30)

Similarly, we have

(3-31)

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

(3-32)

Note that if and are constant matrices in Example 3.2, then the general solution given by .

4. Conclusion

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

Acknowledgments

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

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